A cohomology theory for colored tangles

TL;DR

This paper develops a cohomology theory for colored tangles using sl(2) foam cohomology, categorifying the colored Jones polynomial and enabling efficient computation of colored invariants for knots and links.

Contribution

It introduces a new cohomology framework for colored tangles that extends existing categorifications of the colored Jones polynomial.

Findings

Provides a categorification of the colored Jones polynomial.

Enables efficient computation of colored knot and link invariants.

Defines the theory over Gaussian integers and formal parameters.

Abstract

We employ the sl(2) foam cohomology to define a cohomology theory for oriented framed tangles whose components are labelled by irreducible representations of U_q(sl(2)). We show that the corresponding colored invariants of tangles can be assembled into invariants of bigger tangles. For the case of knots and links, the corresponding theory is a categorification of the colored Jones polynomial, and provides a tool for efficient computations of the resulting colored invariant of knots and links. Our theory is defined over the Gaussian integers Z[i] (and more generally over Z[i][a,h], where a,h are formal parameters), and enhances the existing categorifications of the colored Jones polynomial.