An explicit connected-sum family with unbounded knot Ramsey-stick gap

Abstract

Let $R(K)$ and $s(K)$ denote the Ramsey number and the stick number of a knot $K$. Johnson proved that $R-s$ is unbounded on $(p-1,p)$ torus knots and asked whether other such families exist. We give an affirmative answer. We construct an explicit eight-stick alternating knot $J$ with $c(J)=6$ and $\operatorname{br}(J)=2$. For $K_n=\#^nJ$, the connected-sum formulas yield $R(K_n)\ge 7n+3$ and $s(K_n)\le 5n+3$.