Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and proteins [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. To be precise, I am referring to polyhedra and their higher-dimensional analogues. Might mathematicians have developed such a concept? 

One idea that occurred to me was to use the number of Hamiltonian cycles. If the polytope is embedded in $\mathbb{R}^3$ this allows us to ask questions such as how many polytopes with distinct topology with $n$ vertices have less than $\lambda n!$ Hamiltonian cycles where $\lambda \in [0,1]$. 

Another way of approaching this might be to think about compression algorithms and Kolmogorov Complexity. However, I haven't found much work applying notions of algorithmic complexity to problems in topology. 

## References: 
1. Elizabeth Drellich, Andrew Gainer-Dewar, Heather A. Harrington,
Qijun He, Christine Heitsch, AND Svetlana Poznanovic. Geometric Combinatorics AND Computational Molecular biology: branching polytopes for RNA sequences. 2016. 
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. PNAS. 2018.