A note on Rivin’s interpretation of McShane’s identity as identities for closed geodesics with one self-intersection

Abstract

Rivin interpreted McShane’s identity as an identity for closed geodesics with one self-intersection on a one-cusped hyperbolic torus (cf. [I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math. 231 (2012) 2391–2412, Theorem 3.2]). In this note we point out that only those geodesics of non-hyperelliptic type are included in the interpreted identity, while those of hyperelliptic type are missing, and we give the desired identity for the closed geodesics with one self-intersection of hyperelliptic type. We also remark that the interpretation in [I. Rivin, Geodesics with one self-intersection, and other stories, Adv. Math. 231 (2012) 2391–2412, Theorem 3.7] of generalized McShane’s identity for a one-coned hyperbolic torus with cone angle [Formula: see text] is only partly valid because closed curves with one minimal self-intersection of non-hyperelliptic type are realizable as closed geodesics on the torus if and only if [Formula: see text].