 
Summary: The Alexander and Jones Polynomials Through
Representations of Rook Algebras
Stephen Bigelow
, Eric Ramos
, Ren Yi
October 5, 2011
Abstract
In the 1920's Artin defined the braid group, Bn, in an attempt to understand
knots in a more algebraic setting. A braid is a certain arrangement of strings in
threedimensional 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 re
cent 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 in
cluding 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.
1 Introduction
