On unique decomposition of knotted handlebodies

TL;DR

The paper establishes a uniqueness theorem for a maximal ${P}_{3}$-system decomposing knotted handlebodies in $oldsymbol{S}^3$ under $2$-indecomposability and $oldsymbol{ abla}$-irreducibility, and extends these ideas to understand the topology of ${P}_{3}$-decomposable exteriors. By analyzing arc configurations and index-minimization, it proves that maximal ${P}_{3}$-systems are unique up to isotopy, and shows that exteriors with essential disks, annuli, or tori can be arranged disjoint from the decomposition. It then develops a topology of ${P}_{3}$-decomposable handlebody-knots, derives consequences for genus-two exteriors (including atoroidality) and constructs an infinite family of hyperbolic examples with homeomorphic exteriors. The symmetry analysis yields strong constraints on the mapping class group, determines chirality for specific examples (e.g., $oldsymbol{6_{10}}$), and provides higher-genus classifications, showing that symmetry orders divide the number of ends and, in some cases, are limited to trivial or cyclic groups. Collectively, the results advance the classification and symmetry understanding of higher-genus handlebody-knots via ${P}_{3}$-decompositions and their exteriors.

Abstract

The paper considers the uniqueness question of factorization of a knotted handlebody in the $3$-sphere along decomposing $2$-spheres. We obtain a uniqueness result for factorization along decomposing $2$-spheres meeting the handlebody at three parallel disks. The result is used to examine handlebody-knot symmetry; particularly, the chirality of $6_{10}$ in the handlebody-knot table, previously unknown, is determined. In addition, an infinite family of hyperbolic handlebody-knots with homeomorphic exteriors is constructed.

Figures (8)

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Theorems & Definitions (56)

• Theorem 1.1: Theorem \ref{['teo:uniqueness']}
• Theorem 1.2
• Theorem 1.3: Theorem \ref{['teo:genus_two_symmetry']}
• Theorem 1.4: Theorem \ref{['teo:higher_genus_symmetry']}
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof