# Read e-book online Derivatives of Links: Milnor's Concordance Invariants and PDF By Tim D. Cochran

ISBN-10: 0821824899

ISBN-13: 9780821824894

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Extra resources for Derivatives of Links: Milnor's Concordance Invariants and Massey's Products

Example text

Thus c ( f t ) (and c(<72)) consists of components corresponding to vertices of the tree. But the only loop in the system which has non-zero linking with any component of c(cr2, a n ) is the one corresponding to the vertex

X, y) . . ) ) or [x, [ x , . . , [x, y ] j . . j according to the context. For this proof, let 6(k) denote xx . . xxy containing k occurences of the letter x. 6a). 6e). 6c). 11. 6d). The curves c(S(k)) 2 < k < (n — 2) are then similar to ci(xy) and each bounds a surface of genus one. Hence {c(S(n — 3)),c(yx)} forms a Whitehead link, so c((S(n — 3), yx)) is of weight n and has self-linking —1 as desired. Notice that C2{xy) and c ( 6 ( n - 3 ) ) bound punctured torii in Ss — V(x) — V(y). ,^] so C2{xy) and c(6(n — 3)) lie in G".

Or [x, [ x , . . , [x, y ] j . . j according to the context. For this proof, let 6(k) denote xx . . xxy containing k occurences of the letter x. 6a). 6e). 6c). 11. 6d). The curves c(S(k)) 2 < k < (n — 2) are then similar to ci(xy) and each bounds a surface of genus one. Hence {c(S(n — 3)),c(yx)} forms a Whitehead link, so c((S(n — 3), yx)) is of weight n and has self-linking —1 as desired. Notice that C2{xy) and c ( 6 ( n - 3 ) ) bound punctured torii in Ss — V(x) — V(y). ,^] so C2{xy) and c(6(n — 3)) lie in G".