Download Derivatives of Links: Milnor's Concordance Invariants and by Tim D. Cochran PDF

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By Tim D. Cochran

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The commas are included for visual convenience only. The exact value is ( —l) n_t " 1 . 10). 2n as [y, C l (xy)][y, c 2 (xy)j[[x, y], c ( x x y ) ] . . [S[n - 5)te{6(n - 4))][*(n - 4), c(S(n - 3))] where we have ignored inverses because we are computing modulo 2. xx(6(k)) = 0 for any k > 0 and t6(k){6(k)) = -hi. x(6{n - 4))eS(n -S)yx,xy Here, in the second expression, 6(n — 4) stands for the simple commutator in the free group. = e6{n-*),vz,xV(c(6{n - $))). 4 with a system of weight 2n, a — S(n — 3) and ft = (y, x), c(6(n — 3)) = [[y, x],c(((5(n - 3 ) , ( y , x ) ) ) ] mod C n -fS- Thus the above sequence of congruences continues: = t6(n-s)Vx(c(6{n - 3 ) , (y 5 x ) ) ) .

Each a^ is a surface whose boundary is a sum of 1-manifolds each of wrhich is formed from the intersection of two previously-defined surfaces (actually if (i,j) = ( l , n — l) or (2, n) then some boundary components might be meridians). In our context, surfaces have been indexed by brackets so we shall use the bracket to denote the surface. Thus , in our dualized matrix, on teh diagonal one sees 1-brackets x %i 5 • • • J xin- O n ^ n e off-diagonal entry one sees a formal sum of fc-brackets each of which is the "inner-product" of the corresponding row and column (as in [13]).

0 . Suppose that (o~i,... ,an) is an n-tuple of non-trivial bracketings (not necessarily distinct), in m letters, and suppose that D(L) is an n-component link. ;c(an)}. THEOREM 7 . 1 . , <7n), as long as, if some Gj equals X{ then there are no other occurences of x» as a sub-bracket in any a*. We postpone the proof until after the explicit process of antidifferentiation is explained. Antiderivatives are not unique, but, just as in calculus, the "constants of integration'' are the linking numbers of derived links of lesser complexity.

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