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

作者：Lei, Da; Zhang, Ying^*

来源：Journal of Knot Theory and its Ramifications, 2023, 2350049.

DOI：10.1142/S0218216523500499

• Punctured torus
• simple closed geodesic
• self-intersection
• McShane's identity

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 ?? ? (0, 2p) 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 ?? ? (0, 2/3p).

• 单位
  苏州大学; y