A characterization of quasipositive two-bridge knots

TL;DR

This paper provides a clear criterion for when two-bridge knots are quasipositive, linking continued fractions to knot properties, and demonstrates that smoothly slice two-bridge knots are not quasipositive.

Contribution

It introduces a simple necessary and sufficient condition for quasipositivity of two-bridge knots based on continued fractions, with applications to contact and symplectic topology.

Findings

Characterization criterion for quasipositive two-bridge knots

Proof that smoothly slice two-bridge knots are non-quasipositive

Alternative proof using knot theory methods

Abstract

We prove a simple necessary and sufficient condition for a two-bridge knot K(p,q) to be quasipositive, based on the continued fraction expansion of p/q. As an application, coupled with some classification results in contact and symplectic topology, we give a new proof of the fact that smoothly slice two-bridge knots are non-quasipositive. Another proof of this fact using methods within the scope of knot theory is presented in the Appendix.