arXiv subjects

Stavros Garoufalidis

Publications and source records attributed to Stavros Garoufalidis.

At least 109 records · Page 6Linked to original sources

arXiv2011

The degree of a $q$-holonomic sequence is a quadratic quasi-polynomial

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 holonomic $D$-modules to the case of $q$-holonomic $D$-modules) combined with the Lech-Mahler-Skolem theorem from number theory. En route, we use the Newton polygon of a linear $q$-difference equation, and introduce the notion of regular-singular $q$-difference equation and a WKB basis of solutions of a linear $q$-difference equation at $q=0$. We then use the Lech-Mahler-Skolem theorem to study the vanishing of their leading term. Unlike the case of $q=1$, there are no analytic problems regarding convergence of the WKB solutions.Our proofs are constructive, and they are illustrated by an explicit example.

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arXiv2011

The SL_3 Jones polynomial of the trefoil: a case study of $q$-holonomic sequences

The SL_3 colored Jones polynomial of the trefoil knot is a $q$-holonomic sequence of two variables with natural origin, namely quantum topology. The paper presents an explicit set of generators for the annihilator ideal of this $q$-holonomic sequence as a case study. On the one hand, our results are new and useful to quantum topology: this is the first example of a rank 2 Lie algebra computation concerning the colored Jones polynomial of a knot. On the other hand, this work illustrates the applicability and computational power of the employed computer algebra methods.

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arXiv2011

Asymptotics of quantum spin networks at a fixed root of unity

A classical spin network consists of a ribbon graph (i.e., an abstract graph with a cyclic ordering of the vertices around each edge) and an admissible coloring of its edges by natural numbers. The standard evaluation of a spin network is an integer number. In a previous paper, we proved an existence theorem for the asymptotics of the standard evaluation of an arbitrary classical spin network when the coloring of its edges are scaled by a large natural number. In the present paper, we extend the results to the case of an evaluation of quantum spin networks of arbitrary valency at a fixed root of unity. As in the classical case, our proofs use the theory of $G$-functions of André, together with some new results concerning holonomic and $q$-holonomic sequences of Wilf-Zeilberger.

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arXiv2011

Asymptotics of the instantons of Painleve I

The 0-instanton solution of Painlevé I is a sequence $(u_{n,0})$ of complex numbers which appears universally in many enumerative problems in algebraic geometry, graph theory, matrix models and 2-dimensional quantum gravity. The asymptotics of the 0-instanton $(u_{n,0})$ for large $n$ were obtained by the third author using the Riemann-Hilbert approach. For $k=0,1,2,...$, the $k$-instanton solution of Painlevé I is a doubly-indexed sequence $(u_{n,k})$ of complex numbers that satisfies an explicit quadratic non-linear recursion relation. The goal of the paper is three-fold: (a) to compute the asymptotics of the 1-instanton sequence $(u_{n,1})$ to all orders in $1/n$ by using the Riemann-Hilbert method, (b) to present formulas for the asymptotics of $(u_{n,k})$ for fixed $k$ and to all orders in $1/n$ using resurgent analysis, and (c) to confirm numerically the predictions of resurgent analysis. We point out that the instanton solutions display a new type of Stokes behavior, induced from the tritronquée Painlevé transcendents, and which we call the induced Stokes phenomenon. The asymptotics of the 2-instanton and beyond exhibits new phenomena not seen in 0 and 1-instantons, and their enumerative context is at present unknown.

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arXiv2010

What is a sequence of Nilsson type?

Sequences of Nilsson type appear in abundance in Algebraic Geometry, Enumerative Combinatorics, Mathematical Physics and Quantum Topology. We give an elementary introduction on this subject, including the definition of sequences of Nilsson type and the uniqueness, existence, and effective computation of their asymptotic expansion.

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arXiv2010

q-Terms, singularities and the extended Bloch group

Our paper originated from a generalization of the Volume Conjecture to multisums of $q$-hypergeometric terms. This generalization was sketched by Kontsevich in a problem list in Aarhus University in 2006; \cite{Ko}. We introduce the notion of a $q$-hypergeometric term (in short, $q$-term). The latter is a product of ratios of $q$-factorials in linear forms in several variables. In the first part of the paper, we show how to construct elements of the Bloch group (and its extended version) given a \qterm. Their image under the Bloch-Wigner map or the Rogers dilogarithm is a finite set of periods of weight 2, in the sense of Kontsevich-Zagier. In the second part of the paper we introduce the notion of a special $q$-term, its corresponding sequence of polynomials, and its generating series. Examples of special $q$-terms come naturally from Quantum Topology, and in particular from planar projections of knots. The two parts are tied together by a conjecture that relates the singularities of the generating series of a special $q$-term with the periods of the corresponding elements of the extended Bloch group. In some cases (such as the $4_1$ knot), the conjecture is known.

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arXiv2010

Resurgence of the Kontsevich-Zagier power series

The paper is concerned with the Kontsevich-Zagier formal power series $$ f(q)=\sum_{n=0}^\infty (1-q)... (1-q^n) $$ and its analytic properties. To begin with, we give an explicit formula for the Borel transform of the associated formal power series $F(x)=e^{-1/(24x)}f(e^{-1/x})$ from which its analytic continuation, its singularities and their structure can be manifestly determined. This gives rise to right/left and median summation of the original power series. These sums, which are well-defined in the open right half-plane are expressed by an integral formula involving the Dedekind eta function. The median sum can also be expressed as a series involving the complex error function. Moreover, it is shown using results of Zagier that the limiting values at $-1/(2 πi \a)$ for rational numbers $\a$ coincide with $F(-1/(2 πi \a))$. One motivation for studying the series $f(q)$ is Quantum Topology, which assigns numerical invariants to knotted 3-dimensional objects. Motivated by our results, we formulate a resurgence conjecture for the formal power series of knotted objects, which we prove in the case of the trefoil knot and the Poincare homology sphere, and more generally for torus knots and Seifert fibered 3-manifolds. In a subsequent publication we will study resurgence for a class of power series that includes the quantum invariants of the simplest hyperbolic $4_1$ knot.

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arXiv2010

Knots and tropical curves

Using elementary ideas from Tropical Geometry, we assign a a tropical curve to every $q$-holonomic sequence of rational functions. In particular, we assign a tropical curve to every knot which is determined by the Jones polynomial of the knot and its parallels. The topical curve explains the relation between the AJ Conjecture and the Slope Conjecture (which relate the Jones polynomial of a knot and its parallels to the $\SL(2,\BC)$ character variety and to slopes of incompressible surfaces). Our discussion predicts that the tropical curve is dual to a Newton subdivision of the $A$-polynomial of the knot. We compute explicitly the tropical curve for the $4_1$, $5_2$ and $6_1$ knots and verify the above prediction.

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arXiv2010

The Jones slopes of a knot

The paper introduces Slope Conjecture which relates the degree of the Jones polynomial of a knot and its parallels with the slopes of incompressible surfaces in the knot complement. More precisely, we introduce two knot invariants, the Jones slopes (a finite set of rational numbers) and the Jones period (a natural number) of a knot in 3-space. We formulate a number of conjectures for these invariants and verify them by explicit computations for the class of alternating knots, torus knots, the knots with at most 9 crossings, and the $(-2,3,n)$ pretzel knots.

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arXiv2010

Sum-integral interpolators and the Euler-Maclaurin formula for polytopes

A local lattice point counting formula, and more generally a local Euler-Maclaurin formula follow by comparing two natural families of meromorphic functions on the dual of a rational vector space $V$, namely the family of exponential sums (S) and the family of exponential integrals (I) parametrized by the set of rational polytopes in $V$. The paper introduces the notion of an interpolator between these two families of meromorphic functions. We prove that every rigid complement map in $V$ gives rise to an effectively computable \SI-interpolator (and a local Euler-MacLaurin formula), an \IS-interpolator (and a reverse local Euler-MacLaurin formula) and an \ISo-interpolator. Rigid complement maps can be constructed by choosing an inner product on $V$ or by choosing a complete flag in $V$. The corresponding interpolators generalize and unify the work of Berline-Vergne, Pommersheim-Thomas, and Morelli.

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arXiv2009

Universality and asymptotics of graph counting problems in nonorientable surfaces

Bender-Canfield showed that a plethora of graph counting problems in oriented/unoriented surfaces involve two constants $t_g$ and $p_g$ for the oriented and the unoriented case respectively. T.T.Q. Le and the authors recently discovered a hidden relation between the sequence $t_g$ and a formal power series solution $u(z)$ of the Painlevé I equation which, among other things, allows to give exact asymptotic expansion of $t_g$ to all orders in $1/g$ for large $g$. The paper introduces a formal power series solution $v(z)$ of a Riccati equation, gives a nonlinear recursion for its coefficients and an exact asymptotic expansion to all orders in $g$ for large $g$, using the theory of Borel transforms. In addition, we conjecture a precise relation between the sequence $p_g$ and $v(z)$. Our conjecture is motivated by the enumerative aspects of a quartic matrix model for real symmetric matrices, and the analytic properties of its double scaling limit. In particular, the matrix model provides a computation of the number of rooted quadrangulations in the 2-dimensional projective plane. Our conjecture implies analyticity of the $\mathrm{O}(N)$ and $\mathrm{Sp}(N)$-types of free energy of an arbitrary closed 3-manifold in a neighborhood of zero. Finally, we give a matrix model calculation of the Stokes constants, pose several problems that can be answered by the Riemann-Hilbert approach, and provide ample numerical evidence for our results.

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arXiv2009

Resurgence of the fractional polylogarithms

The fractional polylogarithms, depending on a complex parameter $\a$, are defined by a series which is analytic inside the unit disk. After an elementary conversion of the series into an integral presentation, we show that the fractional polylogarithms are multivalued analytic functions in the complex plane minus 0 and 1. For non-integer values of $\a$, we prove the analytic continuation, compute the monodromy around 0 and 1, give a Mittag-Leffler decomposition and compute the asymptotic behavior for large values of the complex variable. The fractional polylogarithms are building blocks of resurgent functions that are used in proving that certain power series associated with knotted objects are resurgent. This is explained in a separate publication \cite{CG3}. The motivic or physical interpretation of the monodromy of the fractional polylogarithms for non-integer values of $\a$ is unknown to the authors.

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arXiv2009

The non-commutative $A$-polynomial of twist knots

The purpose of the paper is two-fold: to introduce a multivariable creative telescoping method, and to apply it in a problem of Quantum Topology: namely the computation of the non-commutative $A$-polynomial of twist knots. Our multivariable creative telescoping method allows us to compute linear recursions for sums of the form $J(n)=\sum_k c(n,k) \hatJ (k)$ given a recursion relation for $(\hatJ(n))$ a the hypergeometric kernel $c(n,k)$. As an application of our method, we explicitly compute the non-commutative $A$-polynomial for twist knots with -8 and 11 crossings. The non-commutative $A$-polynomial of a knot encodes the monic, linear, minimal order $q$-difference equation satisfied by the sequence of colored Jones polynomials of the knot. Its specialization to $q=1$ is conjectured to be the better-known $A$-polynomial of a knot, which encodes important information about the geometry and topology of the knot complement. Unlike the case of the Jones polynomial, which is easily computable for knots with 50 crossings, the $A$-polynomial is harder to compute and already unknown for some knots with 12 crossings.

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arXiv2009

A new algorithm for the recursion of multisums with improved universal denominator

The purpose of the paper is to introduce two new algorithms. The first one computes a linear recursion for proper hypergeometric multisums, by treating one summation variable at a time, and provides rational certificates along the way. A key part in the search of a linear recursion is an improved universal denominator algorithm that constructs all rational solutions $x(n)$ of the equation $$ \frac{a_m(n)}{b_m(n)}x(n+m)+...+\frac{a_0(n)}{b_0(n)}x(n)= c(n),$$ where $a_i(n), b_i(n), c(n)$ are polynomials. Our algorithm improves Abramov's universal denominator.

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arXiv2009

The C-polynomial of a knot

In an earlier paper the first author defined a non-commutative A-polynomial for knots in 3-space, using the colored Jones function. The idea is that the colored Jones function of a knot satisfies a non-trivial linear q-difference equation. Said differently, the colored Jones function of a knot is annihilated by a non-zero ideal of the Weyl algebra which is generalted (after localization) by the non-commutative A-polynomial of a knot. In that paper, it was conjectured that this polynomial (which has to do with representations of the quantum group U_q(SL_2)) specializes at q=1 to the better known A-polynomial of a knot, which has to do with genuine SL_2(C) representations of the knot complement. Computing the non-commutative A-polynomial of a knot is a difficult task which so far has been achieved for the two simplest knots. In the present paper, we introduce the C-polynomial of a knot, along with its non-commutative version, and give an explicit computation for all twist knots. In a forthcoming paper, we will use this information to compute the non-commutative A-polynomial of twist knots. Finally, we formulate a number of conjectures relating the A, the C-polynomial and the Alexander polynomial, all confirmed for the class of twist knots.

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arXiv2008

Analyticity of the Free Energy of a Closed 3-Manifold

The free energy of a closed 3-manifold is a 2-parameter formal power series which encodes the perturbative Chern-Simons invariant (also known as the LMO invariant) of a closed 3-manifold with gauge group U(N) for arbitrary $N$. We prove that the free energy of an arbitrary closed 3-manifold is uniformly Gevrey-1. As a corollary, it follows that the genus $g$ part of the free energy is convergent in a neighborhood of zero, independent of the genus. Our results follow from an estimate of the LMO invariant, in a particular gauge, and from recent results of Bender-Gao-Richmond on the asymptotics of the number of rooted maps for arbitrary genus. We illustrate our results with an explicit formula for the free energy of a Lens space. In addition, using the Painlevé differential equation, we obtain an asymptotic expansion for the number of cubic graphs to all orders, stengthening the results of Bender-Gao-Richmond.

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arXiv2008

G-functions and multisum versus holonomic sequences

The purpose of the paper is three-fold: (a) we prove that every sequence which is a multidimensional sum of a balanced hypergeometric term has an asymptotic expansion of Gevrey type-1 with rational exponents, (b) we construct a class of $G$-functions that come from enumerative combinatorics, and (c) we give a counterexample to a question of Zeilberger that asks whether holonomic sequences can be written as multisums of balanced hypergeometric terms. The proofs utilize the notion of a $G$-function, introduced by Siegel, and its analytic/arithmetic properties shown recently by André.

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arXiv2008

Chern-Simons theory, analytic continuation and arithmetic

The purpose of the paper is to introduce some conjectures regarding the analytic continuation and the arithmetic properties of quantum invariants of knotted objects. More precisely, we package the perturbative and nonperturbative invariants of knots and 3-manifolds into two power series of type P and NP, convergent in a neighborhood of zero, and we postulate their arithmetic resurgence. By the latter term, we mean analytic continuation as a multivalued analytic function in the complex numbers minus a discrete set of points, with restricted singularities, local and global monodromy. We point out some key features of arithmetic resurgence in connection to various problems of asymptotic expansions of exact and perturbative Chern-Simons theory with compact or complex gauge group. Finally, we discuss theoretical and experimental evidence for our conjecture.

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