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Equivalence of Doubly Periodic Tangles

Ioannis Diamantis, Sofia Lambropoulou, Sonia Mahmoudi

TL;DR

This work develops a complete topological framework for doubly periodic tangles (DP tangles), defined as lifts of knots/links in the thickened torus to the universal cover $\mathbb{E}^2\times I$ and encoded by flat motif diagrams on tori. It shows that DP tangle isotopy reduces to a finite sequence of local motif moves together with global affine operations (translations, scalings, shearings, and torus inflation/deflation) that act on lattices and motifs, yielding DP tangle equivalence independent of lattice choice. The central result proves that two DP tangles are DP isotopic if and only if their flat motif diagrams are connected by surface isotopy, Reidemeister moves, shift equivalence, scale equivalence, torus inflation/deflation, and shear equivalence. The framework is then extended to a broad family of diagrammatic settings—regular, framed, virtual, welded, singular, pseudo, tied, and bonded DP tangles—providing a unified basis for invariants and applications in materials science, textiles, and molecular chemistry.

Abstract

Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots and links in the thickened torus, and which can be chosen in infinitely many ways. DP tangles are used in modeling materials and physical systems of entangled filaments. In this paper we establish the complete mathematical framework of the topological theory of DP tangles. We present an exhaustive analysis of DP tangle isotopies. These are distinguished in local isotopies and global isotopies. Our analysis yields the characterization of DP isotopy as an equivalence relation on the level of their (flat) motifs, called DP tangle equivalence. Along the way we also discuss motif minimality. We further generalize our results to other diagrammatic categories, namely framed, virtual, welded, singular, pseudo, tied and bonded DP tangles, which could be used in novel applications.

Equivalence of Doubly Periodic Tangles

TL;DR

This work develops a complete topological framework for doubly periodic tangles (DP tangles), defined as lifts of knots/links in the thickened torus to the universal cover and encoded by flat motif diagrams on tori. It shows that DP tangle isotopy reduces to a finite sequence of local motif moves together with global affine operations (translations, scalings, shearings, and torus inflation/deflation) that act on lattices and motifs, yielding DP tangle equivalence independent of lattice choice. The central result proves that two DP tangles are DP isotopic if and only if their flat motif diagrams are connected by surface isotopy, Reidemeister moves, shift equivalence, scale equivalence, torus inflation/deflation, and shear equivalence. The framework is then extended to a broad family of diagrammatic settings—regular, framed, virtual, welded, singular, pseudo, tied, and bonded DP tangles—providing a unified basis for invariants and applications in materials science, textiles, and molecular chemistry.

Abstract

Doubly periodic tangles, or DP tangles, are embeddings of curves in the thickened plane that are periodically repeated in two directions. They are defined as universal covers of their generating cells, the flat motifs, which represent knots and links in the thickened torus, and which can be chosen in infinitely many ways. DP tangles are used in modeling materials and physical systems of entangled filaments. In this paper we establish the complete mathematical framework of the topological theory of DP tangles. We present an exhaustive analysis of DP tangle isotopies. These are distinguished in local isotopies and global isotopies. Our analysis yields the characterization of DP isotopy as an equivalence relation on the level of their (flat) motifs, called DP tangle equivalence. Along the way we also discuss motif minimality. We further generalize our results to other diagrammatic categories, namely framed, virtual, welded, singular, pseudo, tied and bonded DP tangles, which could be used in novel applications.
Paper Structure (19 sections, 18 theorems, 8 equations, 32 figures)

This paper contains 19 sections, 18 theorems, 8 equations, 32 figures.

Table of Contents

  1. Introduction
  2. Topological set-up for DP tangles and motifs
  3. DP tangles, motifs and diagrams
  4. Lattices and flat motifs
  5. Minimal motifs
  6. DP tangle isotopy
  7. DP tangle equivalence by local isotopies
  8. DP tangle global equivalence: translation and shift equivalence
  9. DP tangle equivalence: scaling and scale equivalence
  10. DP tangle equivalence: shearing and Dehn equivalence
  11. Unimodular linear transformations
  12. DP tangle isotopy and flat motif equivalence
  13. DP tangle equivalence
  14. A discussion on minimal motifs
  15. DP tangle equivalence in other diagrammatic settings
  16. ...and 4 more sections

Key Result

Proposition 2.3

Two DP tangles $(\tau_\infty, \Lambda)$ and $(\tau'_\infty, \Lambda')$ are DP isotopic if and only if any two DP diagrams $(d_\infty, \Lambda)$ and $(d'_\infty, \Lambda')$ of theirs are related by a sequence of planar isotopies and Reidemeister moves.

Figures (32)

  • Figure 1: A motif (diagram), a corresponding flat motif (diagram) and its DP tangle (DP diagram).
  • Figure 2: A planar isotopy move and the Reidemeister moves.
  • Figure 3: A shearing of a DP tangle realized by $\Delta$-moves.
  • Figure 4: Textile representation of a S-twill and Z-twill.
  • Figure 5: A motif and its corresponding mixed link.
  • ...and 27 more figures

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.3
  • Definition 1.4
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Lemma 3.1
  • Lemma 3.2
  • Proposition 3.3
  • ...and 26 more