Generalized State-Sum Invariants of Knots

Abstract

We present various generalizations of the finite type theory of knots by establishing a relationship between generalized finite type theories of knot-like structures and diagrammatic combinatorial systems for representing those structures. By inventing suitable new combinatorial diagram systems for representing knots and knot-like structures, we can compute generalized finite type theories. We will create a category theoretical formalization of this correspondence, and demonstrate it through three examples: the standard finite type theory and its relationship with clasp diagrams, the finite type theory of delta moves and a new diagram system called looms, and the finite type theory of combinatorial structures we call virtual transverse knots. The finite type theory of delta moves may have applications to unknotting number, and the theory of virtual transverse knots leads to many interesting and difficult conjectures. Finally, we will go beyond finite type theories to study how our formalization can be applied more generally.