# Wydawnictwa / Czasopisma IMPAN / Fundamenta Mathematicae / Wszystkie zeszyty

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

### Tom 184 / 2004

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

#### Streszczenie

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.

#### Autorzy

• 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

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