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Tangle replacement on spatial graphs

Giovanni Bellettini, Giovanni Paolini, Maurizio Paolini, Yi-Sheng Wang

TL;DR

The paper addresses how tangle replacement on a spatial handcuff graph affects neighborhood equivalence classes of the resulting handlebody-knots and the effectiveness of invariants in distinguishing them. The authors analyze the G4_1 handcuff graph, define $ au$-tangle replacement inside a ball, and study the exterior topology via a 3-punctured sphere decomposition to relate tangles to exterior features, including essential annuli. They prove that for atoroidal $(B, au)$, two resulting handlebody-knots are equivalent iff the tangles are equivalent or dual, and they classify when the exterior contains infinitely many annuli based on the tangle being $ rac{1}{3}$-rational (or its negation), yielding corollaries on chirality and symmetry. The results extend to twisted variants $V_ au^{(k)}$, enabling a broad generalization and providing a robust tool to distinguish challenging handlebody-knots that evade traditional invariants.

Abstract

We study tangle replacement in the context of spatial graphs. The main results show that, for certain spatial handcuff graphs, there is a one-to-one correspondence between the neighborhood equivalence classes of the spatial graphs obtained by tangle replacement and the tangles with which the replacement is performed, up to possibly some permutation. As corollaries, we distinguish handlebody-knots difficult to differentiate with computational invariants and determine their chirality and symmetry groups.

Tangle replacement on spatial graphs

TL;DR

The paper addresses how tangle replacement on a spatial handcuff graph affects neighborhood equivalence classes of the resulting handlebody-knots and the effectiveness of invariants in distinguishing them. The authors analyze the G4_1 handcuff graph, define -tangle replacement inside a ball, and study the exterior topology via a 3-punctured sphere decomposition to relate tangles to exterior features, including essential annuli. They prove that for atoroidal , two resulting handlebody-knots are equivalent iff the tangles are equivalent or dual, and they classify when the exterior contains infinitely many annuli based on the tangle being -rational (or its negation), yielding corollaries on chirality and symmetry. The results extend to twisted variants , enabling a broad generalization and providing a robust tool to distinguish challenging handlebody-knots that evade traditional invariants.

Abstract

We study tangle replacement in the context of spatial graphs. The main results show that, for certain spatial handcuff graphs, there is a one-to-one correspondence between the neighborhood equivalence classes of the spatial graphs obtained by tangle replacement and the tangles with which the replacement is performed, up to possibly some permutation. As corollaries, we distinguish handlebody-knots difficult to differentiate with computational invariants and determine their chirality and symmetry groups.

Paper Structure

This paper contains 6 sections, 16 theorems, 4 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Tangle replacement on ${\rm{G}4_1}$
  3. Topology of $V_\tau$
  4. Proof of Main Results
  5. Applications
  6. Extension

Key Result

Theorem 1.3

Suppose $\Gamma$ is ${\rm{G}4_1}$ and $(B,\alpha),(B,\beta)$ are atoroidal. Then $V_\alpha,V_\beta$ are equivalent if and only if $(B,\alpha)$ is equivalent either to $(B,\beta)$ or to $(B,\beta^*)$.

Figures (11)

  • Figure 1: Hard pair in IshKisMorSuz:12.
  • Figure 2: Spine of trivial handlebody-knots.
  • Figure 3: Tangle replacement.
  • Figure 4: Small crossing handcuff spatial graphs.
  • Figure 5:
  • ...and 6 more figures

Theorems & Definitions (32)

  • Theorem 1.3: Theorems \ref{['teo:rational']} and \ref{['teo:irrational']}
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4: KodOzaWan:25ii
  • Lemma 2.5
  • ...and 22 more