A Diagrammatic Multivariate Alexander Invariant of Tangles

TL;DR

This paper introduces a multivariate extension of Bigelow's diagrammatic Alexander polynomial calculation, providing a generalized tangle invariant that up to Reidemeister I, connects to subfactor planar algebras.

Contribution

It presents a novel multivariate tangle invariant based on a diagrammatic approach, extending previous methods to include Reidemeister I moves and linking to subfactor planar algebras.

Findings

Provides a multivariate tangle invariant up to Reidemeister I

Generalizes Bigelow's diagrammatic Alexander polynomial calculation

Suggests a connection to subfactor planar algebras

Abstract

Recently, Bigelow defined a diagrammatic method for calculating the Alexander polynomial of a knot or link by resolving crossings in a planar algebra. I will present my multivariate version of Bigelow's calculation. The advantage to my algorithm is that it generalizes to a multivariate tangle invariant up to Reidemeister I. I will conclude with a possible link to subfactor planar algebras from the work of Jones and Penneys.