The Self-Linking Number in Planar Open Book Decompositions

TL;DR

This paper develops a method to compute the self-linking number of null-homologous transverse links in contact manifolds with planar open book decompositions, extending classical formulas to more general settings.

Contribution

It introduces a construction of Seifert surfaces compatible with planar open books and derives a generalized self-linking number formula.

Findings

Provides a new formula for the self-linking number in planar open books.

Extends Bennequin's classical self-linking number formula.

Enables computation of self-linking numbers in broader contact manifold contexts.

Abstract

We construct a Seifert surface for a given null-homologous transverse link in a contact manifold that is compatible with a planar open book decomposition, then obtain a formula of the self-linking number. It extends Bennequin's self-linking number formula for braids in the standard contact 3-sphere.