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A construction of multiple group racks

Katsunori Arai

TL;DR

The paper tackles the problem of distinguishing spatial surfaces in $S^{3}$ using algebraic colorings by rack-like structures. It introduces a new construction of a multiple group rack from a $G$-family of racks and a normal subgroup $N$ via the semidirect product $G \ltimes N$, yielding $X \times (G \ltimes N)$. When the right action of $N$ on $G$ is nontrivial, the resulting MGR satisfies a property ($\star$) that cannot be realized by associated MGRs or their abelian extensions, and it provides genuine new invariants for spatial surfaces. The paper demonstrates this with an explicit pair of spatial surfaces whose boundaries and regular neighborhoods are ambiently isotopic, yet are distinguishable by the new invariant through colorings; in contrast, invariants from associated MGRs fail to distinguish them. This work broadens the toolkit for spatial-surface invariants by introducing a nontrivial algebraic construction that can separate surfaces beyond the reach of existing methods.

Abstract

A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e., oriented compact surfaces with boundaries embedded in the $3$-sphere $S^{3}$. A $G$-family of racks is a set with a family of binary operations indexed by the elements of a group $G$. There are two known methods for constructing multiple group racks. One is via a $G$-family of racks. The resulting multiple group rack is called the associated multiple group rack of the $G$-family of racks. The other is by taking an abelian extension of a multiple group rack. In this paper, we introduce a new method for constructing multiple group racks by using a $G$-family of racks and a normal subgroup $N$ of $G$. We show that this construction yields multiple group racks that are neither the associated multiple group racks of any $G$-family of racks nor their abelian extensions when the right conjugation action of $G$ on $N$ is nontrivial. As an application, we present a pair of spatial surfaces that cannot be distinguished by invariants derived from the associated multiple group racks of any $G$-family of racks, yet can be distinguished using invariants obtained from a multiple group rack introduced in this paper.

A construction of multiple group racks

TL;DR

The paper tackles the problem of distinguishing spatial surfaces in using algebraic colorings by rack-like structures. It introduces a new construction of a multiple group rack from a -family of racks and a normal subgroup via the semidirect product , yielding . When the right action of on is nontrivial, the resulting MGR satisfies a property () that cannot be realized by associated MGRs or their abelian extensions, and it provides genuine new invariants for spatial surfaces. The paper demonstrates this with an explicit pair of spatial surfaces whose boundaries and regular neighborhoods are ambiently isotopic, yet are distinguishable by the new invariant through colorings; in contrast, invariants from associated MGRs fail to distinguish them. This work broadens the toolkit for spatial-surface invariants by introducing a nontrivial algebraic construction that can separate surfaces beyond the reach of existing methods.

Abstract

A multiple group rack is a rack which is a disjoint union of groups equipped with a binary operation satisfying some conditions. It is used to define invariants of spatial surfaces, i.e., oriented compact surfaces with boundaries embedded in the -sphere . A -family of racks is a set with a family of binary operations indexed by the elements of a group . There are two known methods for constructing multiple group racks. One is via a -family of racks. The resulting multiple group rack is called the associated multiple group rack of the -family of racks. The other is by taking an abelian extension of a multiple group rack. In this paper, we introduce a new method for constructing multiple group racks by using a -family of racks and a normal subgroup of . We show that this construction yields multiple group racks that are neither the associated multiple group racks of any -family of racks nor their abelian extensions when the right conjugation action of on is nontrivial. As an application, we present a pair of spatial surfaces that cannot be distinguished by invariants derived from the associated multiple group racks of any -family of racks, yet can be distinguished using invariants obtained from a multiple group rack introduced in this paper.

Paper Structure

This paper contains 5 sections, 10 theorems, 15 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Diagrams of Spatial surfaces
  3. Multiple group racks
  4. Coloring diagrams by multiple group racks
  5. Main results

Key Result

Theorem 2.1

Two spatial surfaces are equivalent if and only if their diagrams are related by a finite sequence of $\mathrm{R}2$, $\mathrm{R}3$, $\mathrm{R}5$ and $\mathrm{R}6$ moves, as depicted in Fig. Fig:R-moves, and isotopies in $S^2 = \mathbb{R}^{2} \cup \{ \infty \}$. Let $D$ be a diagram of a spatial surface. A Y-orientation of $D$ is an assignment of orientations to all edges of $D$ such that no vert

Figures (9)

  • 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: $X$-coloring conditions
  • ...and 4 more figures

Theorems & Definitions (24)

  • Theorem 2.1: Matsuzaki2021
  • Theorem 2.2: Matsuzaki-Murao2023
  • Definition 3.1: Fenn-Rourke1992Joyce1982Matveev1982
  • Definition 3.2: Ishii-Matsuzaki-Murao2020
  • Definition 3.3: Ishii-Iwakiri-Jang-Oshiro2013
  • Example 3.4
  • Proposition 3.5: Ishii2015'Ishii-Matsuzaki-Murao2020
  • Example 3.6
  • Definition 4.1
  • Theorem 4.2: Ishii-Matsuzaki-Murao2020
  • ...and 14 more