2018/07/18 by Katherine Vance, Vance, Katherine
Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #math.GT
paper · pdf · doi:10.48550/arxiv.1807.07092
arxiv created 2018/07/18 · arxiv updated 2018/07/20
In 2003, Ozsváth and Szabó defined the concordance invariant τ for knots in oriented 3-manifolds as part of the Heegaard Floer homology package. In 2011, Sarkar gave a combinatorial definition of τ for knots in S3 and a combinatorial proof that τ 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 S3 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 τ invariant for balanced spatial graphs generalizing the τ knot concordance invariant. In particular, this defines a τ invariant for links in S3. Using techniques similar to those of Sarkar, we show that our τ invariant gives an obstruction to a link being slice.