Handlebody-knots obtained by tangle replacement
arXiv:2511.07796
The paper investigates tangle replacement for handlebody‑knots, classifying the knots obtained from the prime spatial handcuff graph with four crossings and analyzing their symmetry and chirality.
Abstract
We study tangle replacement in the context of handlebody-knots. The main results classify the handlebody-knots obtained by performing tangle replacement on the prime spatial handcuff graph with four crossings, and determine their symmetry. The work is motivated by the study of small crossing handlebody-knots that are difficult to distinguish with computational invariants and by their chirality problem.
major revision: results improved and more complete; references corrected and expanded; more details provided for the symmetry group computation; figures updated; introduction rewritten