Lectures on SL(3) foams and link homology

TL;DR

This paper introduces the evaluation of SL(3) foams and their role in link homology, connecting topological graph theory with categorification of quantum invariants.

Contribution

It provides an accessible introduction to SL(3) foam evaluation and its application to link homology, including categorification and foam cobordism techniques.

Findings

Review of Kuperberg web categorification

Explanation of SL(3) foam evaluation methods

Connection to topological and quantum invariants

Abstract

These notes are based on the three lectures that one of the authors gave at Tsinghua University in the summer of 2023 as part of the workshop on Geometric Representation Theory and Applications. They contain an introduction to the evaluation of $\mathsf{SL}(3)$ foams and the associated topological theory of trivalent planar graphs and foam cobordisms between them. A categorification of the Kuperberg quantum $\mathfrak{sl}_3$ web and link invariant and the Robert-Wagner $\mathsf{SL}(N)$ foam evaluation are reviewed as well.