Twisted cohomology for hyperbolic three manifolds

Menal-Ferrer, Pere; Porti, Joan

doi:10.48550/arxiv.1001.2242

Cited by 3 publications

(7 citation statements)

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“…More interestingly, the converse holds: the torsions τ(X m , α m ) determine T K (see Theorem 4.5). This latter result follows from work of David Fried [Fri] (see also Hillar [Hil]) and that of Menal-Ferrer and Porti [MFP1].…”

Section: Introductionsupporting

confidence: 54%

“…The torsions τ(X m , α m ) are interesting invariants in their own right. For example, Menal-Ferrer and Porti [MFP1] showed that τ(X m , α m ) is non-zero for any m. Furthermore, Porti [Por1] showed that τ(X 1 , α 1 ) = τ(X , α) = T K (1) is 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 n whose volumes converge to a positive real number, but the numbers T K n (1) converge to zero.…”

Section: Introductionmentioning

confidence: 99%

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Twisted Alexander Polynomials of Hyperbolic Knots

Dunfield

Friedl

Jackson

2012

Experimental Mathematics

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We study a twisted Alexander polynomial naturally associated to a hyperbolic knot in an integer homology 3-sphere via a lift of the holonomy representation to SL(2,C). It is an unambiguous symmetric Laurent polynomial whose coefficients lie in the trace field of the knot. It contains information about genus, fibering, and chirality, and moreover is powerful enough to sometimes detect mutation.We calculated this invariant numerically for all 313,209 hyperbolic knots in S 3 with at most 15 crossings, and found that in all cases it gave a sharp bound on the genus of the knot and determined both fibering and chirality.We also study how such twisted Alexander polynomials vary as one moves around in an irreducible component X 0 of the SL(2,C)-character variety of the knot group. We show how to understand all of these polynomials at once in terms of a polynomial whose coefficients lie in the function field of X 0 . We use this to help explain some of the patterns observed for knots in S 3 , and explore a potential relationship between this universal polynomial and the Culler-Shalen theory of surfaces associated to ideal points.

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Section: Introductionsupporting

confidence: 54%

Section: Introductionmentioning

confidence: 99%

Twisted Alexander Polynomials of Hyperbolic Knots

Dunfield

Friedl

Jackson

2012

Experimental Mathematics

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“…Thus, every representation ρ := ρ(m) of G also defines a flat vector bundle E ρ over X. Now by our assumption (2.2) on Γ and [MePo1,Proposition 2.8], the cohomology H * (X, ρ) never vanishes. Thus in order to define the Reidemeister torsion of E ρ , one needs to fix bases in the homology H * (X, ρ).…”

Section: Introductionmentioning

confidence: 94%

“…However, this is no longer the case for the representations σ k , k ∈ Z − 1 2 Z. Furthermore, Theorem 1.2 can not be applied to a group Γ ′ corresponding to an acyclic spin-structure of X since by [MePo1,Lemma 2.4] such a group never satisfies the assumption (2.2). This assmuption is yet needed for our compuations involving the Selberg trace formula and in order to apply the results about the meromorphic continuation of the zeta functions obtained in [Pf].…”

Section: Introductionmentioning

confidence: 99%

Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

Pfaff

2014

Math. Z.

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For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL 2 (C) to the corresponding Reidemeister torsion. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti. 1 2 JONATHAN PFAFFLet us now turn to the actual setup of this paper. We let X be a hyperbolic 3-manifold which is not compact but of finite volume. If G := SL 2 (C), K := SU(2), then X := G/K can be identified with the hyperbolic 3-space and there exists a discrete, torsion free subgroup Γ of G such that X = Γ\ X. One can identify Γ with the fundamental group of X. Throughout this paper, we assume that Γ satisfies a certain condition, which is formulated in equation (2.2) below. Let ρ be an irreducible finite-dimensional complex representation of G. Restrict ρ to Γ and let E ρ be the associated flat vector-bundle over X. One can equip E ρ with a canonical metric, called admissible metric. The associated Laplace operator ∆ p (ρ) on E ρ -valued p-forms has a continuous spectrum and therefore, the heat operator exp(−t∆ p (ρ)) is not trace class. So the usual zeta function regularization can not be used to define the analytic torsion. However, picking up the concept of the b-trace of Melrose, employed by Park in a similar context, in [MP2] we introduced the regularized trace Tr reg e −t∆p(ρ) of the operators e −t∆p(ρ) and in this way we extended the definition of the analytic torsion to the non-compact manifold X. These definitions will be reviewed in section 4 below. Let T X (ρ) denote the analytic torsion on X associated to ρ.The aim of the present article is to find a suitable generalization of the aforementioned Cheeger-Müller theorems to the specific non-compact situation of the hyperbolic 3-manifold X with cusps. For m ∈ 1 2 N we let ρ(m) denote the 2m-th symmetric power of the standard representation of SL 2 (C). Let X be the Borel-Serre compactifcation of X. We recall that X is a compact smooth manifold with boundary and that X is diffeomorphic to the interior of X. Moreover, X and X are homotopy-equivalent. Thus, every representation ρ := ρ(m) of G also defines a flat vector bundle E ρ over X. Now by our assumption (2.2) on Γ and [MePo1, Proposition 2.8], the cohomology H * (X, ρ) never vanishes. Thus in order to define the Reidemeister torsion of E ρ , one needs to fix bases in the homology H * (X, ρ). However, by [MP2, Lemma 7.3] the bundle E ρ is L 2 -acyclic and thus the metrics on X and E ρ do not give such bases. This fact is a significant difference to the situation on a closed manifold described above and causes additional difficulties. To overcome this problem, we use the normalized Reidemeister torsion which was introduced by Menal-Ferrer and Porti [MePo2]. Recall that the boundary of X is a disjoint union of finitely many tori T i . For each i fix a non-trivial cycle θ i ∈ H 1 (T i...

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“…Fix for each j a nonzero vector ω j (m) ∈ V (m) fixed under ρ(m)(π 1 (T j )). The set of such vectors is a onedimensional complex vector space, see [MePo1,Lemma 2.4]. Fix moreover for each j a non-trivial cylce θ j ∈ H 1 (T j ; Z) and let η j ∈ H 2 (T j ; Z) be a Z-generator of H 2 (T j ; Z).…”

Section: Ruelle Zeta Functions and The Asymptotics Of Normalized Reid...mentioning

confidence: 99%