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

TL;DR

This paper characterizes when simple loops on 2-bridge spheres in Heckoid orbifolds for the trivial knot are null-homotopic, peripheral, or torsion, and when two such loops are homotopic, providing complete criteria.

Contribution

It provides necessary and sufficient conditions for the homotopy and classification of simple loops in these specific orbifolds, advancing understanding of their topological structure.

Findings

Criteria for null-homotopic, peripheral, and torsion loops

Conditions for homotopy equivalence of loops

Complete classification of simple loops in the orbifold

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.