A new obstruction of quasialternating links

Abstract

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.

The classification of quasi-alternating Montesinos links

“…(i) L is homologically thin for both Khovanov homology and knot Floer homology [MO08]. For some further properties see [LO15], [QC15], [Ter15] and [ORS13, Remark after Proposition 5.2].…”

“…Further conditions are described in [CO15] coming from the fact that the double branched cover of a quasi-alternating link is an L-space. Some additional restrictions were found in [QC15].…”

On Khovanov Homology of Quasi-Alternating Links

“…Moreover, we obtain a lower bound for the determinant of any such link in terms of the breadth of its Jones polynomial. This establishes a weaker version of Conjecture 3.8 in [16]. The main tool in obtaining this result is proving the Knight Move Conjecture [2] for the class of quasi-alternating links.…”

“…1]; (5) the reduced odd Khovanov homology group of any quasi-alternating link is σ-thin [14, Remark after Proposition. 5.2]; (6) the determinant of any quasi-alternating link is bigger than the degree of its Q-polynomial [16,Theorem 2.2]. This inequality was sharpened to the determinant minus one is bigger than or equal to the degree of the Q-polynomial with equality holds only for (2, n)-torus links [20,Theorem 1.1].…”

“…On the other hand, we show that some subclasses of quasi-alternating links have no gaps in the differential grading of their Khovanov homology. This leads us to suggest Conjecture 4.11 that implies both Conjecture 2.3 in [5] and Conjecture 3.8 in [16]. This paper is organized as follows.…”