arXiv math/04042362004-04math.GTmath.NTmath.QA

On the Mahler measure of Jones polynomials under twisting

Abhijit Champanerkar, Ilya Kofman

备注：: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-1.abs.html
发表信息：: Algebr. Geom. Topol. 5 (2005) 1-22
DOI：: 10.2140/agt.2005.5.1

英文摘要：

We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a_1,...,a_n), we show that the Mahler measure of the Jones polynomial converges if all a_i tend to infinity, and approaches infinity for a_i = constant if n tend to infinity, just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Mahler measure of the colored Jones polynomial and the volume conjecture

In this note, I will discuss a possible relation between the Mahler measure of the colored Jones polynomial and the volume conjecture. In particular, I will study the colored Jones polynomial of the figure-eight knot on the unit circle. I will also propose a method to prove the volume conjecture for satellites of the figure-eight knot.

2002-06-24math.GT

A cellular basis for the generalized Temperly-Lieb Algebra and Mahler Measure

Just as the Temperley-Lieb algebra is a good place to compute the Jones polynomial, the Kauffman bracket skein algebra of a disk with $2k$ colored points on the boundary, each with color $n$, is a good place to compute the $n^{th}$ colored Jones polynomial. Here, this colored skein algebra is shown to be a cellular algebra and a set of separating Jucys-Murphy elements is provided. This is done by explicitly providing the cellular basis and the JM-elements. Having done this several results of Mathas on such algebras are considered, including the construction of pairwise non-isomorphic irreducib…

2013-11-06math.GT

Twisting Alexander Invariants with Periodic Representations

Twisted Alexander invariants have been defined for any knot and linear representation of its group. The invariants are generalized for any periodic representation of the commutator subgroup of the knot group. Properties of the new twisted invariants are given. Under suitable hypotheses, reciprocality and bounds on the moduli of zeros are obtained. A topological interpretation of the Mahler measure of the invariants is presented. Keywords: Knot, twisted Alexander polynomial, representation shift, Mahler measure.

2010-12-21math.GT

纽结Q多项式的马勒测度与根分布

On the Mahler measure and root distribution of the $Q$-polynomial of links

Osaka Metropolitan University（大阪公立大学）

该研究探讨纽结Q多项式的马勒测度与根分布，证明扭转下Q多项式马勒测度收敛、多数根趋近[-2,2]，对比三类多项式根的差异，给出仅含非零实根的2桥纽结无限族并提出相关猜想。

2026-09-10math.GTmath.NT

Euclidean Mahler measure and twisted links

If the twist numbers of a collection of oriented alternating link diagrams are bounded, then the Alexander polynomials of the corresponding links have bounded euclidean Mahler measure (see Definition 1.2). The converse assertion does not hold. Similarly, if a collection of oriented link diagrams, not necessarily alternating, have bounded twist numbers, then both the Jones polynomials and a parametrization of the 2-variable Homflypt polynomials of the corresponding links have bounded Mahler measure.

2009-03-10math.GT

Infinitely many roots of unity are zeros of some Jones polynomials

Let $N=2n^2-1$ or $N=n^2+n-1$, for any $n\ge 2$. Let $M=\frac{N-1}{2}$. We construct families of prime knots with Jones polynomials $(-1)^M\sum_{k=-M}^{M} (-1)^kt^k$. Such polynomials have Mahler measure equal to $1$. If $N$ is prime, these are cyclotomic polynomials $Φ_{2N}(t)$, up to some shift in the powers of $t$. Otherwise, they are products of such polynomials, including $Φ_{2N}(t)$. In particular, all roots of unity $ζ_{2N}$ occur as roots of Jones polynomials. We also show that some roots of unity cannot be zeros of Jones polynomials.

2021-02-20math.GT

更多相关论文收起更多论文

Non-triviality of the Jones polynomial and the crossing numbers of amphicheiral knots

Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and Whitehead doubles of semiadequate knots, have non-trivial Jones polynomial. We also prove that there are infinitely many positive knots with no positive minimal crossing diagrams. Some relations to the twist number of a link, Mahler measure and the hyperbolic volume are given, for example explicit upper bounds on the volume for Montesinos and 3-braid links in terms …

2007-07-03math.GT

Twisting q-holonomic sequences by complex roots of unity

A sequence $f_n(q)$ is $q$-holonomic if it satisfies a nontrivial linear recurrence with coefficients polynomials in $q$ and $q^n$. Our main theorems state that $q$-holonomicity is preserved under twisting, i.e., replacing $q$ by $ωq$ where $ω$ is a complex root of unity, and under the substitution $q \to q^α$ where $α$ is a rational number. Our proofs are constructive, work in the multivariate setting of $\partial$-finite sequences and are implemented in the Mathematica package HolonomicFunctions. Our results are illustrated by twisting natural $q$-holonomic sequences which appear in quantum …

2012-05-16cs.SCmath.COmath.GT

On The Jones Polynomial of Quasi-alternating Links

We prove that twisting any quasi-alternating link $L$ with no gaps in its Jones polynomial $V_L(t)$ at the crossing where it is quasi-alternating produces a link $L^{*}$ with no gaps in its Jones polynomial $V_{L^*}(t)$. This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other than $(2,n)$-torus links, has no gaps. This would give a new property of quasi-alternating links and a simple obstruction criterion for a link to be quasi-alternating. We prove that the conjecture holds for quasi-alternating Montesinos links as well as quasi-alternating links with …

2018-10-28math.GT

虚拟链环的Jones-Kauffman多项式的交错性

On Alternateness of the Jones-Kauffman Polynomial of Virtual Links

Hassan II University EST（哈桑二世大学）；Yarmouk University（亚尔穆克大学）；Hassan II University（哈桑二世大学）

本文研究虚拟链环的Jones-Kauffman多项式的交错性，定义拟交错虚拟链环类，利用扭转技术构造无限族，并给出棋盘可着色链环的拟行列式的连通状态展开。

2026-10-06math.GT

The Jones polynomial and related properties of some twisted links

Twisted links are obtained from a base link by starting with a $n$-braid representation, choosing several ($m$) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a certain form. The Jones polynomial of full $m$-twisted links have some interesting properties. It is known that when sufficiently many full $m$-twists are added that the coefficients break up into disjoint blocks which are independent of the number of full twists. These blocks are …

2011-08-20math.GT

Behavior of knot invariants under genus 2 mutation

Genus 2 mutation is the process of cutting a 3-manifold along an embedded closed genus 2 surface, twisting by the hyper-elliptic involution, and gluing back. This paper compares genus 2 mutation with the better-known Conway mutation in the context of knots in the 3-sphere. Despite the fact that any Conway mutation can be achieved by a sequence of at most two genus 2 mutations, the invariants that are preserved by genus 2 mutation are a proper subset of those preserved by Conway mutation. In particular, while the Alexander and Jones polynomials are preserved by genus 2 mutation, the HOMFLY-PT p…

2012-02-19math.GTmath.QA