The Jones polynomial and related properties of some twisted links

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.