Strongly quasipositive quasi-alternating links and Montesinos links

Abstract

The aim of this article is to give a characterization of strongly quasipositive quasi-alternating links and detect new classes of strongly quasipositive Montesinos links and non-strongly quasipositive Montesinos links. In this direction, we show that, if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_0$ is alternating (where $L_0$ has the induced orientation), then $L$ is definite if and only if it is strongly quasipositive (up to mirroring). We also show that if $L$ is an oriented quasi-alternating link with a quasi-alternating crossing $c$ such that $L_0$ is fibred or more generally has a unique minimal genus Seifert surface (where $L_0$ has the induced orientation), then $L$ is definite if and only if it is strongly quasipositive (up to mirroring).

19 pages, 12 figures