The Alexander and Jones polynomials through representations of rook algebras

Stephen Bigelow, Eric Ramos, Ren Yi

Research output: Contribution to journalArticlepeer-review

5 Scopus citations

Abstract

In the 1920's Artin defined the braid group, B n, 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.

Original languageEnglish
Article number1250114
JournalJournal of Knot Theory and its Ramifications
Volume21
Issue number12
DOIs
StatePublished - Oct 2012

Keywords

  • Alexander polynomial
  • Hecke algebra
  • Jones polynomial
  • Knot invariants
  • planar rook algebra
  • representations of braid groups

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