The Singular Temperley-Lieb Category

Carmen Caprau,Joel Smith

doi:10.1155/2014/321509

Abstract

We introduce and study the singular Temperley-Lieb category over ℤ[q,q-1], which is a free pivotal category over two self-dual generators and is an extension of the (classical) Temperley-Lieb category. Our construction is motivated by a state model for the sl(2) polynomial of an oriented link and provides a categorical perspective to this link invariant. We also construct a couple of polynomial invariants for oriented tangles from category theory point of view.

Highlights

The Temperley-Lieb algebra has played a central role in the discovery of the Jones polynomial [1] and in the subsequent developments relating to knot theory, topological quantum field theory, and statistical mechanics [2]
The chain homotopy equivalence class of the complex is an invariant of the tangle and specializes in the Khovanov homology when the tangle is a link
We remark that BarNatan [7] used another approach to Khovanov homology for tangles via cobordisms modulo local relations

Summary

Introduction

The Temperley-Lieb algebra has played a central role in the discovery of the Jones polynomial [1] and in the subsequent developments relating to knot theory, topological quantum field theory, and statistical mechanics [2]. −q) via webs, which are oriented bivalent graphs whose vertices are either “sinks” or “sources.” This state model for the sl(2) polynomial for oriented links constitutes the motivation of this paper, and our main goal is to understand this model and the corresponding geometric objects from the category theory point of view. Regarding the vertices of the webs as singularities on diagrams, we call a state associated with a planar tangle diagram a singular flat tangle and we regard it as morphism in a monoidal category STL, which we refer to as the singular Temperley-Lieb category. An oriented edge in a singular flat tangle stands for the standard (2-dimensional) vector representation dual (i.e., isomorphic tVo1itosfdquuaalnrteupmresseln(2ta).tiSoinncVe1∗V),1itisissnelof-t surprising that every object in the singular Temperley-Lieb category STL is self-dual. Throughout the paper, we represent a bivalent vertex with a red triangle pointing toward the preferred edge of that vertex

The Category of Oriented Tangles

The Singular Temperley-Lieb Category

Polynomial Invariants for Tangles

Ribbon Graphs and Singular Flat Tangles