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