Related papers: Generalized knots-quivers correspondence
We consider a large class of $q$-series that have the structure of Nahm sums, or equivalently motivic generating series for quivers. First, we initiate a systematic analysis and classification of classical and quantum A-polynomials…
In an attempt to generalise knot matrix models for non-torus knots, which currently remains an open problem, we derived formulas for the Harer-Zagier transform of the HOMFLY-PT polynomial for some infinite families of twisted hyperbolic…
We inductively define layers of colorings of knot and knotted surface diagrams using ternary quasigroups. Homological invariants from such systems of colorings use shorter differentials and of higher degree than the standard homology…
In this paper, we study the properties of the colored HOMFLY polynomials via HOMFLY skein theory. We prove some limit behaviors and symmetries of the colored HOMFLY polynomial predicted in some physicists' recent works.
Mutant knots, in the sense of Conway, are known to share the same Homfly polynomial. Their 2-string satellites also share the same Homfly polynomial, but in general their m-string satellites can have different Homfly polynomials for m>2. We…
In this paper we discuss the applications of knotoids to modelling knots in open curves and produce new knotoid invariants. We show how invariants of knotoids generally give rise to well-behaved measures of how much an open curve is…
The aim of this paper is to study generalized q-analogs of the well-known q-deformed harmonic oscillators and to connect them with q-Hermite polynomials. We give a construction of the appropriate oscillator-like algebras and show that…
For a given knot $K$ and $w>0$, we construct infinitely many mutually distinct hyperbolic knots $\{K_i\}$ such that the $P$-satellites of $K$ and $K_i$ have the same HOMFLY polynomial up to given $z$-degrees, for all braided patterns $P$…
We first study superpolynomial associated to triply-graded reduced colored HOMFLY-PT homology. We propose conjectures of congruent relations and cyclotomic expansion for it. We prove conjecture of $N=1$ for torus knot case, through which we…
This work reports and classifies the most general construction of rational quantum potentials in terms of the generalized Hermite polynomials. This is achieved by exploiting the intrinsic relation between third-order shape-invariant…
Using the duality between Wilson loop expectation values of SU(N) Chern-Simons theory on $S^3$ and topological open-string amplitudes on the local mirror of the resolved conifold, we study knots on $S^3$ and their invariants encoded in…
This is an expository paper which has two parts. In the first part, we study quiver varieties of affine $A$-type from a combinatorial point of view. We present a combinatorial method for obtaining a closed formula for the generating…
We introduce a Poincar\'{e} polynomial with two-variable $t$ and $x$ for knots, derived from Khovanov homology, where the specialization $(t, x)$ $=$ $(1, -1)$ is a Vassiliev invariant of order $n$. Since for every $n$, there exist…
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…
Knot polynomials colored with symmetric representations of $SL_q(N)$ satisfy difference equations as functions of representation parameter, which look like quantization of classical ${\cal A}$-polynomials. However, they are quite difficult…
We discuss the polynomial representation for long knots and elaborate on how to obtain them with a bound on degrees of the defining polynomials, for any knot-type.
We introduce an invariant of tangles in Khovanov homology by considering a natural inverse system of Khovanov homology groups. As application, we derive an invariant of strongly invertible knots; this invariant takes the form of a graded…
In this paper, we introduce a new method to prove the Lickorish-Millett type formulae for colored HOMFLY-PT polynomials of links.
The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable $q$. The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid…
In this short note, we prove the cyclotomic expansion formula for the colored HOMFLY-PT invariant of double twist knots, it confirms the cyclotomic expansion conjecture for $SU(N)$-invariants proposed in \cite{CLZ}.