Quandles and Lefschetz Fibrations

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.