Handlebody-knots obtained by tangle replacement

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