Geometry and Topology Seminar

Two knots in the 3-sphere are said to be concordant if they differ by connected summing a knot that bounds an embedded disk in the 4-ball. This notion of equivalence leads to the study of the knot concordance group. Heegaard Floer homology gives rise to many nice invariants that offer insight into the structure of the knot concordance group. In this talk, I will review some of these invariants and discuss their behavior under the satellite operation. In particular, I will focus on the Upsilon invariant introduced by Ozsvath, Stipsicz, and Szabo in 2014, and present an inequality that partially tells us its behavior of under the cabling operation, which is a special type of satellite operation. This result is a generalization of Hedden's and Van Cott's work on tau-invariant. As an application, I will show this inequality allows one to easily deduce the existence of an infinite-rank summand of topologically slice knots. Time permitted, I will also discuss the behavior of the Upsilon invariant under general satellite operation and some current open questions.