Mathematics Colloquia and Seminars

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What is the shape of a knot tied tight in rope? The ropelength problem asks us to minimize the length of a knot or link in space, subject to a thickness constraint that keeps a unit tube around the curve embedded. We derive a Balance Criterion giving necessary and sufficient conditions for a space curve to be ropelength-critical. Our approach is modeled on rigidity theory for frameworks and uses a new infinite-dimensional version of the Kuhn-Tucker theorem.

In terms of the core curve, the thickness constraint has two parts: an upper bound on curvature and a self-contact condition. The curvature bound is especially difficult to handle, as the curve may fail to be twice differentiable. In the end, we express thickness as the minimum of a compact family of smooth functions in order to apply Clarke's theorem on the derivative of such a minimum.

Using our balance criterion, we can give explicit descriptions of several tight links. The tight configuration of the Borromean rings, for instance, is piecewise smooth with 42 pieces. Even two simply clasped ropes surprising geometric behavior: there is a slight gap between them when they are pulled tight.

Finally, we consider a two-dimensional analog with a more combinatorial flavor: we require a unit-width ribbon around a knot diagram to be immersed with consistent crossing information. Attempting to characterize critical points for ribbonlength leads us to new results about the medial axis of an immersed disk in the plane, including a certain topological stability for thin disks.