相关论文: Combinatorial structure of colored HOMFLY-PT polyn…

We introduce analogues of the Hopf algebra of Free quasi-symmetric functions with bases labelled by colored permutations. As applications, we recover in a simple way the descent algebras associated with wreath products $\Gamma\wr\SG_n$ and…

We describe the explicit form and the hidden structure of the answer for the HOMFLY polynomial for the figure eight and some other 3-strand knots in representation [21]. This is the first result for non-torus knots beyond (anti)symmetric…

We derive a closed-form expression for the adjoint polynomials of torus knots and investigate their special properties. The results are presented in the very explicit double sum form and provide a deeper insight into the structure of…

We reveal a relationship between the colored Jones polynomial and the A-polynomial for twist knots. We demonstrate that an asymptotics of the $N$-colored Jones polynomial in large $N$ gives the potential function, and that the A-polynomial…

We conjecture a structure formula for the $\mathrm{SU}(r)$ Vafa-Witten partition function for surfaces with holomorphic 2-form. The conjecture is based on $S$-duality and a structure formula for the vertical contribution previously derived…

In this work we demonstrate that the q-numbers and their two-parameter generalization, the q,p-numbers, can be used to obtain some polynomial invariants for torus knots and links. First, we show that the q-numbers, which are closely…

We find a combinatorial setting for the coefficients of the Boros-Moll polynomials $P_m(a)$ in terms of partially 2-colored permutations. Using this model, we give a combinatorial proof of a recurrence relation on the coefficients of…

We suggest a relatively simple and totally geometric conjectural description of uncolored DAHA superpolynomials of arbitrary algebraic knots (conjecturally coinciding with the reduced stable Khovanov-Rozansky polynomials) via the flagged…

The chromatic polynomial is characterized as the unique polynomial invariant of graphs, compatible with two interacting bialgebras structures: the first coproduct is given by partitions of vertices into two parts, the second one by a…

Motivated by some recent works on BPS invariants of open strings/knot invariants, we guess there may be a general correspondence between the Ooguri-Vafa invariants of toric Calabi-Yau 3-folds and cohomologies of Nakajima quiver varieties.…

In these notes we generalize the Ohsawa's results on the Hartogs extension phenomenon in the complement of effective divisors in K\"ahler manifolds with semipositive non-flat normal bundle. Namely, we prove that the Hartogs extension…

$R$-coloured knot polynomials for $m$-strand torus knots $Torus_{[m,n]}$ are described by the Rosso-Jones formula, which is an example of evolution in $n$ with Lyapunov exponents, labelled by Young diagrams from $R^{\otimes m}$. This means…

Using the vertex model approach for braid representations, we compute polynomials for spin-1 placed on hyperbolic knots up to 15 crossings. These polynomials are referred to as 3-colored Jones polynomials or adjoint Jones polynomials.…

We derive a new explicit formula in terms of sums over graphs for the $n$-point correlation functions of general formal weighted double Hurwitz numbers coming from the Kadomtsev-Petviashvili tau functions of hypergeometric type (also known…

We consider the Hurwitz spaces of ramified coverings of $\mathbb{P}^1$ with prescribed ramification profile over the point at infinity. By means of a particular symmetric bidifferential on a compact Riemann surface, we introduce…

Following the suggestion of arXiv:1407.6319 to lift the knot polynomials for virtual knots and links from Jones to HOMFLY, we apply the evolution method to calculate them for an infinite series of twist-like virtual knots and antiparallel…

HOMFLY polynomials are one of the major knot invariants being actively studied. They are difficult to compute in the general case but can be far more easily expressed in certain specific cases. In this paper, we examine two particular…

We study resurgence for some 3-manifold invariants when $G_{\mathbb{C}}=SL(2, \mathbb{C})$. We discuss the case of an infinite family of Seifert manifolds for general roots of unity and the case of the torus knot complement in $S^3$. Via…

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 define a new pair of dual bases that generalize the immaculate and dual immaculate bases to the colored algebras $QSym_A$ and $NSym_A$. The colored dual immaculate functions are defined combinatorially via tableaux, and we present…