Coefficients of Catalan States of Lattice Crossing

Skein modules are algebraic invariants of oriented 3-manifolds motivated by knot theory. The Kauffman bracket skein module is the most extensively studied skein module due to its relation with the Jones polynomial. Results of this dissertation can naturally be regarded as contributions to further development of the theory of skein modules. The lattice crossing was first studied by Dabkowski, Li, and Przytycki in 2015 as a part of an effort to find closed-form formulas for the natural product in the Kauffman bracket skein algebra of a four-punctured sphere. In this dissertation, we focus on finding coefficients of Catalan states obtained from lattice crossing and derive some relations between the coefficients of two different Catalan states. In particular, we develop methods for computing coefficients of Catalan states and find closed-form formulas for the coefficients of Catalan states obtained from lattice crossing with 4 vertical strands. We also examine the unimodality property of the coefficients of Catalan states. The generalized crossing is an n-tangle obtained as a half-twist of n vertical strands related to the lattice crossing. In the last part of this dissertation, we present some results concerning coefficients of Catalan states obtained from generalized crossing.