Knotting and entanglement complexity of open curves in 3-space

A novel framework in knot theory will be introduced that can characterize the complexity of open knots and open curves in 3-space in general. In particular, it will be shown how the Jones polynomial, a traditional topological invariant in knot theory, is a special case of a general Jones polynomial that applies to both open and closed curves in 3-space. Similarly, Vassiliev measures will be generalized to characterize the knotting of open and closed curves. When applied to open curves, these are continuous functions of the curve coordinates instead of topological invariants. Using the second Vassiliev measure, the knotting complexity of open random walks in confinement is studied for the first time without any closure scheme and it is proved that it grows as O(n^2) with the length of the walk. A major problem in Applied Knot Theory is the computational cost of such functions in practice. It will be shown that preliminary theoretical results suggest the existence of closed formulas for the computation of the second Vassiliev measure, which would enable a dramatic improvement in the computational methods for identifying complexity or knotting of open or closed curves in applications. We will apply our methods to proteins and show that these enable us to create a new framework for understanding protein folding, which is validated by experimental data. These methods can thus help us understand biopolymer function and biological material properties in many contexts with the goal of their prediction and design.