arXiv research

We study properties of a generalization of the Mahler measure to elements in group rings, in terms of the Lueck-Fuglede-Kadison determinant. Our main focus is the variation of the Mahler measure when the base group is changed. In particular, we study how to obtain the Mahler measure over an infinite group as limit of M…

The paper confirms a conjecture linking link bipyramid volume and Mahler measure.

Let l be an oriented link of d components in a homology 3-sphere. For any nonnegative integer q, let l(q) be the link of d-1 components obtained from l by performing 1/q surgery on the dth component. Then the Mahler measure of the Alexander polynomial of l(q) converges to the Mahler measure of the Alexander polynomial …

We study a class of 2-variable polynomials called exact polynomials which contains $A$-polynomials of knot complements. The Mahler measure of these polynomials can be computed in terms of a volume function defined on the vanishing set of the polynomial. We prove that the local extrema of the volume function are on the …

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 altern…

The Vol-Det Conjecture relates the volume and the determinant of a hyperbolic alternating link in $S^3$. We use exact computations of Mahler measures of two-variable polynomials to prove the Vol-Det Conjecture for many infinite families of alternating links. We conjecture a new lower bound for the Mahler measure of cer…

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 …

We present computational data and heuristic arguments which suggest that given a hyperbolic knot the volume correlates with its determinant, the Mahler measure of its Alexander polynomial and the Mahler measure of the twisted Alexander polynomial corresponding to the discrete and faithful SL(2,C)-representation.

The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler measure of the Kauffman bracket. Second we find a general form for the Kauffman bra…

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 f…

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 …

Let l be a link of d components. For every finite-index lattice in Z^d there is an associated finite abelian cover of S^3 branched over l. We show that the order of the torsion subgroup of the first homology of these covers has exponential growth rate equal to the logarithmic Mahler measure of the Alexander polynomial …

These are the notes of the three lectures I delivered at the mini-workshop "Knot Theory and Number Theory around the A-Polynomial" at the Instituto Superior Tecnico (IST) in Lisbon in January 2014. The goal of the lectures was to familiarize, both the author and, the audience with the A-polynomials and the connection b…

The (torsion) complexity of a finite edge-weighted graph is defined to be the order of the torsion subgroup of the abelian group presented by its Laplacian matrix. When G is d-periodic (i.e., G has a free action of the rank-d free abelian group by graph automorphisms, with finite quotient) the Mahler measure of its Lap…

The space C of conservative vertex colorings (over a field F) of a countable, locally finite graph G is introduced. The subspace of based colorings is shown to be isomorphic to the bicycle space of the graph. For graphs G with a free Z^d-action by automorphisms, C is a finitely generated module over the polynomial ring…

Uniform systole bounds for arithmetic orbifolds and number fields.

The Mahler measure of the polynomials $t(x^m-1) y - (x^n-1) \in \dC[x,y]$ is essentially the sum of volumes of a certain collection of ideal hyperbolic polyhedra in $\HH^3$, which can be determined a priori as a function on the parameter $t$. We obtain a formula that generalizes some previous formulas given by Cassaign…

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 alg…

Let $p$ be a prime number. We develop a theory of $p$-adic Mahler measure of polynomials and apply it to the study of $\mathbb{Z}$-covers of rational homology 3-spheres branched over links. We obtain a $p$-adic analogue of the asymptotic formula of the torsion homology growth and a balance formula among the leading coe…

In this report I discuss the relations between systoles and volumes of hyperbolic manifolds and a conjecture of Lehmer about the Mahler measure of non-cyclotomic polynomials.

Meyer and Reisner had proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler…

Jones polynomials have infinitely many roots of unity as zeros.

The complexity of a finite connected graph is its number of spanning trees; for a non-connected graph it is the product of complexities of its connected components. If $G$ is an infinite graph with cofinite free ${\mathbb Z}^d$-symmetry, then the logarithmic Mahler measure $m(Δ)$ of its Laplacian polynomial $Δ$ is the …

We prove a conjecture of K. Schmidt in algebraic dynamical system theory on the growth of the number of components of fixed point sets. We also generalize a result of Silver and Williams on the growth of homology torsions of finite abelian covering of link complements. In both cases, the growth is expressed by the Mahl…

Let L be any infinite biperiodic alternating link. We show that for any sequence of finite links that Folner converges almost everywhere to L, their determinant densities converge to the Mahler measure of the 2-variable characteristic polynomial of the toroidal dimer model on an associated biperiodic graph.

New Lehmer constants computed for free groups, improving bounds.

We establish a version of the bottleneck conjecture, which in turn implies a partial solution to the Mahler conjecture on the product $v(K) = (\Vol K)(\Vol K^\circ)$ of the volume of a symmetric convex body $K \in \R^n$ and its polar body $K^\circ$. The Mahler conjecture asserts that the Mahler volume $v(K)$ is minimiz…

We study the growth of the order of torsion subgroups of the homology in a tower of finite abelian coverings. In particular, we prove that it is exponential for when the tower converges to the maximal free abelian cover of a link complement when the first nonzero Alexander polynomial has positive logarithmic Mahler mea…

We continue to investigate the relation between the Mahler measure of certain two variable polynomials, the values of the Bloch--Wigner dilogarithm $D(z)$ and the values $ζ_F(2)$ of zeta functions of number fields. Specifically, we define a class $\A$ of polynomials $A$ with the property that $πm(A)$ is a linear combin…

The Mahler volume of a centrally symmetric convex body K is defined as M(K)= (Vol K)(Vol K^dual). Mahler conjectured that this volume is minimized when K is a cube. We introduce the bottleneck conjecture, which stipulates that a certain convex body K^diamond subset K X K^dual has least volume when K is an ellipsoid. If…

Study of Alexander polynomials of torus knots and links, showing zeros equidistribute on unit circle.

We formulate and prove a profinite rigidity theorem for the twisted Alexander polynomials up to several types of finite ambiguity. We also establish torsion growth formulas of the twisted homology groups in a $\mathbb{Z}$-cover of a 3-manifold with use of Mahler measures. We examine several examples associated to Riley…

We reformulate Lehmer's question from 1933 and a question due to Schinzel and Zassenhaus from 1965 in terms of a comparison of the Mahler measures and the houses, respectively, of monic integer reciprocal and skew-reciprocal polynomials of the same degree. This entails that understanding the difference between orientat…

We study the asymptotic behavior of smooth, origin-symmetric, strictly convex bodies under the centro-affine normal flows. By means of a stability version of the Blaschke-Santaló inequality, we obtain regularity of the solutions provided that initial convex bodies have almost maximum Mahler volume. We prove that suitab…

We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperb…

For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we pr…

The paper studies limits of sub-Riemannian Heisenberg manifolds and proves convergence under certain conditions.

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 prov…

New findings on Mabuchi energy and stability of manifolds.

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T--links are a natural generalization, given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T--links, so Lorenz links and T-…

This paper addresses a long standing open problem due to Lehmer in which the triple 2,3,7 plays a notable role. Lehmer's problem asks whether there is a gap between 1 and the next smallest algebraic integer with respect to Mahler measure. The question has been studied in a wide range of contexts including number theory…

A sequence of rational functions in a variable $q$ is $q$-holonomic if it satisfies a linear recursion with coefficients polynomials in $q$ and $q^n$. We prove that the degree of a $q$-holonomic sequence is eventually a quadratic quasi-polynomial. Our proof uses differential Galois theory (adapting proofs regarding hol…

Complexity of signed graphs linked to Alexander polynomials and Lehmer's question.

The paper explores volume product and slicing conjectures using convex body deformations.

The Funk metric connects billiards, projective geometry, and convex geometry.

The paper contains a combinatorial theorem (the sequence of Newton polygons of a reccurent sequence of polynomials is quasi-linear) and two applications of it in classical and quantum topology, namely in the behavior of the $A$-polynomial and a fixed quantum invariant (such as the Jones polynomial) under filling. Our c…

We explore a natural generalization of systolic geometry to Finsler metrics and optical hypersurfaces with special emphasis on its relation to the Mahler conjecture and the geometry of numbers. In particular, we show that if an optical hypersurface of contact type in the cotangent bundle of the 2-dimensional torus encl…

R. Schwartz's inequality provides an upper bound for the Schwarzian derivative of a parameterization of a circle in the complex plane and on the potential of Hill's equation with coexisting periodic solutions. We prove a discrete version of this inequality and obtain a version of the planar Blaschke-Santalo inequality …