Simple loops on 2-bridge spheres in Heckoid orbifolds for the trivial knot

Abstract

In this paper, we give a necessary and sufficient condition for an essential simple loop on a $2$-bridge sphere in an even Heckoid orbifold for the trivial knot to be null-homotopic, peripheral or torsion in the orbifold. We also give a necessary and sufficient condition for two essential simple loops on a $2$-bridge sphere in an even Heckoid orbifold for the trivial knot to be homotopic in the orbifold.

14 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1402.6870, arXiv:1010.2232, arXiv:1206.0429