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A groupoid rack and spatial surfaces

Katsunori Arai

TL;DR

The paper introduces the groupoid rack as an algebraic framework to define invariants for spatial surfaces embedded in $S^{3}$ via colorings of diagrams derived from spatial trivalent graphs. It defines coloring rules on Y-oriented diagrams using groupoid racks and proves that the number of colorings is invariant under Y-oriented Reidemeister moves, thereby yielding a topological invariant of spatial surfaces. A universality result shows that any coloring scheme satisfying a set of axioms naturally arises from a groupoid-rack structure, unifying prior approaches (e.g., heap racks, multiple group racks) under one formalism. The work provides both a concrete invariant mechanism and a theoretical underpinning that positions groupoid racks as a canonical tool for studying colorings of diagrams of spatial surfaces. This could facilitate classification and comparison of spatial surfaces via combinatorial-algebraic methods.

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.

A groupoid rack and spatial surfaces

TL;DR

The paper introduces the groupoid rack as an algebraic framework to define invariants for spatial surfaces embedded in via colorings of diagrams derived from spatial trivalent graphs. It defines coloring rules on Y-oriented diagrams using groupoid racks and proves that the number of colorings is invariant under Y-oriented Reidemeister moves, thereby yielding a topological invariant of spatial surfaces. A universality result shows that any coloring scheme satisfying a set of axioms naturally arises from a groupoid-rack structure, unifying prior approaches (e.g., heap racks, multiple group racks) under one formalism. The work provides both a concrete invariant mechanism and a theoretical underpinning that positions groupoid racks as a canonical tool for studying colorings of diagrams of spatial surfaces. This could facilitate classification and comparison of spatial surfaces via combinatorial-algebraic methods.

Abstract

A spatial surface is a compact surface embedded in the -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.
Paper Structure (6 sections, 15 theorems, 37 equations, 13 figures)

This paper contains 6 sections, 15 theorems, 37 equations, 13 figures.

Table of Contents

  1. Introduction
  2. A groupoid rack
  3. Diagrams of spatial surfaces
  4. Colorings for diagrams of spatial surfaces
  5. A proof of Theorem \ref{['thm coloring invariant']}
  6. A universality of groupoid racks on colorings

Key Result

Proposition 2.2

Let $X$ be a groupoid rack associated with a groupoid $\mathcal{C}$. Let $\rho : X \to X$ be the map sending $x$ to $x^{-1}$, where $x^{-1}$ is the inverse morphism of $x$ in the groupoid $\mathcal{C}$. Then, $\rho$ is a good involution of $X$. Using the good involution as in Proposition Prop:Good_involution, a groupoid rack $X$ is regarded as a symmetric rack. We show some examples of groupoid r

Figures (13)

  • Figure 1: A construction of a spatial surface from a diagram of a spatial trivalent graph
  • Figure 2: Reidemeister moves for diagrams of spatial surfaces
  • Figure 3: All orientations around trivalent vertices
  • Figure 4: Y-oriented R$5$ and R$6$ moves
  • Figure 5: A Y-oriented R$5$ move
  • ...and 8 more figures

Theorems & Definitions (36)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Example 2.3: A multiple group rack, Ishii-Matsuzaki-Murao2020
  • Example 2.4: A heap rack, Saito-Zappala2024
  • Example 2.5
  • Theorem 3.1: Matsuzaki2021
  • Lemma 3.2
  • proof
  • Theorem 3.3: Matsuzaki-Murao2023
  • ...and 26 more