We compare the run times of two algorithms for calculating Jones polynomials, as currently contained in sage. One goes through computing Markov traces of the Jones representation of braids whose trace closures are the relevant links. This is contained in braid.py
. The other uses the Kauffman bracket of a PD code of the link. This is contained in link.py
. We will refer to these two algorithms as the "Jones representation" and the "State sum" algorithm respectively.
At the end of the day, we want a decision algorithm which takes a link, given either by its PD code or as the closure of a braid, and tells us which of the two algorithms one should use. Below we illustrate that in the case of braid closures, the output of this decision should depend on the number of crossings in the braid, the number of strands, as well as the dynamical complexity of the braid.