Symmetric union diagrams and refined spin models

TL;DR

This paper introduces new invariants derived from topological spin models that can distinguish symmetric union diagrams of links, providing insights into their uniqueness and complementing existing polynomial invariants.

Contribution

It demonstrates that topological spin models yield infinitely many effective invariants for symmetric union diagrams, expanding the tools for studying their equivalence and uniqueness.

Findings

Spin model invariants distinguish infinitely many symmetric union diagrams.

These invariants are distinct from the refined Jones polynomial.

Partial progress on the symmetry representation question for ribbon knots.

Abstract

An open question akin to the slice-ribbon conjecture asks whether every ribbon knot can be represented as a symmetric union. Next to this basic existence question sits the question of uniqueness of such representations. Eisermann and Lamm investigated the latter question by introducing a notion of symmetric equivalence among symmetric union diagrams and showing that inequivalent diagrams can be detected using a refined version of the Jones polynomial. We prove that every topological spin model gives rise to many effective invariants of symmetric equivalence, which can be used to distinguish infinitely many symmetric union diagrams representing the same link. We also show that such invariants are distinct from the refined Jones polynomial and we use them to provide a partial answer to a question left open by Eisermann and Lamm.