Extending quasi-alternating links

Abstract

Champanerkar and Kofman [Twisting quasi-alternating links, Proc. Amer. Math. Soc. 137(7) (2009) 2451-2458] introduced an interesting way to construct new examples of quasi-alternating links from existing ones. Actually, they proved that replacing a quasi-alternating crossing c in a quasi-alternating link by a rational tangle of same type yields a new quasi-alternating link. This construction has been extended to alternating algebraic tangles and applied to characterize all quasi-alternating Montesinos links. In this paper, we extend this technique to any alternating tangle of same type as c. As an application, we give new examples of quasi-alternating knots of 13 and 14 crossings. Moreover, we prove that the Jones polynomial of a quasi-alternating link that is obtained in this way has no gap if the original link has no gap in its Jones polynomial. This supports a conjecture introduced in [N. Chbili and K. Qazaqzeh, On the Jones polynomial of quasi-alternating links, Topology Appl. 264 (2019) 1-11], which states that the Jones polynomial of any prime quasi-alternating link except (2,p)-torus links has no gap.