Knot (unit)

Abstract: Publisher Summary This chapter describes knots and links, and some of their algebraic properties. An edge-connected 4-valent planar map is called a polyhedron, and a polyhedron is basic if no region has just 2 vertices. The term region includes the infinite region, which is regarded in the same light as the others. Knot diagrams can be obtained from polyhedra by substituting tangles for their vertices, for instance, tangles 1 or −1 could always be substituted. A knot diagram K can be obtained by substituting algebraic tangles for the vertices of some nonbasic polyhedron P. There is a polyhedron Q with fewer vertices than P obtained by shrinking some 2-vertex region of P, and K can simply be obtained by substituting algebraic tangles for the vertices of Q. Any knot diagram can be obtained by substituting algebraic tangles for the vertices of some basic polyhedron P in fact P, and the manner of substitution is essentially unique.

Abstract: In 1867, Lord Kelvin proposed that atoms—then considered to be elementary particles—could be described as knotted vortex tubes in either1. For almost two decades, this idea motivated an extensive study of the mathematical properties of knots, and the results obtained at that time by Tait2 remain central to mathematical knot theory3,4. But despite the clear relevance of knots to a large number of physical, chemical and biological systems, the physical properties of knot-like structures have not been much investigated. This is largely due to the absence of a theoretical means for generating stable knots in the nonlinear field equations that can be used to describe such systems. Here we show that knot-like structures can emerge as stable, finite-energy solutions in one such class of equations—local, three-dimensional langrangian field-theory models. Our results point to several experimental and theoretical situations where such structures may be relevant, ranging from defects in liquid crystals and vortices in superfluid helium to the structure-forming role of cosmic strings in the early Universe.