Computations of quandle cocyle invariants of surface-links using marked graph diagrams

TL;DR

This paper introduces a method to compute quandle cocycle invariants of surface-links using marked graph diagrams, extending the cohomology theory of quandles to unoriented and oriented surface-links.

Contribution

It provides a new interpretation and computational method for quandle cocycle invariants of surface-links via marked graph diagrams, including symmetric quandles for unoriented cases.

Findings

Defined quandle cocycle invariants for surface-links using broken surface diagrams.

Extended invariants to unoriented links with symmetric quandles.

Presented a practical method to compute these invariants from marked graph diagrams.

Abstract

By using the cohomology theory of quandles, quandle cocycle invariants and shadow quandle cocycle invariants are defined for oriented links and surface-links via broken surface diagrams. By using symmetric quandles, symmetric quandle cocycle invariants are also defined for unoriented links and surface-links via broken surface diagrams. A marked graph diagram is a link diagram possibly with $4$-valent vertices equipped with markers. S. J. Lomonaco, Jr. and K. Yoshikawa introduced a method of describing surface-links by using marked graph diagrams. In this paper, we give interpretations of these quandle cocycle invariants in terms of marked graph diagrams, and introduce a method of computing them from marked graph diagrams.