A groupoid rack and spatial surfaces

TL;DR

This paper introduces the concept of a groupoid rack to define invariants for spatial surfaces embedded in the 3-sphere, providing a new algebraic tool for studying their properties.

Contribution

It proposes the novel notion of a groupoid rack and demonstrates its universal property in coloring diagrams of spatial surfaces.

Findings

Groupoid racks can be used to color diagrams of spatial surfaces.

The universal property of groupoid racks in this context is established.

A new invariant for spatial surfaces is derived from groupoid rack colorings.

Abstract

A spatial surface is a compact surface embedded in the $3$-sphere. We assume that a spatial surface is oriented and that each connected component of a spatial surface is neither a disk nor without a boundary. A diagram of a spatial surface is a diagram of a spatial trivalent graph that is a spine of the spatial surface. In this paper, we introduce the notion of a groupoid rack, which is used for considering colorings for diagrams of spatial surfaces in order to obtain an invariant of spatial surfaces. Furthermore, we show that a groupoid rack has a universal property on colorings for diagrams of spatial surfaces.