The Kauffman bracket skein module of the quaternionic manifold

Abstract

In this work, we study the structure of the Kauffman bracket skein module of the quaternionic manifold over the field of rational functions. We begin with a brief survey of manifolds whose Kauffman bracket skein modules are known, and proceed in Chapter 2 by recalling the facts from Temperley-Lieb recoupling theory that we use in the proofs. In Chapter 3, using recoupling theory and with Mathematica's assistance, we index an infinite presentation of the skein module, and conjecture that it is five-dimensional. In Chapter 4, using a new set of relations, we prove that the skein module is indeed spanned by five elements, again using Mathematica for the difficult computations. Using the quantum invariants of these skein elements and Z2-homology of the manifold, we determine that they are linearly independent in Chapter 5. In Chapter 6, we conclude with a few brief remarks about future uses and extensions of this work. In the appendices, we present the Mathematica code referenced in Chapters 3 and 4, and we give a proof, due to Paul van Wamelen, of a lemma concerning Gauss sums needed in Chapter 5.