Chromosomal DNA is tightly packed in all organisms. When the chromosomes are circular, or divided into looped domains, the packing geometry has a direct impact on the topology of the chains. Problems of packing motivate our research on unconfined and confined lattice polygons. First, it is interesting to ask what is the minimal length of DNA needed to tie a particular knot type. We pursue analytical and numerical characterization of the minimum length (also called minimum step number) needed to form a particular knot in the simple cubic lattice. Second, suppose that we need to identify the topological type of a population of circular DNA molecules. In this case it is important to distinguish a knot from its mirror image. In general, a simple way of detecting chirality of a knot type is to compute the mean writhe of the polygons; if the mean writhe is non-zero then the knot is chiral. We provide numerical evidence that the sign of the mean writhe behaves as a topological invariant of chiral knots, and we propose a new nomenclature of knots based on the sign of their expected writhes. This nomenclature can be of particular interest to applied scientists. This is joint work with Javier Arsuaga, Reuben Brasher, Kai Ishihara, Juliet Portillo, Rob Scharein and Koya Shimokawa.
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