Divides with cusps, shadows, and transvergent diagrams

TL;DR

This work addresses representing symmetric links in $S^{3}$ via two frameworks: transvergent diagrams and divides with cusps, by leveraging Turaev's shadow theory. It introduces divides with gleams on surfaces and provides explicit algorithms to translate between a transvergent diagram and a divide with cusps, and conversely, enabling constructive proofs of Sugawara's correspondence. Key contributions include a concrete algorithm to obtain a divide with cusps representing a given symmetric link from its transvergent diagram, an algorithm to draw a transvergent diagram from a divide with cusps, and a characterization of freely $2$-periodic links via divides with gleams on $S^2$ with total gleam $\overline{gl}=2$. The results unify divide-based and shadow-based descriptions of symmetric and periodic links in $S^{3}$ and provide practical tools for diagrammatic analysis and isotopy constructions within 4-manifold shadow theory.

Abstract

A link in $S^{3}$ is called a symmetric link if there is an orientation preserving involution of $S^{3}$ with a non-empty fixed point set that preserves the link. Any symmetric link can be depicted by a diagram with a symmetry axis lying on the plane of the diagram, called a transvergent diagram. Recently, Sugawara proved that any symmetric link can be represented by a divide with cusps, which is a generalization of A'Campo's divide that allows a finite number of cusps. In this paper, we introduce a generalization of A'Campo's divide in terms of Turaev's shadow, called a divide with gleams. By using divides with gleams, we provide an algorithm to obtain a divide with cusps that represents a symmetric link from its given transvergent diagram. Conversely, we also provide an algorithm to draw a transvergent diagram of the link of a given divide with cusps. Furthermore, we characterize freely $2$-periodic links through divides with gleams on a $2$-sphere.

Figures (20)

• Figure 1: A transvergent diagram of a symmetric link.
• Figure 2: The transvergent diagram $DL$ defines a divide $P_{DL}$.
• Figure 3: A divide with gleams on a $2$-sphere that satisfies the condition of Theorem \ref{['theo:free_2_period']}.
• Figure 4: A divide with cusps $P$. A region $R$ of $P$, an inner cusp for $R$ and an outer cusp for $R$.
• Figure 5: The divide with cusps $P$ and the link $L_P$ of $P$.

Theorems & Definitions (49)

• Theorem 1
• Theorem 2
• Definition
• Definition
• Theorem 1.1: Sugawara S24
• Example 1
• Definition
• Definition
• Theorem 1.2: Turaev Tur94
• Definition