The equation $[B,(A-1)(A,B)]=0$ and virtual knots and links

Volume 184 / 2004

Stephen Budden, Roger Fenn Fundamenta Mathematicae 184 (2004), 19-29 MSC: 57M25, 57M27. DOI: 10.4064/fm184-0-2

Abstract

Let $A$, $B$ be invertible, non-commuting elements of a ring $R$. Suppose that $A-1$ is also invertible and that the equation $[B,(A-1)(A,B)]=0$ called the fundamental equation is satisfied. Then this defines a representation of the algebra ${\mathcal F}=\{ A, B\mid [B,(A-1)(A,B)]=0\} $. An invariant $R$-module can then be defined for any diagram of a (virtual) knot or link. This halves the number of previously known relations and allows us to give a complete solution in the case when $R$ is the quaternions.

Authors

  • Stephen BuddenSchool of Mathematical Sciences
    University of Sussex
    Falmer, Brighton, BN1 9RH, England
    e-mail
  • Roger FennSchool of Mathematical Sciences
    University of Sussex
    Falmer, Brighton, BN1 9RH, England
    e-mail

Search for IMPAN publications

Query phrase too short. Type at least 4 characters.

Rewrite code from the image

Reload image

Reload image