Constructions of invariants for surface-links via link invariants and applications to the Kauffman bracket

TL;DR

This paper develops new invariants for surface-links in 4-space by extending knot invariants, particularly the Kauffman bracket, leading to more effective tools for distinguishing surface-links.

Contribution

It introduces a novel construction of surface-link invariants from knot invariants and defines a series of new, more effective invariants using skein relations.

Findings

The Kauffman bracket ideal coset invariant for surface-links.

A series of new invariants K_{2n-1} that outperform previous invariants.

Enhanced ability to distinguish different surface-links.

Abstract

In this paper, we formulate a construction of ideal coset invariants for surface-links in $4$-space using invariants for knots and links in $3$-space. We apply the construction to the Kauffman bracket polynomial invariant and obtain an invariant for surface-links called the Kauffman bracket ideal coset invariant of surface-links. We also define a series of new invariants $\{{\mathbf K}_{2n-1}(\mathcal L) | n=2, 3, 4, \ldots\}$ for surface-links $\mathcal L$ by using skein relations, which are more effective than the Kauffman bracket ideal coset invariant to distinguish given surface-links.