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Relative Helicity and Tiling Twist

Boris Khesin, Nicolau C. Saldanha

TL;DR

The paper constructs a geometric realization of 3D domino tilings by associating to each tiling a divergence-free vector field via a five-pipe scheme, linking combinatorial flux to the relative rotation class and twist to relative helicity. It proves that for tilings with zero relative flux, the relative helicity of the associated field is proportional to the tiling twist, precisely $\operatorname{Hel}(\tilde{\xi}_{\mathbf t})=36\varphi^2 \operatorname{Tw}({\mathbf t}) + C$, with $\varphi$ the per-pipe flux and $C$ a constant depending on initial choices; flips leave helicity invariant while trits change it by a fixed amount. The framework extends helicity to non-tangent fields and arbitrary 3-manifolds, introduces isolating shells to manage boundary effects, and connects tiling moves to topological invariants via the relative flux/rotation class correspondence. This work forges a bridge between discrete tiling combinatorics and continuous vector-field topology, enabling a topological interpretation of the twist invariant and offering tools for analyzing 3D dimer-like systems through helicity and linking theory.

Abstract

We consider domino tilings of 3D cubiculated regions. The tilings have two invariants, flux and twist, often integer-valued, which are given in purely combinatorial terms. These invariants allow one to classify the tilings with respect to certain elementary moves, flips and trits. In this paper we present a construction associating a divergence-free vector field $ξ_t$ to any domino tiling $t$, such that the flux of the tiling $t$ can be interpreted as the (relative) rotation class of the field $ξ_t$, while the twist of $t$ is proved to be the relative helicity of the field $ξ_t$.

Relative Helicity and Tiling Twist

TL;DR

The paper constructs a geometric realization of 3D domino tilings by associating to each tiling a divergence-free vector field via a five-pipe scheme, linking combinatorial flux to the relative rotation class and twist to relative helicity. It proves that for tilings with zero relative flux, the relative helicity of the associated field is proportional to the tiling twist, precisely , with the per-pipe flux and a constant depending on initial choices; flips leave helicity invariant while trits change it by a fixed amount. The framework extends helicity to non-tangent fields and arbitrary 3-manifolds, introduces isolating shells to manage boundary effects, and connects tiling moves to topological invariants via the relative flux/rotation class correspondence. This work forges a bridge between discrete tiling combinatorics and continuous vector-field topology, enabling a topological interpretation of the twist invariant and offering tools for analyzing 3D dimer-like systems through helicity and linking theory.

Abstract

We consider domino tilings of 3D cubiculated regions. The tilings have two invariants, flux and twist, often integer-valued, which are given in purely combinatorial terms. These invariants allow one to classify the tilings with respect to certain elementary moves, flips and trits. In this paper we present a construction associating a divergence-free vector field to any domino tiling , such that the flux of the tiling can be interpreted as the (relative) rotation class of the field , while the twist of is proved to be the relative helicity of the field .
Paper Structure (15 sections, 10 theorems, 33 equations, 19 figures)

This paper contains 15 sections, 10 theorems, 33 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Domino tilings, combinatorial flux and local moves
  3. Twist of a tiling
  4. Rotation class and helicity of vector fields
  5. Relative rotation class
  6. Helicity of fields and pipes
  7. Relative helicity for vector fields
  8. Pipes, fluxes, and shells
  9. The five pipes construction
  10. Proof of theorem on relative flux
  11. An isolating shell
  12. Key examples
  13. Proof of the main theorem on twist and helicity
  14. A 3D 'crosses-and-toes' example
  15. Appendix: Helicity via linking and self-linking

Key Result

Theorem 1.1

For two tilings ${\mathbf t}_0$ and ${\mathbf t}_1$ of the same flux and of zero relative flux in a 3D cubiculated region ${\cal R}$, the difference of their twists is proportional to the relative helicity of the associated vector fields $\xi_{{\mathbf t}_0}$ and $\xi_{{\mathbf t}_1}$: where $\varphi \in {\mathbb{R}}$ is the flux in a single pipe of the vector fields $\xi_{{\mathbf t}_i}$.

Figures (19)

  • Figure 1: A tiling of the box $[0,4]\times [0,4]\times [0,2]$. The orientation of ${\mathbb{R}}^3$ is important: the $z$ axis points upward away from the paper. Examples of dominoes in this tiling are $[0,1]\times[0,2]\times[0,1]$, $[0,2]\times[0,1]\times[1,2]$ and $[1,2]\times[1,2]\times[0,2]$. This tiling admits no flips.
  • Figure 2: Five tilings of the box $[0,3]\times[0,3]\times[0,2]$. The first move is a trit, the two other moves are flips. Five more flips take us from the fourth to the fifth tiling. The first tiling has twist $-1$, the others have twist $0$.
  • Figure 3: Two linked tubes: the solid tori are thin tubular neighborhoods of the curves $C_i$. A few cross-sections are shown, while the vector field is transversal to these sections.
  • Figure 4: A domino, five polygonal lines inside it and the respective smooth tubes approximating the lines. Tubes are only very partially drawn in order to keep the figure simple. Lines are always oriented from black to white. At the right, the central square separating the two unit cubes of the domino, crossed by five lines and five tubes.
  • Figure 5: An additional narrow toroidal pipe with a field flux $\varphi$ and directed along the axis of the tile.
  • ...and 14 more figures

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Theorem 3.1
  • Remark 3.3
  • Definition 4.1
  • Definition 4.2
  • ...and 25 more