The Knotting-Unknotting Game played on Sums of Rational Shadows

Details The Knotting-Unknotting Game played on Sums of Rational Shadows The Knotting-Unknotting Game played on Sums of Rational Shadows

2011-07-13

arxiv.org/abs/1107.2635v2

Mathematics

Combinatorics

31 pages, 24 figures; improved quality of figures, reduced file size, and fixed an incorrect definition

Scientific paper

We consider the recently introduced knotting-unknotting game, in which two players take turns resolving crossings in a knot diagram which initially is missing all its crossing information. Once the knot is fully resolved, the winner is decided by whether the knot is equivalent to the unknot. In this paper we determine which player wins under perfect play in the initial positions (knot shadows) that are guaranteed to produce rational knots or sums of rational knots. This is accomplished in part by an analysis of the effects of pseudo Reidemeister moves on positions, and a computer verification of some cases.

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