Thickness and crossing number of knots

The helicity of a smooth vector field defined on a domain in 3-space is the standard measure of the extent to which the field lines wrap and coil around one another; it plays important roles in fluid mechanics, magnetohydrodynamics and plasma physics. In this report we show how the relation between energy and helicity of a vector field is influenced by the geometry and topology of the domain on which it is defined. In particular, we will see that the standard model for the magnetic field in the Crab Nebula (equivalently, the spheromak field of plasma physics) is the unique energyminimizing divergence-free vector field of given nonzero helicity, defined on and tangent to the boundary of a round ball, and that the essential features of this energy-minimizing field persist even as the domain changes topological type. We will also see that when volume-preserving deformation of domain is permitted, the spheromak field is not the absolute energy-minimizing field with given helicity; instead, the round ball on which it is defined can be dimpled in at the poles and expanded out at the equator to further decrease the field energy while preserving helicity. Our numerical computations suggest that this volume-preserving, helicity-preserving, energy-decreasing deformation of domain and field converges to a singular domain, in which the north and south poles have been pressed together at the center, along with a corresponding singular field. Two fundamental problemsWe organize this report by focusing on two fundamental problems:1. Minimize energy among all divergence-free vector fields of given nonzero helicity, defined on and tangent to the boundary of a given domain.2. Find the above minimum over all domains of given volume.Such energy-minimizing vector fields provide models for stable force-free magnetic fields in gaseous nebulae and laboratory plasmas, while the search for them seems to bring out some of the deepest and most useful mathematics connected with helicity. Helicity and writhing numberThe helicity H(V ) of a smooth (meaning C ∞ ) vector field V on the domain Ω in 3-space, defined by the

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“…Similar bounds have been obtained independently in [Buck and Simon, 1998], and also in [Freedman and He, 1991].…”

Section: How the Geometry Of The Domain Influences Helicitysupporting

confidence: 85%

Influence of Geometry and Topology on Helicity

Cantarella¹,

DeTurck²,

Gluck³

et al. 2013

Magnetic Helicity in Space and Laboratory Plasmas

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“…Then, case (i) gives a bound on MCN(L) of L 4/3 ; we established the bound in this case in earlier work, and showed that it is sharp (Buck 1998;Buck & Simon 1999).…”

Section: Interpreting the Entanglement Estimatessupporting

confidence: 57%

The spectrum of filament entanglement complexity and an entanglement phase transition

Buck

Simon

2012

Proc. R. Soc. A.

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DNA, hair, shoelaces, vortex lines, rope, proteins, integral curves, thread, magnetic flux tubes, cosmic strings and extension cords; filaments come in all sizes and with diverse qualities. Filaments tangle, with profound results: DNA replication is halted, field energy is stored, polymer materials acquire their remarkable properties, textiles are created and shoes stay on feet. We classify entanglement patterns by the rate with which entanglement complexity grows with the length of the filament. We show which rates are possible and which are expected in arbitrary circumstances. We identify a fundamental phase transition between linear and nonlinear entanglement rates. We also find (perhaps surprising) relationships between total curvature, bending energy and entanglement.

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“…We will show it inductively on m. Case 1 0 : As m = 0, we prove that there exist some positive constants M i = M i (δ, α, β, f (3) 0,α ) > 0 such that for all s and s ,…”

Section: Proof Of the Main Resultsmentioning

confidence: 97%