Related papers: On rectangular HOMFLY for twist knots

We study factorizations of HOMFLY polynomials of certain knots and oriented links. We begin with a computer analysis of knots with at most 12 crossings, finding 17 non-trivial factorizations. Next, we give an irreducibility criterion for…

We give a topological realization of the (spherical) double affine Hecke algebra $\mathrm{SH}_{q,t}$ of type $A_1$, and we use this to construct a module over $\mathrm{SH}_{q,t}$ for any knot $K \subset S^3$. As an application, we give a…

In recent years, several families of hyperbolic knots have been shown to have both volume and $\lambda_1$ (first eigenvalue of the Laplacian) bounded in terms of the twist number of a diagram, while other families of knots have volume…

This article introduces a natural extension of colouring numbers of knots, called colouring polynomials, and studies their relationship to Yang-Baxter invariants and quandle 2-cocycle invariants. For a knot K in the 3-sphere let \pi_K be…

We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of $N$ and generalizes the previously known "tug-the-hook" symmetry of the colored…

We study some kinds of generalizations of Schr\"oder paths below a line with rational slope and derive the $q$-difference equations that are satisfied by their generating functions. As a result, we establish a relation between the…

We introduce the notion of "special superpolynomials" by putting q=1 in the formulas for reduced superpolynomials. In this way we obtain a generalization of special HOMFLY polynomials depending on one extra parameter t. Special HOMFLY are…

Using the theory of perverse sheaves of vanishing cycles, we define a homological invariant of knots in three-manifolds, similar to the three-manifold invariant constructed by Abouzaid and the second author. We use spaces of SL(2,C) flat…

The alternating knots, links and twists projected on the $S_2$ sphere were identified with the phase space of a Hamiltonian dynamic system of one degree of freedom. The saddles of the system correspond to the crossings, the edges correspond…

We obtain a quiver representation for a family of knots called double twist knots $K(p,-m)$. Particularly, we exploit the reverse engineering of Melvin-Morton-Rozansky(MMR) formalism to deduce the pattern of the charge matrix for these…

The motivation for this work was to construct a nontrivial knot with trivial Jones polynomial. Although that open problem has not yielded, the methods are useful for other problems in the theory of knot polynomials. The subject of the…

We prove a positive combinatorial formula for the Schur expansion of LLT polynomials indexed by a 3-tuple of skew shapes. This verifies a conjecture of Haglund. The proof requires expressing a noncommutative Schur function as a positive sum…

We give a new definition of the knot invariant associated to the Lie algebra su_{N+1}. The knot or link must be presented as the plat closure of a braid. The invariant is then a homological intersection pairing between two submanifolds of a…

We evaluate the differences of HOMFLY-PT invariants for pairs of mutant knots colored with representations of $SL(N)$, which are large enough to distinguish between them. These mutant pairs include the pretzel mutants, which require at…

We use a link invariant defined by Cimasoni-Florens to compute \rho-invariants. This generalizes results of Cochran-Teichner and Friedl on knots to the setting of links. As an application, we prove with only twelve possible exceptions that…

It is known that the colored Jones polynomials of a knot in the 3-dimensional sphere satisfy recursive relations, it is also known that these recursive relations come from recurrence polynomials which have been related, by the AJ…

Khovanov homology for knots has generated a flurry of activity in the topology community. This paper studies the Khovanov type cohomology for graphs with a special attention to torsions. When the underlying algebra is $\mathbb{Z}[x]/(x^2)$,…

Explicit answer is given for the HOMFLY polynomial of the figure eight knot $4_1$ in arbitrary symmetric representation R=[p]. It generalizes the old answers for p=1 and 2 and the recently derived results for p=3,4, which are fully…

We construct an infinite family of topologically slice knots that are not smoothly concordant to their reverses. More precisely, if T denotes the concordance group of topologically slice knots and R is the involution of T induced by string…

We provide a very brief review of the description of colored invariants for the Hopf link in terms of characters, which need to be taken at a peculiar deformation of the topological locus, depending on one of the two representations…