The four-genus of connected sums of torus knots

TL;DR

This paper investigates the four-genus of linear combinations of torus knots, revealing how it can be determined by signature functions, the Upsilon invariant, or a combination of both, depending on the case.

Contribution

It provides a detailed analysis of the four-genus for sums of torus knots, demonstrating the roles of signature functions and the Upsilon invariant in its determination.

Findings

Four-genus determined by signature functions in some cases

Upsilon invariant fully determines four-genus in other cases

Interplay between signatures and Upsilon invariant observed

Abstract

We study the four-genus of linear combinations of torus knots: 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.