Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links

Details Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links Bi-oriented Quantum Algebras, and a Generalized Alexander Polynomial for Virtual Links

2001-12-26

arxiv.org/abs/math/0112280v2

Mathematics

Geometric Topology

LaTeX document, 34 pages, 13 figures, in text LaTeX graphics

Scientific paper

This paper discusses the construction of a generalized Alexander polynomial for virtual knots and links, and the reformulation of this invariant as a quantum link invariant. The algebraic background for the generalized Alexander module is formulated in terms of the biquandle, a generalization of the quandle of David Joyce (which, in turn, is a generalization of the fundamental group of the knot or link). We then introduce the concept of a bi-oriented quantum algebra which provides an algebraic context for the associated quantum invariant.

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