# A Generalization of the Gram determinant of type A

## TL;DR

This paper generalizes the Gram determinant of type A by deriving a closed formula through bilinear forms of non-intersecting connections, and discusses its extension to the Klein bottle, advancing topological invariants.

## Contribution

It introduces a generalized framework for the Gram determinant of type A and provides a closed-form expression, extending the theory to new topological contexts.

## Key findings

- Derived a closed formula for the generalized Gram determinant of type A.
- Evaluated bilinear forms of non-intersecting connections in the annulus.
- Discussed implications for the Gram determinant of type Mb on the Klein bottle.

## Abstract

The Gram determinant of type $A$ was introduced by Lickorish in his work on invariants of 3 - manifolds. We generalize the theory of the Gram determinant of type $A$ by evaluating, in the annulus, a bilinear form of non-intersecting connections in the disc. The main result provides a closed formula for this Gram determinant. We conclude the paper by discussing Chen's conjecture about the Gram determinant of type $Mb$ evaluated in the Klein bottle.