Abstract
One of the best ways to get acquainted with a mathematical object is to play with it. Knots are no exception. In the process of trying to learn more about the properties of knots, a team of undergraduate researchers played around with knot diagrams, changing crossings to see how the knottedness of their knots was affected. It felt like they were playing a game. And so they were! It turns out that the team invented a game that is now called the “Knotting-Unknotting Game,” one of the first topological-combinatorial games of its kind to be studied. Since this initial game was introduced, many more researchers have devised games that can be played with knot and link diagrams. Studying these games to determine their mathematical properties can be just as fun as simply playing them, and in the process, we can learn more about the knotty objects we’re playing with.
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Notes
- 1.
While it was once a conjecture, this Tait Conjecture is now a theorem! In fact it is one of many theorems proven from conjectures made by Peter Guthrie Tait. We still call Tait’s conjectures “conjectures” for historical reasons.
- 2.
Actually, you can prove they’re distinct using the Tait conjecture we learned about in Sect. 1.1, but this conjecture is proven using the Kauffman bracket, a relative of the famous knot and link invariant called the Jones polynomial.
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Acknowledgements
The author would also like to thank the Simons Foundation (#426566) for their support of this work. She would also like to thank Steve Klee and the anonymous reviewer for their valuable suggestions as well as the editors of this volume.
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Henrich, A. (2022). Playing with Knots. In: Goldwyn, E.E., Ganzell, S., Wootton, A. (eds) Mathematics Research for the Beginning Student, Volume 1. Foundations for Undergraduate Research in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-08560-4_9
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