Handlebody-knots obtained by tangle replacement

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.