Physical Knot Theory: An Introduction to the Study of the Influence of Knotting on the Spatial Characteristics of Polymers

Abstract

This paper contains selected topics from four lectures given at the Abdus Salam International Centre for Theoretical Physics in May 2009. We introduce the study of the influence of knotting and linking on the spatial characteristics of linear and ring polymer chains with examples of scientific interest. We describe a few basic concepts of the geometry and topology of knots and measures of the spatial shape of open and closed polymer chains. We then present some fundamental mathematical results concerning them. Next we discuss random sampling methods of collections of open and closed chains that are employed to provide estimates of the spatial properties of the chains. Finally, we discuss implementations of the sampling algorithms, survey consequences of theoretical and experimental results, and discuss some interesting problems deserving further research.

On The Expected Total Curvature of Confined Equilateral Quadrilaterals

“…In this work, we study confined random polygons, using the diameter of a polygon as a measure of its confinement. There are other possible measures as well, such as the confinement radius or the gyration radius of the polygon [8]. Imposing confinement on a random polygon affects the geometry in subtle ways and is a topic of active research.…”