Quaternion Algebras and Invariants of Virtual Knots and Links II: The Hyperbolic Case

Details Quaternion Algebras and Invariants of Virtual Knots and Links II: The Hyperbolic Case Quaternion Algebras and Invariants of Virtual Knots and Links II: The Hyperbolic Case

2006-10-16

arxiv.org/abs/math/0610483v1

Mathematics

Geometric Topology

11 pages, accepted by JKTR

Scientific paper

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 an invariant $R$-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case have been found by Bartholomew, Budden and Fenn. Solutions in the generalised quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of $2\times2$ matrices and the aim of this paper is to provide solutions to the missing cases.

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