Journal of Knot Theory and Its Ramifications

In the spirit of Bar-Natan’s construction of Khovanov homology, we give a categorification of the Vandermonde determinant. Given a sequence of positive integers $\overrightarrow{x}=(x_1,\dots ,x_n)$, we construct a complex of colored smoothings of the two-strand torus link $T_{2,n}$ in the shape of the Bruhat order on $S_n$, and apply a topological quantum field theory (TQFT) to obtain a chain complex whose Euler characteristic is equal to the Vandermonde determinant evaluated at $\overrightarrow{x}$. A generalization to arbitrary link diagrams is given, yielding categorifications of certain generalized Vandermonde determinants.

This work presents formulas for the Kauffman bracket and Jones polynomials of three-bridge knots using the structure of Chebyshev knots and their billiard table diagrams. In particular, these give far fewer terms than in the Skein relation expansion. The subject is introduced by considering the easier case of two-bridge knots, where some geometric interpretation is provided, as well, via combinatorial tiling problems.

We show that relations in Homflypt type skein theory of an oriented $3$-manifold $M$ are induced from a $2$-groupoid defined from the fundamental $2$-groupoid of a space of singular links in $M$. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental $2$-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.

A well-known identity ${\widehat{\mathrm{\Delta }}}_{L_{+}}-{\widehat{\mathrm{\Delta }}}_{L_{-}}=(t^{\frac{1}{2}}-t^{\frac{-1}{2}})\cdot {\widehat{\mathrm{\Delta }}}_{L_0}$ holds for three 1-links $L_{+}$, $L_{-}$, and $L_0$ which satisfy a famous local-move relation, where ${\widehat{\mathrm{\Delta }}}_L$ becomes the Alexander–Conway polynomial of $L$ if we let $z=t^{\frac{1}{2}}-t^{\frac{-1}{2}}$. We prove a new local-move identity for the $\mathbb{Z}[t,t^{-1}]$-Alexander polynomials of 2-links, which is a 2-dimensional analogue of the 1-dimensional one. In the 1-dimensional link case there is a well-known relation between the Alexander–Conway polynomial and the linking number. As its 2-dimensional analogue, we find a relation between the $\mathbb{Z}[t,t^{-1}]$-Alexander polynomials of 2-links and the alinking number of 2-links. We produce high-dimensional analogues of these results. Furthermore, we prove that in the 2-dimensional case we cannot normalize the $\mathbb{Z}[t,t^{-1}]$-Alexander polynomials to be compatible with our identity but that in a high-dimensional case we can do that to be compatible with our new identity.

Conference like this in Dallas is a perfect occasion to give some reminiscences of my 40 years of work as a mathematician, and to muse a little about my mathematical past. It is a rare occasion to be a little sentimental. In this essay, I summarize and expand my birthday talk given at University of Texas at Dallas conference. There, I mostly covered the early years of my mathematical career ending with a detailed description about the smallest volume hyperbolic 3-manifold and the HOMFLYPT polynomial. Thus I seldom venture to the time after 1984 (for example to mention skein modules constructed in April 1987) and only occasionally after 1995 when I settled to work at George Washington University and have had several successful PhD students.

In this paper, we adjust the Yang–Baxter operators constructed by Jones for the HOMFLYPT polynomial. Then we compute the second homology for this family of Yang–Baxter operators. It has the potential to yield 2-cocycle invariants for links.