Dehn quandles of surfaces and their bounded cohomology

TL;DR

This paper introduces new quandle invariants for classifying surfaces, explores their algebraic and metric properties, and computes their bounded cohomology, revealing deep structural insights.

Contribution

It generalizes the classical Dehn quandle to new families, analyzes their properties, and computes their bounded cohomology, advancing surface classification methods.

Findings

Quandles are unbounded with respect to the metric.

Second bounded quandle cohomology is infinite-dimensional.

The natural map induces an injection on bounded cohomology.

Abstract

We introduce new families of quandles that serve as invariants for classifying closed orientable surfaces. These families generalize the classical Dehn quandle and are defined, respectively, on isotopy classes of unoriented closed curves and on integral weighted multicurves. We establish their fundamental algebraic properties and construct a natural quandle covering that relates them. We then analyze their metric properties, showing that these quandles are unbounded with respect to the quandle metric. Next, we compute their second bounded quandle cohomology, proving it to be infinite-dimensional. We also establish a version of the Gromov Mapping Theorem, showing that the natural map from an abelian quandle extension onto the original quandle induces an injection on bounded quandle cohomology in every dimension. Finally, inspired by recent developments in quandle rings, we analyze idempotents in the integral quandle rings arising from the classical Dehn quandle of a surface.