Abstract. We construct an infinite family of quasi-alternating links from a given quasialternating link by replacing a crossing by a product of rational tangles each of which extends that crossing. Consequently, we determine an infinite family of quasi-alternating Montesinos links. This family contains all the classes of quasi-alternating Montesinos links that have been detected by Widmar in [W]. We conjecture that this family contains all quasi-alternating Montesinos links up to mirror image that are not alternating and this will characterize all quasi-alternating Montesinos links.

We prove that the degree of the Q-polynomial of any quasialternating link is less than its determinant. Therefore, we obtain a new and simple obstruction criterion for the link to be quasialternating. As an application, we identify some knots of 12 crossings or less and some links of 9 crossings or less that are not quasialternating. Our obstruction criterion applies also to show that there are only finitely many Kanenobu knots that are quasialternating. Moreover, we identify an infinite family of Montesinos links that are not quasialternating.

We show that the crossing number of any link that is known to be quasi-alternating is less than or equal to its determinant. Based on this, we conjecture that the crossing number of any quasi-alternating link is less than or equal to its determinant. Thus if this conjecture is true, then it gives a new property of quasi-alternating links and easy obstruction to a link being quasi-alternating.

We give recursive relations to compute the rational Khovanov bigraded homology groups of three infinite families of non-quasi-alternating 3-column pretzel links. Based on this computations, we obtain an infinite family of homologically thin but not quasi-alternating knots.

We prove that twisting any quasi-alternating link L with no gaps in its Jones polynomial V L (t) at the crossing where it is quasi-alternating produces a link L * with no gaps in its Jones polynomial V L * (t). This leads us to conjecture that the Jones polynomial of any prime quasi-alternating link, other than (2, n)-torus links, has no gaps. This would give a new property of quasi-alternating links and a simple obstruction criterion for a link to be quasi-alternating. We prove that the conjecture holds for quasi-alternating Montesinos links as well as quasi-alternating links with braid index 3.Date: 23/05/2018.