The four-genus of connected sums of torus knots

We study the four-genus of linear combinations of torus knots: g4(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 Ozsvath–Stipsicz–Szabo determines the four-genus for all a and b; for the third, a surprising interplay between signatures and Upsilon appears.