Cabling, contact structures and mapping class monoids

TL;DR

This paper explores how cabled open book decompositions affect contact structures, introduces rational open books for studying surgeries, and identifies monoids in the mapping class group with contact geometric importance.

Contribution

It introduces rational open book decompositions, demonstrates contact structures supported by non-positive Dehn twist monodromies, and identifies significant monoids in the mapping class group.

Findings

Stein fillable contact structures can have monodromies not expressible as positive Dehn twists.

Rational open books generalize standard open books for surgery studies.

Several monoids in the mapping class group are shown to have contact geometric significance.

Abstract

In this paper we discuss the change in contact structures as their supporting open book decompositions have their binding components cabled. To facilitate this and applications we define the notion of a rational open book decomposition that generalizes the standard notion of open book decomposition and allows one to more easily study surgeries on transverse knots. As a corollary to our investigation we are able to show there are Stein fillable contact structures supported by open books whose monodromies cannot be written as a product of positive Dehn twists. We also exhibit several monoids in the mapping class group of a surface that have contact geometric significance.