Extending Quasi-alternating Links II

TL;DR

This paper investigates conditions under which the method of constructing quasi-alternating links by replacing crossings with tangles can be extended to non-alternating tangles and alternating tangles of opposite types, broadening the applicability of the technique.

Contribution

The authors identify new conditions and specific families of tangles that allow the construction of quasi-alternating links beyond previously known cases.

Findings

Conditions for applying the construction to alternating tangles of opposite types

Identification of families of non-alternating tangles suitable for the construction

Extended applicability of quasi-alternating link construction methods

Abstract

Champanerkar and Kofman [1] introduced an innovative method for constructing quasi-alternating links by substituting a quasi-alternating crossing c in a quasi-alternating link with a rational tangle of the same type. This construction was later extended by Chbili and Kaur [2] for any alternating tangle of same type. However, this approach does not generally apply to non-alternating tangles or alternating tangles of opposite types, although there are examples where it holds for alternating tangles of opposite type and certain non-alternating tangles. This raises a natural question to find the conditions and specific tangles for which this approach is applicable. In this paper, we explore this question by identifying several conditions under which this construction of new quasi-alternating links can still be valid for alternating tangles of opposite types. Additionally, we establish specific families of non-alternating tangles for which this construction is effective in creating new quasi-alternating links.