Curves intersecting in a circuit pattern

TL;DR

This paper demonstrates that the cycle relation among Dehn twists in a circuit pattern can identify whether the circuit bounds an embedded disc, by analyzing the generated group's structure across different surfaces.

Contribution

It introduces a method to detect embedded discs in surface circuits using the isomorphism type of Dehn twist groups, advancing understanding of surface topology.

Findings

Cycle relation detects embedded discs in circuits

Group generated by Dehn twists characterizes circuit properties

Method applies to various surface types

Abstract

We show that the cycle relation between Dehn twists about curves in a circuit detects whether the circuit bounds an embedded disc. This is done by determining the isomorphism type of the group generated by said Dehn twists for various surfaces.