We distinguish the handlebody-knots and 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 and , and show that any two handlebody-knots in each family have homeomorphic complements but they are not equivalent.
"Inequivalent handlebody-knots with homeomorphic complements." Algebr. Geom. Topol. 12 (2) 1059 - 1079, 2012. https://doi.org/10.2140/agt.2012.12.1059