Topological studies have shown that DNA extracted from P4 bacteriophages (P4 DNA) is knotted and the observed knots are very complex (i.e. high average crossing number). Interestingly, the distribution of knots of low average crossing number show an absence of the four crossing knot and a prevalence of the toroidal five crossing knot ($5_1$) over the twist crossing knot ($5_2$).

In this work we use cryo-electron microscopy to determine the layer organization of DNA in bacteriophage P4, and the continuum theory of liquid crystals to model its liquid crystal structure. The model shows that the experimentally observed structure is a minimizer of the proposed energy and that the experimental knot distribution can be reproduced by perturbing the minimizer. We therefore propose a new liquid crystal model based on cryoEM observations that is consistent with topological studies of P4 DNA.

One immediate application of this method is to improve bounds on stick-numbers - both for equilateral polygons and non-equilateral polygons. In particular, we find upper bounds for both the stick and equilateral-stick numbers for all knots of 13 or fewer crossings. In some cases these upper bounds on stick-numbers actually give exact stick-number.

In many cases there remains a gap between the equilateral-stick and stick number bounds. By adapting the move set of BFACF we can try to "equilateralise" polygons. This is sometimes sufficient to infer the existence of an equilateral conformation without actually producing it. We also apply the recent CoBarS method of Cantarella and Schumacher to produce equilateral conformations.

This is work together with Jason Cantarella and Clayton Shonkweiler, building on some earlier work with Nick Beaton and Nathan Clisby. Of course, any errors are entirely my own.