Hosting and Friendship of Knots on Minimal Genus Seifert Surfaces

TL;DR

This paper introduces a new framework for understanding how knots appear on minimal genus Seifert surfaces, revealing that no single knot can universally host all others, but families of knots can.

Contribution

It establishes that universal hosting occurs only at the family level, not for individual knots, and provides explicit examples and descriptions of hosting and friendship among knots.

Findings

Torus knots form a universal host family for non-trivial knots.

No single knot can host all other knots (no universal host at the individual level).

Explicitly describes hosting set of the trefoil knot and friendship phenomena.

Abstract

For a knot $K\subset S^3$, let $S(K)$ denote the set of knot types represented by simple closed curves on a minimal genus Seifert surface of $K$. We study the directed relation $K\to J$ defined by $J\in S(K)$, which we call the \emph{hosting relation}, and call its symmetric part friendship. This gives a new framework for describing how knots appear on minimal genus Seifert surfaces of other knots. A classical result of Lyon implies that the family of torus knots is a universal host family: every non-trivial knot is hosted by some torus knot. In contrast, a central result of this paper is that no knot is a universal host: for every knot $K$, there exists a knot $J$ such that \[ J\notin S(K). \] Thus universal hosting occurs at the level of families, but never at the level of a single knot. We also study explicit examples of hosting and friendship. In particular, we describe the hosting set of the trefoil in terms of primitive slope classes on its once-punctured torus fiber, and use this description to obtain concrete friendship and non-friendship phenomena. For example, we show that $3_1$ and $8_{19}$ are friends, whereas $3_1$ and $4_1$ are not. These results provide a framework for studying universal host phenomena, hosting, and friendship among knots on minimal genus Seifert surfaces, and suggest further connections with graph-theoretic, rigidity, and categorical aspects of knot theory.