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Links and cycles II: families of commutation relations, parametrized by knots, in the mapping class groups and beyond

100%

Zablow J.

Banach Center Publications

2014

tom 103

nr 1

371-387

We generate families of commutation relations in various groups, by examining quandle colorings of knots and their quandle 2-cycles. The colorings are via quandles associated to the given groups.

Extending the Dehn quandle to shears and foliations on the torus

51%

Chamanara R., Hu J., Zablow J.

Fundamenta Mathematicae

2014

tom 225

nr 1

1-22

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.

Ograniczanie wyników

1 Banach Center Publications

1 Fundamenta Mathematicae

2 Zablow J.

1 Chamanara R.

1 Hu J.

2 2014

Wyniki wyszukiwania

Links and cycles II: families of commutation relations, parametrized by knots, in the mapping class groups and beyond

Extending the Dehn quandle to shears and foliations on the torus