Thomas Fleming

Milnor's invariants and self

It has long been known that a Milnor invariant with no repeated index is an invariant of link homotopy. We show that Milnor invariants with repeated indices are invariants not only of isotopy, but also of self Ck-equivalence. Here self C k -equivalence is a natural generalization of link homotopy based on certain degree k clasper surgeries, which provides a filtration of link homotopy classes.

C_{k}

We study intrinsically linked graphs where we require that every embedding of the graph contains not just a non-split link, but a link that satisfies some additional property. Examples of properties we address in this paper are: a two component link with lk(A, L) = k2 r , k 6 0, a non-split n-component link where all linking numbers are even, or an n-component link with components L, Ai where lk(L, Ai) = 3k, k 6 0. Links with other properties are considered as well. For a given property, we prove that every embedding of a certain complete graph contains a link with that property. The size of the complete graph is determined by the property in question. AMS Classification 57M15; 57M25,05C10

-equivalence

Edge-homotopy and vertex-homotopy are equivalence relations on spatial graphs which are generalizations of Milnors link-homotopy. We introduce some edge (resp. vertex)-homotopy invariants of spatial graphs by applying the Sato-Levine invariant for the 2-component constituent algebraically split links and show examples of non-splittable spatial graphs up to edge (resp. vertex)-homotopy, all of whose constituent links are link-homotopically trivial.

Intrinsically linked graphs and even linking number

We introduce a notion of intrinsic linking and knotting for virtual spatial graphs. Our theory gives two filtrations of the set of all graphs, allowing us to measure, in a sense, how intrinsically linked or knotted a graph is; we show that these filtrations are descending and non-terminating. We also provide several examples of intrinsically virtually linked and knotted graphs. As a byproduct, we introduce the {\it virtual unknotting number} of a knot, and show that any knot with non-trivial Jones polynomial has virtual unknotting number at least 2.