The four-genus of connected sums of torus knots

Abstract

AbstractWe study the four-genus of linear combinations of torus knots:g4(aT(p, q) #-bT(p′, q′)). Fixing positivep, q, p′, andq′, our focus is on the behavior of the four-genus as a function of positiveaandb. Three types of examples are presented: in the first, for allaandbthe 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 allaandb; for the third, a surprising interplay between signatures and Upsilon appears.