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Homotopy classification of knotted defects in bounded domains

Yuta Nozaki, David Palmer, Yuya Koda

TL;DR

This work extends the global topology of knotted defects to defects in bounded 3D domains, specifically unknotted handlebodies with boundary data fixed up to homotopy. Defects are encoded by meridional monodromies and planar $G$-colored diagrams, and a central algebraic invariant $(H,N,[g])$ in $\mathcal{S}_{G,\mathrm{Im}(f_0)_*,N_{\beta\gamma}}$ together with a diagrammatic bijection yields a complete classification (Theorem comb_classif). The authors implement a diagrammatic framework for $G$-colored diagrams rel $f_0$ and prove bijectivity of the global classification map $\Phi$, illustrated with explicit biaxial-nematic and octahedral-frame-field examples on the solid torus with order-parameter space $S^3/Q$. This provides a practical, algebraic methodology for predicting and distinguishing global defect configurations in bounded domains, with potential implications for 3D meshing and boundary-controlled defect engineering in soft matter.

Abstract

Nozaki et.~al.\ gave a homotopy classification of the knotted defects of ordered media in three-dimensional space by considering continuous maps from complements of spatial graphs to the order parameter space modulo a certain equivalence relation. We extend their result by giving a classification scheme for ordered media in handlebodies, where defects are allowed to reach the boundary. Through monodromies around meridional loops, global defects are described in terms of planar diagrams whose edges are colored by elements of the fundamental group of the order parameter space. We exhibit examples of this classification in octahedral frame fields and biaxial nematic liquid crystals.

Homotopy classification of knotted defects in bounded domains

TL;DR

This work extends the global topology of knotted defects to defects in bounded 3D domains, specifically unknotted handlebodies with boundary data fixed up to homotopy. Defects are encoded by meridional monodromies and planar -colored diagrams, and a central algebraic invariant in together with a diagrammatic bijection yields a complete classification (Theorem comb_classif). The authors implement a diagrammatic framework for -colored diagrams rel and prove bijectivity of the global classification map , illustrated with explicit biaxial-nematic and octahedral-frame-field examples on the solid torus with order-parameter space . This provides a practical, algebraic methodology for predicting and distinguishing global defect configurations in bounded domains, with potential implications for 3D meshing and boundary-controlled defect engineering in soft matter.

Abstract

Nozaki et.~al.\ gave a homotopy classification of the knotted defects of ordered media in three-dimensional space by considering continuous maps from complements of spatial graphs to the order parameter space modulo a certain equivalence relation. We extend their result by giving a classification scheme for ordered media in handlebodies, where defects are allowed to reach the boundary. Through monodromies around meridional loops, global defects are described in terms of planar diagrams whose edges are colored by elements of the fundamental group of the order parameter space. We exhibit examples of this classification in octahedral frame fields and biaxial nematic liquid crystals.

Paper Structure

This paper contains 14 sections, 6 theorems, 9 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Outline.
  3. Preliminaries
  4. Groups and their presentations
  5. Homotopy groups
  6. Wirtinger presentation
  7. Review of previous work
  8. Mathematical model
  9. Formulation via continuous maps
  10. Diagrammatic description
  11. Main results and examples
  12. Proof of Theorem \ref{['thm:comb_classif']}
  13. Classification example: a system on the solid torus with order parameter space $S^3 / Q$ and no boundary defects
  14. The subgroups of the binary octahedral group $2O$

Key Result

Theorem 2.4

The map defined by $\Phi([f])=\operatorname{Im}(f_\ast\colon \pi_1(\mathbb{R}^3\setminus\Gamma)\to G)$, is bijective.

Figures (26)

  • Figure 1: Integral curves and defects of three different octahedral frame fields on the solid torus. Their defects---all of conjugacy class $\left[(1 + i)/\sqrt{2}\right] \subset 2O$---form a $4$-component unlink (left), a $(4, 2)$-torus link (center), and a $(4, 3)$-torus knot (right).
  • Figure 2: A complete list of defect configurations, up to homotopy, of the biaxial nematic system on the solid torus without boundary defects and subject to the condition that the map $f_0\colon \partial M \to S^3 / Q$ defined for the system satisfies $(f_0)_* (\alpha) = i \in Q$.
  • Figure 3: Relators in a Wirtinger presentation.
  • Figure 4: Handlebodies.
  • Figure 5: The oriented based loops $\beta_1,\dots,\beta_{k}, \gamma_1,\dots,\gamma_n$ on the boundary of the handlebody $W$.
  • ...and 21 more figures

Theorems & Definitions (20)

  • Example 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: NKTK24
  • Definition 3.1
  • Remark 3.2
  • Example 3.3
  • Theorem 3.4
  • Lemma 3.5
  • proof
  • ...and 10 more