On some families of links and new approaches to link homologies

Programa de Doctorado “Matem´aticas” PhD Dissertation On some families of links and new approaches to link homologies Author Marithania Silvero Casanova Supervisors Prof. Dr. Pedro M. Gonz´alez Manch´on Prof. Dr. Juan Gonz´alez-Meneses L´opez December 18th, 2015 A mis padres. Agradecimientos En primer lugar quisiera dar las gracias a mis directores, Juan y Pedro. Por la confianza depositada en m´ı, por su tiempo, por su paciencia, por su implicaci´on. Porque he sentido c´omo me acompa˜nabais en este camino, sin soltar mi mano al tiempo que me dabais libertad. No habr´ıa podido tener unos directores mejores. A Juan, porque fue ´el quien me dio esta oportunidad. Por el d´ıa a d´ıa, por los correos a deshoras y por su plena disposici´on para responder a mis preguntas. Por esforzarse en hacerme sentir c´omoda. Porque, aunque quiz´as t´u no lo sepas, despu´es de alg´un traspi´es matem´atico han sido tus palabras las que me han devuelto la confianza necesaria para continuar intent´andolo. Porque me tranquilizas y sosiegas. A Pedro por aceptar ser mi director aun sin conocerme, y por acogerme en su casa. Fue ´el quien me ense˜n´o qu´e es un nudo. Con infinita paciencia me acompa˜n´o en mis primeros pasos en esta etapa y me transmiti´o las nociones b´asicas con las que poder tener una incipiente autonom´ıa en mi trabajo. Pedro, gracias tambi´en por tus llamadas a la hora de la cena y por aquellos d´ıas en Madrid de almuerzo a las 5 de la tarde, porque yo s´e que han sido por y para m´ı. Para llegar hasta aqu´ı es necesario tener una base lo suficientemente s´olida. Es por ello que mis agradecimientos tambi´en van dirigidos a todos los profesores que he tenido a lo largo de mi vida, porque ellos me han ense˜nado que el esfuerzo y la constancia son fundamentales en el camino que lleva a una meta, cualquiera que sea ´esta. Quisiera destacar mi gratitud a Juan N´u˜nez, pues fue ´el el primero en confiar en m´ı acad´emicamente y la persona con quien escrib´ı mi primer art´ıculo. En estos cuatro a˜nos he tenido la oportunidad de viajar y aprender de otros matem´aticos en congresos y estancias (aprovecho para dar gracias al MINECO por la ayuda econ´omica). He tenido la gran suerte de que personas de la talla de J´ozef Przytycki, Louis Kauffman y Charles Livingston me hayan acogido en sus Universidades y en sus casas. I would like to thank Charles Livingston for his patience and for introducing me to knot concordance. He was always available for answering my questions and inquiries. Thank you also for offering me your home (and your central heating) on my first weekend at Bloomington at −30oC. Thanks also to Louis Kauffman. I have been very lucky to be given the opportunity to work with such a great mathematician, but more importantly, with a very supportive and encouraging person. Thank you, Lou, for sharing your knowledge with me and also for giving me an ¨ Erdos number of 3. I must express my deepest gratitude to J´ozef Przytycki for his continuing support, his advice and for caring so much about my work. Thank you also for all those afternoons at Starbucks speaking about Maths, for your many invitations to conferences and for offering me your house (and also to Teresa for her pierogies). It has been wonderful learning so many things from one of the most influential researchers on knot theory. I am very lucky to have an American supervisor who has taken care of me both mathematically and personally. Dzi¸ekuj¸e bardzo, J´ozef. Quiero dar las gracias a los compa˜neros del departamento de ´ Algebra, principalmente a los becarios, por las tardes de bromas y confidencias, a Tornero y a Lola. A ´el porque siempre me ha tendido una mano cuando lo he necesitado y porque su jazz, si bien no sustituye al Cascanueces, ha amenizado algunas horas de trabajo. A ella por sus palabras de cari˜no, que a veces me sonrojan. Tambi´en a los compa˜neros del IMUS, especialmente a aquellos con los que he compartido despacho y almuerzos. Me permito nombrar a Alba y a Carmen Ana, por ser amigas a la vez que compa˜neras. Gracias por los momentos de risa y por haber sido un apoyo cuando lo he necesitado. Fuera del marco universitario, doy las gracias a mis amigos, porque siempre han estado presentes, a´un en la distancia. A Miguel, porque he podido y s´e que siempre podr´e contar con ´el. A Jos´e, por muchas cosas, sobre todo por la ilusi´on. A mi familia, especialmente a mis padres, a quienes dedico esta tesis. Por estar conmigo en los buenos momentos, y en los menos buenos, por tener la frase precisa, el abrazo en el momento adecuado. Por ense˜narme a luchar por lo que quiero. Por todas las veces que os ha tocado atrasar el reloj 6 horas, por escuchar mis preocupaciones y porque s´e que siempre tendr´e vuestro apoyo, en cualquier situaci´on, cualquiera que sea el camino que elija tomar. A todos, gracias. Resumen Esta tesis se enmarca dentro de la Teor´ıa de Nudos. En los Cap´ıtulos 1 y 2 se usan herramientas cl´asicas para estudiar las propiedades de ciertas familias de enlaces y las relaciones de inclusi´on entre ellas. En los Cap´ıtulos 3 y 4 se estudian nuevas aproximaciones a invariantes homol´ogicos de enlaces introducidos en este siglo. En 1983 Louis Kauffman conjetur´o que las familias de enlaces alternativos y pseudoalternantes eran iguales. En el Cap´ıtulo 1 se prueba esta conjetura para el caso de enlaces cuyo primer n´umero de Betti es menor que 3. Tambi´en se dan dos contraejemplos que muestran que, en general, la conjetura no es cierta. El polinomio de Conway de los enlaces positivos es positivo. En el Cap´ıtulo 2 se extiende esta propiedad a la familia de enlaces fuertemente cuasipositivos con ´ındice de trenza 3. As´ı mismo, se prueba que esta propiedad no puede ser extendida a los enlaces fuertemente cuasipositivos con ´ındice de trenza mayor que 5. El Cap´ıtulo 3 se centra en los intentos de dar una definici´on de la homolog´ıa Knot Floer en t´erminos de los estados FKT de Kauffman. Concretamente, se muestra la no invariancia de una propuesta de Y. Rong bajo el movimiento de Reidemeister II. En el Cap´ıtulo 4 se da una definici´on geom´etrica de la cohomolog´ıa extrema de Khovanov, en t´erminos del grafo de Lando asociado al diagrama de un enlace. Este nuevo punto de vista permite construir una familia de nudos H-gruesos cuya cohomolog´ıa extrema de Khovanov cuenta con tantos grupos no triviales como se desee. Los conceptos y resultados b´asicos asociados a cada cap´ıtulo se encuentran recogidos en sus respectivas introducciones. As´ı mismo, al final de cada uno de ellos se incluyen cuestiones abiertas a las que da lugar el trabajo desarrollado. 7 Abstract This thesis falls within the scope of Knot Theory. In Chapters 1 and 2 we use classical tools for studying the properties and relations among some families of links. In Chapters 3 and 4 we discuss new approaches to homological link invariants introduced in this century. In 1983 Louis Kauffman conjectured that the families of alternative and pseudoalternating links coincide. In Chapter 1 we prove that this conjecture holds for the special case of links whose first Betti number is at most 2. We also disprove the conjecture for the general case by showing two counterexamples. Positive links have positive Conway polynomial. In Chapter 2 we extend this property to the family of strongly quasipositive links whose braid index equals 3. Moreover, we prove that this result cannot be extended when considering strongly quasipositive links whose braid index is greater than 5. Chapter 3 focuses in the attemps of giving a definition of Knot Floer homology in terms of Kauffman FKT-states. Namely, we show that the definition given by Y. Rong is not invariant under the Reidemeister II move. In Chapter 4 we give a geometric realization of the extreme Khovanov cohomology in terms of the Lando graph associated to a link diagram. This new approach allows us to give a family of H-thick knots with any number of nontrivial extreme Khovanov cohomology modules. Each chapter has its own introduction containing its associated motivation and background. Moreover, at the end of each of them we have included open questions related to the results exposed in the chapter. 8 Contents Resumen 7 Abstract 8 Introduction 11 1 Kauffman’s conjecture 15 1.1 Introduction.............................. 15 1.2 Definitions............................... 16 1.2.1 Alternative links . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.2 Homogeneous links . . . . . . . . . . . . . . . . . . . . . . 18 1.2.3 Pseudoalternating links . . . . . . . . . . . . . . . . . . . . 20 1.3 Two counterexamples to the conjecture . . . . . . . . . . . . . . . 22 1.4 The case β(L)≤2 .......................... 27 1.5 Genus zero four-components links . . . . . . . . . . . . . . . . . . 30 2 Strongly quasipositive links and their Conway polynomials 33 2.1 Introduction.............................. 33 2.1.1 Relations and some properties . . . . . . . . . . . . . . . . 36 2.2 Conway polynomial from Burau representation . . . . . . . . . . . 38 2.3 Conway polynomial of 3-braids differing by ∆2........... 39 2.4 Strongly quasipositive links with braid index 3 . . . . . . . . . . . 42 2.4.1 Resolution trees . . . . . . . . . . . . . . . . . . . . . . . . 42 2.4.2 Positivity of Conway polynomial . . . . . . . . . . . . . . . 43 2.4.3 Non-positive but strongly quasipositive links . . . . . . . . 46 2.5 The result cannot be extended . . . . . . . . . . . . . . . . . . . . 47 3 Knot Floer Homology and FKT states 51 3.1 Introduction.............................. 51 3.2 Parities of black and white holes . . . . . . . . . . . . . . . . . . . 52 3.3 Disproving a proposed differential . . . . . . . . . . . . . . . . . . 58 4 A geometric realization of the extreme Khovanov cohomology 69 4.1 Introduction.............................. 69 4.2 Khovanov cohomology . . . . . . . . . . . . . . . . . . . . . . . . 70 9 16 Section 1.2. Definitions paper he referred explicitly to Conjecture 1.1, noting that its veracity would imply the equality between the three classes of links (alternative, homogeneous and pseudoalternating). In his own words, “Kauffman conjectures that the class of alternative links equals the class of pseudoalternating ones, implying that all three classes are equivalent. This is not obvious”. One key point regarding the difference between these three concepts is the fact that alternative and homogeneous links are defined in terms of diagrams, (a link is alternative [homogeneous] if it can be represented by an alternative [homogeneous] diagram), but the definition of pseudoalternating links is more geometric, as they are defined as those links bounding a surface having some properties. There are some examples in the literature of authors using alternative and homogeneous as synonyms [54]. In Section 1.4 we prove that Conjecture 1.1 holds for those links whose first Betti number is at most 2 (this includes the class of knots of genus one). However, it is not true in general when this value increases, as we show in Section 1.3 by finding two counterexamples: a link and a knot whose first Betti numbers equal 3 and 4, respectively. In the way we work with the intermediate family of homogeneous links. Figure 1.1: The implications between the families of alternative, homogeneous and pseudoalternating families. 1.2 Definitions In this section we provide the definitions of the three families of links mentioned in Section 1.1. We remark that the alternative, homogeneous and pseudoalternating characters of a link are orientation dependant, so from now on all links will be oriented (and non-split). 1.2.1 Alternative links Given an oriented diagram Dof a link L, it is possible to smooth every crossing coherently with the orientation of the diagram, (that is, in the only way preserving orientation, as in Figure 1.3). Doing this for all crossings in D, we obtain a set of topological circles called Seifert circles. Following Kauffman, the spaces of the Chapter 1. Kauffman’s conjecture 17 Figure 1.2: The sign convention is shown. It will be used throughout this work. Figure 1.3: The smoothing used in Seifert’s algorithm. diagram Dare the connected components of the complement of its Seifert circles in S2(one can also think on their complement in the plane), not to be confused with the regions of the knot diagram. We can then draw an edge joining two Seifert circles at the place where there was a crossing in D, and label the edge with the sign of the corresponding crossing according to Figure 1.2. We will refer to the resulting set of topological circles and labeled edges as the Seifert diagram of D, because of the analogy of this process to Seifert’s algorithm for constructing an orientable surface spanning a link [50]. Definition 1.2. [27]An oriented diagram Dis alternative if all edges in any given space of Dhave the same sign. An oriented link is alternative if it admits an alternative diagram. Notice that there are non-alternative diagrams representing alternative links, as can be seen in Figure 1.6. Positive and negative links are alternative (a positive/negative diagram leads to a Seifert diagram with all edges having the same sign). Alternating links are alternative. There are nevertheless alternative links which are not alternating: for instance, every positive (hence alternative) non-alternating link, like the knot 819. Although the following result is not written in terms of links but of diagrams, it clarifies the relation between both families: Proposition 1.3. [27, Lemma 9.2] A link diagram is alternating if and only if it is alternative and the sign of the edges in its Seifert diagram changes alternatively when passing through adjacent spaces. Figure 1.4 illustrates the previous Proposition. 18 Section 1.2. Definitions Figure 1.4: An alternating diagram Drepresenting a two components link and its associated Seifert diagram SCD. A dark edge has a positive label; a light edge a negative one. As Dis alternating, it is an alternative diagram and the sign of the edges in SCDalternates when passing through adjacent spaces. Figure 1.5: A connected graph and its associated blocks. The cut vertices are represented by grey dots. 1.2.2 Homogeneous links We consider now the family of homogeneous links, introduced by Peter Cromwell in 1989 [19]. From the Seifert diagram associated to D, we can construct a graph GDas follows: associate a vertex to each Seifert circle and draw an edge connecting two vertices in GDfor each edge joining the associated circles in the Seifert diagram; each edge must be labeled with the sign + or −of its associated crossing in D. The signed graph GDis called the Seifert graph associated to D. GDcan be obtained from the Seifert diagram of Dby collapsing each circle to a vertex. See Figure 1.6. Note that, as Seifert circles are oriented, Seifert diagrams have no edges with both endpoints in the same circle, hence Seifert graphs do not contain loops. Given a connected graph Gwith no loops, a vertex vis a cut vertex if removing it disconnects the graph, that is, if G\{v}is disconnected. A block of Gis a maximal subgraph of Gwith no cut vertices. Blocks of the graph Gcan be thought of in the following way: remove all the cut vertices of G; each remaining connected component together with its adjacent cut vertices is a block of G. In Figure 1.5 we show a connected graph and its associated blocks. Definition 1.4. [19]A Seifert graph is homogeneous if all the edges of a block Chapter 1. Kauffman’s conjecture 19 Figure 1.6: Two diagrams Dand D0of the knot 943, their associated Seifert diagrams and Seifert graphs. A dark edge has a positive label; a light edge a negative one. D shows that 943 is homogeneous, as the diagram in the figure is so; Dis non-alternative, but since D0is so, 943 is alternative. have the same sign, for all blocks in the graph. An oriented diagram Dis homogeneous if its associated Seifert graph GDis homogeneous. An oriented link is homogeneous if it admits a homogeneous diagram. Note that the original diagram Dcan be recovered from its Seifert diagram, as the sign and position of the crossings in the diagram are preserved (see Figure 1.6). However, as the relative position of the circles and the order of the edges are not encoded in the Seifert graph, Dcannot be recovered from GD. In this sense, one can say that there is a loss of information when considering Seifert graphs instead of Seifert diagrams. Proposition 1.5. Alternative links are homogeneous. Proof. The proof lies in the fact that alternative diagrams are homogeneous. The key point is to notice that the circles “touching” positive and negative edges in the Seifert diagram become cut vertices in the Seifert graph (hence, there are no positive and negative edges in the same block). It is clear that the converse is not true in terms of diagrams, that is, homogeneous diagrams are not necessarily alternative (diagram Din Figure 1.6 is such an example). Before introducing pseudoalternating links, let us give an answer to a question posed by Peter Cromwell in [19]. Motivated by the fact that a non-alternating diagram of a prime alternating link cannot have minimal crossing number [38], he wondered if there exists any homogenous link with a non-homogeneous diagram of minimal crossing number. The answer is that there exists such a link. As 20 Section 1.2. Definitions Figure 1.7: Two equivalent diagrams representing Perko pair (Dand D0are the classical diagrams for knots 10161 and 10162 in Rolfsen’s table, respectively). Both of them have minimal crossing number (10), however D0is homogeneous and Dis not. an example, consider the Perko pair given by the equivalent diagrams of knots 10161 ≡10162 [9]. The first one is minimal and non-homogeneous, and the second one is positive hence homogeneous (see Figure 1.7). [Notice that in some updated versions knot 10162 has been removed and subsequent knots have been renumbered, so that knot 10163 is called 10162, and so on.] 1.2.3 Pseudoalternating links Remember that starting from an oriented diagram Dof a link Lone can obtain its associated canonical surface SDby applying Seifert’s algorithm [50]. The graph GDcan also be thought as the spine graph of the corresponding canonical surface. Primitive flat surfaces [36] are those canonical surfaces arising from positive or negative diagrams whose Seifert diagrams have no nested circles. A generalized flat surface is, roughly speaking, an orientable surface obtained by gluing a finite number of primitive flat surfaces along some of their discs, without overlapping bands. (Figure 1.8 helps to understand the idea.) More precisely, given two primitive flat surfaces S1and S2, choose a disc of each one, d1and d2. Now, identify both discs in such a way that there exists a sphere S2⊂S3separating S3into two non empty 3-balls B1and B2such that Si⊂Biand S2∩Si=di, for i= 1,2. Bands starting at d1and d2are not allowed to overlap when identifying d1and d2. This special kind of Stallings plumbing (or Murasugi sum) will be noted by ∗(see Figure 1.8). Generalized flat surfaces are obtained as a finite iteration of this process, plumbing a primitive flat surface at each step. The first Betti number of a surface S,β(S), is the rank of its first homology group. For a primitive flat surface this is just the number of holes in the surface, or equivalently the number of connected components of the complement of its Chapter 1. Kauffman’s conjecture 21 Figure 1.8: S1and S2are primitive flat surfaces, with β(S1) = 1 and β(S2) = 4. By an identification of discs d1and d2one obtains the generalized flat surface S=S1∗S2, having β(S) = 5. The link spanned by Sis a pseudoalternating link. spine graph in the plane minus 1. As the Euler characteristic of one of these surfaces can be computed as its number of discs dSminus its number of bands bS, it follows that β(S) = bS−dS+ 1. Notice that the first Betti number is additive under this special kind of plumbing: each time one plumbs two surfaces both plumbing discs are identified, so the resulting spine graph can be thought as gluing the two original graphs along a vertex. As generalized flat surfaces are orientable, the following definition makes sense: Definition 1.6. [36]An oriented link is said to be pseudoalternating if it is the boundary of a generalized flat surface, with the natural inherited orientation. Proposition 1.7. [36]Let Lbe a pseudoalternating link with associated generalized flat surface SL. Then SLhas maximal Euler characteristic and it is algebraically unknotted. Namely, π1(S3−S) = F2g+µ−1, a free group of rank 2g+µ−1, where gis the genus of SLand µthe number of components of L. The fact that generalized flat surfaces have maximal Euler characteristic (or equivalently, minimal genus) among all the connected Seifert surfaces that span a given pseudoalternating link (Proposition 1.7) implies that the first Betti number of a pseudoalternating link L,β(L) = min{rank(H1(S)) |∂S =L}, is given by any generalized flat surface spanning it. It also follows from a well-known result by Gabai [22, Corollary 6.7]. Proposition 1.8. Homogeneous links are pseudoaltenating. Proof. Let Dbe a homogeneous diagram of a link and SDits associated canonical surface. Let G1, . . . , Gnbe the blocks of the associated homogeneous Seifert graph 22 Section 1.3. Two counterexamples to the conjecture Figure 1.9: The surface Sspans the diagram D. Although Sis a generalized flat surface, Dis not homogeneous, as its associated Seifert graph GDis not homogeneous. GD,Dithe subdiagram of D(together with some arcs) associated to the subgraph Gi⊂GD, and SDi⊂SDthe projection surface constructed from Di. Then, SD=SD1∗. . .∗SDn. Since blocks in a graph do not contain cut vertices, each SDiis a primitive flat surface: all bands are twisted in the same way and the discs on the surface are either not nested, or there exists a single disc containing the other ones (and this situation is isotopic to the previous one). Hence, SDis a generalized flat surface. Although the canonical surface constructed from a homogeneous diagram is a generalized flat surface, the converse is not necessarily true, that is, a diagram spanned by a generalized flat surface is not necessarily homogeneous, as can be seen in Figure 1.9. Remember that something similar happened with alternative and homogeneous diagrams. These two observations together with the next restatement of Conjecture 1.1 are the key points for understanding the difficulties of proving the conjecture, and, at the same time, the “clues” for looking for counterexamples. Propositions 1.5 and 1.8 imply that Conjecture 1.1 can be restated as follows: Conjecture 1.9. [27]The classes of alternative, homogeneous and pseudoalternating links are equal. In the following section we present two pseudoalternating links which are not homogeneous, hence non-alternative; these links are counterexamples to Conjecture 1.9, hence to Conjecture 1.1. 1.3 Two counterexamples to the conjecture The main problem when trying to deal with Kauffman’s conjecture is that identifying whether a link is alternative or pseudoalternating is not easy, as these properties are defined in terms of diagrams. Of course, by finding an alternative diagram one shows the alternativity of a link, but this does not help when the link is not alternative. In this sense, working with the intermediate family of homogeneous links will help us. Chapter 1. Kauffman’s conjecture 23 Figure 1.10: Diagrams D+,D−and D0. Cromwell’s paper [19] is devoted to the study of homogeneous links. He provides some sufficient conditions for determining the non-homogeneous character of a link. These properties help to determine the homogeneity within some families of arborescent links, double knots, and all but 5 knots with crossing number at most 10. In particular, we will use the following result : Theorem 1.10. [19, Corollary 5.1] If Lis a homogeneous link and the leading coefficient of its Conway polynomial ∇(L)is ±1, then the crossing number of L is at most 2·maxdeg ∇(L). The Alexander-Conway polynomial (or just Conway polynomial) was re-discovered in 1969 by J. Conway as a normalized version of the Alexander polynomial. Given an oriented link L, its Conway polynomial ∇(L) is an invariant of Ldefined as ∇(L) = P(1, z), where P(v, z) is the HOMFLYPT polynomial ([21] [42]). More precisely, ∇(L) is the only polynomial in Z[z] satisfying the skein relation ∇(L+)−∇(L−) = z∇(L0), with normalization ∇(unknot) = 1. Here L+,L−and L0are links represented by diagrams D+,D−and D0respectively, which only differ in the neighborhood of a crossing as shown in Figure 1.10. At this point, we are ready to show the result refuting Conjecture 1.9: Theorem 1.11. There exists a pseudoalternating link which is not homogeneous. Proof. Consider the oriented link Lwith two components presented by the oriented diagram Dshown in Figure 1.11. This link is L9n18{1}in [16], which corresponds to the two-components link L9n18 in Thistlethwaite table (or 92 53 in Rolfsen table). Its Conway polynomial is ∇(L) = z3+ 4z, so by Theorem 1.10, if Lwere homogeneous its crossing number would be at most 2 ·3 = 6, yielding a contradiction. Consequently, Lis non-homogeneous. See Figure 1.11. The diagram Dcan be transformed by a Reidemeister III and a Reidemeister I move into the diagram D0. The new diagram D0allows us to see Las the boundary of a certain surface S. The surface Sis clearly isotopic to S0, and S0is the result of performing two Stallings plumbings of three surfaces, S1, S2, S3, using discs d1,d2and d3as “gluing patches”, this is, S=S1∗S2∗S3. Each of these surfaces consists on two discs joined by a pair of bands twisted in a positive way, so S1, S2and S3are primitive flat surfaces; hence Sis a generalized flat surface and Lis a pseudoalternating link. 24 Section 1.3. Two counterexamples to the conjecture Figure 1.11: Dand D0are diagrams of the link L=L9n18{1}in [16]; Sis a Seifert surface for L, and S0is obtained from Sjust by overturning the subsurface S3over the disc d. Finally S0=S1∗S2∗S3. Figure 1.12: Notice that the diagram Din Figure 1.11 becomes negative after reversing orientation of the dark component. Obviously, this diagram is alternative, as its associated Seifert diagram has only edges of negative type. Since every alternative link is homogeneous, the link L9n18{1}, with β(L) = 3, is also a counterexample to Conjecture 1.1. Corollary 1.12. There exists a pseudoalternating link which is not alternative. Remember that alternative, homogeneous and pseudoalternating characters are orientation dependant. In fact, if one changes the orientation of one component of L(for example, reversing the orientation of the dark component in Figure 1.11) the resulting link is negative, hence alternative, as can be seen in Figure 1.12. At this point one can wonder if there exist knots or links which are pseudoalternating and non-alternative with any possible orientation of its components. We will show such an example by finding a knot of genus two with these properties, as Chapter 1. Kauffman’s conjecture 25 Figure 1.13: From D1to D2and from D3to D4just move the grey strand, leaving unchanged the rest of the diagram; from D2to D3perform two Reidemeister III moves on the dotted strand. By performing a Reidemeister I move on the dotted strand in D4, we see that Sis a Seifert surface for K;S0is the result of overturning one of the primitive flat surfaces over the disc d. pseudoalternating, homogeneous and alternative characters are not orientationdependant in the case of knots. This knot, whose first Betti number equals 4, is pseudoalternating and non-alternative. It would be interesting to find a link with more than one component being a counterexample to the conjecture with all possible orientations. Theorem 1.13. There exists a pseudoalternating knot which is not homogeneous. Proof. The proof is analogous to that of Theorem 1.11. Let Kbe the genus two knot 10145 in Rolfsen table, with the orientation given in Figure 1.13. Its Conway polynomial is ∇(K) = z4+ 5z2+ 1, so by Theorem 1.10 we deduce that Kis not homogeneous, as 10 >2·4. See Figure 1.13. By a finite sequence of Reidemeister moves, the classical diagram representing Kin [15], D1, can be transformed into D4by performing the following steps: from D1, we obtain D2by leaving unchanged the diagram except for the grey strand; perform two Reidemeister III moves on the dotted strand of D2in order to get D3; finally, transform D3into D4by moving the grey strand. At this point it is easy to see that the surface Sbounds K;S is a generalized flat surface obtained by performing two Stallings plumbings of three primitive flat surfaces (two of them consist on a pair of discs joined by two bands, and the other one consists on two discs together with three bands, as shown in S0) using twice the same disc, d, as “gluing patch”. As a result, Kis a pseudoalternating knot. 32 Section 1.5. Genus zero four-components links components after each plumbing. Combining both facts in this paragraph, after the first plumbing S1∗S2the situation must be as shown in Figure 1.20. At this point, we need to plumb S3. If the gluing disc in S1∗S2is different from the disc used in the first plumbing, then the resulting link is alternative (one just has to overturn the surfaces as in Figure 1.16 if necessary). Now consider that the gluing disc is used twice. There are three possible options attaching the two bands of S3in the same boundary component, as can be seen in Figure 1.20. The key point is noticing that in all of them any surface can be overturned inside the gluing disc, in case its sign is different from the others, and the resulting diagram would be alternative. Figure 1.20: The surface S1∗S2after the first plumbing in the case of a generalized flat surface constructed as the plumbing of three primitive flat surfaces using the gluing disc twice. The sign of the twists in the bands coming from S1(S2) is ε1(ε2). The number of bands and discs in each of the primitive flat surfaces can be decreased/increased. Chapter 2 Strongly quasipositive links and their Conway polynomials 2.1 Introduction The notion of positivity related to a link has been deeply studied from many different points of view. Maybe the simplest class involving this concept is the family of positive links. Recall that an oriented link is said to be positive if it has a positive diagram, that is, a diagram with all crossings being positive (see Figure 2.4). Many authors have extended this classical notion of positivity by defining new families of links. Frequently, these definitions depend on finding a representative (a diagram, a braid, a Seifert surface...) of the link having some properties. As in general finding such a representative is not easy, giving conditions for the invariants of a link belonging to a specific family becomes useful. A well-known result by J. Alexander is the popularly known as Alexander’s Theorem [2]. It states that every oriented link can be represented as the closure of a braid β∈Bnfor some n. Recall that the braid index of a link is the smallest nsuch that the link can be represented as a closed braid on nstrands (it can also be defined as the minimum number of Seifert circles in any Seifert diagram of the link). In this chapter we give a necessary condition for a link with braid index 3 to be strongly quasipositive. Strongly quasipositive links are intrinsically related to positive braids, so let us recall some notions of positivity related to braids. We remark that in the following cases positivity depends on the choice of presentation of the braid group. Roughly speaking, a braid is positive if there exists a positive word representing it, that is a word with all its letters having positive exponents. Through this chapter we mainly work with two different presentations of the braid group on nstrands, Bn. The first one is the standard presentation due to Artin ([4], [5]): 33 34 Section 2.1. Introduction Figure 2.1: Braid relations in the Artin presentation, the Artin-positive braid α= σ3σ3σ2σ3σ2σ3σ1σ2σ3σ2σ1on 4 strands and its closure bα, an Artin-positive knot. Bn=σ1, σ2, ..., σn−1σiσjσi=σjσiσj|i−j|= 1 σiσj=σjσi|i−j|>1 Attending to the presentation above, a braid is said to be Artin-positive if it can be represented by a positive standard braid word, that is, a braid word where each Artin generator σiappears with positive exponent. The closure of an Artin-positive braid is an Artin-positive link. See Figure 2.1 Artin-positive links are, indeed, positive links. However, the converse is not true. In Proposition 2.2 we show a proof of this well known fact. The braid group Bnadmits another well known presentation due to Birman, Ko and Lee [12]. The so called BKL generators or band generators,σij, are related to Artin-generators by the formula σij = (σj−2. . . σi)−1σj−1(σj−2. . . σi), with i<j. They correspond to a positive crossing of strands in positions iand jpassing in front of the other strands, as shown in Figure 2.2. Bn=σrs,1≤r < s ≤nσqrσst =σstσqr (t−r)(t−q)(s−r)(s−q)>0 σstσrs =σrtσst =σrsσrt 1≤r < s < t ≤n Just as before, a BKL-word having only positive exponents is a BKL-positive word. Chapter 2. Strongly quasipositive links and their Conway polynomials 35 Figure 2.2: Braid relations in the Birman-Ko-Lee presentation, the BKL-positive braid α=σ34σ14σ23σ12σ24σ23σ12 on 4 strands and its closure bα, a BKL-positive (or strongly quasipositive) link. Definition 2.1. A braid is BKL-positive if it can be expressed by a word written in terms of the generators given by Birman, Ko and Lee, with all letters having positive exponents. The closure of a BKL-positive braid is a BKL-positive link. In [46] Rudolph introduced BKL-positive links as the boundaries of certain surfaces, and he called them strongly quasipositive links. Rudolph also proved in [48] that positive links are strongly quasipositive, which is not obvious. The converse is not true. In fact, Baader [6] showed that a link is positive if and only if it is strongly quasipositive and homogeneous; in particular, the link L9n18{1} appearing in the proof of Theorem 1.11 is strongly quasipositive but not positive [51]. A polynomial with real coefficients is said to be positive if all its coefficients are non-negative. In 1983 Van Buskirk [56] proved that Artin positive links have positive Conway polynomial. Six years later Cromwell [19] extended this result to the class of positive links. Artin-positive links are indeed positive, and the latter are strongly-quasipositive. It seems natural to wonder if the positivity of Conway polynomial is preserved under this extension. The main result in this chapter is the following: Theorem 2.8 Strongly quasipositive links with braid index 3 have positive Conway polynomial. This result cannot be generalized to strongly quasipositive links with arbitrary 36 Section 2.1. Introduction Figure 2.3: Implications between the families of Artin-positive, positive and strongly quasipositive (or BKL-positive) links, and their relation with the positivity of Conway polynomial. braid index, as we prove in Proposition 2.12. Namely, we give an example showing that for any n≥6 there exists a BKL-positive braid α∈Bnwhose closure has non-positive Conway polynomial. 2.1.1 Relations and some properties In this section we review some details about the families of Artin-positive, positive and strongly quasipositive links and their relations. Proposition 2.2. There are positive links which are not Artin-positive. Proof. In Figure 2.4 a positive diagram of the knot 52is shown; its braid index is 3, as it is the closure of the braid α=σ−3 2σ−1 1σ2σ−1 1. Suppose now that 52is the closure of a braid γrepresented by a positive Artin-word w. We take wwith minimal length. Let Dbe the associated positive diagram. As projection surfaces constructed from homogeneous diagrams have minimal genus (Propositions 1.7, 1.8) and positive diagrams are homogeneous, g(SD) = g(52) = 1 leads to s+1 = c, where sand care the number of Seifert discs and bands in SD, the projection surface arising from D. Notice that sis the number of strands and cis the number of crossings of γ. Since c≥5, γmust have at least 4 strands, and then some generator σi must appear at most once. All generators must appear, since 52is a knot (a one-component link), so there exists one generator appearing exactly once, and this is a nugatory crossing. This is a contradiction with the minimality of wsince 52is prime. The families of strongly quasipositive links and BKL-positive links are equal. The first name reminds that the link is boundary of a nice kind of surface; the second one brings to our mind the algebraic presentation of the braid group. To be more precise, strongly quasipositive links were introduced as the boundaries of what Rudolph called quasipositive surfaces [46]. A quasipositive surface Chapter 2. Strongly quasipositive links and their Conway polynomials 37 Figure 2.4: We remind the chosen convention of signs and a positive diagram representing the positive knot 52. Figure 2.5: The quasipositive surface associated to the closed BKL-positive braid in Figure 2.2. Its boundary is a strongly quasipositive link (or BKL-positive link). is an orientable surface which is ambient isotopic to a surface consisting of a finite number of parallel discs joined by some bands twisted in a positive way, as in Figure 2.5. Each of these bands corresponds to a BKL-generator. Strongly quasipositive links should not be confused with quasipositive links, which are closures of products of conjugates of Artin-positive braids. We are not going to deal with quasipositive links through this chapter. Quasipositive surfaces satisfy many interesting properties, such as being of minimal genus for the strongly quasipositive links they span, or like the following fact: Theorem 2.3. [47, Theorem 4.3] The Stallings plumbing (or Murasugi sum) of two surfaces S1and S2is quasipositive if and only if both S1and S2are quasipositive. Primitive flat surfaces (see Subsection 1.2.3) with their bands twisted in a positive sense are quasipositive, so their boundaries are strongly quasipositive links. Recall that pseudoalternating links are the boundary of generalized flat surfaces, which are constructed as a finite number of plumbings of primitive flat surfaces (see Definition 1.6). Hence Theorem 2.3 implies that a given pseudoalternating link is strongly quasipositive if and only if there exists an associated generalized 38 Section 2.2. Conway polynomial from Burau representation flat surface coming from the plumbing of positive primitive flat surfaces. The link L9n18{1}is such an example (see Figure 1.11 in Chapter 1). 2.2 Conway polynomial from Burau representation Burau [14] introduced a linear representation of the braid group on nstrands, Bn, by squared matrices of order nover the ring of Laurent polynomials Z[t, t−1]. This representation has been widely studied, being its faithfulness one of the remaining open problems (it is known to be faithful when n≤3 and unfaithful when n≥5, but the case n= 4 is still unsolved). We will use the reduced Burau representation ψ:Bn→GL(n−1,Z[s, s−1]) defined by the formula ψ(σi) = Ai, where A1=  −s20 0 1 1 0 0 0 In−3  , An−1=  In−30 0 0 1 s2 0 0 −s2  and for 1 < i < n −1 Ai=       Ii−20 0 0 0 0 1 s20 0 0 0 −s20 0 0 0 1 1 0 0 0 0 0 In−i−2       with Ikbeing the identity matrix or order k. In [25] the reduced Burau representation is noted ψr n:Bn→GL(n−1,Z[t, t−1]), and it is equal to the above definition after the substitution t=s2. Then Lemma 3.12 and Theorem 3.13 in [25] can be restated to give the following well-known presentation of the Conway polynomial in terms of the reduced Burau representation: Theorem 2.4. [25]Let α∈Bnbe a braid and bαthe link obtained as the closure of α. Then the Conway polynomial of bαis given by ∇(bα)(z)=(−1)n+1 s−eα [n]|ψ(α)−In−1| after the substitution s−1−s=z, where [n] = s−n−sn s−1−sand eα∈Zdenotes the image of αunder the homomorphism Bn→Zsending each generator σito 1. Note that eαcoincides with the exponent sum of any braid word representing the braid α, which is invariant under the (homogeneous) relations of the braid group. Moreover, eαis invariant under conjugation. Chapter 2. Strongly quasipositive links and their Conway polynomials 39 In the proof of Theorem 2.6 we study the case of braids with three strands, so we specify the case n= 3. The reduced Burau representation of B3is given by the matrices B1=−s20 1 1 and B2=1s2 0−s2. Since for a square matrix Aof order two |A−xI2|=x2−xTr(A) + |A|, one has |ψ(α)−I2|= 1−Tr(ψ(α))+|ψ(α)|when working with a braid with 3 strands. Note that, as |ψ(B1)|=|ψ(B2)|=−s2, then |ψ(α)|= (−s)2eα,∀α∈B3. At this point, we find useful to show a slight modification of a result by P. V. Koseleff and D. Pecker [30] which simplifies the substitution s−1−s=z in the formula above by using Fibonacci polynomials. Recall that Fibonacci polynomials are defined by the recurrence relation Fn(z) = z Fn−1(z) + Fn−2(z) for n≥2, starting with F0(z) = 0 and F1(z) = 1. There exist closed combinatorial formulas to express the Fibonacci polynomials; however, we will not use them through this chapter. For our purposes we need to extend these polynomials to the case when the subindex is negative; we do it in the natural way, by defining F−n(z) = (−1)n+1Fn(z). Now we are ready to state our adapted version of Lemma 4.1 in [30]: Lemma 2.5. [30]Let Fn(z)be the nth Fibonacci polynomial. After the substitution z=s−1−s, the identity (s−1)n+(−s)n=Fn+1(z)+Fn−1(z)holds for every integer n. The following sections in this chapter are fully devoted to present our new results. 2.3 Conway polynomial of 3-braids differing by ∆2 From now on, we consider the braid group on 3 strands B3, unless otherwise stated. Although the results contained in this section are interesting on their own, they become really useful in Section 2.4 when proving Corollary 2.10, which completes the proof of the main theorem in this chapter (Therorem 2.8). Recall that the Garside element ∆ ∈Bnis defined as the half twist of the strands, ∆ = (σ1σ2. . . σn−1)(σ1. . . σn−2). . . (σ1σ2)σ1. Since ∆σi=σn−i∆ it follows that ∆2is in the center of Bn. Moreover, the whole center of Bnis generated by ∆2[18]. The following result provides a relation between the Conway polynomials of two closed braids differing in an even power of the Garside element ∆ = σ1σ2σ1 in B3. See Figure 2.6 for getting an example of two braids αand βas the ones on the statement for the case k= 1. 40 Section 2.3. Conway polynomial of 3-braids differing by ∆2 Figure 2.6: The Garside element in B3, ∆ = σ1σ2σ1, the (dark) braid α= σ−1 2σ1σ1σ2σ−1 1σ2σ1∈B3and the braid β= ∆2α. Theorem 2.6. Let α, β ∈B3with β= ∆2kαand k > 0. Then the difference between the Conway polynomials of their closures is given by ∇(b β)− ∇(bα) = z k−1 X i=0 (Feα+6i+4 +Feα+6i+2) with Fnbeing the nth Fibonacci polynomial for any integer n. Proof. We proceed by induction on k. We start with the case k= 1, that is, the case when the braid βis obtained after multiplying the braid αby a full twist. As the Garside element in B3is ∆ = σ1σ2σ1, it follows that eβ=eα+6. Since ψ(∆2) = s6I2, we have that Tr(ψ(β)) = s6·Tr(ψ(α)). We now combine these facts with Theorem 2.4 for computing the Conway polynomial of both closed braids. Taking into account the substitution s−1−s=z, we have ∇(bα) = s−eα [3] |ψ(α)−I2|=s−eα [3] (1 −Tr(ψ(α)) + |ψ(α)|) =1 [3] s−eα−s−eαTr(ψ(α)) + (−1)eαseα and ∇(b β) = s−eβ [3] |ψ(β)−Id|=s−eβ [3] (1 −Tr(ψ(β)) + |ψ(β)|) =1 [3] s−eα−6−s−eαTr(ψ(α)) + (−1)eαseα+6. Now we compute the difference: Chapter 2. Strongly quasipositive links and their Conway polynomials 41 ∇(b β)− ∇(bα) = 1 [3] s−eα−6−s−eα+ (−1)eα(seα+6 −seα) =(−s)eα+4 + (s−1)eα+4−(−s)eα+2 + (s−1)eα+2. The second equality comes from the fact that s−1−s·s−eα−6−s−eα+ (−1)eα(seα+6 −seα) =s−3−s3·(−s)eα+4 +s−eα−4−(−s)eα+2 −s−eα−2. Applying twice Lemma 2.5 we obtain ∇(b β)− ∇(bα) = (Feα+5 +Feα+3)−(Feα+3 +Feα+1) = (Feα+5 −Feα+3)+(Feα+3 −Feα+1) =z(Feα+4 +Feα+2). Now, suppose the statement true for 1,2, . . . , k −1, and let β= ∆2kα. If we write γ= ∆2(k−1)α, then by induction ∇(b β)− ∇(bα) = ∇(b β)− ∇(bγ) + ∇(bγ)− ∇(bα) =zFeγ+4 +Feγ+2+z k−2 X i=0 (Feα+6i+4 +Feα+6i+2) =z k−1 X i=0 (Feα+6i+4 +Feα+6i+2) where the third equality holds since eγ=eα+ 6(k−1). With some extra work, the theorem above can be deduced from work by Murasugi in [37, Proposition 4.1]. He compares the normalized Alexander polynomial of two closed braids differing in an even power of ∆ and provides an expression for their difference, with an indeterminacy on a power of −t. We think that the formula we present in Theorem 2.6 is quite simpler even in the case of the Alexander polynomial (that is, just before the change of variables s−1−s=z). As a consequence of Theorem 2.6 we get an interesting corollary. The result for the even case was proved by Joan Birman in [11]; as far as we know there is no reference for the odd case. Corollary 2.7. Let α∈B3with eα=−3kand k > 0, and consider β= ∆2kα. We have that: •If kis even, then ∇(b β) = ∇(bα). 48 Section 2.5. The result cannot be extended Figure 2.11: The BKL-positive braid α=σ16σ16σ46σ35σ24σ13σ25. Its closure is a strongly quasipositive link having non-positive Conway polynomial. has positive Conway polynomial. The following result gives a negative answer to this question by showing a counterexample: Proposition 2.12. There are strongly quasipositive links having non-positive Conway polynomial. Proof. Consider the BKL-positive braid on 6 strands α=σ16σ16σ46σ35σ24σ13σ25. Its closure (Figure 2.11) is a strongly quasipositive link, whose Conway polynomial is ∇(bα) = −z2+ 1, which is non-positive. (We computed ∇(bα) by using a C++ version of the program br9z.p, developed by Short and Morton in 1985: http://www.liv.ac.uk/ ∼su14/knotprogs.html). In [49], in the proof of Corollary 88, Rudolph stated that every Seifert matrix of a given link can be obtained as the Seifert matrix of a strongly quasipositive link. As a consequence, given a link Lwith Conway polynomial ∇(L), there would exist a strongly quasipositive link L0having the same Conway polynomial, that is, satisfying ∇(L) = ∇(L0). He also gives a procedure for constructing L0 as a closed BKL-positive braid, starting from a braid diagram of L. This result would provide an infinite family of examples of strongly quasipositive links having non-positive Conway polynomial. However, we think that there is a problem with the proof of this result: it is claimed that the procedure for obtaining L0starting from L(a sequence of doubled-delta moves, also called trefoil insertion) preserves the Seifert matrix. After applying this move to the braid β=σ1σ1σ−1 1∈B2, one obtains β0=σ16σ16σ25σ13σ24σ35σ46. The closure of βis the trivial knot, hence ∇(b β) = 1; however ∇(b β0)=7z2+ 1. This contradicts the fact that doubled-delta moves preserve the Seifert matrix. It follows from Proposition 2.12 that for any n≥6 there exist BKL-positive braids β∈Bnwhose closures have non-positive Conway polynomial. To get such Chapter 2. Strongly quasipositive links and their Conway polynomials 49 an example, it suffices to take the braid in the proof of Proposition 2.12 and perform iterated positive stabilizations (that is, replace α∈B6with βσ6σ7. . . σn−1∈ Bn). Despite applying some well-known results for giving a boundary on the braid index of a link, we do not know whether the braid index of the closure of the braid αin the proof of Proposition 2.12 is 4, 5 or 6. At Knotinfo [15] one can find 103 and 90 strongly quasipositive knots having braid index equal to 4 and 5 respectively; all of these knots have positive Conway polynomial. Using the computer program above, we have also checked many examples of links being the closure of BKL-positive braids on 4 strands and all of them have positive Conway polynomial. Hence, determining if the result in Theorem 2.8 can be extended to the case of 4 and 5 strands is an open question. 50 Section 2.5. The result cannot be extended Chapter 3 Knot Floer Homology and FKT states 3.1 Introduction In this chapter we present our contribution to a project by some authors, aiming to find a combinatorial definition of Knot Floer homology in terms of the states defined in Formal Knot Theory [27] (from now on FKT states), in a similar way as that given by Viro [57] for the Khovanov homology in terms of Jones states (see Section 4.2). More specifically, taking FKT states as the generators of the chain complex the goal is to try to find an appropriate differential in such a way that the resulting homology matches with Knot Floer homology. The aim of this chapter is not to give an extensive formal definition of Knot Floer homology, but to explain its relations with the classical concepts introduced in FKT. Knot Floer homology is a relatively new link invariant developed independently by Ozsv´ath and Szab´o [40] and Rasmussen [43]. There has been an intensive study of this theory, as it provides important geometric information about a link, such as its genus or its fibredness, and even characterizes the trivial knot [31]. One of the main properties about Knot Floer Homology is that it categorifies the Alexander polynomial. To be more specific, Knot Floer homology is a bigraded homology whose Euler characteristic is the Alexander polynomial, that is: X i,j (−1)j·rank(Hi,j(K)) ·ti= ∆K(t), where Hi,j(K) is the Knot Floer homology module of the knot Kand ∆K(t) is the normalized version of the Alexander-Conway polynomial of Kas defined in Sections 1.3 and 3.2. There are several approaches for proving this fact. For example, in [10] 51 52 Section 3.2. Parities of black and white holes M. Benheddi and D. Cimasoni used the purely combinatorial version of Knot Floer Homology (the one based on grid diagrams) for verifying it. But this version is not directly related to the FKT model. For our purposes, the most interesting proof of this categorification is the one given by Ozsv´ath and Szab´o in [39,41], as they identify the graded Euler characteristic of Knot Floer homology theory as a state summation related to the Formal Knot Theory model. There is a technical difference in that they sum over different local weights and use a different sign convention. They claim that the resulting state sum is the Alexander-Conway polynomial by reference to the FKT model (Theorem 11.3 in [41]). In section 3.2 we include both definitions of the polynomial and prove their equality by providing a proof using classical concepts. The key point in this proof is Proposition 3.8, where we relate the parity of the white and black holes for the states in the FKT model. These results are contained in a joint paper with Louis Kauffman [26]. Going back to the problem of finding a definition of Knot Floer Homology in terms of FKT states, there have been many attempts to find such a differential. In particular, we want to present a differential proposed by Y. Rong [45]. As far as we know, they could prove neither its validity nor find an example where its invariance fails; we found such an example, revealing that this differential is not correct. In Section 3.3 we present this counterexample and give an extended explanation of the ideas behind Rong’s differential, as his approach could be the starting point for finding the “good differential”. 3.2 Parities of black and white holes At the level of combinatorial knot theory, this section shows that a reformulation of the Alexander-Conway polynomial following the FKT model using white hole rather than black hole counts gives the same topological result. In proving this we obtain an interesting result (Proposition 3.8) relating the parities of the white and black holes for the states in the FKT model. We include in this section some of the results contained in a joint paper with Louis Kauffman [26]. We start by explaining classical notions such as marker, region or state. Given an oriented non-split link Lin S3, fix an associated diagram D, and write v1, . . . , vnfor its crossings. If one forgets the under-over information, one obtains a planar diagram PDdividing the plane into n+ 2 regions, one of them unbounded. Choose two adjacent regions and mark each of them with a star. Definition 3.1. In the previous conditions, a (FKT) state Sassociated to Dis an assignment of markers to each vi(that is, a choice of an adjacent unstarred region for each crossing) so that no region of PDcontains more than one marker. Let Sdenote the set of all possible states in D. In Figure 3.1 we present two types of labels for the crossings of an oriented link Chapter 3. Knot Floer Homology and FKT states 53 Figure 3.1: The left part of the first (second) row shows the Alexander-Conway labeling defining h· | ·i ([· | ·]); the rightmost part shows the state marker producing a black (white) hole. diagram. The labeling shown in the first row will be called the Alexander-Conway labeling. It gives rise, as we will see in Theorem 3.2, to a normalized version of the Alexander polynomial and gives a state summation for the Alexander-Conway polynomial in z=t1/2−t−1/2. We call these Alexander-Conway labels because they are different from labels that derive directly from Alexander’s original paper on his polynomial [3]. In [27] Kauffman reformulates the Alexander polynomial as a state summation by using the Alexander-Conway labels. More precisely, given a state S, let hD|Sidenote the product of the Alexander- Conway labels associated to each marker in S(see Figure 3.1). A marker is called a black hole if it occurs between two ingoing lines to the corresponding crossing. A marker is called a white hole if it occurs between outgoing lines at a crossing. Write BS(WS) for the set of double points whose markers are black (white) holes. Theorem 3.2. [27, Theorem 4.3] Let Dbe a diagram representing an oriented link L. Then the polynomial in Z[t1 2, t−1 2] ∆L(t) = X S∈S (−1)]BShD|Si. is independent of the choice of the diagram D, and it is in fact, up to unit ±tn, the original Alexander polynomial of Las defined in [3]. In particular ∆(t) = 1 and if L+, L−and L0are three links with identical diagrams D+,D−and D0 except in a small neighborhood where they differ as shown in Figure 1.10, then they satisfy the skein relation ∆L+(t)−∆L−(t) = (t1/2−t−1/2)∆L0(t). In [41] an alternative definition of the Alexander polynomial is given in the following terms: 54 Section 3.2. Parities of black and white holes ΘL(t) = X S∈S (−1)]WS[D|S], where Dis a diagram representing the oriented link Land the square bracket is defined in terms of the labels in the second row of Figure 3.1. This definition is a state sum using white holes and labels that are essentially the inverses of the ones in Theorem 3.2. We will show that this alternative definition of the Alexander polynomial is, in fact, equal to the Alexander-Conway polynomial as expressed in Theorem 3.2. Before proving that fact, we will show an interesting result relating the parities of white and black holes in a given state in Proposition 3.8. Definition 3.3. A flat Reidemeister move is a move performed in a planar diagram representing a link (a link diagram with double points, that is with no specification of over or under crossings). Such diagrams are called universes in [27]. Flat Reidemeister moves are diagrammatically the same as the usual Reidemeister moves, but have no constraint about patterns of over and under crossings, as shown in Figure 3.2. Lemma 3.4. Let Dbe a planar diagram representing an oriented link L; write s(D)and c(D)for the number of (Seifert) circles and edges in the Seifert diagram associated to D(defined in Subsection 1.2.1). The parity of e(D) = c(D)−s(D)+ 1is invariant under flat Reidemeister moves. Hence, the parity is a link invariant which will be called E(L). Proof. The proof is given in Figure 3.2. Corollary 3.5. Let Lbe an oriented link with µcomponents. Then the parity of µis opposite to E(L). Proof. By a finite sequence of flat Reidemeister moves, any planar diagram representing Lcan be transformed into the trivial link with µcomponents, which satisfies e(Uµ) = 1 −µ. Lemma 3.6. Let Dbe a planar diagram of an oriented link L. There always exists an FKT state Sfor Dsatisfying ]BS+]WS=c(D)−s(D)+1, where BS and WSare the sets of black and white holes in Srespectively. Proof. We give an algorithm for constructing such an state, following some ideas in [27]. Starting from D, construct its associated Seifert diagram. Choose an innermost Seifert circle siand choose one of the edges joining sito another circle sj. Joint the two circles in the place of the edge by considering the opposite smoothing as the one given by Seifert’s algorithm. The resulting configuration has one fewer circle, and after repeating this procedure s(D)−1 times using each Chapter 3. Knot Floer Homology and FKT states 55 Figure 3.2: The proof of Lemma 3.4 for first and second Reidemeister moves with all possible orientations and Seifert circles configurations are shown in this picture (D0and Dare the diagrams before and after the corresponding Reidemeister move, respectively). For the third Reidemeister move the many different possibilities can be checked as in the example. * * * * Figure 3.3: The figure shows the trail Tassociated to the planar diagram Dtogether with a choice of the starred regions and the two rooted trees. It also shows the associated Kauffman state S. 56 Section 3.2. Parities of black and white holes Figure 3.4: A clock move transposition and the lattice obtained from the 41classic diagram with a fixed choice of unmarked regions. edge at most once, we obtain a simple closed curve or Jordan trail T, as shown in Figure 3.3. Kauffman proved that there exists a bijection between the collection of states of Dsharing a fixed choice of adjacent stars and the collection on all Jordan trails arising from D. Choose two adjacent regions in Dand draw an star in each of them; let Sbe the Kauffman state associated to Tconstructed in the following way: construct two trees, each one rooted at one of the stars, in such a way that their branches visit each region once (see Figure 3.3). If we consider the tree starting at the root, draw in each vertex a marker in the place where the branch enters a new region. Let us see that Ssatisfies the condition in the statement. The markers coming from a re-smoothing are neither black nor white holes. Each marker coming from a smoothing preserving orientation is either a black or a white hole. As we did s(D)−1 re-smoothings, it follows that ](BS) + ](WS) = c(D)−s(D) + 1. Aclock move is a modification of two adjacent markers as shown in the leftmost part of Figure 3.4. In [27] Kauffman proved that the collection of possible states arising from a given planar diagram with a fixed choice of adjacent unmarked regions is a lattice, where all the states are related by a finite sequence of clock moves. In Figure 3.4 we show the example of knot 41. Chapter 3. Knot Floer Homology and FKT states 57 Corollary 3.7. Let Dbe a planar diagram of an oriented link L. Every FKT state Sfor Dsatisfies ]BS+]WS=c(D)−s(D)+1, where BSand WSare the sets of black and white holes in Srespectively. Proof. Notice that clock moves preserve the number of black plus white holes. This remark together with Lemma 3.6 leads to the fact that every state Sarising from a diagram Dsatisfies ]BS+]WS=c(D)−s(D)+1. Corollary 3.5 together with Corollary 3.7 leads to the following result: Proposition 3.8. Let Lbe an oriented link with µcomponents and Sany FKT state of a diagram Drepresenting L. Then, the parities of the number of black and white holes in Sare equal when µis odd, and opposite when µis even. That is, (−1)]WS= (−1)µ+1(−1)]BS. The following result shows that the definition of Knot Floer Homology given by Ozsv´ath and Szab´o in [41] categorifies the Alexander-Conway polynomial via the Formal Knot Theory state sum shown in [27]. Theorem 3.9. Both versions of the Alexander polynomial, that given by Ozsv´ath and Szab´o [41] and that by Kauffman [27] coincide. Precisely, ΘL(t)=∆L(t). Proof. Let Ldenote the mirror image of L. Since ∆L(t) satisfies the skein relation in Theorem 3.2, after substituting z=t1 2−t−1 2and using an inductive argument, it follows ∆L(z) = ∆L(−z) = (−1)µ+1∆L(z). Hence ∆L(z) = (−1)µ+1∆L(z). Note that the labels used in [·|·] for positive (negative) crossings are equal to those used in h·|·i for negative (positive) crossings. As a consequence ∆L(z) = X S∈S (−1)]BShD|Si=X S∈S (−1)]BS[D|S]. Combining these facts with Proposition 3.8 one gets: ∆L(t) = (−1)µ+1∆L(t) = = (−1)µ+1 X S∈S (−1)]BS[D|S] = X S∈S (−1)µ+1(−1)]BS[D|S] = =X S∈S (−1)]WS[L|S] = ΘL(t). 64 Section 3.3. Disproving a proposed differential Hence the non-trivial homology groups associated to Dare H15 2,6(D)≃H13 2,5(D)≃H11 2,4(D)≃H9 2,3(D)≃H7 2,2(D)≃Z2. In fact, this computation matches with the one expected for knot 51: since this knot is alternating, its Knot Floer homology is known to have five non-trivial groups disposed in a diagonal (that is, appearing in Hi,i+kwith kbeing a constant). Now consider the diagram D0. According to Rong’s model its chain complex, which has 49 generators, is the following: Chapter 3. Knot Floer Homology and FKT states 65 66 Section 3.3. Disproving a proposed differential If one computes its homology the non-trivial groups are H9,7(D0)≃H9,6(D0)≃H5,3(D0)≃Z2, H8,6(D0)≃H8,5(D0)≃H7,4(D0)≃Z2 2, H7,5(D0)≃H6,4(D0)≃Z3 2. The computations of the homology groups in these examples are summarized in Figure 3.9. One can see that both homologies are not equivalent, even when allowing some shifting on the gradings. Chapter 3. Knot Floer Homology and FKT states 67 Figure 3.9: The first (second) board summarizes Hi,j(D) (Hi,j(D0)). A cell indexed by (i, j) represents Hi,j. Blank cells correspond to trivial groups. Although we have shown that this model is not accurate enough, we think that there are some interesting ideas in Rong’s approach which could be used as a start point for finding an appropriate differential. Focusing on the obstructions found in Example 3.12, our idea is to extend the definition of adjacency between two different states, that is, allowing other moves in addition to those shown in Figure 3.7. I have been discussing these ideas with Louis Kauffman and this is the topic of an ongoing joint project. 68 Section 3.3. Disproving a proposed differential Chapter 4 A geometric realization of the extreme Khovanov cohomology 4.1 Introduction The Khovanov cohomology of knots and links was introduced by Mikhail Khovanov at the end of last century (see [28]) and nicely explained by Bar-Natan in [8]. In [57] Viro interpreted it in terms of enhanced states of diagrams. This chapter is devoted to explain a new approach to the extreme Khovanov cohomology of a link in terms of the independence complex of a graph. More precisely, using Viro’s point of view, in Theorem 4.4 we prove that the hypothetical extreme Khovanov cohomology of a link is equal to the cohomology of the independence simplicial complex of its Lando graph. The Lando graph [17] of a link diagram was studied by Morton and Bae in [7], where they proved that the hypothetical extreme coefficient of the Jones polynomial is equal to certain numerical invariant of the graph, named in Section 4.3 independence number. Hence, on one hand the Jones polynomial can be seen as the bigraded Euler characteristic of the Khovanov cohomology, and on the other hand the formula for the independence number certainly suggests the formula of an Euler characteristic. Both ideas together gave us the key for understanding the extreme Khovanov cohomology in terms of Lando graph. This new approach to Khovanov cohomology is detailed in Section 4.4. There are many other interesting constructions of graphs starting with a link diagram (see for example [20]), not to be confused with the Lando graph. In addition, there are other very interesting ways of trying to understand the Khovanov cohomology as the cohomology of something. For example in [32] Lipshitz and Sarkar construct, in an explicit and combinatorial way, a chain bicomplex that produces the Khovanov cohomology. In [34] it was shown that the independence number can take any value, hence there are links (in fact knots) with arbitrary extreme coefficients. We also extended this idea to Khovanov cohomology in Theorem 4.16, by proving that there 69 70 Section 4.2. Khovanov cohomology Figure 4.1: The smoothing of a crossing according to its Aor B-label. A-chords (B-chords) are represented by dark (light) segments. are links (in fact knots) with an arbitrary number of non-trivial extreme Khovanov cohomology modules. These knots are examples of H-thick knots (see [29]). The basic piece of these examples is a link with exactly two non-trivial extreme Khovanov cohomology modules, constructed in Theorem 4.13. Not only this result but the construction of the example is interesting in itself, as it shows an interesting relation between the homology of a simplicial complex and the extreme Khovanov cohomology of a link. We develop these results in Sections 4.5 and 4.6. The next two sections are mainly devoted to review concepts as Khovanov cohomology, independence complex or Lando graph, in order to fix the conventions about signs, gradings and notation that we are going to use through the chapter. 4.2 Khovanov cohomology In this section we review Khovanov cohomology in a concise way by following Viro’s approach in terms of enhanced states [57] and precise some concepts like extreme complexes and extreme cohomology modules. Let Dbe an oriented diagram of a link Lwith ccrossings and writhe w=p−n, with pand nbeing the number of positive and negative crossings in D, according to the sign convention shown in Figure 1.2. A state sassigns a label, Aor B, to each crossing of D. Let Sbe the collection of 2cpossible states of D. For s∈ S assigning a(s)A-labels and b(s)B-labels, write σ=σ(s) = a(s)−b(s). The result of smoothing each crossing of Daccording to its label following Figure 4.1 is a collection sD of disjoint circles embedded in the plane together with some A and B-chords (segments joining the circles in the site where there was a crossing). We represent A-chords as dark segments, and B-chords as light ones. See Figure 4.2. Enhance the state swith a map ewhich associates a sign i=±1 to each of the |s|circles in sD. Unless otherwise stated, we will keep the letter sfor the enhanced state (s, e) to avoid cumbersome notation. Write τ=τ(s) = P|s| i=1 i, Chapter 4. A geometric realization of the extreme Khovanov cohomology 71 Figure 4.2: The first row shows a diagram Drepresenting 31, a state sand sD. Here |s|= 2. The four related enhanced states are shown in the second row. From left to right the values of τare 2,−2,0,0. and define, for the enhanced state s, the integers i=i(s) = w−σ 2, j =j(s) = w+i+τ. The enhanced state tis adjacent to sif the following conditions are satisfied: 1. i(t) = i(s) + 1 and j(t) = j(s). 2. The labels assigned by tare identical to those assigned by sexcept in one (change) crossing x=x(s, t), where sassigns an A-label and taB-label. 3. The signs assigned to the common circles in sD and tD are equal. Note that the circles which are not common to sD and tD are those touching the crossing x. In fact, passing from sD to tD can be realized by melting two circles into one, or splitting one circle into two. The different possibilities according to the previous points are shown in Figure 4.3. Figure 4.3: All possible enhancements when melting two circles are: (++ → +),(+− → −),(−+→ −). The possibilities for the splitting are: (− → −−),(+ → +−) or (+ → −+). We review now the chain complex and cohomology modules associated to a diagram Drepresenting a link as presented by Viro in [57]. 72 Section 4.3. Lando graph and its cohomology Let Rbe a commutative ring with unit. Let Ci,j(D) be the free module over R generated by the set of enhanced states sof Dwith i(s) = iand j(s) = j. Order the crossings in D. Now fix an integer jand consider the ascendant complex . . . →Ci,j(D)di −→ Ci+1,j(D)−→ . . . with differential di(s) = P(s:t)t, where (s:t) = 0 if tis not adjacent to sand (s:t) = (−1)kotherwise, with kbeing the number of B-labeled crossings coming after the change crossing x. It turns out that di+1 ◦di= 0 and the corresponding cohomology modules over R Hi,j(D) = ker(di) im(di−1) are independent of the diagram Drepresenting the link Land the ordering of its crossings, that is, these modules are link invariants. They are the Khovanov cohomology modules Hi,j(L) of L([28], [8]), as presented by Viro in [57] in terms of enhanced states. Let jmin =jmin(D) = min{j(s)/ s is a enhanced state of D}. We will refer to the complex {Ci,jmin (D), di}as the extreme Khovanov complex, and to the corresponding modules Hi,jmin (D) as the (hypothetical) extreme Khovanov modules. Indeed, there are analogous definitions for a certain jmax. Note that the integers jmin and jmax depend on the diagram D, and may differ for two different diagrams representing the same link, as can be seen in Figure 4.4. Figure 4.4: Two diagrams Dand D0representing knot 41.jmin(D) = −5 and jmin(D0) = −7. (Proposition 4.1 makes these computations easier.) 4.3 Lando graph and its cohomology Let Gbe a graph. A set σof vertices of Gis said to be independent if no vertices in σare adjacent. The independence number of Gis defined to be I(G) = X σ (−1)|σ| where the sum is taken all over the independent sets of vertices of G. The empty set is considered as an independent set of vertices with |∅| = 0. A point has independence number 0, an hexagon 2. Chapter 4. A geometric realization of the extreme Khovanov cohomology 73 Figure 4.5: A diagram Drepresenting a link with 4 trivial components, sADand the corresponding Lando graph GD. Starting from a link diagram Dwe review now the construction of its Lando graph ([7], [34]). An Aor B-chord of sD is admissible if its ends are both in the same circle of sD. Let sAbe the state assigning A-labels to all the crossings of D. The Lando graph associated to Dis constructed from sADby considering a vertex for every admissible A-chord, and an edge joining two vertices if the ends of the corresponding A-chords alternate in the same circle. Figure 4.5 exhibits a diagram D, the corresponding sADand its Lando graph GD. In [7] it was proved that the coefficient of the hypothetical lowest degree monomial of the Jones polynomial of a link Lis, up to sign, the independence number I(GD) of the Lando graph GDof D, where Dis a diagram of L. In [34] it was shown that I(GD) can take arbitrary values. The Jones polynomial can be seen as the bigraded Euler characteristic of the Khovanov cohomology [28]. The formula for the independence number suggests that each extreme coefficient of the Jones polynomial is the Euler characteristic of a certain cohomology given in terms of independent sets of vertices of the Lando graph. Combining both facts was the key for thinking in the approach of the extreme Khovanov cohomology in terms of Lando graph that we present in Section 4.4. Let XDbe the independence simplicial complex of the graph GD. By definition, the simplices σof XDare the independent subsets of vertices of GD. Let Ci(XD) be the free module over Rgenerated by the simplices of dimension i, where the dimension of a simplex σis the number of its vertices minus one. Associated to this simplicial complex we have the (standard) cochain complex . . . →Ci(XD)δi −→ Ci+1(XD)→. . . with differential δi(σ) = Pv(−1)kσ∪ {v}where vruns over the set of vertices of GDwhich are not adjacent to any vertex of σ, and k=k(σ, v) is the number of vertices of σcoming after vin the predetermined order of the vertices of GD. It turns out that δi+1 ◦δi= 0 and the corresponding reduced cohomology modules of XDare Hi(XD) = ker(δi) im(δi−1). 80 Section 4.5. Lando cohomology as homology of a simplicial complex σis a simplex of YDif and only if for any admissible black arc there is at least an admissible white arc which is not in σwhose ends alternate with the ends of the black arc in the same circle of sAD. Note that there are two different choices when coloring the regions; in order to get the simplest ground set of YD, choose colors in such a way that white regions contain a lower number of admissible A-chords than black regions. We are now interested in reversing the process above, namely, starting with any simplicial complex, we will construct a bipartite graph with an associated independence simplicial complex whose cohomology is equal to the homology of the original simplicial complex shifted by some degree. Again, the key point is the Alexander duality theorem together with the following result by Jonsson: Theorem 4.12. [24, Theorem 3.2] Let Xbe a simplicial complex. Then there is a bipartite graph Gwhose independence complex is homotopy equivalent to the suspension of X. The bipartite graph Gcan be constructed taking as set of vertices the disjoint union of the ground set Vof the complex X, and the set Mof maximal faces of X. The edges of Gare all pairs {v, µ}such that v∈V,µ∈Mand v6∈ µ. We want to remark that, although Theorem 4.12 holds for any simplicial complex, the graph obtained by the above procedure is not necessarily the Lando graph associated to a link diagram. A graph Gis said to be realizable if there is a link diagram Dsuch that G=GD(in [34] these graphs were originally called convertible). In the next theorem we show a realizable graph whose associated independence complex has two non-trivial cohomology groups, by starting with a simplicial complex whose homology has the same property. As the graph is realizable, Theorem 4.4 implies that there exists a link Lwhose extreme Khovanov cohomology Hi,jmin is non-trivial for two different values of i. In the following section this example will be a basic piece to obtain interesting families of H-thick knots. Theorem 4.13. There exist oriented link diagrams whose extreme Khovanov cohomology modules are non-trivial for two different values of i, that is, Hi,jmin (D) is non-trivial for two different values of i. Proof. Although our argument is equally valid for any commutative ring Rwith unit, just for convenience set R=Z, the ring of integers. Let X={∅,1,2,3,4,5,12,23,34,41}be a simplicial complex with ground set V={1,2,3,4,5}. Its topological realization is the disjoint union of a point and a square, as shown in Figure 4.9, hence its reduced homology is e H0(X)≈e H1(X)≈Z(the other homology groups being trivial). Chapter 4. A geometric realization of the extreme Khovanov cohomology 81 Figure 4.9: The topological realization of the simplicial complex Xand the graph G. The independence complex of Gis homotopy equivalent to the suspension of X∗. Figure 4.10: The graph Gis realizable, the correspondence between its vertices and the A-chords in sADis shown. Consider now the Alexander dual of X,X∗={∅,1,2,3,4,5,12,13,14,15,23,24, 25,34,35,45,123,124,134,135,234,245}. Note that |X|+|X∗|= 32 = 2|V|. Applying the combinatorial Alexander duality leads to e Hi(X)≈e H|V|−i−3(X∗) = e H2−i(X∗), which implies that e H2(X∗)≈e H1(X∗)≈Z, the other groups being trivial. Applying Theorem 4.12 (and the construction described right after) to the simplicial complex X∗leads to a graph Gconsisting in two hexagons sharing a common vertex as shown in Figure 4.9, whose independence complex XGis homotopy equivalent to the suspension of X∗. In particular, e Hi−1(X∗)≈e Hi(S(X∗)) ≈e Hi(XG). Hence, e H3(XG)≈e H2(XG)≈Zare the only non-trivial groups in the reduced cohomology of XG. In fact, as the indices are different from zero, this is still true for the (non-reduced) cohomology, so H2(XG)≈H3(XG)≈Z. An important point now is the fact that the graph Gis realizable. In fact, G=GDwith Dbeing the link diagram in Figure 4.11. Indeed, Figure 4.10 shows the correspondence between the vertices of Gand the A-chords in sAD. Consider now the link Lrepresented by the diagram Doriented as shown in Figure 4.11. Then applying Corollary 4.2 we get that jmin =c−3n− |sA|= 11−3·3−1 = 1, and by Theorem 4.4 one gets Hi,1(L)≈Zfor i= 0,1, the other cohomology groups being trivial. This concludes the proof. 82 Section 4.6. Families of H-thick knots Figure 4.11: The oriented diagram Drepresenting the link L. Each of its three components has been drawn in a different way. For the example in the previous proof we have checked with computer assistance that the ranks of the chain groups Ci=Ci(XD) and differentials δi are 1(1)11(10)43(33)73(39)52(12)13(1)1, where the rank of the differentials are parenthesized. This means that the Lando ascendant complex is 0−→ C−1δ−1 −→ C0δ0 −→ C1δ1 −→ C2δ2 −→ C3δ3 −→ C4δ4 −→ C5δ5 −→ 0, with rk(C−1) = 1, rk(δ−1) = 1, rk(C0) = 11, rk(δ0) = 10 and so on. Hence rk(H2(XD)) = 73 −39 −33 = 1 and rk(H3(XD)) = 52 −12 −39 = 1. Remark 4.14. The proof of Theorem 4.13 does not work if, for example, we start with the simplicial complex whose topological realization is a point plus a triangle. Although one gets again a graph Gsuch that XGhas two non-trivial cohomology groups, Gconsists of two hexagons with four common consecutive edges (a total of eight vertices), which is no longer a realizable graph. 4.6 Families of H-thick knots Citing Khovanov [29], there are 249 prime unoriented knots with at most 10 crossings (not counting mirror images). It is known that for all but 12 of these knots the non-trivial Khovanov cohomology groups lie on two adjacent diagonals, in a matrix where rows are indexed by jand columns by i. Such knots are called H-thin. An H-thick knot is a knot which is not H-thin. For example, any non-split alternating link is H-thin, and in the opposite direction, any adequate non-alternating knot is H-thick (see [29], Theorem 2.1 and Proposition 5.1). Up to eleven crossings, there are no knots with more that one non-trivial cohomology group in the rows corresponding to the hypothetical extreme jmax or jmin obtained from the associated diagrams in [9]. There are examples which Chapter 4. A geometric realization of the extreme Khovanov cohomology 83 seem to contradict this statement. For example knot 10132, whose Khovanov cohomology groups are trivial for j > −1 and has two non-trivial groups for j=−1, but for the diagram of 10132 taken from [9]jmax(D) = −c+ 3p+|sB|= −10 + 3 ·3 + 2 = 1. We do not know if there exists a diagram Drepresenting the knot 10132 with jmax(D) = −1. In this section we show examples of H-thick knots having any arbitrary number of non-trivial cohomology groups in the non-trivial row of smallest possible index. More precisely, we will provide a diagram Dwhose row indexed by jmin(D) is non-trivial, and hence corresponds to the non-trivial row of smallest possible index. Moreover, this row has as many non-trivial cohomology groups as desired. The basic piece in our construction is the link given in the proof of Theorem 4.13. We want to remark that Theorem 4.4 allows us to compute the extreme Khovanov cohomology of any link diagram Dby considering independently each of the circles appearing in sAD, as the non-admissible A−chords do not take part in the construction of the simplicial complex YDdescribed in Section 4.5. More precisely, let Dbe a link diagram and c1,...cnthe circles of sAD. Write Cifor the circle citogether with the admissible A−chords having both ends in the circle ci, and let Dibe the diagram reconstructed from Ciby reversing the corresponding smoothings. Then, from the construction right after Theorem 4.11 it follows that YD=YD1∗. . . ∗YDn, with ∗being the join of simplicial complexes. Recall that the join X∗Yof two simplicial complexes Xand Yis defined as the simplicial complex whose simplices are the disjoint unions of simplices of Xand Y. The reduced homology of the join of two simplicial complexes can be computed directly from the reduced homology of each of the complexes, namely e Hi(X∗Y) = X r+s=i−1e Hr(X)⊗e Hs(Y)⊕X r+s=i−2 Tor( e Hr(X),e Hs(Y)). Taking copies of the example in the proof of Theorem 4.13 one obtains a link which, by Theorem 4.11 and the above formula, has any number of non-trivial extreme Khovanov cohomology groups. Although one could also use the general formula for the Khovanov cohomology of a split link ([28, Corollary 12]), we think that our techniques are more useful in order to make computations. Even more, our understanding of extreme Khovanov cohomology in terms of Lando cohomology allows us to slightly modify a link in such a way that one obtains a knot with the same extreme Khovanov cohomology. We explain this construction in detail in Theorem 4.16 and Remark 4.17. We need first the following result: Proposition 4.15. Let ∗nXbe the join of ncopies of the simplicial complex X={∅,1,2,3,4,5,12,23,34,41}. Then e Hi(∗nX)≈Z(n i−n+1)if n−1≤i≤2n−1, and it is trivial otherwise. Proof. By induction on n. In the proof of Theorem 4.13 we saw that e H0(X)≈ e H1(X)≈Z, which is the case n= 1. For n > 1 we apply the formula for the 84 Section 4.6. Families of H-thick knots homology of a join (torsion terms do not appear in any case): e Hi(∗nX)≈Lr+s=i−1he Hr(∗n−1X)⊗e Hs(X)i ≈e Hi−1(∗n−1X)Le Hi−2(∗n−1X) ≈Z(n−1 (i−1)−(n−1)+1)LZ(n−1 (i−2)−(n−1)+1) ≈Z(n−1 i−n+1)LZ(n−1 i−n) ≈Z(n i−n+1). Theorem 4.16. For every n > 0there exists an oriented knot diagram Dwith exactly n+1 non-trivial extreme Khovanov integer cohomology groups Hi,jmin (D). Proof. Let Lbe the oriented link represented by the diagram Din Figure 4.11. Considering as ground set the chords in the unbounded region of sAD, the associated simplicial complex YDis the simplicial complex Xappearing in the proof of Theorem 4.13, whose topological realization is the disjoint union of a point and a square (Figure 4.9). Hence it has two non-trivial reduced homology groups, e H0(YD)≈e H1(YD)≈Z. Now consider the link Lnconsisting of the split union of ncopies of L. It can be represented by Dn, the disjoint union of ncopies of D, so sADnis the disjoint union of ncopies of sAD, shown in Figure 4.10. Hence its associated simplicial complex is YDn=∗nYD=∗nX, that is, the join of ncopies of X. Applying Proposition 4.15 to ∗nXone gets e Hi(YDn)≈Z(n i−n+1) for n−1≤i≤2n−1. This fact together with Theorem 4.11 shows that the extreme Khovanov cohomology of Lnhas n+ 1 non-trivial groups. Now we will construct a knot having the same extreme Khovanov cohomology groups as Ln(the value of jmin changes in general). Starting from the diagram Din Figure 4.11, add four crossings, as shown in Figure 4.12, in such a way that the resulting diagram D0has one component. Note that sAD0is obtained from sADby adding two circles with four A-chords. Consider now ncopies of D0and join them as shown in Figure 4.13. The resulting diagram D0 nis a knot diagram. Since sAD0 njust adds 5n−1 non-admissible A-chords to sADn, both diagrams Dn and D0 nshare the same Lando graph. Hence D0 nrepresents a knot having n+ 1 non-trivial groups in its extreme Khovanov cohomology. Chapter 4. A geometric realization of the extreme Khovanov cohomology 85 Figure 4.12: The first row shows Dand D0. The corresponding sADand sAD0are shown in the second row. Note that D0has one component. Figure 4.13: D0 nand sAD0 nare shown for the case n= 3. 86 Section 4.6. Families of H-thick knots Figure 4.14: This transformation reduces the number of components of the link by one, and the extreme Khovanov cohomology is preserved. The proof of Theorem 4.16 provides a family of knots having as many nontrivial extreme Khovanov cohomology groups as desired. These are examples of H-thick knots as far of being H-thin as desired. Remark 4.17. Every link Lwith µcomponents can be turned into a knot preserving its extreme Khovanov cohomology (jmin can change). One just needs to consider a diagram Dof Land add two extra crossings melting two different components into one, as shown in Figure 4.14. Since sAD0just adds two nonadmissible A-chords to sAD, both diagrams share the same Lando graph. After repeating this procedure µ−1times, the link Lis transformed into a knot. The approach to extreme Khovanov cohomology given in Theorem 4.4 leads some open questions. A first problem is studying the realization of other degrees (not just the extreme ones) of Khovanov cohomology by using similar techniques, replacing the Lando graph, which is related to the state where all the crossings are labeled with an A-label, with a graph related to states having all but one crossings marked with an A-label. An open conjecture states that the extreme Khovanov cohomology of a link diagram is homotopy equivalent to a wedge of spheres. Theorem 4.4 leads to a new approach to study this conjecture. More precisely, this conjecture can be restated in the following way Conjecture 4.18. The independence simplicial complex of the Lando graph of any link diagram is homotopy equivalent to a wedge of spheres. In this sense, giving some properties of even characterizing the possible Lando graphs which can be obtained from any link diagram seems to be an interesting problem. Chapter 4. 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