Related papers: Generalizing the relation between the Kauffman bra…
For any n>1 we define an isotopy invariant, <Gamma>_n, for a certain set of n-valent ribbon graphs Gamma in R^3, including all framed oriented links. We show that our bracket coincides with the Kauffman bracket for n=2 and with the…
We use recoupling theory to study the Kauffman bracket skein module of the quaternionic manifold over Z[A,A^{-1}] localized by inverting all the cyclotomic polynomials. We prove that the skein module is spanned by five elements. Using the…
This paper resolves the problem of comparing the skein modules defined using the skein relations discovered by R. Kirby and P. Melvin that underlie the Reshetikhin-Turaev model for $SU(2)$ Chern-Simons theory to the Kauffman bracket skein…
We extend the results of our previous paper from knots to links by using a formula for the Jones polynomial of a link derived recently by N. Reshetikhin. We illustrate this formula by an example of a torus link. A relation between the…
We construct an infinite family of homology theories of framed links in thickened surfaces, as well as a homology theory whose graded Euler characteristic is exactly the Kauffman bracket of the link in the surface. Both theories are based…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
We construct a simple topological invariant of certain 3-manifolds, including quotients of the 3-sphere by finite groups, based on the fact that the tangent bundle of an orientable 3-manifold is trivialisable. This invariant is strong…
This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is…
Let $G$ be a connected reductive algebraic group over a field of positive characteristic $p$ and denote by $\mathcal T$ the category of tilting modules for $G$. The higher Jones algebras are the endomorphism algebras of objects in the…
Based on different views on the Jones polynomial we review representation theoretic categorified link and tangle invariants. We unify them in a common combinatorial framework and connect them via the theory of Soergel bimodules. The…
The W-polynomial is applied in two ways to questions involving the Kauffman bracket of some families of links. First we find a geometric property of a link diagram, which is less than or equal to the twist number, that bounds the Mahler…
We construct a bigraded cohomology theory of links whose Euler characteristic is the Jones polynomial.
For a 3-manifold $M$ with boundary, we study the Kauffman module with indeterminate equal to $-1+\epsilon$ where $\epsilon^2=0$. We conjecture an explicit relation between this module and the Reidemeister torsion of $M$ which we prove in…
We introduce a quantum trace map for an ideally triangulated hyperbolic knot complement $S^3\backslash \mathcal{K}$. The map assigns a quantum operator to each element of Kauffmann Skein module of the 3-manifold. The quantum operator lives…
Just as the Temperley-Lieb algebra is a good place to compute the Jones polynomial, the Kauffman bracket skein algebra of a disk with $2k$ colored points on the boundary, each with color $n$, is a good place to compute the $n^{th}$ colored…
We consider two Laurent polynomials in two variables associated to a braid, given by {\em graded intersections} between {\em fixed Lagrangians in configuration spaces}. In order to get link invariants, we notice that we have to quotient by…
In a recent paper Jones introduced a correspondence between elements of the Thompson group $F$ and certain graphs/links. It follows from his work that several polynomial invariants of links, such as the Kauffman bracket, can be…
A Kauffman bracket on a surface is an invariant for framed links in the thickened surface, satisfying the Kauffman skein relation and multiplicative under superposition. This includes representations of the skein algebra of the surface. We…
This paper is an introduction to Khovanov homology, starting with the Kauffman bracket state summation, emphasizing the Bar-Natan Canopoloy and tangle cobordism approach. The paper discusses a simplicial approach to Khovanov homology and a…
Let M be a closed 3-manifold and S(M) the skein module of M at some odd root of unity. Using the Frobenius morphism, we can see S(M) as the space of global sections of a coherent sheaf over the SL2 character scheme of M. We prove that when…