Amicable Knots on Minimal Genus Seifert Surfaces

Abstract

For a knot $K\subset S^3$, let $S(K)$ denote the set of non-trivial knot types represented by simple closed curves on a minimal genus Seifert surface of $K$. We study the relation $J\in S(K)$ and its symmetric part, which leads to the notion of \emph{amicable knots}: knots $K$ and $J$ are called amicable if each is represented by a simple closed curve on a minimal genus Seifert surface of the other. A classical result of Lyon implies that the family of torus knots is universal for this realization problem: for every non-trivial knot type $J$, there exists a torus knot $T$ such that $J\in S(T)$. In contrast, one of the main results of this paper is that no single knot is universal: for every knot $K$, there exists a knot $J$ such that $J\notin S(K)$. We also study explicit examples, keeping track of chirality throughout. Writing $3_1^+=T(2,3)$ and $8_{19}^+=T(3,4)$ for the right-handed positive torus knots, we show that $3_1^+$ and $8_{19}^+$ are amicable, whereas $3_1^+$ and the figure-eight knot $4_1$ are not. We also describe the hosting sets of both chiralities of the trefoil in terms of primitive slope classes on their once-punctured torus fibers.