Paul Kirk and Charles Livingston
Type 1 Knot Invariants in 3-Manifolds

The general theory of knotting in 3-manifolds has recently seen significant progress. One important aspect of this has been the effort toward generalizing the notion of finite type invariants from S³ to arbitrary 3-manifolds. Here we will present a new class of finite type invariants, defined in arbitrary orientable 3-manifolds, that are both simple to define and to compute. They will be seen to be of both practical utility, in distinguishing large families of knots, and also of theoretical interest, giving access to subtle unknotting results.