Related papers: Stable pairs and the HOMFLY polynomial
The main goal is to find the Homfly polynomial of a link formed by decorating each component of the Hopf link with the closure of a directly oriented tangle. Such decorations are spanned in the Homfly skein of the annulus by elements…
This paper concerns the recent Virasoro conjecture for the theory of stable pairs on a 3-fold proposed by Oblomkov, Okounkov, Pandharipande and the author in arXiv:2008.12514. Here we extend the conjecture to 3-folds with…
The (untwisted) oriented cube of resolutions for knot Floer homology assigns a complex $C_{F}(S)$ to a singular resolution $S$ of a knot $K$. Manolescu conjectured that when $S$ is in braid position, the homology $H_{*}(C_{F}(S))$ is…
The theory of the Kauffman bracket, which describes the Jones polynomial as a sum over closed circles formed by the planar resolution of vertices in a knot diagram, can be straightforwardly lifted from sl(2) to sl(N) at arbitrary N -- but…
We prove that the HOMFLYPT polynomial of a link, colored by partitions with a fixed 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…
Theory of motivic superpolynomials is developed, including its extension to algebraic links colored by rows, relations to $L$-functions of plane curve singularities, the justification of the motivic versions of Weak Riemann Hypothesis, and…
We introduce the notion of P-critical connections for hermitian holomorphic vector bundles over compact balanced manifolds: integrable hermitian connections whose curvature solves a polynomial equation. Such connections include HYM and dHYM…
We illustrate from the viewpoint of braiding operations on WZNW conformal blocks how colored HOMFLY polynomials with multiplicity structure can detect mutations. As an example, we explicitly evaluate the (2,1)-colored HOMFLY polynomials…
Many knots and links in S^3 can be drawn as gluing of three manifolds with one or more four-punctured S^2 boundaries. We call these knot diagrams as double fat graphs whose invariants involve only the knowledge of the fusion and the…
The Vassiliev conjecture states that the Vassiliev invariants are dense in the space of all numerical link invariants in the sense that any link invariant is a pointwise limit of Vassiliev invariants. In this article, we prove that the…
We study various specializations of the colored HOMFLY-PT polynomial. These specializations are used to show that the multivariable link invariants arising from a complex family of sl(m|n) super-modules previously defined by the authors…
A celebrated result of F. Jaeger states that the Tutte polynomial of a planar graph is determined by the HOMFLY polynomial of an associated link. Here we are interested in the converse of this result. We consider the question `to what…
A classical result by Pachner states that two $d$-dimensional combinatorial manifolds with boundary are PL homeomorphic if and only they can be connected by a sequence of shellings and inverse shellings. We prove that for balanced, i.e.,…
We relate the stability of knot invariants under twisting a pair of strands to the stability of symmetric quivers under unlinking (or linking) operation. Starting from the HOMFLY-PT skein relations, we confirm the stable growth of…
We study 1-parameter families of holomorphic curves with Lagrangian boundary in Calabi-Yau 3-folds. We show that the expected codimension one phenomena can be organized to match the HOMFLYPT skein relations from quantum topology. It follows…
We study relations between cluster algebra invariants and link invariants. First, we show that several constructions of positroid links (permutation links, Richardson links, grid diagram links, plabic graph links) give rise to isotopic…
Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes…
Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the…
We revisit methods of proof of the Adams Conjecture in order to correct and supplement earlier efforts to prove analogous conjectures in the stable homotopy category. We utilize simplicial schemes over an algebraically closed field of…
Using the correspondence between Chern-Simons theories and Wess-Zumino-Witten models we present the necessary tools to calculate colored HOMFLY polynomials for hyperbolic knots. For two-bridge hyperbolic knots we derive the colored HOMFLY…