A subtle new invariant for framed knots and links

TL;DR

The paper introduces a new facial state sum invariant for framed knots and links, which is simple yet effective in distinguishing mirror pairs and complements existing invariants like Jones and Kauffman polynomials.

Contribution

It presents a novel invariant based on face colorings that remains unchanged under Reidemeister moves, providing a new tool for knot distinction.

Findings

Successfully distinguishes mirror pairs of links.

Proves that knot 9_42 is not equivalent to its mirror image.

Complements existing invariants by distinguishing cases they cannot.

Abstract

We produce a facial state sum on plane diagrams of a knot or a link which admits an invariant specialization under Polyak's recent set of generating of 4 Reidemeister moves. Thus an isotopy invariant of framed links is obtained. Each state is a complete coloring of the faces of the diagram into white and black faces so that no two black faces share an edge. Each state induces a monomial in a ring of 16 variables. The sum of the states, properly specialized defines the new invariant. In despite of its simplicity it complements Jones invariant in distinguishing mirror pairs of links. In particular it proves that $9_{42}$ is distinct from its mirror image. For this pair of knots both the Jones Polynomial and Kauffman 2-variable polynomial fail.