Anchored foams and annular homology

TL;DR

This paper develops a new foam evaluation framework for equivariant SL(2) and SL(3) homology of links in the solid torus, incorporating anchor points that enhance the algebraic structure.

Contribution

It introduces a novel foam evaluation method for equivariant SL(2) and SL(3) homology in the solid torus using anchor points and boundary intersections.

Findings

Defined generators of state spaces via foams with boundary

Extended foam evaluation to include anchor points

Handled both oriented and unoriented SL(3) foams

Abstract

We describe equivariant SL(2) and SL(3) homology for links in the solid torus via foam evaluation. The solid torus is replaced by 3-space with a distinguished line in it. Generators of state spaces for annular webs are represented by foams with boundary that may intersect the distinguished line; intersection points, called anchor points, contribute additional terms, reminiscent of square roots of the Hessian, to the foam evaluation. Both oriented and unoriented SL(3) foams are treated in the paper.