Jun O'Hara's Energy of Knots and Conformal Geometry PDF

By Jun O'Hara

ISBN-10: 9812383166

ISBN-13: 9789812383167

ISBN-10: 9812795308

ISBN-13: 9789812795304

Power of knots is a thought that was once brought to create a "canonical configuration" of a knot — a gorgeous knot which represents its knot sort. This booklet introduces a number of forms of energies, and reviews the matter of even if there's a "canonical configuration" of a knot in each one knot style. It additionally considers this difficulties within the context of conformal geometry. The energies provided within the publication are outlined geometrically. They degree the complexity of embeddings and feature functions to actual knotting and unknotting via numerical experiments.

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Additional resources for Energy of Knots and Conformal Geometry

Sample text

We On E<2> 38 remark that the asymptote lx of Kx has —h'(s) as its direction vector. By (a) \ix'{y)\ l \x-y\ and \dy\ = \d(Ix(y))\ = \Ix'(y)\2\dy\ = My| x-y\2' Therefore V2){K-x)= T-^-T2= Jp+(e) \X-y\2 \dy\ Jp+(e) is equal to the arc-length of ii"x (p + (e),p_(e)) = Ix(h(s + e, 1 + s — e)) which is a subarc of the inverted open knot Kx between p+ (e) and p_ (e). Let k = \h"(s)\ be the curvature of K at x and n = h"(s) be the unit principal normal vector of K = /i(5'1) at x = h(s). Put n = 0 when |/i"(s)| = 0.

Then u(y) is a positive tangent vector to C(y,x\u) at y. (2) Being a unit vector is not a conformally invariant condition. Our definition will make the inner product conformally invariant. Namely, we have I T*v,T*u(T(y)) J = (v,u(y)) for any Mobius transformation T of M3 U {oo} and for any u G TXR3 and v € TyR3. 5 Existence of E^ minimizers A composite knot is a knot which is a connected sum of two non-trivial knots. A prime knot is a knot which is not a composite knot. 1 of a prime knot. ([FHW]) There is an minimizer for any knot type Outline of proof.

2) Let i . j e l 3 (x ^ y), u e TXR3, and v G TVR3. The conformal angle 9 = 6(u,v) between u and v is given by the angle between u(y) and v. It is preserved under any Mobius transformation. 2 (1) Let C(y,x;u) denote the oriented circle that is tangent to u at x and that passes through y whose orientation comes from that of u. Then u(y) is a positive tangent vector to C(y,x\u) at y. (2) Being a unit vector is not a conformally invariant condition. Our definition will make the inner product conformally invariant.

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Energy of Knots and Conformal Geometry by Jun O'Hara


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