Polynomial knot and link invariants from the virtual biquandle

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.

12 pages; version 3 includes corrected figures

References in corpus (1)

• Parity Biquandles

Cited by in corpus (10)

• Alexander invariants for virtual knots
• On possible existence of HOMFLY polynomials for virtual knots
• Milnor's concordance invariants for knots on surfaces
• Virtual concordance and the generalized Alexander polynomial
• Prime Decomposition and Non-Commutativity in the Monoid of Long Virtual Knots
• Quotient Quandles and the Fundamental Latin Alexander Quandle
• Psyquandles, Singular Knots and Pseudoknots
• Biquasiles and Dual Graph Diagrams
• Extensions of Tong-Yang-Ma representation
• Virtual Knot Groups