An L<sup>2</sup>–Alexander–Conway Invariant for Knots and the Volume Conjecture

Li, Weiping; Zhang, Weiping

doi:10.1142/9789812772527_0025

Cited by 13 publications

(13 citation statements)

References 21 publications

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“…The

L^{2}

‐torsion function

ρ^{(2)} false(\overset{M}{true} false)

which is determined by

ρ_{u}^{(2)} false(\overset{M}{true} false) \in {prefixWh}^{w} false(π false)

and whose value at

t = 1

is the

L^{2}

‐torsion

ρ^{(2)} false(\overset{M}{true} false)

. This

L^{2}

‐torsion function is studied for instance in . Moreover,

{trueprefixlim sup}_{t \to \infty} ρ^{false(2 false)} (\tilde{M}) (t) / prefixln false(t false)

and

{trueprefixlim inf}_{t \to 0} ρ^{false(2 false)} (\tilde{M}) (t) / prefixln false(t false)

exist as real numbers and their difference is called the degree of the

L^{2}

‐torsion function.…”

Section: Universal L2‐torsion For Cw‐complexes and Manifoldsmentioning

confidence: 99%

“…In Section 3.4 we will see that the universal

L^{2}

‐torsion of a 3‐manifold

M

determines the

L^{2}

‐torsion and more generally the

L^{2}

‐torsion function (also called

L^{2}

‐Alexander polynomial and

L^{2}

‐Alexander torsion) that recently was intensively studied, see for example, . The former invariant is determined by the volumes of the hyperbolic pieces in the Jaco–Shalen–Johannson decomposition of

M

, see [, Theorem 0.7].…”

Section: Introductionmentioning

confidence: 99%

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Universal L2-torsion, polytopes and applications to 3-manifolds

Friedl

Lück

2017

Proc. London Math. Soc.

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Given an L2‐acyclic connected finite CW‐complex, we define its universal L2‐torsion in terms of the chain complex of its universal covering. It takes values in the weak Whitehead group prefixWhwfalse(Gfalse). We study its main properties such as homotopy invariance, sum formula, product formula and Poincaré duality. Under certain assumptions, we can specify certain homomorphisms from the weak Whitehead group prefixWhwfalse(Gfalse) to abelian groups such as the real numbers or the Grothendieck group of integral polytopes, and the image of the universal L2‐torsion can be identified with many invariants such as the L2‐torsion, the L2‐torsion function, twisted L2‐Euler characteristics and, in the case of a 3‐manifold, the dual Thurston norm polytope.

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“…The

L^{2}

‐torsion function

ρ^{(2)} false(\overset{M}{true} false)

which is determined by

ρ_{u}^{(2)} false(\overset{M}{true} false) \in {prefixWh}^{w} false(π false)

and whose value at

t = 1

is the

L^{2}

‐torsion

ρ^{(2)} false(\overset{M}{true} false)

. This

L^{2}

‐torsion function is studied for instance in . Moreover,

{trueprefixlim sup}_{t \to \infty} ρ^{false(2 false)} (\tilde{M}) (t) / prefixln false(t false)

and

{trueprefixlim inf}_{t \to 0} ρ^{false(2 false)} (\tilde{M}) (t) / prefixln false(t false)

exist as real numbers and their difference is called the degree of the

L^{2}

‐torsion function.…”

Section: Universal L2‐torsion For Cw‐complexes and Manifoldsmentioning

confidence: 99%

“…In Section 3.4 we will see that the universal

L^{2}

‐torsion of a 3‐manifold

M

determines the

L^{2}

‐torsion and more generally the

L^{2}

‐torsion function (also called

L^{2}

‐Alexander polynomial and

L^{2}

M

, see [, Theorem 0.7].…”

Section: Introductionmentioning

confidence: 99%

Universal L2-torsion, polytopes and applications to 3-manifolds

Friedl

Lück

2017

Proc. London Math. Soc.

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“…The L 2 -Alexander torsion is a generalization of the L 2 -Alexander polynomial introduced by Li-Zhang [LZ06a], [LZ06b], [LZ08] and has been studied recently by many authors Dubois-Wegner [DW10], [DW13], Ben Aribi [BA13], [BA16] Dubois-Friedl-Lück [DFL14a], [DFL14b], [DFL15], Friedl-Lück [FL15] and Liu [Li15].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The $$L^2$$ L 2 -Alexander torsion for Seifert fiber spaces

Herrmann

2017

Arch. Math.

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“…In this section, we will study the L 2 -Alexander torsion for knots; in particular we will relate it to the L 2 -Alexander invariant that was introduced by Li-Zhang [37][38][39]. We will also prove a relationship between L 2 -Alexander torsions and the classical Alexander polynomial of a knot.…”

Section: The L 2 -Alexander Torsion For Knotsmentioning

confidence: 99%

“…In these papers, it is also implicitly proved that the function Δ (2) K , as an invariant of K, is well-defined up to multiplication by a function of the form t → |t| n with n ∈ Z. Furthermore, Li-Zhang [37,38] implicitly, and Dubois-Wegner [15,Proposition 3.2] explicitly showed that, for any t ∈ C \ {0}, we have Δ…”

Section: The L 2 -Alexander Invariant Of Li-zhangmentioning

confidence: 99%

TheL²-Alexander torsion of 3-manifolds

Dubois¹,

Friedl

Lück³

2016

J Topology

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We introduce L 2 -Alexander torsions for 3-manifolds, which can be viewed as a generalization of the L 2 -Alexander invariant of Li-Zhang. We state the L 2 -Alexander torsions for graph manifolds and we partially compute them for fibered manifolds. We furthermore show that, given any irreducible 3-manifold there exists a coefficient system such that the corresponding L 2 -torsion detects the Thurston norm.

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An L²–Alexander–Conway Invariant for Knots and the Volume Conjecture

Cited by 13 publications

References 21 publications

Universal L2-torsion, polytopes and applications to 3-manifolds

Universal L2-torsion, polytopes and applications to 3-manifolds

The $$L^2$$ L 2 -Alexander torsion for Seifert fiber spaces

TheL²-Alexander torsion of 3-manifolds

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An L2–Alexander–Conway Invariant for Knots and the Volume Conjecture

Cited by 13 publications

References 21 publications

Universal L2-torsion, polytopes and applications to 3-manifolds

Universal L2-torsion, polytopes and applications to 3-manifolds

The $$L^2$$ L 2 -Alexander torsion for Seifert fiber spaces

TheL2-Alexander torsion of 3-manifolds

Contact Info

Product

Resources

About

An L²–Alexander–Conway Invariant for Knots and the Volume Conjecture

TheL²-Alexander torsion of 3-manifolds