Numerical computations in cobordism categories

TL;DR

This paper explores the connections between combinatorial sequences, language density, and cobordism categories in topology, providing methods to translate between these perspectives in the context of 1+1 dimensional cobordism.

Contribution

It introduces a framework linking combinatorial, algebraic, and topological interpretations of cobordism categories in low dimensions.

Findings

Derived the sequence's interpretation in language density and algebraic quotient contexts.

Established a method to transition between combinatorial and topological viewpoints.

Connected the sequence to the fundamental group of a cobordism category.

Abstract

The sequence 2,5,15,51,187,... with the form $(2^n+1)(2^{n-1}+1)/3$ has two interpretations in terms of the density of a language with four letters and the cardinality of the quotient of $\ZZ_2^n\times \ZZ_2^n$ under the action of the special linear group $\op{SL}(2,\ZZ)$. The last interpretation follows the rank of the fundamental group of the $\ZZ_2^n$-cobordism category in dimension 1+1. This article presents how to pass from one side to another between these two approaches.