knots, applied knot theory
Traditional topological knot theory mainly studies knotting within closed curves. However, much of what people consider knotting in everyday life, e.g. in shoelaces, involve tangled open rope-like materials with two free ends. Furthermore, knotting has been discovered in open DNA and protein chains. Several notions of open knotting have been defined and applied in different situations. In this paper, we survey the different techniques used for measuring knotting in open chains, including some discussion of different properties of the techniques. We present the knotting fingerprint, a means for visualizing the knotting of subchains of open and closed chains via image matrices. Finally, we show image matrices for some random closed knots and relate features of the matrices with features of the configurations.
New Directions in Geometric and Applied Knot Theory