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[in Journal: Journal of Knot Theory and Its Ramifications] AND [Keyword: Quaternions] (3)	27 Mar 2025	Run
[in Journal: Journal of Hyperbolic Differential Equations] AND [Keyword: Convergence] (6)	27 Mar 2025	Run
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articleNo Access
QUATERNION ALGEBRAS AND INVARIANTS OF VIRTUAL KNOTS AND LINKS I: THE ELLIPTIC CASE
- ROGER FENN
Journal of Knot Theory and Its Ramifications01 Mar 2008
Preview Abstract
In this paper, we show how generalized quaternions including some 2 × 2 matrices, can be used to find solutions of the equation
These solutions can then be used to find polynomial invariants of virtual knots and links. The remaining 2 × 2 matrices will be considered in a later paper.
articleNo Access
QUATERNION ALGEBRAS AND INVARIANTS OF VIRTUAL KNOTS AND LINKS II: THE HYPERBOLIC CASE
- STEPHEN BUDDEN and
- ROGER FENN
Journal of Knot Theory and Its Ramifications01 Mar 2008
Preview Abstract
Let A, B be invertible, non-commuting elements of a ring R. Suppose that A - 1 is also invertible and that the equation
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 generalized quaternion case have been found by Fenn in an earlier paper. These latter solutions are only partial in the case of 2 × 2 matrices and the aim of this paper is to provide solutions to the missing cases.
articleNo Access
BIQUANDLES AND THEIR APPLICATION TO VIRTUAL KNOTS AND LINKS
- ROGER FENN
Journal of Knot Theory and Its Ramifications01 Jun 2009
Preview Abstract
In this paper, based on a talk given at the Oberwolfach research centre in May 2008 I will describe how biquandles and their big brother, biracks, can be used to differentiate isotopy classes of virtual (and welded) knots and links.