# Invariants in Quantum Geometry

## TL;DR

This paper explores invariants in quantum geometry involving loops, surfaces, and regions in four-dimensional space, establishing a causality-related equivalence relation linked to linking numbers.

## Contribution

It introduces a new ambient isotopic equivalence relation for triples in quantum geometry that encodes causality and linking properties.

## Key findings

- Defines an ambient isotopic equivalence relation for triples in $\,\mathbb{R}^4$
- Connects invariants to linking numbers and causality
- Provides a framework for classifying quantum geometric configurations

## Abstract

In quantum geometry, we consider a set of loops, a compact orientable surface and a solid compact spatial region, all inside $\mathbb{R} \times \mathbb{R}^3 \equiv \mathbb{R}^4$, which forms a triple. We want to define an ambient isotopic equivalence relation on such triples, so that we can obtain equivalence invariants. These invariants describe how these submanifolds are causally related to or `linked' with each other, and they are closely associated with the linking number between links in $\mathbb{R}^3$. Because we distinguish the time-axis from spatial subspace in $\mathbb{R}^4$, we see that these equivalence relations, will also imply causality.