A Burau-Alexander 2-functor on tangles

TL;DR

This paper introduces a weak 2-functor from the bicategory of oriented tangles to Lagrangian cospans, unifying several classical invariants like the Burau representation and Alexander modules within a higher categorical framework.

Contribution

It constructs a novel weak 2-functor that extends the Burau representation, Turaev's tangle generalization, and Alexander modules, linking them through a categorical approach.

Findings

Unified tangle invariants via 2-functor construction

Extension of Burau representation to tangles

Connection between Alexander modules and Lagrangian cospans

Abstract

We construct a weak 2-functor from the bicategory of oriented tangles to a bicategory of Lagrangian cospans. This functor simultaneously extends the Burau representation of the braid groups, its generalization to tangles due to Turaev and the first-named author, and the Alexander module of 1 and 2-dimensional links.