The Link Smoothing Game

TL;DR

The paper introduces the Link Smoothing Game, a topological combinatorial game played on link diagram shadows, analyzing outcomes through graph-theoretic methods related to link shadows and checkerboard colorings.

Contribution

It presents a new combinatorial game on link shadows and advances the classification of link shadows into outcome classes using graph-theoretic techniques.

Findings

Progress towards classifying link shadows by outcome class

Establishes a relationship between link shadows and associated planar graphs

Provides a framework for analyzing link shadow connectivity through game moves

Abstract

We introduce a topological combinatorial game called the Link Smoothing Game. The game is played on the shadow of a link diagram and legal moves consist of smoothing precrossings. One player's goal is to keep the diagram connected while the other player's goal is to disconnect the shadow. We make significant progress towards a complete classification of link shadows into outcome classes by capitalizing on the relationship between link shadows and the planar graphs associated to their checkerboard colorings.