Dice Question Streamline Icon: https://streamlinehq.com

Rasmussen invariants of twisted positive Whitehead doubles

Establish that for every integer n ≥ 2 and every field F, the Rasmussen invariant of the (n^2 − 1)-twisted positive Whitehead double W_n of the torus knot T(n, n+1) satisfies s_F(W_n) = 2 if Char F divides n and s_F(W_n) = 0 otherwise.

Information Square Streamline Icon: https://streamlinehq.com

Background

This conjectural input would imply the optimality conjecture about the Rasmussen-based bound for Gordian distance by enabling constructions of knots with prescribed s-invariants across characteristics.

The statement is attributed to prior conjectures in the literature (Schütz, Lewark–Zibrowius, and others), but here the authors explicitly identify it as a conjectural condition under which they can prove the optimality claim.

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), Subsection "Relationship with the s-invariant"