Joachim Hyam Rubinstein

We study the minimum ribbonlength for immersed planar ribbon knots and links. Our approach is to embed the space of such knots and links into a larger more tractable space of disk diagrams. When length minimisers in disk diagram space are ribbon, then these solve the ribbonlength problem. We also provide examples when minimisers in the space of disk diagrams are not ribbon and state some conjectures. We compute the minimal ribbonlength of some small knot and link diagrams and certain infinite families of link diagrams. Finally, we present a bound for the number of crossings for a diagram yielding the minimum ribbonlength of a knot or link amongst all diagrams.

A shape of a combinatorial polytope is a convex embedding into Euclidean space. We provide necessary and sufficient conditions for a piecewise linear map between two shapes of the same polytope to be a compression (respectively a weak compression), meaning a distance decreasing (respectively a distance non-increasing) map between distinct pairs of points. We establish that there is a partial order on the space of shapes given by the relation of having a weak compression map between pairs of shapes. Finally, we construct a compression metric on the projective shape space of a polytope; the space of convex Euclidean realizations modulo rigid motions and homothety. For the projective space shape of a simplex, we show that the compression metric is complete. For general polytopes, we establish that the projective shape space has a natural completion given by the projective shape space of weakly convex realizations.

Let M be a compact 3-manifold with boundary a single torus. We present upper and lower complexity bounds for closed 3-manifolds obtained as even Dehn fillings of M. As an application, we characterise some infinite families of even Dehn fillings of M for which our method determines the complexity of their members up to an additive constant. The constant only depends on the size of a chosen triangulation of M , and the isotopy class of its boundary. We then show that, given a triangulation T of M with 2-triangle torus boundary, there exist infinite families of even Dehn fillings of M for which we can determine the complexity of the filled manifolds with a gap between upper and lower bounds of at most 13jT j C 7. This result is bootstrapped to obtain the gap as a function of the size of an ideal triangulation of the interior of M , or the number of crossings of a knot diagram. We also show how to compute the gap for explicit families of fillings of knot complements in the 3-sphere. The practicability of our approach is demonstrated by determining the complexity up to a gap of at most 10 for several infinite families of even fillings of the figure-eight knot, the pretzel knot P. 2; 3; 7/, and the trefoil.

A procedure is presented for finding the shortest network connecting three given undirected points, subject to a curvature constraint on both the path joining two of the points and the path that connects to the third point. The problem is a generalisation of the Fermat–Torricelli problem and is related to a shortest curvature-constrained path problem that was solved by Dubins. The procedure has the potential to be applied to the optimal design of decline networks in underground mines.