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Enumeration of planar Tangles

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Abstract

A planar Tangle is a smooth simple closed curve piecewise defined by quadrants of circles with constant curvature. We can enumerate Tangles by counting their dual graphs, which consist of a certain family of polysticks. The number of Tangles with a given length or area grows exponentially, and we show the existence of their growth constants by comparing Tangles to two families of polyominoes.

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Acknowledgements

Many thanks go to the author’s son, Gabriel, whose love of toy trains inspired this paper. Thanks also to Ron Taylor, who introduced the author to existing literature on Tangles, to Julian Fleron for providing a copy of [9], and to the anonymous referee for several useful comments.

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  1. Department of Mathematical Sciences, Piedmont College, P.O. Box 10, 1021 Central Avenue, Demorest, GA, 30535, USA

    Douglas A Torrance

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  1. Douglas A Torrance
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Correspondence to Douglas A Torrance.

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Communicating Editor: Sukanta Pati

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Torrance, D.A. Enumeration of planar Tangles. Proc Math Sci 130, 50 (2020). https://doi.org/10.1007/s12044-020-00575-7

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  • DOI: https://doi.org/10.1007/s12044-020-00575-7

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