Counting Unlabelled Chord Diagrams of Maximal Genus

TL;DR

This paper develops a new enumeration formula for unlabelled maximal chord diagrams by establishing a bijection with certain symmetric diagrams, extending previous work on rotation-based enumeration.

Contribution

It introduces a novel bijection between rooted one-vertex one-face maps on surfaces and symmetric chord diagrams, enabling enumeration of unlabelled maximal chord diagrams.

Findings

Derived an explicit enumeration formula for unlabelled maximal chord diagrams.

Extended Cori and Marcus's results from rotation-based to isomorphism-based enumeration.

Established a bijection linking maps on surfaces to symmetric chord diagrams.

Abstract

Maximal chord diagrams up to all isomorphisms are enumerated. The enumerating formula is based on a bijection between rooted one-vertex one-face maps on locally orientable surfaces and a certain class of symmetric chord diagrams. This result extends the one of Cori and Marcus regarding maximal chord diagrams enumerated up to rotations.