Tau invariants of spatial graphs
Vance, Katherine Rose Poulsen
Doctor of Philosophy
In 2003, Ozsvath and Szabo defined the concordance invariant tau for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of tau for knots in S^3 and a combinatorial proof that tau gives a lower bound for the slice genus of a knot. Recently, Harvey and O’Donnol defined a relatively bigraded combinatorial Heegaard Floer homology theory for transverse spatial graphs in S^3 which extends knot Floer homology. We define a Z-filtered chain complex for balanced spatial graphs whose associated graded chain complex has homology determined by Harvey and O’Donnol’s graph Floer homology. We use this to show that there is a well-defined tau invariant for balanced spatial graphs generalizing the tau knot concordance invariant. In particular, this defines a tau invariant for links in S^3. Using techniques similar to those of Sarkar, we show that our tau invariant gives an obstruction to a link being slice.
Knot theory; Link concordance; Spatial graphs; Heegaard Floer homology