On rack colorings for surface-knot diagrams without branch points

TL;DR

This paper investigates rack colorings for surface-knot diagrams without branch points, establishing their invariance for $S^2$-knots and relating them to quandles and diagram equivalences.

Contribution

It proves rack colorings are invariants for $S^2$-knots and links them to quandle theory, expanding understanding of surface-knot invariants.

Findings

Rack colorings are invariants for $S^2$-knots.

Rack colorings relate to quandle structures.

Discussion on regular-equivalences of surface-knot diagrams.

Abstract

Racks do not give us invariants of surface-knots in general. For example, if a surface-knot diagram has branch points (and a rack which we use satisfies some mild condition), then it admits no rack colorings. In this paper, we investigate rack colorings for surface-knot diagrams without branch points and prove that rack colorings are invariants of $S^2$-knots. We also prove that rack colorings for $S^2$-knots can be interpreted in terms of quandles, and discuss a relationship with regular-equivalences of surface-knot diagrams.