Thang T. Q. Le

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.

Integrality and symmetry of quantum link invariants

0. Introduction. Quantum invariants of framed links whose components are colored by modules of a simple Lie algebra g are Laurent polynomials in v1/D (with integer coefficients), where v is the quantum parameter and D an integer depending on g. We show that quantum invariants, with a suitable normalization, are Laurent polynomials in v2. We also establish two symmetry properties of quantum link invariants at roots of unity. The first asserts that quantum link invariants, at rth roots of unity, are invariant under the action of the affine Weyl group Wr , which acts on the weight lattice. A fundamental domain of Wr is the fundamental alcove Cr , a simplex. Let G be the center of the corresponding simply connected complex Lie group. There is a natural action of G on Cr . The second symmetry property, in its simplest form, asserts that quantum link invariants are invariant under the action ofG if the link has zero linking matrix. The second symmetry property generalizes symmetry principles of Kirby and Melvin (the sl2 case) and Kohno and Takata (the sln case) to arbitrary simple Lie algebra.

Nahm sums, stability and the colored Jones polynomial

Nahm sums are q-series of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory - we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among q-series.MSCPrimary 57N10; Secondary 57M25.

The quantum MacMahon Master Theorem

We state and prove a quantum generalization of MacMahons celebrated Master Theorem and relate it to a quantum generalization of the boson–fermion correspondence of physics.

ASYMPTOTICS OF THE COLORED JONES FUNCTION OF A KNOT

To a knot in 3-space, one can associate a sequence of Laurent polynomials, whose nth term is the nth colored Jones polynomial. The paper is concerned with the asymptotic behavior of the value of the nth colored Jones polynomial at e�/n, whenis a fixed complex number and n tends to infinity. We analyze this asymptotic behavior to all orders in 1/n whenis a sufficiently small complex number. In addition, we give upper bounds for the coefficients and degree of thenth colored Jones polynomial, with applications to upper bounds in the Generalized Volume Conjecture. Work of Agol-Dunfield-Storm-W.Thurston implies that our bounds are asymptotically optimal. Moreover, we give results for the Generalized Volume Conjecture when � is near 2�i. Our proofs use crucially the cyclotomic expansion of the colored Jones function, due to Habiro.

Homology torsion growth and Mahler measure

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 Mahler measure of the first non-zero Alexander polynomial of the corresponding modules. We use the notion of pseudo-isomorphism, and also tools from commutative algebra and algebraic geometry, to reduce the conjectures to the case of torsion modules. We also describe concrete sequences which give the expected values of the limits in both cases. For this part we utilize a result of Bombieri and Zannier (conjectured before by A. Schinzel) and a result of Lawton (conjectured before by D. Boyd).

Strong integrality of quantum invariants of 3-manifolds

We prove that the quantum SO(3)-invariant of an arbitrary 3-manifold M is always an algebraic integer if the order of the quantum parameter is co-prime with the order of the torsion part of H 1 (M,Z). An even stronger integrality, known as cyclotomic integrality, was established by Habiro for integral homology 3-spheres. Here we also generalize Habiros result to all rational homology 3-spheres.

On Kauffman bracket skein modules at roots of unity

We reprove and expand results of Bonahon and Wong on central elements of the Kauffman bracket skein modules at root of 1 and on the existence of the Chebyshev homomorphism, using elementary skein methods.

The colored HOMFLYPT function is

We prove that the HOMFLYPT polynomial of a link, colored by partitions with a xed number of rows is a q-holonomic function. Specializing to the case of knots colored by a partition with a single row, it proves the existence of an (a;q) super-polynomial of knots in 3-space, as was conjectured by string theorists. Our proof uses skew Howe duality that reduces the evaluation of web diagrams and their ladders to a Poincare-Birkho- Witt computation of an auxiliary quantum group of rank the number of strings of the ladder diagram.

q

We calculate the Kauffman bracket skein module (KBSM) of the complement of all two-bridge links. For a two-bridge link, we show that the KBSM of its complement is free over the ring