The Jones polynomial and related properties of some twisted links

TL;DR

This paper investigates the properties of the Jones polynomial for twisted links, especially focusing on full twists, and characterizes the polynomial behavior for specific classes like Lorenz and T-links with braid index three.

Contribution

It provides explicit constants for the Jones polynomial of two-strand twists in three-braids and identifies patterns and properties for various twisted link classes, including Lorenz and T-links.

Findings

Constants for the Jones polynomial when two strands of a three-braid are twisted are determined.

Patterns in coefficient blocks of the Jones polynomial for full twists are characterized.

Jones polynomial formulas for Lorenz and T-links with braid index three are derived.

Abstract

Twisted links are obtained from a base link by starting with a $n$-braid representation, choosing several ($m$) adjacent strands, and applying one or more twists to the set. Various restrictions may be applied, e.g. the twists may be required to be positive or full twists, or the base braid may be required to have a certain form. The Jones polynomial of full $m$-twisted links have some interesting properties. It is known that when sufficiently many full $m$-twists are added that the coefficients break up into disjoint blocks which are independent of the number of full twists. These blocks are separated by constants which alternate in sign. Other features are known. This paper presents the value of these constants when two strands of a three-braid are twisted. It also discloses when this pattern emerges for either two or three strand twisting of a three-braid, along with other properties. Lorenz links and the equivalent T-links are positively twisted links of a special form. This paper presents the Jones polynomial for such links which have braid index three. Some families of braid representations whose closures are identical links are given.