Videos

SL(4) Foam Evaluation and 6-Valent Seams

Presenter
May 13, 2025
Abstract
A crucial step to categorification of link (or web) polynomials via the `tautological construction' is the evaluation of closed foams. Robert-Wagner's `SL(n)' foam evaluations provide a purely combinatorial method for computing SL(n) link homology. In practice, the foams they use are actually GL(n) foams, which have facets of weights 1 through 4. Inspired by the recent development on rotational-invariant SL(4) web basis by Gaetz-Pechenik- Pfannerer-Striker-Swanson, we modify the Robert-Wagner foam evaluation formula to accommodate the use of SL(4) foams, which now feature only facets of weights 1 and 2, at the cost of introducing new types of defect lines and vertices. Additionally, we introduce a categorification of the `hexagon web relation' for both GL(4) and SL(4) foams. In this talk, I will review background material on webs and foams and talk about the modified SL(4) foam construction. If time permits, I will also mention its connections to SL(4) web algebra and SL(4) affine building. This talk is based on upcoming joint work with Jernej Grlj, Mikhail Khovanov, and Melissa Zhang.