Crossing-number lower bounds for satellite knots

Establish whether every satellite knot K with companion knot C and wrapping number w satisfies the lower bounds c(K) w^2 c(C) and, at minimum, c(K) c(C).

Background

The paper studies how crossing number behaves under satellite operations. A satellite knot K is formed by embedding a pattern in a solid torus whose core is knotted as a companion knot C, and the wrapping number w measures the minimum intersection of the pattern with a meridian disc. The standard diagrammatic construction produces clusters of at least w2 crossings for each crossing of a companion diagram, motivating substantially stronger lower bounds than those currently known in general.

The authors prove only a nonlinear bound, c(K) (2/5)(w2c(C))1/11, described in the abstract as partial progress on Problem 1.2 of the K3 Problem List. Thus the conjectured quadratic and linear lower bounds remain unresolved in the general case.

References

For this reason one might believe that $c(K)$ must always be at least $w2c(C)$, or less optimistically at least $c(C)$. These conjectures appear as Problem 1.2 in the K3 problem list .

— The Crossing Number and Arc Index of Satellite Knots  (2609.20900 - Ketchell, 17 Sep 2026) in Section 1, Introduction

One could also conjecture that $c(K)$ should depend on $c(P)$ (suitably defined) in some way. Writing down a neat formula is difficult because of interactions between the framing of the solid torus and the writhe of the diagram of $C$, but loosely speaking we might expect $c(K)$ to be ``something like $w2c(C)+c(P)$".

— The Crossing Number and Arc Index of Satellite Knots  (2609.20900 - Ketchell, 17 Sep 2026) in Section 1, Introduction