Gram determinant of planar curves

TL;DR

This paper studies the Gram determinant associated with curves on a planar surface, specifically a disk with two holes, revealing divisibility properties and providing explicit calculations and conjectures on its factorization.

Contribution

It introduces new divisibility results for Gram determinants in complex surfaces and computes explicit examples, advancing understanding of their algebraic structure.

Findings

Divisibility of Gram determinants when increasing the number of curves

Explicit calculations of Gram determinants for specific cases

Conjectures on the complete factorization of these determinants

Abstract

We investigate the Gram determinant of the bilinear form based on curves in a planar surface, with a focus on the disk with two holes. We prove that the determinant based on $n-1$ curves divides the determinant based on $n$ curves. Motivated by the work on Gram determinants based on curves in a disk and curves in an annulus (Temperley-Lieb algebra of type $A$ and $B$, respectively), we calculate several examples of the Gram determinant based on curves in a disk with two holes and advance conjectures on the complete factorization of Gram determinants.