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The groups $G_{n}^{k}$ and $2n$-gon tilings

Vassily Olegovich Manturov, Seongjeong Kim

TL;DR

Problem: understand how sequences of flips on rhombile tilings of zonogons encode invariants valued in the group $G_{n}^{3}$ and interact with line- and plane- configuration frameworks. Approach: develop a geometric–combinatorial pipeline linking tilings, cube stacking, dual diagrams, Desargues configurations, and Manturov–Nikonov indices to realize elements of $G_{n}^{3}$ and study their triviality on different surfaces. Contributions: (i) a realisable subgroup $RG_{n}^{3}$ is extracted from closed flip-paths, with nontrivial RP$^2$ images; (ii) a third Reidemeister move interpretation via dual diagrams and octagon relations; (iii) construction of MN-indices yielding $( obreakmath{ obreak Z}_{2})^{*2^{2(n-3)}}$-type invariants; (iv) extension to projective-plane tilings and connections to braid actions. Significance: links tiling dynamics to $3$-manifold topology and braid invariants, offering new tools for nontriviality detection and potential higher-dimensional generalizations.

Abstract

In the present paper we discuss four ways of looking at rhombile tilings: stacking 3-dimensional cubes, elements of groups, and configurations of lines and points.

The groups $G_{n}^{k}$ and $2n$-gon tilings

TL;DR

Problem: understand how sequences of flips on rhombile tilings of zonogons encode invariants valued in the group and interact with line- and plane- configuration frameworks. Approach: develop a geometric–combinatorial pipeline linking tilings, cube stacking, dual diagrams, Desargues configurations, and Manturov–Nikonov indices to realize elements of and study their triviality on different surfaces. Contributions: (i) a realisable subgroup is extracted from closed flip-paths, with nontrivial RP images; (ii) a third Reidemeister move interpretation via dual diagrams and octagon relations; (iii) construction of MN-indices yielding -type invariants; (iv) extension to projective-plane tilings and connections to braid actions. Significance: links tiling dynamics to -manifold topology and braid invariants, offering new tools for nontriviality detection and potential higher-dimensional generalizations.

Abstract

In the present paper we discuss four ways of looking at rhombile tilings: stacking 3-dimensional cubes, elements of groups, and configurations of lines and points.
Paper Structure (12 sections, 13 theorems, 12 equations, 19 figures)

This paper contains 12 sections, 13 theorems, 12 equations, 19 figures.

Table of Contents

  1. Introduction
  2. rhombile tilings, flips and stacking cubes
  3. The realisable counterpart of $G_{n}^{3}$
  4. Rhombile tilings and the third Reidemeister moves
  5. Rhombile tiling of $\mathbb{R}P^{2}$ and realisable elements
  6. Configurations of planes and points
  7. Manturov-Nikonov indices and the group $G_{n}^{3}$
  8. Further research
  9. Volumes of 3-manifolds and $G_{n}^{3}$
  10. Manturov-Nikonov indices and cubes
  11. Groups $G_{n}^{k}$ and k-dimensional manifolds
  12. Rhombile tilings of a zonogon and braid invariants

Key Result

Proposition 1.1

Let $C, C_{1},C_{2}$ be three cubics in $\mathbb{R}P^{2}$. If $C$ goes through eight of the nine intersection points of $C_{1},C_{2}$ then $C$ goes through the ninth intersection point as well.

Figures (19)

  • Figure 1: Cubes in 3-space or rhombile tiling of a plane
  • Figure 2: A flip and a stacking cube
  • Figure 3: Rhombile tiling and dual diagram
  • Figure 4: A flip and the corresponding move for dual diagrams
  • Figure 5: A formula associated with an inversion
  • ...and 14 more figures

Theorems & Definitions (24)

  • Proposition 1.1
  • Proposition 2.1: HenriquesSpeyer
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4: Far-commutativity
  • Lemma 2.5: Octagon relation
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3: HenriquesSpeyer
  • Proposition 3.4: HenriquesSpeyer
  • ...and 14 more