Extending the Dehn quandle to shears and foliations on the torus

• Autorzy: Reza Chamanara , Jun Hu , Joel Zablow
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 17Dxx: Other nonassociative rings and algebras
• 57Mxx: Low-dimensional topology

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2014
• Tom: 225
• Numer: 1
• Strony: 1-22

Daty

wydano

2014

Twórcy

autor

Reza Chamanara

• Department of Mathematics, Brooklyn College, Brooklyn, NY 11210, U.S.A.

autor

Jun Hu

• Department of Mathematics, Brooklyn College, Brooklyn, NY 11210, U.S.A.
• Ph.D. Program in Mathematics, Graduate Center, CUNY, 365 5th Ave., New York, NY 10016, U.S.A.

autor

Joel Zablow

• Department of Mathematics, Long Island University, Brooklyn Campus, 1 University Plaza, Brooklyn, NY 11201, U.S.A.

Identyfikatory

DOI

10.4064/fm225-1-1