The Ozsváth-Szabó tau-invariant of braided satellites

Abstract

We give formulas for the $ τ$ and $ \varepsilon $ concordance invariants of satellite knots whose patterns are braided, meaning they wind around the solid torus without reversing. Our methods lead us to define the class of squeezed patterns, analogous to squeezed knots as defined by Feller-Lewark-Lobb. We show that all braided patterns are squeezed, and we give a $ τ$ formula for squeezed patterns as well. Towards a conjecture of Hedden, we show that no squeezed pattern, and thus no braided pattern, with winding number at least 2 induces a homomorphism on the concordance group.