# The topological complexity of polytopes

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.