A presentation for the category of $sl(3)$ webs and an extension of the quantum $sl(3)$-invariant to tangles

TL;DR

This paper extends the $sl(3)$-polynomial invariant from links to tangles by defining a category of $sl(3)$ webs, establishing a functor from tangles to webs, and recovering the invariant for links.

Contribution

It introduces a new categorical framework for $sl(3)$ webs and extends the quantum $sl(3)$-invariant to tangles, generalizing previous link invariants.

Findings

Defined a category of $sl(3)$ webs with generators and relations

Constructed a functor from tangles to $sl(3)$ webs

Extended the $sl(3)$-polynomial invariant to tangles

Abstract

We extend the $sl(3)$-polynomial invariant for links to tangles. Motivated by Kuperberg's construction of this invariant via planar trivalent graphs, we first define a category of $sl(3)$ webs and its sister linear category, and describe them via generators and relations. Then we define a functor from the category of oriented tangles to the linear category of $sl(3)$ webs, which yields an invariant for tangles and allows us to recover the $sl(3)$-polynomial invariant for links.