Diagram Systems and Generalized Finite Type Theories

TL;DR

This paper generalizes finite type invariants using category theory, linking diagram systems with topological theories, and explores examples like clasp diagrams, delta moves, and virtual transverse knots.

Contribution

It introduces a categorical framework connecting finite type theories with diagram systems, including new systems like looms and applications to virtual knots.

Findings

Established correspondence between finite type theories and diagram systems

Introduced the 'looms' diagram system for generalized theories

Identified potential applications to unknotting and conjectures in virtual knots

Abstract

We present a category theoretical generalization of the Goussarov theorem for finite type invariants, relating generating sets for generalized finite type theories with diagrams systems for the corresponding topological objects. We will demonstrate this correspondence through a few examples including 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.