Secondary Upsilon invariants of knots

The knot invariant Upsilon, defined by Ozsváth, Stipsicz and Szabó, induces a homomorphism from the smooth knot concordance group to the group of piecewise linear functions on the interval [0, 2]. Here we define a set of related secondary invariants, each of which assigns to a knot a piecewise linear function on [0, 2]. These secondary invariants provide bounds on the genus and concordance genus of knots. Examples of knots for which Upsilon vanishes, but which are detected by these secondary invariants are presented.