The girth of a Heegaard splitting

Details The girth of a Heegaard splitting The girth of a Heegaard splitting

2005-06-28

arxiv.org/abs/math/0506558v1

Mathematics

Geometric Topology

Ph.D. dissertation, 73 pages, 22 figures (bitmapped EPS, as the arXiv cannot yet accept the vector-based PDF source)

Scientific paper

We 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.

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