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 VL(t) at the crossing where it is quasi-alternating produces a link L⁎ with no gaps in its Jones polynomial VL⁎(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.