Download Algebraic Topology (Colloquium Pbns. Series, Vol 27) by Solomon Lefschetz PDF

By Solomon Lefschetz

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Additional resources for Algebraic Topology (Colloquium Pbns. Series, Vol 27)

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We shall return to the specifics of the construction of the Jones polynomial via the bracket polynomial model in section 5, but it is now possible to indicate this model as an amplitude by specifying its matrices. They are (Mab) = (Mab) = M where M is the 2 x 2 matrix (with ii=-1) M = 0 iA iA10 Note that MM = I where I is the identity matrix . amplitude for the circle is E MabMab = (iA)2 + = Note also that the EMabMab = E (Mab)2 (-iA-1)2 = - A2 - A-2. The matrix R is then defined by the equation Rabcd = AMabMcd + AIacIbd , or diagrammatically by R=A^ +A-1 Since, diagrammatically , we identify R with the crossing this equation can be written symbolically as Taken together with the loop value of -A2 - A-2, this equation can be regarded as a recursive algorithm for computing the amplitude.

That is, we assume that following sum is equal to one, for each k from 1 to n-1: Ic(k)1> ... where the sum is taken over all i(k) ranging between 1 and m, and k ranging between 1 and n. Each term in this sum can be construed as a combinatorial path from a to b in the two dimensional space of the x-y plane. Thus the amplitude for going from a to b is seen as a summation of contributions from all the "paths" connecting a to b.

Approximate the change according to Fact 1, and regard the point x as the place of curvature evaluation . Let b denote the change in the value of the line. b is given by the formula o = dxrdxsFars ( x)Ta. This is the first order approximation to the change in the Wilson line. In this formula it is understood that the Lie algebra matrices Ta are to be inserted into the Wilson line at the point x , and that we are summing over repeated indices . This means that each Ta is a new Wilson line obtained from the original line by leaving 26 the form of the loop unchanged, but inserting the matrix Ta into that loop at the point x.

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