Knots and Links in Fluid Mechanics

Polyhedral Knots and Links

This paper contains a survey of different methods for generating knots and links based on geometric polyhedra, suitable for applications in chemistry, biology, architecture, sculpture (or jewelry). We describe several ways of obtaining 4-valent polyhedral graphs and their corresponding knots and links from geometrical polyhedra: midedge construction, cross-curve and double-line covering, and ed...

Virtual Knots and Links

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach that if a connected sum of two virtual knots K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm, using Haken-Matveev technique, for recognizing...

Solitons, Links and Knots

Using numerical simulations of the full nonlinear equations of motion we investigate topological solitons of a modified O(3) sigma model in three space dimensions, in which the solitons are stabilized by the Hopf charge. We find that for solitons up to charge five the solutions have the structure of closed strings, which become increasingly twisted as the charge increases. However, for higher c...

Knots, Links and Tangles

We start with some terminology from differential topology [1]. Let  be a circle and  ≥ 2 be an integer. An immersion  :  → R is a smooth function whose derivative never vanishes. An embedding  :  → R is an immersion that is oneto-one. It follows that () is a manifold but () need not be ( is only locally one-to-one, so consider the map that twists  into a figure eight). A knot is a s...

Knots , Links and Tangles Steven