Extending the Dehn quandle to shears and foliations on the torus

Reza Chamanara, Jun Hu, and Joel Zablow. "Extending the Dehn quandle to shears and foliations on the torus." Fundamenta Mathematicae 225.0 (2014): 1-22. <http://eudml.org/doc/282865>.

@article{RezaChamanara2014,
abstract = {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.},
author = {Reza Chamanara, Jun Hu, Joel Zablow},
journal = {Fundamenta Mathematicae},
keywords = {quandle; Dehn quandle},
language = {eng},
number = {0},
pages = {1-22},
title = {Extending the Dehn quandle to shears and foliations on the torus},
url = {http://eudml.org/doc/282865},
volume = {225},
year = {2014},
}

TY - JOUR
AU - Reza Chamanara
AU - Jun Hu
AU - Joel Zablow
TI - Extending the Dehn quandle to shears and foliations on the torus
JO - Fundamenta Mathematicae
PY - 2014
VL - 225
IS - 0
SP - 1
EP - 22
AB - 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.
LA - eng
KW - quandle; Dehn quandle
UR - http://eudml.org/doc/282865
ER -