arXiv research

New method for knot closures from 1-tangles and annulus twists.

Given an invariant J(K) of a knot K, the corresponding (1,1)-tangle invariant J'(K)=J(K)/J(U) is defined as the quotient of J(K) by its value J(U) on the unknot U. We prove here that J' is always an integer 2-variable Laurent polynomial when J is the Homfly satellite invariant determined by decorating K with any eigenv…

A genus-1 tangle G is an arc properly embedded in a standardly embedded solid torus S in the 3-sphere. We say that a genus-1 tangle embeds in a knot K in S^3 if the tangle can be completed by adding an arc exterior to the solid torus to form the knot K. We call K a closure of G. An obstruction to embedding a genus-1 ta…

Given a compact oriented 3-manifold M in S^3 with boundary, an (M,2n)-tangle T is a 1-manifold with 2n boundary components properly embedded in M. We say that T embeds in a link L in S^3 if T can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of T. We define …

This paper refines previous work by the first author. We study the question of which links in the 3-sphere can be obtained as closures of a given 1-manifold in an unknotted solid torus in the 3-sphere (or genus-1 tangle) by adjoining another 1-manifold in the complementary solid torus. We distinguish between even and o…

Khovanov homology for links in S^3 via 1-tangle diagrams in annulus.

We generalize our previous work on categorification of Kauffman bracket skein module of surfaces, by extending our homology to tangles in cylinders over surfaces, F x [0,1]. Our homology of 0-tangles and 1-tangles in D^3 coincides (up to normalization) with Khovanov link homology and the reduced Khovanov link homology.…

Tangle replacements help in understanding knot properties.

We introduce and study in detail an invariant of (1,1) tangles. This invariant, derived from a family of four dimensional representations of the quantum superalgebra U_q[gl(2|1)], will be referred to as the Links-Gould invariant. We find that our invariant is distinct from the Jones, HOMFLY and Kauffman polynomials (de…

A tangle is an oriented 1-submanifold of the cylinder whose endpoints lie on the two disks in the boundary of the cylinder. Using an algebraic tool developed by Lescop, we extend the Burau representation of braids to a functor from the category of oriented tangles to the category of Z[t,t^{-1}]-modules. For (1,1)-tangl…

Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q(sl_2)$ denote the quantum group with large center corresponding to the lie algebra $sl_2$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q(sl_2)$ is an $N$-dimensional irreducible representation $ρ:U_q(sl_2)\rightarrow M_N(\mathbb{C}…

Random hyperbolic surfaces are mostly tangle-free, with geometric implications.

We explore a knot invariant derived from colorings of corresponding $1$-tangles with arbitrary connected quandles. When the quandle is an abelian extension of a certain type the invariant is equivalent to the quandle $2$-cocycle invariant. We construct many such abelian extensions using generalized Alexander quandles w…

We study relationships between the restricted unrolled quantum group $\overline{U}_q^H(\mathfrak{sl}_2)$ at $2r$-th root of unity $q=e^{πi/r}, r \geq 2$, and the singlet vertex operator algebra $\mathcal M(r)$. We use deformable families of modules to efficiently compute $(1, 1)$-tangle invariants colored with projecti…

Paper provides criteria to detect non-admissible quandles via coloring.

The knot coloring polynomial defined by Eisermann for a finite pointed group is generalized to an infinite pointed group as the longitudinal mapping invariant of a knot. In turn this can be thought of as a generalization of the quandle 2-cocycle invariant for finite quandles. If the group is a topological group then th…

Researchers link knot Floer homology, Burau representation, and quantum gl(1|1).

The abstract describes a strategy to construct reduced Khovanov homology for links in lens spaces.

The ``Links-Gould invariant'' is a two-variable Laurent polynomial invariant of oriented (1,1) tangles, which is derived from the representation of the braid generator associated with the one-parameter family of four dimensional representations with highest weights (0,0|a) of the quantum superalgebra U_q[gl(2|1)]. We u…