In this paper and its sequel, we give a necessary and sufficient conditionfor two essential simple loops on a 2-bridge sphere in an even Heckoid orbifoldfor a 2-bridge link to be homotopic in the orbifold. We also give a necessaryand sufficient condition for an essential simple loop on a 2-bridge sphere inan even Heckoid orbifold for a 2-bridge link to be peripheral or torsion in theorbifold. This paper treats the case when the 2-bridge link is a $(2,p)$-toruslink, and its sequel will treat the remaining cases.
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