Annular Rasmussen invariants: Properties and 3-braid classification

TL;DR

This paper investigates the properties of annular Rasmussen invariants and knot Floer invariants, demonstrating finiteness results for fixed braid index and genus, and explicitly computing invariants for all 3-braid closures.

Contribution

It establishes finiteness of possible invariant shapes for fixed braid index and genus, and provides explicit calculations for all 3-braid closures.

Findings

Finiteness of annular Rasmussen $d_t$ invariant shapes for fixed braid index.

Finiteness of $$ invariant possibilities for fixed concordance genus.

Explicit computation of invariants for all 3-braid closures.

Abstract

We prove that for a fixed braid index there are only finitely many possible shapes of the annular Rasmussen $d_t$ invariant of braid closures. Applying the same perspective to the knot Floer invariant $\Upsilon_K(t)$, we show that for a fixed concordance genus of $K$ there are only finitely many possibilities for $\Upsilon_K(t)$. Focusing on the case of 3-braids, we compute the Rasmussen $s$ invariant and the annular Rasmussen $d_t$ invariant of all 3-braid closures. As a corollary, we show that the vanishing/non-vanishing of the $\psi$ invariant is entirely determined by the $s$ invariant and the self-linking number.