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The girth of a Heegaard splitting

Christopher Jerdonek
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Abstract

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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Christopher Jerdonek. 2005-06-28. The girth of a Heegaard splitting. https://arxiv.org/abs/math/0506558

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