Tangle replacement on spatial graphs
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.
Cite
@article{arxiv.2511.07796,
title = {Tangle replacement on spatial graphs},
author = {Giovanni Bellettini and Giovanni Paolini and Maurizio Paolini and Yi-Sheng Wang},
journal= {arXiv preprint arXiv:2511.07796},
year = {2025}
}
Comments
Figures updates (11 figures); a missing reference on rational tau-tangles, where it is originally introduced, added (page 3); a remark added after Theorem 1.4 (Theorem 1.3 in the previous version): the theorem also follows from a more general result via 3-decomposition of handlebody-knots