A groupoid rack and spatial surfaces

Katsunori Arai

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arXiv · 2310.06423

A groupoid rack and spatial surfaces

Katsunori Arai

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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.

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Katsunori Arai. 2023-10-10. A groupoid rack and spatial surfaces. https://arxiv.org/abs/2310.06423

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