Generalized Lantern Relations and Planar Line Arrangements

TL;DR

This paper establishes a connection between planar line arrangements and Dehn twist relations, providing new proofs for generalized lantern and daisy relations through combinatorial methods.

Contribution

It introduces a novel approach linking planar line arrangements to Dehn twist relations, offering alternative proofs for known generalized lantern and daisy relations.

Findings

Derived relations from line arrangements correspond to Dehn twist relations

Provided an alternative proof for Wajnryb's generalized lantern relations

Extended the understanding of relations in planar line arrangements

Abstract

In this paper we show that to each planar line arrangement defined over the real numbers, for which no two lines are parallel, one can write down a corresponding relation on Dehn twists that can be read off from the combinatorics and relative locations of intersections. This leads to an alternate proof of Wajnryb's generalized lantern relations, and of Endo, Mark and Horn-Morris' daisy relations.