Tangle replacement on spatial graphs

TL;DR

This paper investigates tangle replacement in spatial graphs, establishing a correspondence between neighborhood equivalence classes and tangles, and applying these results to distinguish complex handlebody-knots and analyze their symmetries.

Contribution

It introduces a novel correspondence between tangle replacements and neighborhood classes in spatial graphs, aiding in knot differentiation and symmetry analysis.

Findings

One-to-one correspondence between neighborhood classes and tangles for certain graphs.

Ability to distinguish handlebody-knots difficult for computational invariants.

Determination of chirality and symmetry groups of complex knots.

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.