Strongly quasipositive links with braid index 3 have positive Conway polynomial

TL;DR

This paper establishes that strongly quasipositive links with braid index 3 necessarily have positive Conway polynomials, providing a new criterion for such links, but shows this does not extend to higher braid indices.

Contribution

It proves a necessary condition linking strong quasipositivity and positive Conway polynomial for braid index 3, and demonstrates the limitation of this condition for higher indices.

Findings

Strongly quasipositive links with braid index 3 have positive Conway polynomial.

The positive Conway polynomial condition does not hold for higher braid indices.

A counterexample is provided for braid indices greater than 3.

Abstract

Strongly quasipositive links are those links which can be seen as closures of positive braids in terms of band generators. In this paper we give a necessary condition for a link with braid index 3 to be strongly quasipositive, by proving that in that case it has positive Conway polynomial (that is, all its coefficients are non-negative). We also show that this result cannot be extended to a higher number of strands, as we provide a strongly quasipositive braid whose closure has non-positive Conway polynomial.