Polynomial of an oriented surface-link diagram via quantum A_2 invariant

TL;DR

This paper defines a polynomial invariant for marked graph diagrams of surface-links using the quantum A_2 invariant and explores its behavior under Yoshikawa moves, with applications to ribbon 2-knots.

Contribution

It introduces a new polynomial invariant for surface-link diagrams based on the quantum A_2 invariant, enhancing tools for studying ribbon 2-knots.

Findings

The polynomial changes predictably under Yoshikawa moves.

The polynomial is useful for classifying ribbon 2-knots.

A new notion of ribbon marked graph is introduced.

Abstract

It is known that every surface-link can be presented by a marked graph diagram, and such a diagram presentation is unique up to moves called Yoshikawa moves. G. Kuperberg introduced a regular isotopy invariant, called the quantum A_2 invariant, for tangled trivalent graph diagrams. In this paper, a polynomial for a marked graph diagram is defined by use of the quantum A_2 invariant and it is studied how the polynomial changes under Yoshikawa moves. The notion of a ribbon marked graph is introduced to show that this polynomial is useful for an invariant of a ribbon 2-knot.