Average crossing number and writhe of knotted random polygons in confinement

In this paper we study the average crossing number and writhe of random freely-jointed polygons in spherical confinement. Specifically, we use numerical studies to investigate how these geometric quantities are affected by confinement and by knot complexity within random polygons. We report and compare our results with previously published results on knotted random polygons that are unconfined. While some of the results fall in line with what have been observed in studies of unconfined random polygons, some surprising results have emerged from our study, showing properties that are unique due to the effect of confinement. For example, under tight confinement, the average crossing number and the squared writhe grow proportional to the polygon length squared. However, the squared writhe of polygons with a fixed knot type (such as the trefoil) grows much slower than the squared writhe of all polygons. We also observe that while the writhe values at a given length and confinement radius are normally distributed, the distribution of the average crossing number values around their mean are not normal, but rather log-normal.