Polynomial knot and link invariants from the virtual biquandle

Details Polynomial knot and link invariants from the virtual biquandle Polynomial knot and link invariants from the virtual biquandle

2011-10-06

arxiv.org/abs/1110.1371v2

Mathematics

Geometric Topology

12 pages; version 2 includes a new example and a simplified coefficient ring

Scientific paper

The Alexander biquandle of a virtual knot or link is a module over a 2-variable Laurent polynomial ring which is an invariant of virtual knots and links. The elementary ideals of this module are then invariants of virtual isotopy which determine both the generalized Alexander polynomial (also known as the Sawollek polynomial) for virtual knots and the classical Alexander polynomial for classical knots. For a fixed monomial ordering $<$, the Gr\"obner bases for these ideals are computable, comparable invariants which fully determine the elementary ideals and which generalize and unify the classical and generalized Alexander polynomials. We provide examples to illustrate the usefulness of these invariants and propose questions for future work.

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