Related papers: A trivial tail homology for non $A$-adequate links
We show that the head and tail functions of the colored Jones polynomial of adequate links are the product of head and tail functions of the colored Jones polynomial of alternating links that can be read-off an adequate diagram of the link.…
It is known that the colored Jones polynomial of a $+$-adequate link has a well-defined tail consisting of stable coefficients, and that the coefficients of the tail carry geometric and topological information on the $+$-adequate link…
C. Armond, S. Garoufalidis and T.Le have shown that a unicolored Jones polynomial of a B-adequate link has a stable tail at large colors. We categorify this tail by showing that Khovanov homology of a unicolored link also has a stable tail,…
We use categorical annular evaluation to give a uniform construction of both $\mathfrak{sl}_n$ and HOMFLYPT Khovanov-Rozansky link homology, as well as annular versions of these theories. Variations on our construction yield…
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 prove an explicit formula for the tail of the colored Jones polynomial for a class of arborescent links in terms of a product of theta functions and/or false theta functions. We also provide numerical evidence towards a classification of…
It is still unknown whether there is a nontrivial knot with Jones polynomial equal to that of the unknot. Tanaka shows that if an amphichiral knot is a symmetric union of the unknot with one twist region, then its Jones polynomial is…
We use the machinery of categorified Jones-Wenzl projectors to construct a categorification of a type A Reshetikhin-Turaev invariant of oriented framed tangles where each strand is labeled by an arbitrary finite-dimensional representation.…
We prove that the Jones diameter of a link is twice its crossing number whenever the breadth of its Jones polynomial equals the difference between the crossing number and the Turaev genus. This implies that such link is adequate, as per the…
We prove that the bigraded colored Khovanov-Rozansky type A link and tangle invariants are functorial with respect to link and tangle cobordisms.
Using an involved study of the Jones polynomial, we determine, as our main result, the crossing numbers of (prime) amphicheiral knots. As further applications, we show that several classes of links, including semiadequate links and…
The goal of this paper is to address A. Shumakovitch's conjecture about the existence of $\Z_2$-torsion in Khovanov link homology. We analyze torsion in Khovanov homology of semi-adequate links via chromatic cohomology for graphs which…
We give infinitely many $2$-component links with unknotted components which are topologically concordant to the Hopf link, but not smoothly concordant to any $2$-component link with trivial Alexander polynomial. Our examples are pairwise…
We investigate the coefficients of the highest and lowest terms (also called the head and the tail) of the colored Jones polynomial and show that they stabilize for alternating links and for adequate links. To do this we apply techniques…
We show that Khovanov homology and Hochschild homology theories share common structure. In fact they overlap: Khovanov homology of a $(2,n)$-torus link can be interpreted as a Hochschild homology of the algebra underlining the Khovanov…
We prove that the length of any gap in the differential grading of the Khovanov homology of any quasi-alternating link is one. As a consequence, we obtain that the length of any gap in the Jones polynomial of any such link is one. This…
In the cabling procedure for HOMFLY polynomials colored HOMFLY polynomials of a knot are obtained from ordinary HOMFLY of the cabled knot with extra twists added. Thus colored polynomials can be seen as relation between HOMFLYs of cabled…
We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate…
We define a family of formal Khovanov brackets of a colored link depending on two parameters. The isomorphism classes of these brackets are invariants of framed colored links. The Bar-Natan functors applied to these brackets produce…
We use a simple geometric argument and small cancellation properties of link groups to prove that alternating links are non-trivial. This proof uses only classic results in topology and combinatorial group theory.