# Extending the Dehn quandle to shears and foliations on the torus

Reza Chamanara; Jun Hu; Joel Zablow

Fundamenta Mathematicae (2014)

- Volume: 225, Issue: 0, page 1-22
- ISSN: 0016-2736

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topReza 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 -

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