Volume of 4-polytopes from bivectors

TL;DR

This paper derives a formula for calculating the volume of 4-polytopes using face bivectors and boundary graph crossings, establishing volume as an invariant of bivector-coloured graphs in three-dimensional space.

Contribution

It introduces a novel volume formula for 4-polytopes based on bivectors and boundary graph crossings, linking geometric invariants to graph theory.

Findings

Volume formula expressed via face bivectors and crossings

Volume invariant under bivector-coloured graph transformations

Connection between 4-polytope volume and graph invariants

Abstract

In this article we prove a formula for the volume of 4-dimensional polytopes, in terms of their face bivectors, and the crossings within their boundary graph. This proves that the volume is an invariant of bivector-coloured graphs in $S^3$.