arXiv subjects

Stavros Garoufalidis

Publications and source records attributed to Stavros Garoufalidis.

At least 127 records · Page 7Linked to original sources

arXiv2008

An ansatz for the asymptotics of hypergeometric multisums

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singulatiries in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive $K$-theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.

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arXiv2007

An extended version of additive K-theory

There are two infinitesimal (i.e., additive) versions of the $K$-theory of a field $F$: one was introduced by Cathelineau, which is an $F$-module, and another one introduced by Bloch-Esnault, which is an $F^*$-module. Both versions are equipped with a regulator map, when $F$ is the field of complex numbers. In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group. Our results were motivated by two unrelated sources: Neumann's work on the extended Bloch group (which is isomorphic to indecomposable $K_3$ of the complex numbers), and the study of singularities of generating series of hypergeometric multisums. Final version.

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arXiv2007

Algebraic G-functions associated to matrices over a group-ring

Given a square matrix with elements in the group-ring of a group, one can consider the sequence formed by the trace (in the sense of the group-ring) of its powers. We prove that the corresponding generating series is an algebraic $G$-function (in the sense of Siegel) when the group is free of finite rank. Consequently, it follows that the norm of such elements is an exactly computable algebraic number, and their Green function is algebraic. Our proof uses the notion of rational and algebraic power series in non-commuting variables and is an easy application of a theorem of Haiman. Haiman's theorem uses results of linguistics regarding regular and context-free language. On the other hand, when the group is free abelian of finite rank, then the corresponding generating series is a $G$-function. We ask whether the latter holds for general hyperbolic groups. This version has an expanded introduction following suggestions from Lehner, Voiculescu and others.

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arXiv2007

Resurgence of the Euler-MacLaurin summation formula

The Euler-MacLaurin summation formula relates a sum of a function to a corresponding integral, with a remainder term. The remainder term has an asymptotic expansion, and for a typical analytic function, it is a divergent (Gevrey-1) series. Under some decay assumptions of the function in a half-plane (resp. in the vertical strip containing the summation interval), Hardy (resp. Abel-Plana) prove that the asymptotic expansion is a Borel summable series, and give an exact Euler-MacLaurin summation formula. Using a mild resurgence hypothesis for the function to be summed, we give a Borel summable transseries expression for the remainder term, as well as a Laplace integral formula, with an explicit integrand which is a resurgent function itself. In particular, our summation formula allows for resurgent functions with singularities in the vertical strip containing the summation interval. Finally, we give two applications of our results. One concerns the construction of solutions of linear difference equations with a small parameter. And another concerns the problem of proving resurgence of formal power series associated to knotted objects.

math.CA↗

arXiv2007

Gevrey series in quantum topology

Our aim is to prove that two formal power series of importance to quantum topology are Gevrey. These series are the Kashaev invariant of a knot (reformulated by Huynh and the second author) and the Gromov norm of the LMO of an integral homology 3-sphere. It follows that the power series associated to a simple Lie algebra and a homology sphere is Gevrey. Contrary to the case of analysis, our formal power series are not solutions to differential equations with polynomial coefficients. The first author has conjectured (and in some cases proved, in joint work with Costin) that our formal power series have resurgent Borel transform, with geometrically interesting set of singularities.

math.GT↗

arXiv2007

Difference and differential equations for the colored Jones function

The colored Jones function of a knot is a sequence of Laurent polynomials. It was shown by TTQ. Le and the author that such sequences are $q$-holonomic, that is, they satisfy linear $q$-difference equations with coefficients Laurent polynomials in $q$ and $q^n$. We show from first principles that $q$-holonomic sequences give rise to modules over a $q$-Weyl ring. Frohman-Gelca-LoFaro have identified the latter ring with the ring of even functions of the quantum torus, and with the Kauffman bracket skein module of the torus. Via this identification, we study relations among the orthogonal, peripheral and recursion ideal of the colored Jones function, introduced by the above mentioned authors. In the second part of the paper, we convert the linear $q$-difference equations of the colored Jones function in terms of a hierarchy of linear ordinary differential equations for its loop expansion. This conversion is a version of the WKB method, and may shed some information on the problem of asymptotics of the colored Jones function of a knot.Final version. To appear in the Journal of Knot Theory and its Ramifications.

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arXiv2006

Asymptotics of q-difference equations

In this paper we develop an asymptotic analysis for formal and actual solutions of q-difference equations, under a regularity assumption. In particular, evaluations of regular solutions of regular q-difference equations have an exponential growth rate which can be computed from the q-difference equation. The motivation for the paper comes from the Hyperbolic Volume Conjecture, which states that a specific evaluation of the colored Jones function has an exponential growth rate, which is proportional to the volume of the knot complement. The connection of the Hyperbolic Volume Conjecture with the paper comes from the fact that the colored Jones function of a knot is a solution of a q-difference equation, as was proven by T.T.Q. Le and the author. Updated references. To appear in the JAMI Proceedings, Contemporary Math, AMS.

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arXiv2006

Is the Jones polynomial of a knot really a polynomial?

The Jones polynomial of a knot in 3-space is a Laurent polynomial in $q$, with integer coefficients. Many people have pondered why is this so, and what is a proper generalization of the Jones polynomial for knots in other closed 3-manifolds. Our paper centers around this question. After reviewing several existing definitions of the Jones polynomial, we show that the Jones polynomial is really an analytic function, in the sense of Habiro. Using this, we extend the holonomicity properties of the colored Jones function of a knot in 3-space to the case of a knot in an integer homology sphere, and we formulate an analogue of the AJ Conjecture. Our main tools are various integrality properties of topological quantum field theory invariants of links in 3-manifolds, manifested in Habiro's work on the colored Jones function.Revised version updating references on Witten-Reshetikhin-Turaev invariants.

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arXiv2006

On Chern-Simons Matrix Models

The contribution of reducible connections to the U(N) Chern-Simons invariant of a Seifert manifold $M$ can be expressed in some cases in terms of matrix integrals. We show that the U(N) evaluation of the LMO invariant of any rational homology sphere admits a matrix model representation which agrees with the Chern-Simons matrix integral for Seifert spheres and the trivial connection.

math.GT↗

arXiv2005

The colored Jones function is q-holonomic

A function of several variables is called holonomic if, roughly speaking, it is determined from finitely many of its values via finitely many linear recursion relations with polynomial coefficients. Zeilberger was the first to notice that the abstract notion of holonomicity can be applied to verify, in a systematic and computerized way, combinatorial identities among special functions. Using a general state sum definition of the colored Jones function of a link in 3-space, we prove from first principles that the colored Jones function is a multisum of a q-proper-hypergeometric function, and thus it is q-holonomic. We demonstrate our results by computer calculations.

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arXiv2005

Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot

Loosely speaking, the Volume Conjecture states that the limit of the n-th colored Jones polynomial of a hyperbolic knot, evaluated at the primitive complex n-th root of unity is a sequence of complex numbers that grows exponentially. Moreover, the exponential growth rate is proportional to the hyperbolic volume of the knot. We provide an efficient formula for the colored Jones function of the simplest hyperbolic non-2-bridge knot, and using this formula, we provide numerical evidence for the Hyperbolic Volume Conjecture for the simplest hyperbolic non-2-bridge knot.

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arXiv2005

The quantum MacMahon Master Theorem

We state and prove a quantum-generalization of MacMahon's celebrated Master Theorem, and relate it to a quantum-generalization of the boson-fermion correspondence of Physics.

math.QA↗

arXiv2005

An analytic version of the Melvin-Morton-Rozansky Conjecture

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Volume Conjecture for small angles states that the value of the $n$-th colored Jones polynomial at $e^{\a/n}$ is a sequence of complex numbers that grows subexponentially, for a fixed small complex angle $\a$. In an earlier publication, the authors proved the Volume Conjecture for small purely imaginary angles, using estimates of the cyclotomic expansion of a knot. The goal of the present paper is to identify the polynomial growth rate of the above sequence to all orders with the loop expansion of the colored Jones function. Among other things, this provides a strong analytic form of the Melvin-Morton-Rozansky conjecture. The resubmission corrects a misspelling of the first name of the second author.

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arXiv2005

A non-commutative formula for the colored Jones function

The colored Jones function of a knot is a sequence of Laurent polynomials that encodes the Jones polynomial of a knot and its parallels. It has been understood in terms of representations of quantum groups and Witten gave an intrinsic quantum field theory interpretation of the colored Jones function as the expectation value of Wilson loops of a 3-dimensional gauge theory, the Chern-Simons theory. We present the colored Jones function as an evaluation of the inverse of a non-commutative fermionic partition function. This result is in the form familiar in quantum field theory, namely the inverse of a generalized determinant. Our formula also reveals a direct relation between the Alexander polynomial and the colored Jones function of a knot and immediately implies the extensively studied Melvin-Morton-Rozansky conjecture, first proved by Bar-Natan and the first author about ten years ago.

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arXiv2005

On the volume conjecture for small angles

Given a knot in 3-space, one can associate a sequence of Laurrent polynomials, whose $n$th term is the $n$th colored Jones polynomial. The Generalized Volume Conjecture states that the value of the $n$-th colored Jones polynomial at $\exp(2 πi \a/n)$ is a sequence of complex numbers that grows exponentially, for a fixed real angle $\a$. Moreover the exponential growth rate of this sequence is proportional to the volume of the 3-manifold obtained by $(1/\a,0)$ Dehn filling. In this paper we will prove that (a) for every knot, the limsup in the hyperbolic volume conjecture is finite and bounded above by an exponential function that depends on the number of crossings. (b) Moreover, for every knot $K$ there exists a positive real number $\a(K)$ (which depends on the number of crossings of the knot) such that the Generalized Volume Conjecture holds for $\a \in [0, \a(K))$. Finally, we point out that a theorem of Agol-Storm-W.Thurston proves that the bounds in (a) are optimal, given by knots obtained by closing large chunks of the weave.

math.GT↗

arXiv2004

Non-triviality of the A-polynomial for knots in S^3

The A-polynomial of a knot in S^3 defines a complex plane curve associated to the set of representations of the fundamental group of the knot exterior into SL(2,C). Here, we show that a non-trivial knot in S^3 has a non-trivial A-polynomial. We deduce this from the gauge-theoretic work of Kronheimer and Mrowka on SU_2-representations of Dehn surgeries on knots in S^3. As a corollary, we show that if a conjecture connecting the colored Jones polynomials to the A-polynomial holds, then the colored Jones polynomials distinguish the unknot

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arXiv2004

Whitehead doubling persists

The operation of (untwisted) Whitehead doubling trivializes the Alexander module of a knot (and consequently, all known abelian invariants), and converts knots to topologically slice ones. In this note we show that Whitehead doubling does not trivialize the rational function that equals to the 2-loop part of the Kontsevich integral.

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arXiv2004

On the characteristic and deformation varieties of a knot

The colored Jones function of a knot is a sequence of Laurent polynomials in one variable, whose n-th term is the Jones polynomial of the knot colored with the n-dimensional irreducible representation of SL(2). It was recently shown by TTQ Le and the author that the colored Jones function of a knot is q-holonomic, ie, that it satisfies a nontrivial linear recursion relation with appropriate coefficients. Using holonomicity, we introduce a geometric invariant of a knot: the characteristic variety, an affine 1-dimensional variety in C^2. We then compare it with the character variety of SL_2(C) representations, viewed from the boundary. The comparison is stated as a conjecture which we verify (by a direct computation) in the case of the trefoil and figure eight knots. We also propose a geometric relation between the peripheral subgroup of the knot group, and basic operators that act on the colored Jones function. We also define a noncommutative version (the so-called noncommutative A-polynomial) of the characteristic variety of a knot. Holonomicity works well for higher rank groups and goes beyond hyperbolic geometry, as we explain in the last chapter.

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