Knots with Hopf crossing number at most one

TL;DR

This paper classifies knots with diagrams derived from the Hopf map that have at most one crossing, exploring their properties and identifying which are algebraic, thus advancing understanding of simple knot types.

Contribution

It provides a complete classification of knots with Hopf crossing number at most one and characterizes their algebraic properties, addressing a specific open problem.

Findings

Knots with Hopf crossing number ≤ 1 are classified.

Properties of these knots are characterized.

Identification of algebraic knots among them.

Abstract

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler.