Polytopes arise naturally when modelling fundamental structures in Biology such as RNA and protein folding [1,2]. Recently, it occurred to me that a complexity measure on the topology of polytopes might be useful. Might there be mathematicians that have tried to develop 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 POZNANOVIĆ. GEOMETRIC COMBINATORICS AND COMPUTATIONAL. 2016. 
2. Paul M. Dodda, Pablo F. Damascenob and Sharon C. Glotzera. Universal folding pathways of polyhedron nets. 
MOLECULAR BIOLOGY: BRANCHING POLYTOPES FOR RNA
SEQUENCES. PNAS. 2018.