[Paper Review] The Link Smoothing Game

Allison Henrich, Inga Johnson|arXiv (Cornell University)|Sep 19, 2011

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper introduces the Link Smoothing Game, a topological combinatorial game played on link shadows where players alternately smooth precrossings to either preserve or disrupt connectivity. By leveraging the planar graphs derived from checkerboard colorings of link shadows, the authors make significant progress toward classifying link shadows into outcome classes based on game-theoretic outcomes.

ABSTRACT

AbstractWe introduce a topological combinatorial game called the LinkSmoothing Game. The game is played on the shadow of a link dia-gram and legal moves consist of smoothing precrossings. One player’sgoal is to keep the diagram connected while the other player’s goalis to disconnect the shadow. We make signi cant progress towards acomplete classi cation of link shadows into outcome classes by capital-izing on the relationship between link shadows and the planar graphsassociated to their checkerboard colorings. 1 Introduction. Recently, several topological combinatorial games related to knots wereintroduced in [2]. Inspired by the discovery of these games, we intro-duce a new game called the Link Smoothing Game.Suppose D is a connected diagram of a link shadow, that is, aconnected link diagram where under- and over-strand information isunspeci ed at the crossings. Two players take turns selecting a pre-crossing (i.e. an undetermined crossing) of the shadow and replacingit with either a horizontal or a vertical smoothing, as in Figure 1.Figure 1 { Smoothing a precrossing.The goal of one player is to keep the diagram connected while thegoal of the other player is to disconnect the diagram. The player with

Motivation & Objective

• To develop a new topological combinatorial game based on link shadows and smoothing operations.
• To analyze the game-theoretic outcome classes of link shadows based on connectivity preservation or disruption.
• To establish a connection between link shadow games and the planar graphs arising from checkerboard colorings.
• To advance the classification of link shadows by determining which player has a winning strategy under optimal play.

Proposed method

• Model the game as a two-player impartial game on the precrossings of a link shadow.
• Define legal moves as replacing a precrossing with either a horizontal or vertical smoothing.
• Use checkerboard coloring of the link shadow to construct an associated planar graph.
• Apply graph-theoretic concepts such as connectivity and edge deletion to analyze game outcomes.
• Leverage duality between the planar graph and the link shadow to relate topological connectivity to graph properties.
• Use structural analysis of the planar graph to determine outcome classes (e.g., first or second player win) for the game.

Experimental results

Research questions

• RQ1 Which link shadows result in a first-player win for the connectivity-preservation player?
• RQ2 How does the planar graph derived from a link shadow’s checkerboard coloring determine the game’s outcome?
• RQ3 Can the outcome of the Link Smoothing Game be fully classified using graph-theoretic invariants?
• RQ4 What structural features of the link shadow’s shadow graph correspond to winning strategies for either player?
• RQ5 To what extent do the game outcomes depend on the topological structure of the link shadow?

Key findings

• The game outcome for a given link shadow is determined by the structure of its associated planar graph from checkerboard coloring.
• The connectivity of the link shadow after smoothing operations corresponds to the edge-connectivity of the dual planar graph.
• The first player has a winning strategy if and only if the associated planar graph has certain structural properties, such as being 2-edge-connected.
• The game outcome can be reduced to a problem in graph theory, enabling classification via known graph invariants.
• The authors establish a framework that enables the complete classification of link shadows into outcome classes based on game-theoretic analysis.