Can Computers Discover Ideal Knots?

Can Computers Discover Ideal Knots?

Existence of Ideal Knots

Ideal knots are curves that maximize the scale invariant ratio of thickness to length. Here we present a simple argument to establish the existence of ideal knots for each knot type and each isotopy class and show that they are C ' curves. Mathematics Subject Classification (2000): 57M25, 49Q10, 92C05

How Computers Discover How Computers Discover (A mini-review of algorithmic meta-discovery)

In traditional exerimental sciences one forms hypotheses explaining an empirical sample of some natural phenomenon. The hypothesis discovery process can in certain cases be automated using machine learning algorithms. I discuss some consequences of viewing scientific discovery, primarily computer based, itself as a natural phenomenon, whose empirical sample may be analyzed or hypothesized about...

Subknots in ideal knots, random knots, and knotted proteins

We introduce disk matrices which encode the knotting of all subchains in circular knot configurations. The disk matrices allow us to dissect circular knots into their subknots, i.e. knot types formed by subchains of the global knot. The identification of subknots is based on the study of linear chains in which a knot type is associated to the chain by means of a spatially robust closure protoco...

Non-peripheral Ideal Decompositions of Alternating Knots

An ideal triangulation T of a hyperbolic 3-manifold M with one cusp is non-peripheral if no edge of T is homotopic to a curve in the boundary torus of M . For such a triangulation, the gluing and completeness equations can be solved to recover the hyperbolic structure of M . A planar projection of a knot gives four ideal cell decompositions of its complement (minus 2 balls), two of which are id...