On a parameterization of (1,1)-knots

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 [Formula: see text]. A new parameterization of [Formula: see text]-knots distinct from the classical ones is introduced. This parameterization is obtained from minimum length representatives of homotopy classes of arcs in the multipunctured 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 [Formula: see text]-knots for any [Formula: see text] is proposed.