Annular Rasmussen Invariants: Properties and 3-Braid Classification

Abstract

We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen ${\mathit{d}}_{\mathit{t}}$ invariant of braid closures. Focusing on the case of 3-braids, we compute the Rasmussen s-invariant and the annular Rasmussen ${\mathit{d}}_{\mathit{t}}$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the ψ invariant is entirely determined by the s-invariant and the self-linking number for 3-braid closures.