Non-convex 4d polytopes in Spin Foam Models

In this article we consider non-convex $4d$ polytopes in $\mathbb{R}^4$. The paper consist of two parts: Firstly, we extend the proof of the formula for the $4d$ volume in terms of $2d$ face bivectors and boundary graph crossings from convex to non-convex polytopes. Secondly, we consider the EPRL-FK spin foam model, and demonstrate that there exists boundary data which leads to non-convex $4d$ polytopes in the asymptotic analysis of the vertex amplitude.