On Alexander-Conway Polynomials for Virtual Knots and Links
J. Sawollek
TL;DR
This paper introduces a new polynomial invariant for virtual links, derived from surface link invariants, which can detect properties like chirality and non-invertibility, and satisfies a Conway-type skein relation.
Contribution
It defines a novel polynomial invariant for virtual links that extends classical invariants and explores its properties and applications.
Findings
The polynomial can detect chirality of virtual knots.
It can identify non-invertibility of virtual links.
The polynomial satisfies a Conway-type skein relation.
Abstract
A polynomial invariant of virtual links, arising from an invariant of links in thickened surfaces introduced by Jaeger, Kauffman, and Saleur, is defined and its properties are investigated. Examples are given that the invariant can detect chirality and even non-invertibility of virtual knots and links. Furthermore, it is shown that the polynomial satisfies a Conway-type skein relation - in contrast to the Alexander polynomial derived from the virtual link group.
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Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Combinatorial Mathematics · Homotopy and Cohomology in Algebraic Topology