Jones and 𝑄 polynomials for 2-bridge knots and links

Abstract

It is known that the Q Q polynomial of a 2 2 -bridge knot or link can be obtained from the Jones polynomial. We construct arbitrarily many 2 2 -bridge knots or links with the same Q Q polynomial but distinct Jones polynomials.

A bridge between Conway–Coxeter friezes and rational tangles through the Kauffman bracket polynomials

“…There are many known formulae to compute Kauffman bracket polynomials of rational tangles and Jones polynomial of rational links [14,21,22,18,24]. In 1995 Shuji Yamada also found a unique formula to compute such polynomials by using Farey neighbors up to units [26].…”

The Jones polynomial of rational links

“…For example, the authors of [4,13] give an explicit formula for the Conway (Alexander) polynomial invariant of rational links independently. Moreover, the authors of [3,6,10,11,12,14,15,18] have studied the Jones polynomial of rational links either directly or indirectly through studying another polynomial invariant that reduces to the Jones polynomial after some special normalization using different techniques.…”