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NREUP 2009

St. Mary's College of Maryland

Title: Untangle: Knots and Games

Directors:

  • Sandy Ganzell
  • Alex Meadows

Email:

Dates of Program: May 25 - July 3, 2009

Summary: The students will be introduced to knot theory, a currently vibrant research area, as well as the theory of combinatorial games, which has seen an increase in activity in recent years, especially with new links to computer science. In their recent paper [1], Ganzell and Meadows introduced a new combinatorial game called Untangle, in which players take turns performing Reidemeister moves on a projection of the unknot. The game is related to some well-known open questions in knot theory, most notably concerning the number of moves required to untangle an unknot. The analysis of the game is also interesting in its own right. The students will generate data for currently unanswered questions about this game, generate conjectures, and develop mathematical induction as a method for proving their statements. The following two problems are of particular interest: 1. Classify the twisted loop positions from which the second player has a winning strategy. 2. Develop a reduction or decomposition of a class of games into smaller components. Problem 1 is especially appealing as an introductory project since there are many cases that can be attacked individually with computer experiments and arguments by induction. Problem 2 is related to interesting properties of Untangle that distinguish it from most combinatorial games.

Student Researchers Supported by MAA:

  • Lydia Garcia
  • Gustavo Lara
  • Kristi Mitchell
  • Kylie Robillard

More Information:

www.smcm.edu/mathcs/extras.html

Program Contacts:

Bill Hawkins
MAA SUMMA
bhawkins@maa.org
202-319-8473

Michael Pearson
MAA Programs & Services
pearson@maa.org
202-319-8470

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