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.