Upsilon Torsion and barcode complexes

Abstract: Recent work by a number of authors has found that torsion of a certain type in the homology of the curved Heegaard Floer knot complex provides a constraint on the number of maxima and minima in cobordisms between knots and, in particular, on the number of minima of slice disks for knots. One such invariant is Ord_v(K) described by Juhasz-Miller-Zemke. We construct a generalization which we call the Upsilon Torsion function: a piecewise linear function on [0,2] having derivative equal to Ord_v(K) near 0. This provides new constraints on cobordisms between knots. In ongoing work with with Charles Livingston, Misha Tyomkin, and C.M. Michael Wong, we consider the barcode complex associated to the Heegaard Floer knot complex of a knot. This can be thought of (with an appropriate viewpoint) as a generalization of the Upsilon Torsion function. The barcode complex provides constraints on cobordisms between knots and has interesting applications to the detection of prime knots from their Heegaard Floer knot complexes. In this talk, I will briefly discuss the Upsilon Torsion function and and give a preliminary report on the barcode complex. This talk is based on joint work with Charles Livingston and with Charles Livingston, Misha Tyomkin, and C.M. Michael Wong.

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