The Self-Linking Number in Annulus and Pants Open Book Decompositions

TL;DR

This paper derives a formula for the self-linking number of null-homologous transverse links in contact manifolds with annulus or pants open book decompositions, extending classical results to new topological settings.

Contribution

It introduces a generalized self-linking number formula applicable to contact manifolds with annulus or pants open book decompositions, expanding the scope of classical braid theory.

Findings

Derived a self-linking number formula for annulus and pants open book decompositions

Extended Bennequin's classical self-linking formula to new topological contexts

Applicable to null-homologous transverse links in contact manifolds

Abstract

We find a self-linking number formula for a given null-homologous transverse link in a contact manifold that is compatible with either an annulus or a pair of pants open book decomposition. It extends Bennequin's self-linking formula for a braid in the standard contact $3$-sphere.