Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links

Following Riley’s work, for each 2-bridge link $K(r)$ of slope $r\in \text{QQ}$ and an integer or a half-integer $n$ greater than 1, we introduce the {\it Heckoid orbifold $\text{orbs}(r;n)$} and the {\it Heckoid group $\text{Hecke}(r;n)={\pi }_1(\text{orbs}(r;n))$ of index $n$ for $K(r)$}. When $n$ is an integer, $\text{orbs}(r;n)$ 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 $\text{PConway}$ in $\text{orbs}(r;n)$ determined by the 2-bridge sphere of $K(r)$, when is it null-homotopic in $\text{orbs}(r;n)$? (2) For two distinct essential simple loops on $\text{PConway}$, when are they homotopic in $\text{orbs}(r;n)$? We also announce applications of these results to character varieties, McShane’s identity, and epimorphisms from 2-bridge link groups onto Heckoid groups.