BPS counting for knots and combinatorics on words

We discuss relations between quantum BPS invariants defined in terms of a product decomposition of certain series, and difference equations (quantum A-polynomials) that annihilate such series. We construct combinatorial models whose structure is encoded in the form of such difference equations, and whose generating functions (Hilbert-Poincar\'e series) are solutions to those equations and reproduce generating series that encode BPS invariants. Furthermore, BPS invariants in question are expressed in terms of Lyndon words in an appropriate language, thereby relating counting of BPS states to the branch of mathematics referred to as combinatorics on words. We illustrate these results in the framework of colored extremal knot polynomials: among others we determine dual quantum extremal A-polynomials for various knots, present associated combinatorial models, find corresponding BPS invariants (extremal Labastida-Mari\~no-Ooguri-Vafa invariants) and discuss their integrality.Comment: 41 pages, 1 figure, a supplementary Mathematica file attache

Combinatorial structure of colored HOMFLY-PT polynomials for torus knots

We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Mari\~no spectral curve for the colored HOMFLY-PT polynomials of torus knots. This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials. We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data.Comment: 40 pages; section 10 addressing the quantum curve was added, as well as some remarks regarding Meixner polynomials thanks to T.Koornwinde

Equivalence of two diagram representations of links in lens spaces and essential invariants

In this paper we study the relation between two diagrammatic representations of links in lens spaces: the disk diagram and the grid diagram and we find how to pass from one to the other. We also investigate whether the HOMFLY-PT invariant and the Link Floer Homology are essential invariants, that is, we try to understand if these invariants are able to distinguish links in $L(p,q)$ covered by the same link in $\mathbf{S}^3$. In order to do so, we generalize the combinatorial definition of Knot Floer Homology in lens spaces to the case of links and we analyze how both the invariants change when we switch the orientation of the link.Comment: 42 pages, 24 figure

Geometrical relations and plethysms in the Homfly skein of the annulus

The oriented framed Homfly skein C of the annulus provides the natural parameter space for the Homfly satellite invariants of a knot. It contains a submodule C+ isomorphic to the algebra of the symmetric functions. We collect and expand formulae relating elements expressed in terms of symmetric functions to Turaev's geometrical basis of C+. We reformulate the formulae of Rosso and Jones for quantum sl(N) invariants of cables in terms of plethysms of symmetric functions, and use the connection between quantum sl(N) invariants and C+ to give a formula for the satellite of a cable as an element of C+. We then analyse the case where a cable is decorated by the pattern which corresponds to a power sum in the symmetric function interpretation of C+ to get direct relations between the Homfly invariants of some diagrams decorated by power sums.Comment: 28 pages, 15 figure

Matrix Factorizations and Kauffman Homology

The topological string interpretation of homological knot invariants has led to several insights into the structure of the theory in the case of sl(N). We study possible extensions of the matrix factorization approach to knot homology for other Lie groups and representations. In particular, we introduce a new triply graded theory categorifying the Kauffman polynomial, test it, and predict the Kauffman homology for several simple knots.Comment: 45 pages, harvma

A topological introduction to knot contact homology

This is a survey of knot contact homology, with an emphasis on topological, algebraic, and combinatorial aspects.Comment: 38 pages, based on lectures at the Contact and Symplectic Topology Summer School in Budapest, July 2012; v2: fixed typos, updated references, version to be publishe