Extending Quasi-Alternating Links

TL;DR

This paper generalizes a method for constructing quasi-alternating links by replacing crossings with tangles, leading to new examples and supporting a conjecture about the Jones polynomial's properties.

Contribution

It extends the construction technique to all alternating tangles of the same type, broadening the class of known quasi-alternating links.

Findings

New quasi-alternating knots of 13 and 14 crossings identified.

Jones polynomial of constructed links has no gap if original link's polynomial has none.

Supports the conjecture that prime quasi-alternating links (except certain torus links) have gapless Jones polynomials.

Abstract

Champanerkar and Kofman 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 arXiv:1810.11773 [math.GT], which states that Jones polynomial of any prime quasi-alternating link except (2; n)-torus link has no gap.