  Following Riley's work, for each 2-bridge link $K(r)$ of slope $r\in\QQ$ and
an integer or a half-integer $n$ greater than 1, we introduce the {\it Heckoid
orbifold $\orbs(r;n)$} and the {\it Heckoid group
$\Hecke(r;n)=\pi_1(\orbs(r;n))$ of index $n$ for $K(r)$}. When $n$ is an
integer, $\orbs(r;n)$ is called an {\it even} Heckoid orbifold; in this case,
the underlying space is the exterior of $K(r)$, and the singular set is the
lower tunnel of $K(r)$ with index $n$. The main purpose of this note is to
announce answers to the following questions for even Heckoid orbifolds. (1) For
an essential simple loop on a 4-punctured sphere $\PConway$ in $\orbs(r;n)$
determined by the 2-bridge sphere of $K(r)$, when is it null-homotopic in
$\orbs(r;n)$? (2) For two distinct essential simple loops on $\PConway$, when
are they homotopic in $\orbs(r;n)$? We also announce applications of these
results to character varieties, McShane's identity, and epimorphisms from
2-bridge link groups onto Heckoid groups.
