HOMOLOGY THEORIES FOR TANGLES AND BORDERED 3-MANIFOLDS
Heegaard Floer homology is a new and powerful invariant of 3-manifolds and knots in them.  Among many other benefits, it detects the knot genus (the least complicated surface whose boundary is the knot), and in particular, whether a knot is non-trivial.  It is a homology theory, and like many homology theories the Euler characteristic is interesting: for knots, the Euler characteristic is one of the oldest knot invariants, the Alexander polynomial.  Heegaard Floer homology is part of a family of recent homology theories whose Euler characteristic gives other knot polynomials.  But it can be ha