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

Optimality of the Rasmussen-based lower bound for Gordian distance

Determine whether the quantity max{0, max over fields F of s_F(K1,K2)} + max{0, max over fields F of s_F(K2,K1)} is the optimal lower bound for the Gordian distance u(K1, K2) obtainable from the collection of Rasmussen invariants across all fields. More precisely, for any functions a1, a2 on the set C = {0} ∪ {primes} with a_i(c) = a_i(0) for all but finitely many c, construct knots K1 and K2 such that s_F(K_i) = 2 a_i(Char F) for every field F and u(K1, K2) equals max{0, max_F s_F(K1,K2)} + max{0, max_F s_F(K2,K1)}.

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

Background

The paper proves general inequalities relating their Khovanov-homological invariants (λ and λ−) to Gordian distances and Rasmussen invariants across fields. In particular, inequality (eq:rasmussen_u_bound2) bounds the Gordian distance u(K1,K2) below by a sum of maxima of Rasmussen differences over all fields.

The authors propose that this bound is not merely a lower bound but, in an appropriate sense, optimal among all lower bounds derivable from the entire family of Rasmussen invariants. They make this precise by prescribing the values of s_F(K_i) according to functions of the characteristic of F and asking for knots that achieve equality.

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), Subsection "Relationship with the s-invariant" (around equation (eq:rasmussen_u_bound2))