Doctor of Philosophy (PhD)
Skein modules arise naturally when mathematicians try to generalize the Jones polynomial of knots. In the first part of this work, we study properties of skein modules. The Temperley-Lieb algebra and some of its generalizations are skein modules. We construct a bases for these skein modules. With this basis, we are able to compute some gram determinants of bilinear forms on these skein modules. Also we use this basis to prove that the Mahler measures of colored Jones polynomial of a sequence of knots converges to the Mahler measure of some two variable polynomial. The topological quantum field theory constructed by Blanchet, Habegger, Mas- baum and Vogel can be considered as a generalization of quantum invariants. It assigns modules to surfaces and linear maps to cobordisms. In particular, it assigns the ground ring to empty surface and constants to cobordisms of empty surface to itself, which are closed 3-manifolds. In this way, we get quantum invariants of 3-manifolds back. In the second part of the work, knot invariants are constructed using topological quantum field theory from quantum invariants of tangles. We prove that this is another way to compute the Turaev-Viro polynomial of knots and related invariants.
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Cai, Xuanting, "Skein theory and topological quantum field theory" (2013). LSU Doctoral Dissertations. 4070.