Quantum invariants and their connections with Thompson's groups F and T

This thesis concerns extensions and applications of the Jones polynomial invariant of oriented links, and the closely related Witten-Reshetikhin-Turaev (WRT) 3-manifold invariants. We calculate and establish number-theoretic properties of recently developed (t,q)-series invariants of negative definite plumbed 3-manifolds, leading to the development of new invariants which may be thought of as zeta-deformed WRT invariants, for zeta any root of unity. This thesis also presents an extension of recent work by Vaughan Jones, who constructed links in the 3-sphere from Thompson's group F, to the setting of annular links and Thompson's group T. We also present a construction of (n,n)-tangles from F, and show that the oriented subgroup F admits the structure of a lax group action on a category of Khovanov chain complexes associated to tangles.