Skein Modules, Skein Algebras, and Their Ramifications
Open AccessSkein modules are invariants of 3-manifolds which were introduced by Przytycki in 1987 as generalisations of the Jones, HOMFLYPT, and Kauffman bracket polynomial link invariants in S3 to arbitrary 3-manifolds. Over time, skein modules have evolved into one of the most important objects in knot theory and quantum topology having strong ties with many fields of mathematics, such as algebraic geometry via SL(2, C) character varieties, hyperbolic geometry via quantum Teichmüller spaces, quantum cluster algebras, the Witten-Reshetikhin-Turaev 3-manifold invariants, and Topological Quantum Field Theories, to name a few. The Kauffman bracket skein module (KBSM) is the most extensively studied skein module of all. However, computing the KBSM of a 3-manifold is known to be notoriously hard, especially over the polynomial ring of Laurent polynomials. With the goal of finding a definite structure of the KBSM over this ring, several conjectures and theorems were stated over the years. In my dissertation I will show that some of these conjectures, and even theorems, are not true. Marché's generalisation of Witten's finiteness conjecture plays a central role in this discussion. It is well known that skein modules reflect the geometry and topology of 3-manifolds. Indeed, the q-homology skein module, the q-deformation of the signed skein module, and the Kauffman bracket skein module all detect the presence of non-separating 2-spheres and tori embedded in the 3-manifold. Interestingly enough, this presence manifests itself in the form of torsion in the skein module. In my dissertation I investigate the properties of the framing skein module of an oriented 3-manifold M and compute its exact structure, showing that it detects the presence of non-separating 2-spheres embedded in M via the Dirac trick. Certain skein modules, like the KBSM of the product of an oriented surface and the interval, can be endowed with an algebra structure by defining a multiplication operation between the elements of the module. In 1999, Frohman and Gelca found an elegant product-to-sum formula for the multiplication of elements in the Kauffman bracket skein algebra (KBSA) of the thickened torus and showed that this algebra is isomorphic to a subalgebra of the noncommutative torus. They achieved this by decorating the basis of the KBSA using Chebyshev polynomials of the first kind. A question regarding the positivity of the bases of skein algebras, which arose naturally in their work on cluster algebras, was first posed by Fock and Goncharov in 2006. In 2014, D. Thurston conjectured that modified Chebyshev polynomials of the first kind form a positive basis for the KBSA of the thickened surface. Motivated by Frohman, Gelca, and D. Thurston's work, my dissertation gives a complete description of the multiplicative structure of the KBSA of the thickened four-punctured sphere, essentially finding a product-to-sum formula for infinite families of links in this manifold and showing that, for these, the Chebyshev basis is positive.The relative Kauffman bracket skein module (RKBSM) also plays an important role in quantum topology. In 1989, Witten proposed the existence of invariants of links and 3-manifolds using quantum field theory. Later, Reshetikhin and Turaev gave a mathematical realization of Witten's work using representations of the Lie group SU(2). Lickorish gave a simpler, more combinatorial approach to the construction of these invariants using the RKBSM of the disc and a special bilinear form defined on its elements. The determinant of the matrix given by this bilinear form is known as the Gram determinant of type A. A similar bilinear form over the RKBSM of the thickened annulus gives rise to the Gram determinant of type B. This Gram determinant was investigated by combinatorialist Simion in her work on chromatic joins and has connections with statistical mechanics. It is noteworthy that the closed formulae for both these Gram determinants involve Chebyshev polynomials. In my dissertation, the generalised type A bilinear form is introduced and a closed formula for the determinant of the matrix given by this bilinear form is proved. The dissertation also investigates a bilinear form on the RKBSM of the twisted I-bundle over the Möbius band and proves some results which corroborate Chen's conjecture regarding the closed formula for the Gram determinant of type Mb.The last part of my dissertation focuses on self-distributive and non-associative algebraic structures known as quandles, whose axioms arise naturally in knot theory and correspond to the three Reidemeister moves. In 1990, Rourke, Fenn, and Sanderson introduced rack homology which was later enhanced to quandle homology by Carter, Jelsovsky, Seiichi Kamada, Langford, and Saito in order to define cocycle invariants of classical and higher-dimensional knots. My dissertation research in quandles is concerned with studying the properties of quandle and rack homology and identifying and expanding on possible connections to other algebraic structures. In particular, I express the second homology of Alexander quandles in terms of exterior algebras and present a self-contained proof of its structure.
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