Vertex distortion of lattice knots

The vertex distortion of a lattice knot is the supremum of the ratio of the distance between a pair of vertices along the knot and their distance in the [Formula: see text]-norm. Inspired by Gromov, Pardon and Blair–Campisi–Taylor–Tomova, we show that results about the distortion of smooth knots hold for vertex distortion: the vertex distortion of a lattice knot is 1 only if it is the unknot, and there are minimal lattice-stick number knot conformations with arbitrarily high distortion.

A new characterization of markets that don't replicate any option through minimal-lattice subspaces: A computational approach

In this paper the notion of strongly resolving markets with respect to the positive basis of a minimal lattice-subspace Y of Rm is defined. It is proved that if the number of securities is less than half the dimension of Y, then not a single (non-trivial) option can be replicated. This result extends already known results regarding the notion of a market being strongly resolving. Both theoretical and computational methods are provided in order to establish criteria for the characterization of markets that do not replicate any option.