Length minima for an infinite family of filling closed curves on a one-holed torus

We explicitly find the minima as well as the minimum points of the geodesic length functions for the family of filling (hence non-simple) closed curves, \(a^2b^n\) (\(n\ge 3\)), on a complete one-holed hyperbolic torus in its relative Teichm\"uller space, where \(a, b\) are simple closed curves on the one-holed torus which intersect exactly once transversely. This provides concrete examples for the problem to minimize the geodesic length of a fixed filling closed curve on a complete hyperbolic surface of finite type in its relative Teichm\"uller space.