We study the behavior of the Ozsváth-Szabó and Rasmussen knot concordance invariants and s on K m;n , the .m; n/-cable of a knot K where m and n are relatively prime. We show that for every knot K and for any fixed positive integer m, both of the invariants evaluated on K m;n differ from their value on the torus knot T m;n by fixed constants for all but finitely many n > 0. Combining this result together with Hedden's extensive work on the behavior of on .m; mr C 1/-cables yields bounds on the value of on any .m; n/-cable of K . In addition, several of Hedden's obstructions for cables bounding complex curves are extended. 57M25

We compare the values of the nonorientable three genus (or, crosscap number) and the nonorientable four genus of torus knots. In particular, we show that the difference between these two invariants can be arbitrarily large. This contrasts with the orientable setting. Seifert proved that the orientable three genus of the torus knot T (p, q) is

Recent advances in understanding slicing properties of Bing doubles of knots have depended on properties of iterated covering links. We expand and refine this covering link calculus. Our main application here is a simplified proof of the following result of Cha and Kim: If the iterated Bing double of a knot K is slice, then K is algebraically slice. Further applications are included in joint work with Livingston studying 4-genera of Bing doubles. The techniques also appear in the work of Levine studying mixed Bing-Whitehead doubles.

The nonorientable 4-genus γ 4 (K) of a knot K is the smallest first Betti number of any nonorientable surface properly embedded in the 4-ball, and bounding the knot K. We study a conjecture proposed by Batson about the value of γ 4 for torus knots, which can be seen as a nonorientable analogue of Milnor's Conjecture for the orientable 4-genus of torus knots. We prove the conjecture for many infinite families of torus knots, by relying on a lower bound for γ 4 formulated by Ozsváth, Stipsicz, and Szabó. As a side product we obtain new closed formulas for the signature of torus knots.

We study the four-genus of linear combinations of torus knots: g 4 (aT (p, q)# −bT (p , q )). Fixing positive p, q, p , and q , our focus is on the behavior of the four-genus as a function of positive a and b. Three types of examples are presented: in the first, for all a and b the four-genus is completely determined by the Tristram-Levine signature function; for the second, the recently defined Upsilon function of Ozsváth-Stipsicz-Szabó determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.