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Rasmussen invariants of twisted positive Whitehead doubles across characteristics

Determine, for each n ≥ 2 and each field F, the Rasmussen invariant s_F(W_n), where W_n denotes the (n^2−1)-twisted positive Whitehead double of the torus knot T(n, n+1), and prove that s_F(W_n) equals 2 if Char F divides n and 0 otherwise.

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Background

This conjecture is cited as a key input that would imply the optimality conjecture about Rasmussen bounds. It has been formulated in prior works (referenced by the authors) and concerns how the Rasmussen invariant depends on the characteristic of the ground field for a specific family of Whitehead doubles.

Confirming this statement would provide the building blocks needed to realize prescribed patterns of Rasmussen invariants across characteristics and to match the conjectured optimal Gordian distance bound.

References

In fact, we can prove this—provided we accept the following second conjecture as true: For $n \geq 2$, let $W_n$ be the $(n2-1)$-twisted positive Whitehead double $W_n$ of the torus knot $T(n,n+1)$. Then $s_{\mathbb{F}(W_n)$ is expected to be equal to $2$ if $\Char\mathbb{F}$ divides~$n$, and $0$ otherwise (cf.Conj.~6.2 and 6.3, Conj.~6.9, Conj.~1.3).

— Khovanov homology and refined bounds for Gordian distances (2409.05743 - Lewark et al., 9 Sep 2024) in Remark following Proposition thm:s_invariant_and_graded_lambda, Section “Relationship with the s-invariant”