# On symmetric equivalence of symmetric union diagrams

## TL;DR

This paper advances the understanding of symmetric union diagrams by providing a complete solution to their symmetric equivalence problem, using refined topological invariants.

## Contribution

It introduces a new approach and invariants for symmetric equivalence, fully resolving an open question posed by Eisermann and Lamm.

## Key findings

- Complete characterization of symmetric equivalence
- New invariants based on refined topological spin models
- Resolution of the open question on symmetric union diagrams

## Abstract

Eisermann and Lamm introduced a notion of symmetric equivalence among symmetric union diagrams and studied it using a refined form of the Jones polynomial. We introduced invariants of symmetric equivalence via refined versions of topological spin models and provided a partial answer to a question left open by Eisermann and Lamm. In this paper we adopt a new approach to the symmetric equivalence problem and give a complete answer to the original question left open by Eisermann and Lamm.