A polynomial pair invariant of alternating knots and links

TL;DR

This paper introduces a new polynomial pair invariant for alternating knots and links, derived from their checkerboard graphs, which effectively distinguishes all knots up to 10 crossings and compares favorably with classical invariants.

Contribution

The paper presents a novel polynomial pair invariant for alternating knots and links, demonstrating its effectiveness in knot distinction and its relation to existing invariants.

Findings

Invariant distinguishes all knots up to 10 crossings

Values computed for knots in existing tables

Comparison shows the invariant's strength over classical invariants

Abstract

We introduce an invariant of alternating knots and links (called here WRP), namely a pair of integer polynomials associated with their two checkerboard planar graphs from their minimal diagram. We prove that the invariant is well-defined and give its values obtained from calculations for some knots in the tables. This invariant is strong enough to distinguish all knots in the tables with up to 10 crossings (including their mirror images). We compare the strength of the new invariant with classical invariants, including the three-variable Kauffman bracket.