HL-homotopy of handlebody-links and Milnor's invariants

Abstract

Abstract. A handlebody-link is a disjoint union of embeddings of handlebodies in S 3 and an HL-homotopy is an equivalence relation on handlebody-links generated by self-crossing changes. The second author and Ryo Nikkuni classified the set of HLhomotopy classes of 2-component handlebody-links completely using the linking numbers for handlebody-links. In this paper, we construct a family of invariants for HL-homotopy classes of general handlebody-links, by using Milnor's µ-invariants. Moreover, we give a bijection between the set of HL-homotopy classes of almost trivial handlebody-links and tensor product space modulo some general linear actions, especially for 3-or more component handlebody-links. Through this bijection we construct comparable invariants of HL-homotopy classes.

Link-homotopy classes of 4-component links and claspers

“…We mention that Nikkuni and the authors also used the clasper theory to study HLhomotopy classes for handlebody-links [12,6].…”