Inequivalent handlebody-knots with homeomorphic complements

Abstract

We distinguish the handlebody-knots $5_1,6_4$ and $5_2,6_{13}$ in the table, due to Ishii et al, of irreducible handlebody-knots up to six crossings. Furthermore, we construct two infinite families of handlebody-knots, each containing one of the pairs $5_1,6_4$ and $5_2,6_{13}$, and show that any two handlebody-knots in each family have homeomorphic complements but they are not equivalent.