Enumeration of Essential Curves in a Restricted Class of Curve Systems on a Torus With One Puncture and One Boundary Component

We present a closed-form formula for computing the number of essential simple closed curves in a restricted class of curve systems on a torus with exactly one puncture and one boundary component, under a collection of structural assumptions described in the paper. Existing approaches in the literature generally rely on combinatorial enumeration, curve parametrizations, or Dehn–Thurston coordinates, all of which require iterative or algorithmic procedures. The proposed formula bypasses these intermediate steps and yields the count directly. It is expressed as an explicit piecewise closed-form expression. This work addresses a gap in the current body of research, as explicit formulas for this particular surface type are absent. The result is expected to be useful in the study of mapping class groups, hyperbolic geometry, and computational topology.