Abstract: In 2014, Vaughan Jones introduced methods to construct unitary representations of Thomp- son groups F, T, V (from planar algebras or Cuntz algebras) and another to produce knots from their elements. These unitary representations have been instrumental in understanding the subgroup structure of F, while the knot construction allows for the initiation of a new theory analogous to that of braids and knots, but with the Thompson groups replacing the braid groups. On another topic, several C∗-algebras have been introduced by Cuntz and collaborators. The p-adic ring C∗-algebra is the universal C∗-algebra Qp generated by an isometry sp and a unitary u such that spu = upsp and  ∑{i=0}{p-1} uis2s2u−i = 1. This algebra turns out to be simple and purely infinite. 
Inside Qp, the isometries T0 = sp, T1 = usp, . . . , T p − 1 = up−1sp generate a natural copy of the Cuntz algebra Op. I will report on some results regarding them.