On the Jones polynomial of quasi-alternating links

Abstract

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.

On Khovanov Homology of Quasi-Alternating Links

“…Finally, we enclose our discussion with the following conjecture that is supported by the above results and implies both Conjecture 2.3 in [5] and Conjecture 3.8 in [16]. In particular, it implies that the Jones polynomial of any quasi-alternating link that is not a (2, n)-torus link has no gap and the breadth of the Jones polynomial of such a link is a lower bound of the determinant of this link.…”

“…As a consequence, we obtain that the length of any gap in the Jones polynomial of any such link is one. This establishes a weaker version of Conjecture 2.3 in [5]. Moreover, we obtain a lower bound for the determinant of any such link in terms of the breadth of its Jones polynomial.…”

“…Proof. The result follows as a result of Corollary 4.9 and the fact that the Jones polynomial of all quasi-alternating Montesinos links and quasi-alternating links of braid index 3 have no gap (Theorem 3.10 and Theorem 4.3 in [5], respectively).…”

“…We start by stating two corollaries. The first of which establishes a weaker version of Conjecture 2.3 in [5] and the second one establishes a weaker version of Conjecture 3.8 in [16]. Proof.…”

“…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.…”