Narrow Results

Vaughan Jones was my mentor and friend. I provide here a few personal recollections including the “knotted antennas” project which brought me to Berkeley.

There is a one-to-one correspondence between 2-dimensional Topological quantum field theories and Frobenius extensions. Therefore each Frobenius extension defines an invariant of surfaces. We explore these invariants for a family of Frobenius extensions. In addition, we investigate the skein module of surfaces embedded in 3-manifolds corresponding to this family of Frobenius extensions.

In this paper, we compute the graph skein algebra of the punctured disk with two holes. Then, we apply the graph skein techniques developed here to establish necessary conditions for a spatial graph to have a symmetry of order p, where p is a prime. The obstruction criteria introduced here extend some results obtained earlier for symmetric spatial graphs.

We compute the Dubrovnik skein module of the lens spaces $L_{p,1}$, $p>1$, as well as the Kauffman two variables skein module when $p$ is odd. We also show that there is torsion in the Kauffman skein module when $p$ is even.

The Chebyshev polynomials appear somewhat mysteriously in the theory of the skein modules. A generalization of the Chebyshev polynomials is proposed so that it includes both Chebyshev and Fibonacci and Lucas polynomials as special cases. Then, since it requires relaxation of a condition for traces of matrix powers and matrix representations, similar relaxation leads to a generalization of the Jones polynomial via reinterpretation of the Kauffman bracket construction. Moreover, the Witten’s approach via counting solutions of the Kapustin–Witten equation to get the Jones polynomial is simplified in the trivial knots case to studying solutions of a Laplace operator. Thus, harmonic ideas may be of importance in knot theory. Speculations on extension(s) of the latter approach via consideration of spin structures and related operators are given.

In this paper we present two new bases, $B_{H_2}^{\prime }$ and ${{ℬ}}_{H_2}$, for the Kauffman bracket skein module of the handlebody of genus 2 $H_2$, KBSM$\left(H_2\right)$. We start from the well-known Przytycki-basis of KBSM$\left(H_2\right)$, $B_{H_2}$, and using the technique of parting we present elements in $B_{H_2}$ in open braid form. We define an ordering relation on an augmented set $L$ consisting of monomials of all different “loopings” in $H_2$, that contains the sets $B_{H_2}$, $B_{H_2}^{\prime }$ and ${{ℬ}}_{H_2}$ as proper subsets. Using the Kauffman bracket skein relation we relate $B_{H_2}$ to the sets $B_{H_2}^{\prime }$ and ${{ℬ}}_{H_2}$ via a lower triangular infinite matrix with invertible elements in the diagonal. The basis $B_{H_2}^{\prime }$ is an intermediate step in order to reach at elements in ${{ℬ}}_{H_2}$ that have no crossings on the level of braids, and in that sense, ${{ℬ}}_{H_2}$ is a more natural basis of KBSM$\left(H_2\right)$. Moreover, this basis is appropriate in order to compute Kauffman bracket skein modules of closed–connected–oriented (c.c.o.) 3-manifolds $M$ that are obtained from $H_2$ by surgery, since isotopy moves in $M$ are naturally described by elements in ${{ℬ}}_{H_2}$.

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.

We determine the action of the Kauffman bracket skein algebra of the torus on the Kauffman bracket skein module of the complement of the 3-twist knot. The point is to study the relationship between knot complements and their boundary tori, an idea that has proved very fruitful in knot theory. We place this idea in the context of Chern–Simons theory, where such actions arose in connection with the computation of the noncommutative version of the A-polynomial that was defined by Frohman, the first author, and Lofaro, but they can also be interpreted as quantum mechanical systems. Our goal is to exhibit a detailed example in a part of Chern–Simons theory where examples are scarce.

This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by Melvin and Kirby that underlie the quantum group-based Reshetikhin–Turaev model for $\mathrm{\text{SU}}(2)$ Chern–Simons theory to the Kauffman bracket skein modules. Several applications and examples are presented.

We compute the action of the ${𝔤𝔩}_2$-skein algebra of the torus on the ${𝔤𝔩}_2$-skein module of the solid torus. As a result, we show that the ${𝔤𝔩}_2$-skein modules of lens spaces is spanned by $(\lfloor \frac{p}{2}\rfloor +1)(2\lfloor \frac{p}{2}\rfloor +1)$ elements.