The Alexander and Jones Polynomials Through Representations of Rook Algebras

TL;DR

This paper explores how the Jones and Alexander polynomials, key knot invariants, can be derived from representations of the braid group via the planar rook algebra, linking algebraic and topological knot properties.

Contribution

It introduces a novel approach to obtain knot invariants using representations of the braid group through the planar rook algebra, expanding the algebraic tools for knot theory.

Findings

Jones and Alexander polynomials recovered from rook algebra representations

Trace calculations link algebraic representations to topological invariants

New connections established between diagrammatic algebras and knot invariants

Abstract

In the 1920's Artin defined the braid group in an attempt to understand knots in a more algebraic setting. A braid is a certain arrangement of strings in three-dimensional space. It is a celebrated theorem of Alexander that every knot is obtainable from a braid by identifying the endpoints of each string. Because of this correspondence, the Jones and Alexander polynomials, two of the most important knot invariants, can be described completely using the braid group. There has been a recent growth of interest in other diagrammatic algebras, whose elements have a similar topological flavor to the braid group. These have wide ranging applications in areas including representation theory and quantum computation. We consider representations of the braid group when passed through another diagrammatic algebra, the planar rook algebra. By studying traces of these matrices, we recover both the Jones and Alexander polynomials.