UC San Diego

Link homotopy has been an active area of research for knot theorists since its introduction by Milnor in the 1950s. We will examine several generalizations of link homotopy, produce invariants, and apply these invariants to study the behavior of links and spatial graphs under these generalized link homotopy equivalences. We first study links under self C_k-equivalence. This relation is based on certain degree k clasper surgeries. It is known that a Milnor number with no repeated index is an invariant of link homotopy. We show that Milnor's numbers with k- repeated indices are invariants not only of isotopy, but also of self C_k-moves. We will also consider versions of link homotopy for spatial graphs. We introduce a new equivalence relation on spatial graphs called component homotopy, which reduces to link homotopy in the classical case. Unlike previous attempts at generalizing link homotopy to spatial graphs, our new relation allows analogues of some standard link homotopy results and invariants. Researchers have previously studied edge homotopy and vertex homotopy as generalizations of link homotopy to spatial graphs. We introduce some new invariants of these relations and use these invariants to produce examples of non-splittable spatial graphs all of whose constituent sublinks are homotopically trivial