# The relationship between generalized finite type theories and diagram systems

# The relationship between generalized finite type theories and diagram systems

**In-Person Talk**

We will discuss a relationship between finite type theories and certain kinds of combinatorial representation systems for knots. Using this relationship, we show that the finite type theory of delta moves is finitely generated, where a delta move is the operation of passing a strand through a clasp. We achieve this by inventing a corresponding diagram system for knots called 'looms'. We conjecture that the finite type theory of delta moves might give strong lower bounds on unknotting number, but there are computational difficulties due to rapidly exploding combinatorial complexity that must be resolved before that can be numerically verified. Finally, we will discuss how this approach towards computing generalized finite type theories can be applied more broadly.