Stated Skein Theory and Double Affine Hecke Algebra Representations

In this thesis, we explore the representation theory of double affine Hecke algebras (DAHAs) through the lens of stated skein theory. Over the past decade, there have been several works establishing robust connections between skein algebras and DAHAs. Particularly, Samuelson proved that a spherical subalgebra of the type $A_1$ DAHA can be realized as a quotient of the Kauffman bracket skein algebra of the torus with boundary, $K_q(T^2 \setminus D^2)$. Since the $A_1$ double affine Hecke algebra is Morita equivalent to its spherical subalgebra, discovering modules for $K_q(T^2 \setminus D^2)$ immediately provides us with modules for the $A_1$ DAHA. Stated skein theory enhances traditional Kauffman bracket skein theory by incorporating the boundary components of manifolds, thereby offering additional properties such as excision that enrich the algebraic structure. Furthermore, Kauffman bracket skein algebras embed into their stated counterparts, showing that stated skein algebras are extensions of Kauffman bracket skein algebras. We use this extended framework to further develop the representation theory of the $A_1$ DAHA. After identifying generators for the stated skein algebra of $T^2 \setminus D^2$, we embed this algebra into a quantum $6$-torus and leverage the nice representation-theoretic properties of quantum tori to construct a module of Laurent polynomials. Additionally, as $T^2$ is the boundary of any knot complement, we discuss how to construct a more topologically-defined module from various knots and provide an explicit example for the unknot. This approach builds upon the ideas of Berest and Samuelson, who showed that there exists a natural DAHA action on the Kauffman bracket skein module of knot complements.