Do Link Polynomials Detect Causality In Globally Hyperbolic Spacetimes?

TL;DR

This paper investigates whether link polynomials, specifically the Jones and Conway polynomials, can detect causality in certain 2+1-dimensional spacetimes, providing empirical evidence and theoretical insights into their capabilities.

Contribution

It introduces a new tangle invariant related to the Conway polynomial and analyzes its effectiveness in detecting causality in globally hyperbolic spacetimes.

Findings

Jones polynomial shows potential in detecting causality

Conway polynomial does not detect the connected sum of two Hopf links

New tangle invariant related to Conway polynomial introduced

Abstract

Let $X$ be a $(2+1)$-dimensional globally hyperbolic spacetime with a Cauchy surface $\Sigma$ whose universal cover is homeomorphic to $\mathbb{R}^2$. We provide empirical evidence suggesting that the Jones polynomial detects causality in $X$. We introduce a new invariant of certain tangles related to the Conway polynomial, and prove that the Conway polynomial does not detect the connected sum of two Hopf links among relevant 3-component links, which suggests that the Conway polynomial does not detect causality in the scenario described.