Twisted Alexander polynomials of hyperbolic knots

0.43 MB·English

by  Nathan M. Dunfield

TWISTEDALEXANDERPOLYNOMIALSOFHYPERBOLICKNOTS
NATHANM.DUNFIELD,STEFANFRIEDL,ANDNICHOLASJACKSON
2
1
0 ABSTRACT. WestudyatwistedAlexanderpolynomialnaturallyassociatedto
2
ahyperbolicknotinanintegerhomology3-sphereviaaliftoftheholonomy

n representationtoSL(2,C). ItisanunambiguoussymmetricLaurentpolyno-
a mialwhosecoefficientslieinthetracefieldoftheknot. Itcontainsinforma-
J tionaboutgenus,fibering,andchirality,andmoreoverispowerfulenoughto

9 sometimesdetectmutation.
1 Wecalculatedthisinvariantnumericallyforall313,209hyperbolicknotsin
S3withatmost15crossings,andfoundthatinallcasesitgaveasharpbound
]
onthegenusoftheknotanddeterminedbothfiberingandchirality.
T
WealsostudyhowsuchtwistedAlexanderpolynomialsvaryasonemoves
G
aroundinanirreduciblecomponentX0oftheSL(2,C)-charactervarietyofthe
h. knotgroup. Weshowhowtounderstandallofthesepolynomialsatoncein
t termsofapolynomialwhosecoefficientslieinthefunctionfieldofX0. We
a usethistohelpexplainsomeofthepatternsobservedforknotsinS3,and
m
explore a potentialrelationshipbetween thisuniversalpolynomialandthe
[ Culler-Shalentheoryofsurfacesassociatedtoidealpoints.

3
v
5
4
0 CONTENTS
3
1. Introduction 2
.
8
1.1. Basicproperties 3
0
1.4. Topologicalinformation:genusandfibering 4
1
1 1.6. Experimentalresults 4
: 1.8. Topologicalinformation:Chiralityandmutation 5
v
i 1.9. Adjointtorsionpolynomial 5
X
1.10. Charactervarieties 5
r
a 1.14. Otherremarksandopenproblems 6
2. Twistedinvariantsof3–manifolds 7
2.1. Torsionofbasedchaincomplexes 7
2.2. Twistedhomology 7
2.3. Eulerstructures,homologyorientationsandtwistedtorsionofCW
complexes 8
2.5. Twistedtorsionof3-manifolds 9
2.6. Twistedtorsionpolynomialofaknot 9
2.8. TheSL(2,F)torsionpolynomialofaknot 10
2.12. CalculationoftorsionpolynomialsusingFoxcalculus 11
2.20. ConnectiontoWada’sinvariant 14
3. Twistedtorsionofcycliccovers 15
4. Torsionpolynomialsofhyperbolicknots 17
1
2 NATHANM.DUNFIELD,STEFANFRIEDL,ANDNICHOLASJACKSON
4.1. ThediscreteandfaithfulSL(2,C)representations 18
4.3. Thehyperbolictorsionpolynomial 18
5. Example:TheConwayandKinoshita–Terasakaknots 20
5.2. Theadjointrepresentation 22
6. Knotswithatmost15crossings 22
6.1. Genus 22
6.2. Fibering 23
6.3. Chirality 23
6.4. KnotswiththesameT 23
K
6.5. Otherpatterns 24
6.6. Adjointpolynomial 24
6.7. Computationaldetails 25
7. Twistedtorsionandthecharactervarietyofaknot 25
7.3. Thedistinguishedcomponent 27
7.6. 2-bridgeknots 28
8. Charactervarietyexamples 29
8.1. Example:m003 29
8.2. Example:m006 30
8.4. m037 31
8.5. Theroleofidealpoints. 31
8.6. Idealpointsofm006 31
8.7. Idealpointsofm037 32
8.8. Generalpictureforidealpoints 32
References 34
1. INTRODUCTION
A fundamentalinvariant of a knot K in an integral homology 3-sphere Y is
itsAlexanderpolynomial∆ . While∆ containsinformationaboutgenusand
K K
fibering,itisdeterminedbythemaximalmetabelianquotientofthefundamen-
talgroupofthecomplementM Y \K,andsothistopologicalinformationhas
=
clear limits. In 1990, Lin introduced the twistedAlexander polynomial associ-
atedto K anda representation α: π (M) GL(n,F), where F is a field. These
1
→
twistedAlexanderpolynomialsalsocontaininformationaboutgenusandfiber-
ingandhavebeenstudiedbymanyauthors(seethesurvey[FV3]).Muchofthis
work has focused on those α which factor through a finite quotient of π (M),
1
whichiscloselyrelatedtostudyingtheordinaryAlexanderpolynomialinfinite
coversofM. Incontrast,westudyhereatwistedAlexanderpolynomialassoci-
atedtoarepresentationcomingfromhyperbolicgeometry.
SupposethatK ishyperbolic,i.e.thecomplement M hasacomplete hyper-
bolicmetricoffinitevolume,andconsidertheassociatedholonomyrepresen-
tationα: π (M) Isom (H3). SinceIsom (H3) PSL(2,C),therearetwosim-
1 + + ∼
→ =
ple ways toget a linear representation so we can consider thetwisted Alexan-
der polynomial: compose α with the adjoint representation to get π (M)
1
→
TWISTEDALEXANDERPOLYNOMIALSOFHYPERBOLICKNOTS 3
Aut(sl C) SL(3,C),oralternativelyliftαtoarepresentationπ (M) SL(2,C).
2 1
≤ →
TheformerapproachisthefocusoftherecentpaperofDuboisandYamaguchi
[DY];thelattermethodiswhatweuseheretodefineaninvariantT (t) C[t 1]
K ±
∈
calledthehyperbolictorsionpolynomial.
The hyperbolic torsion polynomial T is surprisingly little studied. To our
K
knowledgeithasonlypreviouslybeenlookedatfor2-bridgeknotsin[Mor,KM1,
HM,SW].Hereweshowthatitcontainsagreatdealoftopologicalinformation.
In fact, we show thatT determinesgenus andfibering for all 313,209 hyper-
K
bolicknotsinS3withatmost15crossings,andweconjecturethisisthecasefor
allknotsinS3.
1.1. Basic properties. More broadly, we consider here knots in Z -homology
2
3-spheres. The ambient manifold Y containing the knot K will always be ori-
ented,notjust orientable,andT dependsonthatorientation. Following Tu-
K
raev,weformulateT asaReidemeister-Milnortorsionasthisreducesitsambi-
K
guity;inthatsetting,weworkwiththecompactcoreofM,namelytheknotex-
teriorX : Y \int(N(K))(seeSection2fordetails).Byfixingcertainconventions
=
for lifting the holonomy representation α: π (X) PSL(2,C) to α: π (X)
1 1
→ →
SL(2,C), we construct in Section 4 a well-defined symmetric polynomial T
K
∈
C[t 1].Thefirsttheoremsummarizesitsbasicproperties:
±
1.2.Theorem. LetK beahyperbolicknotinanorientedZ -homology3-sphere.
2
ThenT hasthefollowingproperties:
K
(a) T isanunambiguouselementofC[t 1]whichsatisfiesT (t 1) T (t).
K ± K − K
=
ItdoesnotdependonanorientationofK.
(b) ThecoefficientsofT lieinthetracefieldofK.IfK hasintegraltraces,
K
thecoefficientsofT arealgebraicintegers.
K
(c) T (ξ)isnon-zeroforanyrootofunityξ.Inparticular,T 0.
K K
6=
(d) IfK denotesthemirrorimageofK,thenT (t) T (t),wherethe
∗ K K
∗ =
coefficientsofthelatterpolynomialarethecomplexconjugatesofthose
ofT .
K
(e) IfK isamphichiralthenT isarealpolynomial.
K
(f) ThevaluesT (1)andT ( 1)aremutationinvariant.
K K
−
Moreover,T bothdeterminesandisdeterminedbyasequenceofC-valued
K
torsions of finite cyclic covers of X. Specifically, let X be the m–fold cyclic
m
covercomingfromthefreeabelianizationofH (X;Z). Fortherestrictionα of
1 m
αtoπ (X ),weconsiderthecorrespondingC-valuedtorsionτ(X ,α ).Astan-
1 m m m
dardargumentshowsthatT determinesalltheτ(X ,α )(seeTheorem3.1).
K m m
More interestingly, the converse holds: the torsions τ(X ,α ) determine T
m m K
(seeTheorem4.5). ThislatterresultfollowsfromworkofDavidFried[Fri](see
alsoHillar[Hil])andthatofMenal–FerrerandPorti[MFP1].
1.3. Remark. Conjecturally, the torsions τ(X ,α ) can be expressed as ana-
m m
lytic torsions and as Ruelle zeta functions defined using the lengths of prime
geodesics. See Ray–Singer[RS], Cheeger [Che1, Che2], Müller [Mu1] and Park
4 NATHANM.DUNFIELD,STEFANFRIEDL,ANDNICHOLASJACKSON
[Par]fordetailsandbackgroundmaterial. Wehopethatthispointofviewwill
behelpfulinthefurtherstudyofT .
K
Thetorsionsτ(X ,α )areinterestinginvariantsin theirown right. Forex-
m m
ample, Menal–Ferrer and Porti [MFP1] showed that τ(X ,α ) is non–zero for
m m
any m. Furthermore, Porti [Por1] showed that τ(X ,α ) τ(X,α) T (1) is
1 1 K
= =
not obviously related to hyperbolic volume. More precisely, using a variation
on [Por2, Théorème 4.17] one can show that there exists a sequence of knots
K whosevolumesconvergetoapositiverealnumber,butthenumbersT (1)
n Kn
convergetozero.See[Por2,MFP1,MFP2]forfurtherresults.
1.4. Topological information: genus and fibering. We define x(K) to be the
Thurston norm of a generator of H (X,∂X;Z) Z; if K is null-homologous in
2 ∼
=
Y, then x(K) 2 genus(K) 1, wheregenus(K)istheleast genusofallSeifert
= · −
surfacesboundingK.Also,wesaythatK isfiberedifX fibersoverthecircle.
AkeypropertyoftheordinaryAlexanderpolynomial∆ isthat
K
x(K) deg(∆ ) 1.
K
≥ −
When K is fibered, this is an equality and moreover the lead coefficient of ∆
K
is1(here,wenormalize∆ sothattheleadcoefficientispositive). Aswithany
K
twistedAlexander/torsionpolynomial,wegetsimilarinformationoutofT :
K
1.5.Theorem. LetK beaknotinanorientedZ -homologysphere.Then
2
1
x(K) deg(T ).
K
≥ 2
IfK isfibered,thisisanequalityandT ismonic,i.e.hasleadcoefficient1.
K
Theorem1.5isanimmediateconsequencethedefinitionsbelowandof[FK1,
Theorem1.1](forthegenusbound)andoftheworkofGoda,KitanoandMori-
fuji[GKM](forthefiberedcase);seealsoCha[Cha],KitanoandMorifuji[KM2],
Pajitnov[Paj],Kitayama[Kit2],[FK1]and[FV3,Theorem6.2].
1.6. Experimentalresults. Thecalculationsin[Cha,GKM,FK1]gaveevidence
thatwhenonecanfreelychoosetherepresentationα,thetwistedtorsionpoly-
nomialisverysuccessful atdetectingboth x(K)andnon-fiberedknots. More-
over[FV1]showsthatcollectivelythetwistedtorsionpolynomialsofrepresenta-
tionscomingfromhomomorphismstofinitegroupsdeterminewhetheraknot
isfibered. However, it isnotknown if alltwistedtorsion polynomials together
alwaysdetectx(K).
Insteadofconsideringmanydifferentrepresentationsasintheworkjustdis-
cussed,wefocushereonasingle,albeitcanonical,representation.Despitethis,
we find that T alone is a very powerful invariant. In Section 6, we describe
K
computationsshowingthattheboundonx(K)issharpforall313,209hyperbolic
knotswithatmost15crossings;incontrasttheboundfrom∆ isnotsharpfor
K
2.8%oftheseknots. Moreover,amongsuchknotsT wasmoniconlywhenthe
K
knotwasfibered,whereas4.0%oftheseknotshavemonic∆ butaren’tfibered.
K
(Here we computed T numerically to a precision of 250 decimal places, see
K
Section6.7fordetails.)
TWISTEDALEXANDERPOLYNOMIALSOFHYPERBOLICKNOTS 5
Givenallthisdata,wearecompelledtoproposethefollowing,eventhoughon
itsfaceitfeelsquiteimplausible,giventhegeneralsquishynatureofAlexander-
typepolynomials.
1.7.Conjecture. ForahyperbolicknotK inS3,thehyperbolictorsionpolyno-
mialT determinesx(K),orequivalentlyitsgenus. Moreover,theknotK is
K
fiberedifandonlyifT ismonic.
K
We have not done extensive experiments for knots in manifolds other than
S3,butsofarwehavenotencounteredanyexampleswhereT doesn’tcontain
K
perfectgenusandfiberinginformation.
1.8. Topologicalinformation:Chiralityandmutation. WhenK isanamphichi-
ral,T isarealpolynomial(Theorem1.2(e)). Thisturnsouttobeanexcellent
K
wayto detect chirality. Indeed, among hyperbolic knots in S3 with at most 15
crossings,the353knotswhereT isrealareexactlytheamphichiralknots(Sec-
K
tion6.3).
Also,hyperbolicinvariantsoftendonotdetectmutation,forexamplethevol-
ume[Rub],theinvarianttracefield[MR, Corollary5.6.2], andthebirationality
type of the geometric component of the character variety [CL, Til1, Til2]. The
ordinary Alexander polynomial ∆ is also mutation invariant for knots in S3.
K
However,x(K)canchangeundermutation,andgiventhatx(K)determinesthe
degreeofT forall15crossingknots,itfollowsthatT canchangeundermu-
K K
tation;wediscussmanysuchexamplesinSection6.4.However,sometimesmu-
tationdoespreserveT ,andwedon’tknowofanyexamplesoftwoknotswith
K
thesameT whicharen’tmutants.
K
1.9. Adjointtorsionpolynomial. Aswementionedearlier,thereisanothernat-
uralwaytogetatorsionpolynomialfromtheholonomyα: π (M) PSL(2,C),
1
→
namelybyconsideringtheadjointrepresentationofPSL(2,C)onitsLiealgebra.
The corresponding torsion polynomial was studied by Dubois andYamaguchi
[DY], partlybuilding onwork of Porti[Por2]. Werefertothisinvarianthereas
theadjointtorsionpolynomialanddenoteitTadj. Wealsonumericallycalcu-
K
latedthisinvariantforallknotswithatmost15crossings. Unlikewhatwefound
forT ,thedegreeofTadj wasnot determinedbythegenusfor8,252ofthese
K K
knots. Moreover,wefound12knotswherethegenusboundfromTadj wasnot
K
sharpevenafteraccountingforthefactthatx(K)isnecessarilyanoddinteger.
Thedifferingbehaviorsofthesetwopolynomialsseemsverymysterioustous;
understanding what’s behind it might shed light on Conjecture 1.7. See Sec-
tions5.2and6.6forthedetailsonwhatwefoundforTadj.
K
1.10. Charactervarieties. Sofar,wehavefocusedonthetwistedtorsionpoly-
nomialof(aliftof)theholonomyrepresentationofthehyperbolicstructureon
M. However,thisrepresentationisalwayspartofacomplexcurveofrepresen-
tationsπ (M) SL(2,C),anditisnaturaltostudyhowthetorsionpolynomial
1
→
changes as we vary the representation. In Sections 7 and 8, we describe how
6 NATHANM.DUNFIELD,STEFANFRIEDL,ANDNICHOLASJACKSON
tounderstandallofthesetorsionpolynomialsatonce,andusethistohelpex-
plainsomeofthepatternsobservedinSection6.Forthespecialcaseof2-bridge
knots, Kim and Morifuji [Mor, KM1] had previously studied how the torsion
polynomialvarieswiththerepresentation,andourresultshereextendsomeof
theirworktomoregeneralknots.
ConsiderthecharactervarietyX(K): Hom π (M),SL(2,C) //SL(2,C),which
1
=
isanaffinealgebraicvarietyoverC.Weshowin¡Section7thate¢achχ X(K)has
anassociatedtorsionpolynomialT χ.TheseTχvaryinanunderstan∈dableway,
K K
intermsofapolynomialwithcoefficientsintheringofregularfunctionsC[X ]:
0
1.11.Theorem. LetX beanirreduciblecomponentofX(K)whichcontainsthe
0
characterofanirreduciblerepresentation.ThereisauniqueTKX0∈C[X0][t±1]so
thatforallχ∈X0onehasTKχ(t)=TKX0(χ)(t).Moreover,TKX0isitselfthetorsion
polynomialofacertainrepresentationπ (M) SL(2,F),andthushasallthe
1
→
usualproperties(symmetry,genusbound,etc.).
1.12.Corollary. LetK beaknotinanintegralhomology3-sphere.Then
(a) Theset χ X(K) deg(T χ) 2x(K) isZariskiopen.
∈ K =
(b) Theset©χ X(K) ¯Tχismonic isZªariskiclosed.
∈ ¯ K
© ¯ ª
It is naturalto focus on¯the component X of X(K) which contains the (lift
0
of)theholonomyrepresentationofthehyperbolicstructure,whichwecallthe
distinguishedcomponent.InthiscaseX isanalgebraiccurve,andweshowthe
0
followingconjectureisimpliedbyConjecture1.7.
1.13.Conjecture. LetK beahyperbolicknotinS3,andX bethedistinguished
0
componentofitscharactervariety.Then2x(K) deg(TX0)andTX0ismonicif
= K K
andonlyifK isfibered.
Attheveryleast,Conjecture1.13istrueformany2-bridgeknotsaswediscuss
in Section 7.6. We also give several explicit examples of TX0 in Section 8 and
K
usethesetoexploreapossible avenueforbringingtheCuller-Shalentheoryof
surfacesassociatedtoidealpointsofX(K)tobearonConjecture1.13.
1.14. Otherremarksandopenproblems. Forsimplicity,werestrictedourselves
heretothestudyofknots,especiallythoseinS3.However,weexpectthatmany
of the results and conjectures are valid for more general 3-manifolds. In the
broadersettings, theappropriatequestioniswhetherthetwistedtorsionpoly-
nomialdetectstheThurstonnormandfiberedclasses(see[FK1,FK2]and[FV2]
formoredetails).
Weconcludethisintroductionwithafewquestionsandopenproblems:
(a) DoesT determinethevolumeofthecomplementofK?Somecalcula-
K
tionalevidenceisgivenin[FJ]andinSection6.4inthispaper.
(b) IftwoknotsinS3havethesameT ,aretheynecessarilymutants? See
K
Section6.4formoreonthis.
(c) DoestheinvariantT containinformationaboutsymmetriesoftheknot
K
besidesinformationonchirality?
TWISTEDALEXANDERPOLYNOMIALSOFHYPERBOLICKNOTS 7
(d) DoesthereexistahyperbolicknotwithT (1) 1?
K
=
(e) IfT isarealpolynomial,isK necessarilyamphichiral?
K
(f) Foranamphichiralknot,isthetopcoefficientofT alwayspositive?
K
(g) Forfiberedknots,whyisthesecondcoefficientofT sooftenreal?This
K
coefficient is the sum of the eigenvalues of the monodromy acting on
thetwistedhomologyofthefiber.SeeSection6.5formore.
(h) Why is T ( 1) T (1) for 99.99% of the knots considered in Sec-
K K
| − | > | |
tion6.5?
Acknowledgments. WethankJérômeDubois,TaeheeKim,TakahiroKitayama,
Vladimir Markovic, Jinsung Park, Joan Porti, DanSilver, AlexanderStoimenow,
SusanWilliamsandAlexandruZaharescuforinterestingconversationsandhelp-
fulsuggestions.WeareparticularlygratefultoJoanPortiforsharinghisexpertise
andearlydraftsof[MFP1,MFP2]withus. Wealsothanktherefereeforhelpful
comments.DunfieldwaspartiallysupportedbyUSNSFgrantDMS-0707136.
2. TWISTED INVARIANTS OF3–MANIFOLDS
In this section, we review torsions of twisted homology groups and explain
how they are used to define the twisted torsion polynomial of a knot together
witharepresentationofitsfundamentalgrouptoSL(2,C). Wethensummarize
the basic properties of these torsion polynomials, including how to calculate
them.
2.1. Torsionofbasedchaincomplexes. LetC bea finitechaincomplex over
∗
afieldF. SupposethateachchaingroupC isequippedwithanorderedbasis
i
c andthateachhomology group H (C )isalsoequippedwithanorderedba-
i i
∗
sis h . Thenthereis anassociated torsioninvariant τ(C ,c ,h ) F : F\{0}
i ×
∗ ∗ ∗ ∈ =
asdescribedinthevariousexcellentexpositions[Mil,Tur3,Tur4,Nic]. Wewill
followtheconventionofTuraev, whichisthemultiplicativeinverseofMilnor’s
invariant.IfthecomplexC isacyclic,thenwewillwriteτ(C ,c ): τ(C ,c , ).
∗ ∗ ∗ = ∗ ∗ ;
2.2. Twistedhomology. Fortherestofthissection,fixafiniteCW–complex X
andsetπ: π (X).Considerarepresentationα: π GL(V),whereV isafinite-
1
= →
dimensionalvectorspaceoverF. WecanthusviewV asaleftZ[π]–module. To
define the twisted homology groups Hα(X;V), consider the universalcover X
∗
of X. Regardingπasthegroupofdecktransformationsof X turnsthecellular
e
chaincomplexC (X): C (X;Z)intoa leftZ[π]–module. WethengiveC (X)
arightZ[π]–mod∗ulest=ruct∗ureviac g : g 1 c forc C (Xe)andg π,w∗hich
e e · = − · ∈ ∗ ∈ e
allowsustoconsiderthetensorproduct
e
Cα(X;V): C (X) V.
Z[π]
∗ = ∗ ⊗
NowCα(X;V)isafinitechaincomplexofveectorspaces,andwedefineHα(X;V)
tobeit∗shomology. ∗
We call two representations α: π GL(V) and β: π GL(W) conjugate if
→ →
thereexistsanisomorphism Ψ: V W such thatα(g) Ψ 1 β(g) Ψforall
−
→ = ◦ ◦
g π.NotethatsuchaΨinducesanisomorphismofHα(X;V)withHβ(X;W).
∈ ∗ ∗
8 NATHANM.DUNFIELD,STEFANFRIEDL,ANDNICHOLASJACKSON
2.3. Eulerstructures,homologyorientationsandtwistedtorsionofCWcom-
plexes. To define the twisted torsion, we first need to introduce certain addi-
tionalstructureson which it depends. (In our final application, most of these
willcomeoutinthewash.)First,fixanorientationofeachcellofX.Thenchoose
anorderingofthecellsof X sowecanenumeratethemasc ;onlytherelative
j
orderofcellsofthesamedimensionwillberelevant,butitisnotationallycon-
venienttohaveonlyonesubscript.
AnEulerliftforX associatestoeachcellc ofX acellc˜ ofX whichcoversit.
j j
Ifc˜ isanotherEulerlift,thenthereareuniqueg πsothatc˜ g c˜ . Wesay
′j j ∈ e′j = j· j
thesetwoEulerliftsareequivalentif
g(j−1)dim(cj) representsthetrivialelementinH1(X;Z).
Yj
An equivalence class of Euler lifts is called an Eulerstructureon X. The set of
EulerstructuresonX,denotedEul(X),admitsacanonicalfreetransitiveaction
by H (X;Z); see[Tur2,Tur3,Tur4,FKK]formoreontheseEulerstructures. Fi-
1
nally, ahomologyorientationfor X isjust anorientationoftheR-vectorspace
H (X;R).
∗
Nowwecandefinethetorsionτ(X,α,e,ω)associatedto X,arepresentation
α,anEulerstructuree,andahomologyorientationω.IfHα(X;V) 0,wedefine
τ(X,α,e,ω): 0, andso now assume Hα(X;V) 0. Up to∗sign, th6=etorsion we
seekwillbet=hatofthetwistedcellularc∗hainco=mplexCα(X;V)withrespectto
the following ordered basis. Let {v } be any basis of V,∗and {c˜ } any Euler lift
k j
representinge. Orderthebasis{c˜ v }forCα(X;V)lexicographically, i.e.c˜
j k j
v c˜ v if either j j or bot⊗h j j an∗d k k . We thus have a base⊗d
k j k ′ ′ ′
< ′⊗ ′ < = <
acycliccomplexCα(X;V)andwecanconsider
∗
τ(Cα(X;V),c v ) F×.
∗ ∗⊗ ∗ ∈
When dim(V) is even, this torsion is in fact independent of all the choices in-
volved, butwhen dim(V) isodd we needtoaugmentit as follows toremove a
signambiguity.
Pickanorderedbasish forH (X;R)representingourhomologyorientation
i
∗
ω.SincewehaveorderedthecellsofX,wecanconsiderthetorsion
τ C (X;R),c ,h R×.
∗ ∗ ∗ ∈
¡ ¢
We define β (X) i dim H (X;R) and γ (X) i dim C (X;R) , and
i = k 0 k i = k 0 k
thensetN(X) iβPi(X=) γi(X¡).Follow¢ing[FKK],whicPhg=enerali¡zestheid¢easof
= ·
Turaev[Tur1,TuPr2],wenowdefine
τ(X,α,e,ω):
=
dim(V)
( 1)N(X)·dim(V) τ Cα(X;V),c v sign τ C (X;R),c ,h .
− · ¡ ∗ ∗⊗ ∗¢· ³ ¡ ∗ ∗ ∗¢´
TWISTEDALEXANDERPOLYNOMIALSOFHYPERBOLICKNOTS 9
A straightforward calculation using the basic properties of torsion shows that
thisinvariantdoesnotdependonanyofthechoicesinvolved,i.e.itisindepen-
dentoftheorderingandorientationofthecellsof X,thechoiceofrepresenta-
tivesfore andω,andtheparticularbasisforV. Similarelementaryarguments
prove the following lemma. Here ω denotes theopposite homology orienta-
−
tiontoω,andnotethat(det α):π FfactorsthroughH (X;Z).
1
◦ →
2.4.Lemma. Ifβisconjugatetoα,thengivenh H (X;Z)andǫ { 1,1},one
1
∈ ∈ −
has
τ(X,β,h e,ǫ ω) ǫdim(V) (det α)(h) −1 τ(X,α,e,ω).
· · = · ◦ ·
¡ ¢
2.5. Twisted torsion of 3-manifolds. Let N be a 3-manifold whose boundary
is empty or consists of tori. We first recall some facts about Spinc-structures
on N; see [Tur4, Section XI.1] for details. The set Spinc(N) of such structures
admits a free and transitive action by H (N;Z). Moreover, there exists a map
1
c : Spinc(N) H (N;Z)whichhasthepropertythatc (h s) 2h c (s)forany
1 1 1 1
h H (N;Z)a→nds Spinc(N). · = +
1
∈ ∈
Now consider a triangulation X of N. By [Tur4, Section XI] there exists a
canonicalbijectionSpinc(N) Eul(X)whichisequivariantwithrespecttothe
→
actions by H (N;Z) H (X;Z). Given a representation α: π (N) GL(V), an
1 1 1
elements Spinc(N)=,andahomologyorientationωforN,wedefin→e
∈
τ(N,α,s,ω): τ(X,α,e,ω)
=
wheree istheEulerstructureonX correspondingtos. Itfollowsfromthework
ofTuraev[Tur1,Tur2]thatτ(N,α,s,ω)isindependentofthechoiceoftriangula-
tionandhencewell-defined.Seealso[FKK]formoredetailsaboutτ(N,α,s,ω).
2.6. Twistedtorsionpolynomialofaknot. LetK beaknotinarationalhomol-
ogy3-sphereY. Throughout,wewriteX : Y \int(N(K))fortheknotexterior,
K
=
whichisacompactmanifoldwithtorusboundary. Wedefineanorientationof
K to be a choice of orientedmeridian µ ; if Y is oriented, instead of just ori-
K
entable,thisisequivalenttotheusualnotion. SupposenowthatK isoriented.
Let π : π (X ) and take φ : π Z to be the unique epimorphism where
K 1 K K K
= →
φ(µ ) 0.Thereisacanonicalhomologyorientationω forX asfollows:take
K K K
>
apointasabasisforH (X ;R)andtake{µ }asthebasisforH (X ;R). Wewill
0 K K 1 K
dropK fromthesenotationsiftheknotisunderstoodfromthecontext.
Forarepresentationα: π GL(n,R)overacommutativedomainR,wede-
→
fineatorsionpolynomialasfollows. ConsidertheleftZ[π]–modulestructureon
Rn R[t 1] R[t 1]n givenby
R ± ∼ ±
⊗ =
g (v p): α(g) v tφ(g)p
· ⊗ = · ⊗
¡ ¢ ¡ ¢
for g πand v p Rn R[t 1]. Putdifferently,we getarepresentationα
R ±
∈ ⊗ ∈ ⊗ ⊗
φ: π GL(n,R[t 1]). We denote byQ(t) the field of fractions of R[t 1]. The
± ±
→
representationα φallowsustoviewR[t 1]n andQ(t)n asleftZ[π]–modules.
±
Givens Spinc(X⊗)wedefine
∈
τ(K,α,s): τ(X ,α φ,s,ω ) Q(t)
K K
= ⊗ ∈
10 NATHANM.DUNFIELD,STEFANFRIEDL,ANDNICHOLASJACKSON
tobethetwistedtorsionpolynomialoftheorientedknotK correspondingtothe
representationαandtheSpinc-structures. Callingτ(K,α,s)apolynomialeven
thoughitisdefinedasarationalfunctionisreasonablegivenTheorem2.11be-
low.
2.7.Remark. Thestudyoftwistedpolynomialinvariantsofknotswasintroduced
by Lin [Lin]. The invariant τ(K,α,s) can be viewed as a refinedversion of the
twisted Alexander polynomial of a knot and of Wada’s invariant. We refer to
[Wada,Kit1,FV3]formoreontwistedinvariantsofknotsand3-manifolds.
2.8. The SL(2,F) torsion polynomial of a knot. Our focus in this paper is on
2-dimensional representations, and we now give a variant of τ(K,α,s) which
does not depend on s or the orientation of K. Specifically, for an unoriented
knotK inaQHSandarepresentationα: π SL(2,F)wedefine
→
(2.9) Tα: tφ(c1(s)) τ(K,α,s)
K = ·
foranys Spinc(X)andchoiceoforientedmeridianµandshow:
∈
2.10.Theorem. Forα: π SL(2,F),theinvariantTαisawell-definedelement
→ K
of F(t)whichissymmetric,i.e.Tα(t 1) Tα(t).
K − = K
WewillcallTα F(t)thetwistedtorsionpolynomialassociatedtoK andα.
K ∈
Proof. Thattheright-handsideof(2.9)isindependentofthechoiceofsfollows
easily from Lemma 2.4 using the observation that det (α φ)(g) t2φ(g) for
⊗ =
g πandthepropertiesofc giveninSection2.5. ¡ ¢
1
∈
The choice of meridian µ affectsthe right-handside of (2.9)in twoways: it
isusedtodefinethehomology orientationωandthehomomorphismφ: π
→
Z. Thefirst doesn’t matterbyLemma 2.4, butswitching φto φ is equivalent
−
toreplacing t with t 1. Hence beingindependentofthechoice of meridianis
−
equivalenttothefinalclaimthatTαissymmetric.
K
Now any SL(2,F)-representation preserves the bilinear form on F2 given by
(v,w) det(vw).Usingthisobservationitisshownin[FKK,Theorem7.3],gen-
7→
eralizing[HSW,Corollary3.4]andbuildingonworkofTuraev[Tur1,Tur2],that
inourcontextwehave
τ(K,α,s) t−1 t2φ(c1(s)) τ(K,α,s)
= ·
¡ ¢
whichestablishesthesymmetryTαandhencethetheorem. (cid:3)
K
WhileingeneralTα isarationalfunction,itisfrequentlyaLaurentpolyno-
K
mialorcanbecomputedintermsoftheordinaryAlexanderpolynomial∆ .
K
2.11.Theorem. LetK beaknotinaQHSandletα:π SL(2,F)bearepresen-
K
→
tation.
(a) Ifαisirreducible,thenT αliesinF[t 1].
K ±
(b) Ifαisreducible,thenTα Tβwhereβisthediagonalpartofα,i.e.a
K = K
diagonalrepresentationwheretr β(g) tr α(g) forallg π.
= ∈
¡ ¢ ¡ ¢