POLYNOMIAL KNOT AND LINK INVARIANTS FROM THE VIRTUAL BIQUANDLE

Author:

CRANS ALISSA S.1,HENRICH ALLISON2,NELSON SAM3

Affiliation:

1. Department of Mathematics, Loyola Marymount University, One LMU Drive, Suite 2700, Los Angeles CA 90045, USA

2. Mathematics Department, Seattle University, 901 12th Avenue, PO Box 222000, Seattle WA 98122, USA

3. Department of Mathematical Sciences, Claremont McKenna College, 850 Columbia Ave., Claremont CA 91711, USA

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 Gröbner 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.

Publisher

World Scientific Pub Co Pte Lt

Subject

Algebra and Number Theory

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