Abstract
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.
Citation
Joel Zablow. "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
Information