Algorithmic Invariants for Alexander Modules

ALGORITHMIC INVARIANTS FOR ALEXANDER MODULES J. GAGO-VARGAS, M.I. HARTILLO-HERMOSO, AND J.M. UCHA-ENR´ IQUEZ Abs ac . Le Gbe a g oup gi en by gene a o s and ela ions. I is possible o compu e a p esen a ion ma ix o a module o e a ing h ough Fox’s di - e en ial calculus. We show how o use G ¨obne bases as an algo i hmic ool o compa e he chains o elemen a y ideals de ined by he ma ix. We apply his echnique o classical examples o g oups and o compu e he elemen a y ideals o Alexande ma ix o kno s up o 11 c ossings wi h he same Alexande polynomial. 1. In oduc ion Le G=hx: ibe a g oup gi en by gene a o s and ela ions, whe e x= (x1, . . . , xn) is a base o he ee g oup Fand = ( 1, . . . , m) a e he ela ions. Th ough Fox’s di e en ial calculus [C owell e al.(1977)] i is possible o compu e he p esen a ion ma ix o he Alexande module o he g oup. We e iew b ie ly hese concep s. We build om G he ing o he g oup ZG. A de i a ion o e he g oup ing is a map D:ZG→ZGsuch ha D(ν1+ν2) = Dν1+Dν2, D(ν1ν2) = (Dν1) (ν2) + ν1Dν2, whe e is he i ialize and ν1, ν2∈ZG. Fo elemen s in G, he second condi ion is D(g1g2) = Dg1+g1Dg2. Then a de i a ion can be seen as he unique linea ex ension o ZGo a map D:G→ZG ha e i ies he p e ious condi ion. I is known ha each gene a o xjin he g oup Gde ines a unique de i a ion Dj=∂/∂xjin ZG, such ha ∂xi ∂xj =δij. Le Hbe he abelianized g oup o G. Conside ing he g oup ings we ha e he composi ion o maps ZFDj −→ ZFγ −→ ZGa −→ ZH, whe e γis he p ojec ion and ais he abelianize . The Alexande ma ix om G is A= (aij), whe e aij =aγ∂ j ∂xi. 2000 Ma hema ics Subjec Classi ica ion. P ima y 13P10, 57M05; Seconda y 57M27. Key wo ds and ph ases. G ¨obne bases, Elemen a y ideals, in a ian s o kno s. All au ho s pa ially suppo ed by MTM2004-01165 and FQM-333. 1 2 J. GAGO-VARGAS, M.I. HARTILLO-HERMOSO, AND J.M. UCHA-ENR´ IQUEZ No e ha his ma ix is he ansposed o he ma ix de ined by [C owell e al.(1977)]. The Alexande ma ix p esen s a module o e he ing ZH. I wo g oups a e iso- mo phic hen he modules a e isomo phic. A ini e p esen a ion o Mis an exac sequence Rnα →RmΦ →M→0 whe e Rnand Rma e ee R-modules wi h espec i e bases 1,..., nand e1, . . . em. I αis ep esen ed by he ma ix Awi h espec o hese bases hen he m×nma ix Ais a p esen a ion ma ix o M. Theo em 1. [Licko ish(1998), Thm. 6.1] I A1and A2a e p esen a ion ma ices o a module M hen hey a e ela ed by a sequence o ma ix ans o ma ions o he ollowing o m and hei in e ses: (1) Pe mu a ion o ows and columns. (2) Replacemen o he ma ix A1by A10 0 1 . (3) Addi ion o an ex a column o ze os o he ma ix A1. (4) Addi ion o a scala mul iple o a ow (column) o ano he ow (column). We say ha A1and A2a e Fi ing equi alen s. De ini ion 1. Le Mbe a Rmodule, wi h an m×np esen a ion ma ix A. The - h elemen a y ideal F o Mis he ideal gene a ed by all he (m− +1)×(m− +1) mino s o A. By con en ion, F (M) = Rwhen > m and F (M) = 0 i ≤0. They o m an ascending chain Fk(M)⊂Fk+1(M). The elemen a y ideals a e independen o he p esen a ion ma ix chosen o e alua e hem. 2. Algo i hms in he ing g oup The ing ZHis commu a i e, because His an abelian g oup, and i has a special o m. P oposi ion 1. The ing ZHis isomo phic o Z[x± 1, . . . , x± n]/J, whe e Jis he ideal gene a ed by he ela ions 1, . . . , munde commu a i i y. P oo . Th ough he abelianize , all he ela ions ha e he o m Qxei i= 1, so Jis gene a ed by he elemen s Qxei i−1.  Co olla y 1. The e is an algo i hm o compa e ideals in ZH. P oo . Th ough he bijec ion be ween ideals in Z[x± 1, . . . , x± n]/J and ideals in R= Z[x± 1, . . . , x± n] ha con ains J, he p oblem is educed o compa e ideals in R. In his ing we can compu e G ¨obne bases [Sims(1994), Paue e al.(1999)], o by he isomo phism R≃Z[x1, . . . , xn, w]/hx1· · · xnw−1i[Adams e al.(1994)].  The e is no known algo i hm o decide whe he wo ma ices p esen isomo phic modules. The e a e o he in a ian s as he ideal ow (column) class [Fox e al.(1964)] o he Nakanishi index [Kawauchi(1996)]. Howe e we do no know algo i hms o compu e hem and ad hoc a gumen s a e needed o gi e hei alues o speci ic ma ices [Fox e al.(1964), Kea on e al.(2003)]. ALGORITHMIC INVARIANTS FOR ALEXANDER MODULES 3 Example 1.Le conside he g oups gi en by he p esen a ions D8=hx, y|x4= 1, y2= 1, yxy−1=x−1i, Q8=hx, y|x4= 1, x2=y2, y−1xy =x−1i. D8is he dihed al g oup o o de 8 (symme y g oup o he squa e) and Q8is he qua e nion g oup. A classical exe cise in g oup heo y is o show ha hese wo g oups a e no isomo phic. Le see how can his be accomplished wi h elemen a y ideals. Le ibe he he ela ions in D8: 1:x4= 1, 2:y2= 1, 3:yxy−1x= 1. Then ∂ 1 ∂x = 1 + x+x2+x3,∂ 2 ∂x = 0,∂ 3 ∂x =y+x−1, ∂ 1 ∂y = 0,∂ 2 ∂y = 1 + y, ∂ 3 ∂y = 1 −x−1 In he abelianized g oup we add he ela ion xy =yx, so x2= 1, y2= 1. The Alexande module o he g oup has a p esen a ion ma ix M(D8) = 2+2x0x+y 0y+ 1 1 −x, o e he ing Z[x±, y±]/hx2−1, y2−1i. We p oceed in an analogous way wi h Q8. We w i e he ela ions s1:x4= 1, s2:x2y−2= 1, s3:xy−1xy = 1, and ∂s1 ∂x = 1 + x+x2+x3,∂s2 ∂x = 1 + x, ∂s3 ∂x = 1 + xy−1, ∂s1 ∂y = 0,∂s2 ∂y =−x2(1 + y),∂s3 ∂y =−xy−1+y−1 As be o e, in he abelianized g oup he ela ions a e educed o x2= 1, y2= 1 and a p esen a ion ma ix o he Alexande module is M(Q8) = 2+2x1 + x1 + xy 0−1−y−xy +y o e he ing Z[x±, y±]/hx2−1, y2−1i. We compu e a G ¨obne basis in Z[x±, y±] o he second elemen a y ideal. Adding he polynomials x2−1, y2−1, we ge F2(M(D8)) = h4,1 + y, 1−xi, F2(M(Q8)) = h2,1 + x, 1 + yi. They a e di e en so he g oups a e no isomo phic. Example 2.In [Kanenobu(1986)] i is de ined a class o kno s Kp,q, wi h p, q ∈N, ha has he Alexande ma ix Ap,q = 2−3 + 1 (p−q) 0 2−3 + 1 . Lemma 2 o [Kanenobu(1986)] asse s ha Kp,q and Kp0,q0ha e isomo phic Alexan- de modules i and only i |p−q|=|p0−q0|. Le us show how o apply ou ap- p oach o gi e a new p oo o his lemma. I he modules a e isomo phic hen he second elemen a y ideals F2mus coincide. A G ¨obne basis o he ideal is equal o { 2−3 + 1, p −q}, so F2(Ap,q) = F2(Ap0,q0) i and only i |p−q|=|p0−q0|. 4 J. GAGO-VARGAS, M.I. HARTILLO-HERMOSO, AND J.M. UCHA-ENR´ IQUEZ 3. An applica ion o kno heo y One o he main in a ian s in kno heo y is he undamen al g oup o he kno complemen . The Alexande ma ix can be compu ed om he Sei e ma ix, and wi h he ables lis ed in [Bu de e al.(1985)] and [Li ings on(2004)] we can ge he Alexande ma ix o kno s up o 11 c ossings. As an applica ion o he algo i hm de- sc ibed be o e we gi e a lis o kno s wi h he same Alexande polynomial (g ouped by boxes) and whe e he elemen a y ideals gi e mo e in o ma ion o dis inguish kno s (see Table 1). Fo example, om Table 1 we deduce ha 11a102 and 11a181 a e di e en , bu we canno say any hing abou 11a102 and 11a199. The i s s ep was o compu e he Alexande ma ix o he kno and educe i h ough he ans o ma ions gi en by Theo em 1. In all cases we ha e go a mos a 2 ×2 p esen a ion ma ix, so F3is always equal o R. The G ¨obne bases we e compu ed o e he ing Z[ , w], adjoining o he ideals he polynomial w −1. Re e ences Adams e al.(1994). W.W. Adams, P. Lous aunau, An in oduc ion o G ¨obne bases, olume 3 o G adua e S udies in Ma hema ics, Ame ican Ma hema ical Socie y, P o idence, RI, 1994. Bu de e al.(1985). G. Bu de, H. Zieschang, Kno s, olume 5 o de G uy e S udies in Ma hema - ics, Wal e de G uy e , Be lin and New Yo k, 1985. C owell e al.(1977). R.H. C owell, R.H. Fox, In oduc ion o Kno Theo y, olume 57 o G adua e Tex s in Ma hema ics, Sp inge -Ve lag, New Yo k, 1977. Fox e al.(1964). R.H. Fox, N. Smy he, An ideal class in a ian o kno s, P oc. Ame . Ma h. Soc., 15:707–709, 1964. Kanenobu(1986). T. Kanenobu, In ini ely many kno s wi h he same polynomial in a ian , T ans. Ame . Ma h. Soc., 97:158–162, 1986. Kawauchi(1996). A. Kawauchi, A su ey o kno heo y, Bi kh¨ause Ve lag, Basel, 1996. Kea on e al.(2003). C. Kea on, S.M.J. Wilson, Kno modules and he Nakanishi index, P oc. Ame . Ma h. Soc., 131:655–663, 2003. Licko ish(1998). W.B.R. Licko ish, An in oduc ion o kno heo y, olume 175 o G adua e Tex s in Ma hema ics, Sp inge -Ve lag, New Yo k, 1998. Li ings on(2004). C. Li ings on, Table o kno in a ian s a h p://www.indiana.edu/~kno in o/. Paue e al.(1999). F. Paue , A. Un e ki che , G ¨obne Bases o Ideals in Lau en Polynomials Rings and hei Applica ion o Sys ems o Di e ence Equa ions, Appl. Algeb a Eng g. Comm. Compu ., 9:271–291, 1999. Sims(1994). C.C. Sims, Compu a ion wi h ini ely p esen ed g oups, olume 48 o Encyclopedia o Ma hema ics and i s Applica ions, Camb idge Uni e si y P ess, Camb idge, 1994. Dep o. de ´ Algeb a, Uni e sidad de Se illa. Apdo. 1160, E-41080 Se illa (Spain) E-mail add ess:[email p o ec ed] Dep o. de Ma em´ a icas, Uni e sidad de C´ adiz. Apdo. 40, E-11510 Pue o Real (Spain) E-mail add ess:[email p o ec ed] Dep o. de ´ Algeb a, Uni e sidad de Se illa. Apdo. 1160, E-41080 Se illa (Spain) E-mail add ess:[email p o ec ed] ALGORITHMIC INVARIANTS FOR ALEXANDER MODULES 5 Table 1. Elemen a y ideals Name Elem. ideal F2 11n100 R 937 h3, −2i 11n97 R 61R 946 h3, + 1i 11a102 R 11a181 h3, −2i 11a199 R 10113 R 11a107 h2, 2+ + 1i 11a347 h2, 2+ + 1i 11a187 R 11a249 h3, −2i 11a38 R 11a8R 11a132 h2, 2+ + 1i 11a352 h3, 2− + 1i 11a6R 1065 h2, 2+ + 1i 1077 R 11n71 h− 2+ −1i 11n75 h− 2+ −1i 10103 h5, + 1i 1040 R Name Elem. ideal F2 10140 h2, 2− + 1i 11n73 h 2− + 1i 11n74 h 2− + 1i 11n116 R 11n49 h2, 2+ + 1i 11n1R 948 h3, + 1i 10155 h + 1,5i 11n37 R 89R 11n164 h 2− + 1i 11n85 R 818 h 2− + 1i 924 R 10163 h2, 2+ + 1i 11n87 R 928 R 929 R 1059 R 11n66 R 940 h 2−3 + 1i 1042 R 1075 h3, + 1i Name Elem. ideal F2 1060 R 11n165 h2, 2+ + 1i 11a223 R 11n148 h5, 2+ 2 + 1i 11a108 R 11a139 R 11a231 h 2− + 1i 11a57 h 2− + 1i 11a88 R 11a109 R 11a44 h 2− + 1i 11a47 h 2− + 1i 10123 h 4−3 3+ 3 2−3 + 1i 11a28 R 1087 R 1098 h1− + 2i 11a165 h2,1− + 2i 11a58 R 11n72 h1− + 2i 10144 h2,1− + 2i 11n99 R 11n83 h2, 2+ + 1i 941 h7,1 + i 6 J. GAGO-VARGAS, M.I. HARTILLO-HERMOSO, AND J.M. UCHA-ENR´ IQUEZ Name Elem. ideal F2 11n162 h2, 2+ + 1i 939 R 11a31 R 11a317 h + 1,5i 1067 R 1074 h3, + 1i 11n68 R 11a157 h2, 4+ 2+ 1i 11a264 R 11a305 R 11a80 R 1063 h2, 2+ + 1i 938 R 11a277 h + 1,3i 11a99 R 11a196 h7, + 1i 11a216 R 11a286 R