Vol. 14, No. 2, 2021

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The mathematics of tie knots

Elizabeth Denne, Corinne Joireman and Allison Young

Vol. 14 (2021), No. 2, 241–270

In 2000, Thomas Fink and Yong Mao studied neck ties and, with certain assumptions, found 85 different ways to tie a neck tie. They gave a formal language which describes how a tie is made, giving a sequence of moves for each neck tie. The ends of a neck tie can be joined together, which gives a physical model of a mathematical knot that we call a tie knot. In this paper we classify the knot type of each of Fink and Mao’s 85 tie knots. We describe how the unknot, left and right trefoil, twist knots and (2,p) torus knots can be recognized from their sequence of moves. We also view tie knots as a family within the set of all knots. Among other results, we prove that any tie knot is prime and alternating.

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knots, neck ties, torus knot, twist knot, alternating knot, prime knot
Mathematical Subject Classification
Primary: 57K10
Received: 28 May 2020
Revised: 30 October 2020
Accepted: 25 November 2020
Published: 6 April 2021

Communicated by Colin Adams
Elizabeth Denne
Department of Mathematics
Washington & Lee University
Lexington, VA
United States
Corinne Joireman
Washington & Lee University
Lexington, VA
United States
Allison Young
University of Virginia
Charlottesville, VA
United States