Simplicial Complexes are Game Complexes

TL;DR

This paper explores the relationship between simplicial complexes and strong placement games, demonstrating how invariants like game value are reflected in the combinatorial structures and establishing connections for specific subclasses.

Contribution

It introduces the concept of invariant SP-games, shows the correspondence between simplicial complexes and game trees, and analyzes flag complexes in the context of combinatorial games.

Findings

Every simplicial complex encodes an invariant SP-game.

Isomorphic simplicial complexes correspond to isomorphic game trees and equal game values.

For flag complexes, there exists a game with that complex regardless of the board.

Abstract

Strong placement games (SP-games) are a class of combinatorial games whose structure allows one to describe the game via simplicial complexes. A natural question is whether well-known invariants of combinatorial games, such as "game value", appear as invariants of the simplicial complexes. This paper is the first step in that direction. We show that every simplicial complex encodes a certain type of SP-game (called an "invariant SP-game") whose ruleset is independent of the board it is played on. We also show that in the class of SP-games isomorphic simplicial complexes correspond to isomorphic game trees, and hence equal game values. We also study a subclass of SP-games corresponding to flag complexes, showing that there is always a game whose corresponding complex is a flag complex no matter which board it is played on.