The Dehn quandle, Q, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, 𝕋², we generalize this to show the existence of a quandle Q̂ extending Q and whose elements are measured geodesic foliations. The quandle action in Q̂ is given by applying a shear along such a foliation to another foliation. We extend some results which related Dehn quandle homology to the monodromy of Lefschetz fibrations. We apply certain quandle 2-cycles to yield factorizations of elements of SL₂(ℝ) fixing specified vectors (circles, foliations) and give examples. Using these, we show the quandle homology of Q̂ is nontrivial in all dimensions.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm225-1-1,
author = {Reza Chamanara and Jun Hu and Joel Zablow},
title = {Extending the Dehn quandle to shears and foliations on the torus},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {1-22},
zbl = {1302.57015},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm225-1-1}
}
Reza Chamanara; Jun Hu; Joel Zablow. Extending the Dehn quandle to shears and foliations on the torus. Fundamenta Mathematicae, Tome 227 (2014) pp. 1-22. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm225-1-1/