On Skein Modules and Homology Theories Related to Knot Theory Open Access
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Classical knot theory is the study of simple closed curves in three dimensional space. The dissertation studies algebraic structures related to and motivated by knot theory. Since the first half of 1990, several homology theories have been defined for quandles and, more generally, for shelves whose axioms are motivated by the Reidemeister moves in knot theory. The second chapter of the dissertation studies these homology theories. For rack homology, one of the theorems assert that any finite Abelian group can appear as a torsion subgroup in the rack homology of some quandle. Furthermore, the second quandle homology of quasigroup Alexander quandles is expressed in terms of exterior algebras. Finally, the role of the associativity axiom in the category of self distributive algebraic structures is studied from the viewpoint of the homology theories. One of the results in this discussion explicitly computes the rack homology of shelves with a right fixed element.Associative shelves form a very useful collection of algebraic structures when comparing the homology theories for shelves and semigroups. Examples of such homology theories include group homology, Hochschild homology, and multi term homology. However, from the perspective of self distributivity, rack and one term homology prove not to be very useful as demonstrated in Chapter two. The third chapter of the dissertation introduces a new homology theory to take care of this issue. In fact, this homology theory works for the family of the so-called quasibands which contain the collection of associative shelves. A theorem in this part of the dissertation computes the second homology group of finite semilattices which are also examples of quasibands. Moreover, the relation between this homology theory and Temperley-Lieb algebras is established.The third part of the dissertation focuses on Khovanov homology. In particular, the main goal in this part is to study torsion in Khovanov homology. Z2 torsion in Khovanov homology is much better understood than any other torsion. Three years ago, only a finite number of examples of knots and prime links with non Z2 torsion in their Khovanov homology were known. It was also known that knots and links with thin Khovanov homology can contain only Z2 torsion. The work on Khovanov homology in the dissertation introduces the first known examples of infinite families of knots and prime links with non Z2 torsion in their Khovanov homology. More specifically, infinite families of knots and prime links containing Z3, Z4, Z5, Z7, and Z8 torsion in their Khovanov homology are introduced. The aforementioned results assert that given some reasonably small torsion group, there are `many' examples of knots and links with the given torsion subgroup in their Khovanov homology. The latter half of this study addresses the `large' side, meaning, methods of finding knots and links with `large' torsion groups in their Khovanov homology. To this end, links with even torsion of order 2s, for 0 < s < 24, in their Khovanov homology are introduced. In the case of odd order, knots and links with Z9, Z25, and Z27 torsion in their Khovanov homology are introduced. These results and discoveries are used to resolve all but one part of the Przytycki-Sazdanovic braid conjecture.The final part of the dissertation is motivated by the elegant product-to-sum formula of Frohman and Gelca for multiplying curves in the Kauffman bracket skein algebra of the thickened torus. After comparing the Kauffman bracket skein algebra of the thickened four-holed sphere with the Kauffman bracket skein algebra of the thickened torus, formulas for multiplying curves in some special families are discussed. Following this, an algorithm to multiply any two curves in the Kauffman bracket skein algebra of the thickened four-holed sphere is described.