College of Liberal Arts & Sciences

# Topology Seminar

**Abstract:** In 2003, Ozsvath and Szabo defined the concordance invariant tau for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. Moreover, to each link in $S^3$ they associated a knot in a connected sum of copies of $S^1\!\times S^2$, and then took the tau invariant of the corresponding knot, giving a tau invariants for links. 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 $S^3$, extending HFK for knots. 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 $S^3$ which is potentially different from the tau invariant for links defined by Ozsvath and Szabo. Using techniques similar to those of Sarkar, we show that our tau invariant is an obstruction to a link being slice.