Chillingworth found an algorithm for determining whether a given element of the fundamental group of a surface contains simple closed curves. We extend the theory to ‘open’ curves on a punctured surface.
"An algorithm for determining the simplicity of (i, j)-curves on a surface." Hiroshima Math. J. 36 (1) 61 - 76, March 2006. https://doi.org/10.32917/hmj/1147883397