K-color region select game

TL;DR

This paper generalizes the region select game played on knot diagrams with crossings colored with two colors to a broader setting where crossings can have any number of colors from 2 to infinity, expanding its theoretical framework.

Contribution

The paper introduces a generalized version of the region select game on knot diagrams with multiple colors at crossings, extending previous two-color models.

Findings

Generalization to k-color crossings for 2 ≤ k ≤ ∞

Potential new unknotting operations based on multi-color configurations

Framework for analyzing knot diagrams with multi-color crossings

Abstract

The region select game, introduced by Ayaka Shimizu, Akio Kawauchi and Kengo Kishimoto, is a game that is played on knot diagrams whose crossings are endowed with two colors. The game is based on the region crossing change moves that induce an unknotting operation on knot diagrams. We generalize the region select game to be played on a knot diagram endowed with $k$-colors at its vertices for $2 \leq k \leq \infty$.