Rasmussen invariants of Whitehead doubles and other satellites

Abstract We prove formulae for the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mn>2</m:mn> </m:msub> </m:math> \mathbb{F}_{2} -Rasmussen invariant of satellite knots of patterns with wrapping number 2, using the multicurve technology for Khovanov and Bar-Natan homology developed by Kotelskiy, Watson, and the second author. A new concordance homomorphism, which is independent of the Rasmussen invariant, plays a central role in these formulae. We also explore whether similar formulae hold for the Ozsváth–Szabó invariant 𝜏.