A note on the concordance invariants Upsilon and phi

Abstract

Dai, Hom, Stoffregen and Truong defined a family of concordance invariants [Formula: see text]. The example of a knot with zero Upsilon invariant but nonzero epsilon invariant previously given by Hom also has nonzero phi invariant. We show there are infinitely many such knots that are linearly independent in the smooth concordance group. In the opposite direction, we build infinite families of linearly independent knots with zero phi invariant but nonzero Upsilon invariant. We also give a recursive formula for the phi invariant of torus knots.