Annular SL(2) and SL(3) web algebras

TL;DR

This paper introduces equivariant annular SL(2) and SL(3) web algebras using annular foam TQFTs, establishing invariants for tangles and exploring algebraic properties and bijections with weight lattice paths.

Contribution

It defines new annular web algebras and constructs tangle invariants via complexes of bimodules, extending planar web-bijections to the annular setting.

Findings

Defined equivariant annular SL(2) and SL(3) web algebras.

Constructed tangle invariants as complexes of bimodules.

Established a bijection between non-elliptic annular SL(3) webs and closed paths in the SL(3) weight lattice.

Abstract

We use annular foam TQFTs introduced by the first two authors to define equivariant $SL(2)$ and $SL(3)$ web algebras in the annulus. To a diagram of a tangle in the thickened annulus we assign a complex of bimodules over these algebras whose chain homotopy type is an invariant of the tangle. Several properties of algebras and bimodules are established. An essential technical part of the paper provides a bijective correspondence between non-elliptic annular $SL(3)$ webs and closed paths in the $SL(3)$ weight lattice. This generalizes an analogous bijection in the planar setting.