相关论文: Towards tangle calculus for Khovanov polynomials

We look at how evolution method deforms, when one considers Khovanov polynomials instead of Jones polynomials. We do this for the figure-eight-like knots (also known as 'double braid' knots, see arXiv:1306.3197) -- a two-parametric family…

We conjecture explicit evolution formulas for Khovanov polynomials for pretzel knots in some regions in the windings space. Our description is exhaustive for genera 1 and 2. As previously observed, evolution at T != -1 is not fully smooth:…

We compute explicitly the Khovanov polynomials (using the computer program from katlas.org) for the two simplest families of the satellite knots, which are the twisted Whitehead doubles and the two-strand cables. We find that a quantum…

$R$-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…

We consider braids with repeating patterns inside arbitrary knots which provides a multi-parametric family of knots, depending on the "evolution" parameter, which controls the number of repetitions. The dependence of knot (super)polynomials…

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…

We extend recent work by Howie, Mathews and Purcell to simplify the calculation of A-polynomials for any family of hyperbolic knots related by twisting. The main result follows from the observation that equations defining the deformation…

An elementary introduction to Khovanov construction of superpolynomials. Despite its technical complexity, this method remains the only source of a definition of superpolynomials from the first principles and therefore is important for…

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…

The main goal of the present paper is to construct new invariants of knots with additional structure by adding new gradings to the Khovanov complex. The ideas given below work in the case of virtual knots, closed braids and some other cases…

In this thesis we work with Khovanov homology of links and its generalizations, as well as with the homology of graphs. Khovanov homology of links consists of graded chain complexes which are link invariants, up to chain homotopy, with…

We begin the systematic study of knot polynomials for the twist satellites of a knot, when its strand is substituted by a 2-strand twist knot. This is a generalization of cabling (torus satellites), when the substitute of the strand was a…

In this paper, we use skein-theoretic techniques to classify all virtual knot polynomials and trivalent graph invariants with certain smallness conditions. The first half of the paper classifies all virtual knot polynomials giving…

We explore properties of braids such as their fractional Dehn twist coefficients, right-veeringness, and quasipositivity, in relation to the transverse invariant from Khovanov homology defined by Plamenevskaya for their closures, which are…

We briefly summarise our results on jumps in the analytic formulas for the Khovanov(-Rozansky) polynomials. We conclude from the empiric data that there are ``regular'' and ``weird'' catastrophes, which drastically differ by form of the…

We show that embedding calculus invariants $ev_n$ are surjective for long knots in an arbitrary $3$-manifold. This solves some remaining open cases of Goodwillie--Klein--Weiss connectivity estimates, and at the same time confirms one half…

Twisting a given knot $K$ about an unknotted circle $c$ a full $n \in \mathbb{N}$ times, we obtain a "twist family" of knots $\{ K_n \}$. Work of Kouno-Motegi-Shibuya implies that for a non-trivial twist family the crossing numbers…

In order to obtain a Markov theorem without stabilization, Birman and Menasco introduced the notion of exchange related braids. In this paper I study the way the Fiedler polynomial distinguishes conjugacy classes of some particular braided…

Computing the Jones polynomial of general link diagrams is known to be $\#$P-hard, while restricting the computation to braid closures on fixed number of strands allows for a polynomial time algorithm. We investigate polynomial time…

We elaborate on the recent observation that evolution for twist knots simplifies when described in terms of triangular evolution matrix ${\cal B}$, not just its eigenvalues $\Lambda$, and provide a universal formula for ${\cal B}$,…