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BIRS Workshop Lecture Videos

Accumulated knot probability Rawdon, Eric


Many knots in nature are open knots, not the closed knots from knot theory. There are several definitions of knotting in open curves, each of which have their own advantages and disadvantages. The speaker's favorite open knot definition involves extending rays to infinity in a common direction from the endpoints to create a closed knot for each such direction. In such a case, the knotting in an open chain is classified as the distribution of knot types seen over the different directions of closure. In most cases, there is a knot type that appears in over 50% of the closure directions, in which case we might all be able to agree that the open knot has the essence of that closed knot type. However, there are many cases where there is no knot type that appears in over 50% of the closure directions, especially near transitions between different knot types. We present the accumulated knot probability as a way of making sense of these more ambiguous situations. The short story is that, for a given knot type K, we compute the probability that the closures are a knot type which "includes" K in some sense. In this talk, we use the partial ordering on knots developed by Diao, Ernst, and Stasiak based on crossing changes in minimal knot diagrams, which creates a sort of family tree of knots. However, any sort of family tree could be substituted here depending on what one is trying to model. We show how some of the knotting classifications change for some proteins and tight knot configurations.

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