2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/75598We construct simple curves from immersed curves in the setting of handlebodies and Heegaard splittings. We define a measure of complexity we call girth for closed curves in a handlebody. We extend this complexity to Heegaard splittings and pose a conjecture about all Heegaard splittings. We prove a test case of this conjecture. Let S be a compact surface embedded in the boundary of a handlebody H. Then the minimum girth over all curves in S can be achieved by a simple closed curve. We also present algorithms to compute the girth of curves and surfaces.Ph.D. dissertation, 73 pages, 22 figures (bitmapped EPS, as the arXiv cannot yet accept the vector-based PDF source)Geometric Topology57N10 (Primary) 57M50 (Secondary)text