5.1 Basic properties
The next proposition collects some basic properties of the volume formωK.
Proposition 5.1 LetK be an knot inS3. The volume formωK satisfies the following assertions:
(1) the orientation onReg(K) induced by the1–volume form ωK is the orientation defined in [10],
(2) the volume formωK does not depend on the orientation ofK.
(3) if K∗ denotes the mirror image of K, then the canonical isomorphism Λ: Reg(K)→ Reg(K∗) satisfiesΛ∗(ωK∗)=−ωK.
Proof Items (2) and (3) are immediate consequence of Corollary 3.5 (3) and (4) respectively.
Let us prove the first one. Take a 2n–plat presentation ˆζ of the knotKand letOζˆ denote the orientation onReg( ˆζ) defined in [10]. Precisely the orientationOζˆ is defined [10, Definition 3.5] by the rule Reg( ˆζ)=(−1)n
Qb1∩Qb2
. We easily observe that the orientation induced byvbRTi(Bi) (resp.vRbS(S)) on bRTi(Bi) (resp.bRS(S)) corresponds to the one defined in [10, Sections 3 & 4.1] (resp. in [10, Section 4.1]). Formula7implies thatOζˆ is precisely the orientation induced byωζˆ.
5.2 A connected sum formula
The aim of this subsection is to prove a connected sum formula to compute the volume formωK1]K2 associated to a composite knot K1]K2 in terms of the volume formsωK1 andωK2. The beginning of our study consists in a technical lemma in which we describe the regular part of the representation space of the group ofK1]K2 in terms of the regular parts of the groups ofK1 andK2.
Recall that an abelian representation of GK is conjugate to one and only one of the ϕθ: GK → SU(2) given by ϕθ(m) = cos(θ)+sin(θ)i with 0 6 θ 6 π. The abelian representation ϕθ is called regular if e2θi is not a zero of the Alexander polynomial ofK. For such representation, E. Klassen proved in [12, Theorem 19] that Hϕ0
θ(MK)∼=H0(MK;R)∼=RandHϕ1
θ(MK)∼=H1(MK;R)∼=R.
LetK=K1]K2 be a composite knot. Letmi (resp. Gi) denote the meridian (resp. the group) ofKi, fori=1,2. Letmdenote the meridian of K. The groupGK of K is the amalgamated productG1∗UG2. Here the subgroup Uof amalgamation is the normal closure ofm1m−12 in the free productG1∗G2 (see [4, Proposition 7.10]). Observe that inGK we have m=m1 =m2.
To each representationρ∈R(GK) corresponds the restrictionsρi=ρ|Gi ∈R(Gi) for i = 1,2. We write ρ = ρ1∗ρ2. If ρi ∈ R(Gi) then we can form the “composite"
representationρ =ρ1∗ρ2 of GK if and only if ρ1(m1)= ρ2(m2). In particular, to each representationρ1∈R(G1) (resp.ρ2 ∈R(G2)) corresponds, up to conjugation, a unique abelian representationα2: G2 →SU(2) (resp. α1: G1 → SU(2)) such that ρ1(m1)=α2(m2) (resp.α1(m1)=ρ2(m2)). Hence, we can consider the following two maps:
ι1: bR(MK1)→bR(MK), [ρ1]7→[ρ1∗α2], ι2: bR(MK2)→bR(MK), [ρ2]7→[α1∗ρ2].
Using this notation, we prove:
Lemma 5.2 LetK =K1]K2be the connected sum of the knotsK1 andK2. IfR1(resp.
R2) denotes the space of conjugacy classes of regular representationsρ1: G1→SU(2) (resp.ρ2: G2 → SU(2)) such that the associated abelian representation α2: G2 → SU(2)(resp.α1: G1→SU(2)) is also regular, then:
(1) R1 (resp. R2) is an open submanifold ofReg(K1) (resp.Reg(K2)), (2) Reg(K)=ι1(R1)∪ι2(R2),
(3) ι1: R1→ Reg(K)(resp.ι2: R2→ Reg(K)) is an immersion.
Proof First observe thatRi is obtained formReg(Ki) by removing a finite number of points. Precisely the removed points coincide with the regular representations whose associated abelian representation corresponds to a zero of the Alexander polynomial of K3−i for i=1,2. This is sufficient to prove the first assertion.
We next prove the second item. In [12, Proposition 12], E. Klassen gives the structure of the irreducible representation space ofGK. Any irreducible representationρ∈R(Ge K) is conjugate to one of the following “composite" representations: ρ1∗α2, α1∗ρ2 or ρ1∗ρ2, where ρi ∈ eR(Gi) and αi ∈ A(Gi), i = 1,2. InFigure 4 we illustrate our observations and we give a picture of the representation space of the group of K=K1]K2, whereK1 is the torus knot of type (2,5) andK2 is the trefoil knot (ie, the torus knot of type (2,3)). The proof of the second assertion is based on the following two claims.
Claim 5.3 Ifρi∈eR(Gi), then ρ=ρ1∗ρ2 is not regular.
Proof of the claim Corresponding to the representation ρ = ρ1 ∗ρ2 and to the amalgamated groupGK =G1∗UG2 is the Mayer–Vietoris sequence
0 //R δ //Hρ1(GK) κ∗ //Hρ1
Proof of the claim Corresponding to the irreducible representationρ=ρ1∗α2 and to the amalgamated group GK =G1∗UG2, the Mayer–Vietoris sequence in twisted cohomology reduces to
Assume that ρ1 is not regular or that α2 is not abelian regular. Thus we have dimHρ2
1(G1)>2 or dimH2α
2(G2)>1. Hence, in each case, dimH2ρ(GK)>2, which proves thatρ1∗α2 is not regular.
Now, if ρ1 is assumed to be regular and if α2 is assumed to be abelian regular, then dimHρ1
Figure 4: Irreducible representations of the group ofK=K1]K2.
Let us establish the last point. The epimorphism G1∗UG2 ////G1∗UU∼=G1 induces the injection H1ρ1(G1),→Hρ1(GK). Hence,ι1: R1 → Reg(K) is an immersion. The same argument proves thatι2: R2→ Reg(K) is also an immersion.
Proposition 5.5 Let K =K1]K2 be a composite knot. With the same notation as in Lemma 5.2, we have
ι∗1(ωK)=ωK1 andι∗2(ωK)=ωK2.
Proof Assume that K1 = ζˆ1 is presented as a 2n–plat and K2 = ζˆ2 as a 2m–plat.
If δn−1 denotes the (n−1)–shift operator, then ζ1\δn−1ζ2 is a 2(n+m−1)–plat
presentation of the connected sumK1]K2 (seeFigure 5). Lemma 5.2gives the form of the regular representations ofGK in terms of the regular ones ofG1 andG2. To prove ι∗1(ωK) = ωK1, we use the same method as inLemma 4.3. Consider the punctured 2–sphere S0 =S ∪ {s2n+1, . . . ,s2n+2(m−1)} as inFigure 5. This sphere allows us to split the exterior of the knotK. UseSection 3.4.3and let
f: (−2,2)×(S2)2n→(−2,2)×(S2)2n+2(m−1) be the map defined by
f((2 cos(θ),P1, . . . ,P2n))=(2 cos(θ),P1, . . . ,P2n,
ε1P2n,−ε1P2n, . . . , εm−1P2n,−εm−1P2n).
Hereεi ∈ {±1} depends on the braidζ2. As in the proof ofLemma 4.3, f induces a transformationef: ReS(S)→ReS0(S0).
The same ideas as in the proof ofLemma 4.3imply thatef induces a volume preserving diffeomorphism (−1)nQb1∩bQ2→(−1)n+m−1Qb01∩Qb02in a neighbourhood of each regular representation ofG1. We prove equalityι∗2(ωK)=ωK2 by the same arguments.
1
2
D1ın 12
S0
Figure 5: Connected sum of plats.
5.3 An explicit computation
In this subsection, we give without a proof an explicit formula for the volume form ωK2,q associated to the torus knot K2,q of type (2,q). Here q > 3 denotes an odd integer. Other explicit computations of the volume form ωK are given in [5] for fibered knots—and in particular for all torus knots—using the non abelian Reidemeister torsion. Recall that the group of the torus knotK2,q admits the well-known presentation GK2,q =hx,y|x2=yqi.
Proposition 5.6 LetK2,qbe the (left-handed) torus knot of type(2,q). Each irreducible representation ofGK2,q intoSU(2)is conjugate to one and only one of theρ`,t: GK2,q →
Equation14is due to E Klassen [12]. One can find a proof ofProposition 5.6in [8].
5.4 Remarks on the volume of the manifold Reg(K)
In the introduction, we mentioned a result of Witten about the symplectic volume of the moduli space of a Riemannian surface. To close this paper we collect some remarks about the volume ofReg(K) (with respect to the canonical volume form ωK).
Because of the non-compactness ofReg(K), it is unclear ifR
Reg(K)ωK always exists in general. In the case of the torus knot of type (2,q), the integral exists and we have, usingProposition 5.6,
This formula is obtained by an elementary computation based on the knowledge of explicit descriptions of the representation space of the group of the torus knotK2,q and of the volume formωK2,q.
Suppose thatR
Reg(K)ωK exists, thenR
Reg(K∗)ωK∗ also exists and usingProposition 5.1 (3), we haveR
The author wishes to express his gratitude to Michael Heusener, Joan Porti, Daniel Lines, Vladimir Turaev, Michel Boileau, Jean-Yves LeDimet for helpful discussions related to this paper.
The author also would like to thank the referee for his careful reading of the paper and for his remarks which have contributed to making it clearer.
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Section de Math´ematiques, Universit´e de Gen`eve CP 64 2–4 Rue du Li`evre, CH-1211 Gen`eve 4, Switzerland Jerome.Dubois@math.unige.ch
Received: 24 September 2004 Revised: 25 August 2005