On a parameterization of $(1,1)$-knots

TL;DR

This paper introduces a new parameterization for (1,1)-knots in the 3-sphere based on minimal-length arc representatives, proving its uniqueness for satellite knots and proposing a generalization to higher genus cases.

Contribution

A novel parameterization of (1,1)-knots is developed, distinct from classical methods, with proofs of uniqueness in certain cases and a proposed extension to (g,1)-knots.

Findings

The new parameterization is essentially unique for satellite (1,1)-knots.

It is obtained from minimal-length representatives of homotopy classes.

A generalization to (g,1)-knots for any g ≥ 1 is proposed.

Abstract

A $(1,1)$-knot in the 3-sphere is a knot that admits a 1-bridge presentation with respect to a Heegaard torus in $\mathbb{S}^{3}$. A new parameterization of $(1,1)$-knots distinct from the classical ones is introduced. This parameterization is obtained from minimal-length representatives of homotopy classes of arcs in the mutipunctured plane. In the particular case of satellite $(1,1)$-knots, it is proven that the introduced parameterization is essentially unique. A generalization of this parameterization to the family of $(g,1)$-knots for any $g\geq 1$ is proposed.