Homotopy classification of knotted defects in ordered media

TL;DR

This work provides a complete homotopy-theoretic classification of global defects in ordered media by turning texture data into maps into an order-parameter space and reducing the problem to finitely generated subgroups of a target group. A bijection is established between equivalence classes of maps (under physically motivated local moves) and subgroups of G via the image of the fundamental group, with the quaternion group Q giving a six-type classification for biaxial nematics. The authors introduce G-colored spatial graph diagrams and a robust move system that preserves the generated subgroup, enabling a purely combinatorial route to classification and extending naturally to other groups such as the tetrahedral group. The framework clarifies how the structure of subgroups (normal vs non-normal) affects the distinction between basepoint-preserving and basepoint-free homotopy classifications, and it includes explicit diagrams and examples illustrating the classifications.

Abstract

We give a homotopy classification of the global defects in ordered media, and explain it via the example of biaxial nematic liquid crystals, i.e., systems where the order parameter space is the quotient of the $3$-sphere $S^3$ by the quaternion group $Q$. As our mathematical model we consider continuous maps from complements of spatial graphs to the space $S^3/Q$ modulo a certain equivalence relation, and find that the equivalence classes are enumerated by the six subgroups of $Q$. Through monodromy around meridional loops, the edges of our spatial graphs are marked by conjugacy classes of $Q$; once we pass to planar diagrams, these labels can be refined to elements of $Q$ associated to each arc. The same classification scheme applies not only in the case of $Q$ but also to arbitrary groups.

Figures (15)

• Figure 1: Representatives of each class of global defects in biaxial nematic liquid crystals in $\mathbb{R}^3$ or $S^3$, up to equivalence. Labels (with directions) indicate local structure (see Section \ref{['sec:Colored spatial graph diagrams']}).
• Figure 2: A positive crossing, negative crossing, the loop that corresponds to an arc, and a diagram of the trefoil knot.
• Figure 3: Half-edges around a vertex and an example of a $G$-colored diagram.
• Figure 8: Sequence of moves that resolves a part in a colored diagram where two edges are locally hooked to each other. In this figure, $b_1 = c_1 c_2 c_1 c_2^{-1} c_1^{-1}$, $b_2 = c_1 c_2 c_1^{-1}$, and $c = c_1 c_2 c_1^{-1} c_2^{-1}$.
• Figure 9: Equivalent defects corresponding to (the conjugacy class of) the subgroup $\langle i \rangle$ of $Q$.

Theorems & Definitions (22)

• Theorem 1.1
• Lemma 3.1
• proof
• Definition 3.2
• Theorem 3.3
• Remark 3.4
• Definition 4.1
• Example 4.2
• Remark 4.3
• Theorem 5.1