The Knotting-Unknotting Game played on Sums of Rational Shadows

TL;DR

This paper analyzes the knotting-unknotting game played on rational knot shadows, determining winning strategies for perfect play and exploring how moves affect game positions, with computational verification.

Contribution

It provides a classification of winning strategies for the knotting-unknotting game on rational and sum of rational knots, including effects of pseudo Reidemeister moves.

Findings

Identifies which player wins under perfect play for initial rational knot shadows.

Analyzes the impact of pseudo Reidemeister moves on game positions.

Uses computer verification to support theoretical results.

Abstract

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.