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Optimality of the Rasmussen-invariant lower bound for Gordian distance

Establish that the quantity max{0, max_F s_F(K_1,K_2)} + max{0, max_F s_F(K_2,K_1)} is the optimal lower bound for the Gordian distance u(K_1, K_2) obtainable from all Rasmussen invariants across fields. More precisely, construct knots K_1 and K_2 for any functions a_1, a_2 from C = {0} ∪ (all primes) to Z that are eventually constant, such that for every field F with characteristic in C one has s_F(K_i) = 2 a_i(Char F), and u(K_1, K_2) equals the aforementioned quantity.

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Background

The paper proves that the sum of the positive parts of the maximal Rasmussen-invariant differences over all fields is a lower bound for the Gordian distance and for the invariants introduced. The authors then propose that this bound is in fact optimal among all bounds obtainable from Rasmussen invariants, i.e., it can be realized as the exact Gordian distance for suitably chosen knots.

They formalize the conjecture in terms of prescribing the values of Rasmussen invariants over all characteristics via functions a_1 and a_2 that are eventually constant, and ask for knots realizing those values and the bound as equality for u(K_1, K_2).

References

We conjecture that the left-hand side of eq:rasmussen_u_bound2 is the optimal lower bound for $u(K_1, K_2)$ provided by the set of all Rasmussen invariants. Let us make this precise. Let $C = {0,2,3,\ldots}$ be the set of zero and all primes. Let $a_1, a_2\colon C \to \mathbb{Z}$ be given such that $a_i(c) = a_i(0)$ for almost all $c\in C$. Then we conjecture that there exist knots $K_1, K_2$ such that $s_{\mathbb{F}(K_i) = 2a_i(\Char \mathbb{F})$, and $u(K_1, K_2)$ equals the left-hand side of eq:rasmussen_u_bound2.

— 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”