By M.M. Cohen

ISBN-10: 3540900551

ISBN-13: 9783540900559

Cohen M.M. A path in simple-homotopy thought (Springer, [1973)(ISBN 3540900551)

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3 Self-intersections 39 The piecewise linear case. The classification of generic self-intersections is similar in the piecewise linear case in which M is given by a finite triangulation, K. However, in contrast to the smooth case, the enumeration of the generic types is elementary. Since M is a 2-manifold, the triangles that contain a vertex form a disk. It is not difficult to see that imposing this condition on the vertices suffices to guarantee that K triangulates a 2-manifold without boundary. On the other hand, requiring that each edge belong to exactly two triangles is not sufficient.

We use this to express the sum of squared distances from a set of planes in matrix form. Letting H be a finite set of planes, this gives a function EH : R3 → R defined by d2 (x, hi ) EH (x) = hi ∈H = hi ∈H = xT · (xT · ui )(uTi · x) hi ∈H Hence EH (x) = xT · Q · x, where Q = hi ∈H (ui · uTi ) = ⎡ ui · uTi A P ⎢ P B ⎢ ⎣ Q R U V · x. Q R C W ⎤ U V ⎥ ⎥ W ⎦ Z 46 II Surfaces is a symmetric, four-by-four matrix that we refer to as the fundamental quadric of the map EH . Writing xT = (x1 , x2 , x3 ), we get EH (x) = Ax21 + Bx22 + Cx23 + 2(P x1 x2 + Qx1 x3 + Rx2 x3 ) + 2(U x1 + V x2 + W x3 ) + Z.

Integrating the above relation over all directions of the sphere gives DWr(κ, u0 ) minus Wr(κ) on the left and W (κ, u0 ) minus the average winding number on the right. Hence, Wr(κ) = DWr(κ, u0 ) − W (κ, u0 ) + 1 4π W (κ, u) du. u∈S2 Bibliographic notes. Knots and links have been studied for centuries, and there are a number of excellent books on the subject, including the text by Adams [2]. Motivation for studying the writhing number of a space curve and the twisting number of a ribbon is derived from the double-helix structure of DNA whose discovery is comparably recent [154].

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