Quandles and Lefschetz Fibrations

TL;DR

This paper introduces a quandle structure on simple closed curves in surfaces using Dehn twists and demonstrates how Lefschetz fibration monodromy can be encoded as a quandle homomorphism, opening avenues for new invariants.

Contribution

It establishes a novel quandle framework for surface curves and encodes Lefschetz fibration monodromy as quandle homomorphisms, linking knot theory and surface topology.

Findings

Defined the Dehn quandle for oriented surfaces.

Encoded Lefschetz fibration monodromy as a quandle homomorphism.

Discussed potential for new topological invariants.

Abstract

We show that isotopy classes of simple closed curves in any oriented surface admit a quandle structure with operations induced by Dehn twists, the Dehn quandle of the surface. We further show that the monodromy of a Lefschetz fibration can be conveniently encoded as a quandle homomorphism from the knot quandle of the base as a manifold with a codimension 2 subspace (the set of singular values) to the Dehn quandle of the generic fibre, and discuss prospects for construction of invariants arising naturally from this description of the monodromy.