Polynomial knot and link invariants from the virtual biquandle

Authors

Alissa S. Crans, Loyola Marymount UniversityFollow
Allison Henrich, Seattle University
Sam Nelson, Claremont McKenna College

Document Type

Article - post-print

Publication Date

2013

Abstract

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.

Comments

This is the post-print version of the article.

Original Publication Citation

Crans, A.; Henrich, A.; and Nelson, S. “Knot and link invariants from the Alexander virtual biquandle.” Journal of Knot Theory and its Ramifications, Vol. 22 (2013), No. 4.

Digital Commons @ LMU & LLS Citation

Crans, Alissa S.; Henrich, Allison; and Nelson, Sam, "Polynomial knot and link invariants from the virtual biquandle" (2013). Mathematics, Statistics and Data Science Faculty Works. 60.
https://digitalcommons.lmu.edu/math_fac/60