Isotopy classes of circles on an orientable surface of genus form a quandle under the operation of Dehn twisting about such circles. We derive certain fundamental relations in the Dehn quandle and then consider a homology theory based on this quandle. We show how certain types of relations in the quandle translate into cycles and homology representatives in this homology theory, and characterize a large family of 2–cycles representing homology elements. Finally we draw connections to Lefschetz fibrations, showing isomorphism classes of such fibrations over a disk correspond to quandle homology classes in dimension 2, and discuss some further structures on the homology.
"On relations and homology of the Dehn quandle." Algebr. Geom. Topol. 8 (1) 19 - 51, 2008. https://doi.org/10.2140/agt.2008.8.19