Algebraic Structure of Knot Modules by J. P. Levine PDF

By J. P. Levine

ISBN-10: 3540097392

ISBN-13: 9783540097396

ISBN-10: 354038555X

ISBN-13: 9783540385554

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Extra info for Algebraic Structure of Knot Modules

Example text

Sequences: {0 + Ai+ 1 + A. ÷ A i ÷ A i+l + 0} i where = Ak+ 1 A k+l = 0, we first construct {0 ÷ Bi+ 1 ÷ B i + B i + B i+l + 0} with By exists #kB induction = 0 we may whose assume n-primary there sequences {0 ÷ Bi+ I + B I• + B i + B i+l + 0} . 1, let C be an k = i, with elementary of d e g r e e k (#C) 0 ~ C O . 3. consisting and induced map We define of all then elements C O = A k. = 0. We may consider Then Let (~C) 0 = C O = A k ÷ B 0 A =(C ( ~ B ~ D , of the form c E ~C. ~k+iA with to we have By L e m m a in §ii.

X, y Let satisfies ~, ~, x, y, etc. X, U, x, y, etc. residue ~ T ~", B" e T N ~k+is. {~', ~'}, then that We will replace for suitable by which then A) in their X, U, x, y ~ S in or A, R. such that i~+ We have an obvious U B ~ ~S short exact sequence: 0 + K ÷ R~R~ where R~R Dedekind, rank one. another ÷ I is the map the sequence The pair pair (B, -~) (X 0, ~0 ) The fourelements splits which, I + 0 (X, ~) ~ ~a + ~B rood ~S. and K is an R-torsion obviously together belongs with ~, B, ~0' ~0' generate to K; (B, -~), R, Since ~ is free R-module of thus there is generates K.

The proof. " isomorphism class homogeneous of degree Proposition is modules domain, because are of the Definition: Bi + A i of Consider easy to see that o f whose i d e a l s with is #i which is again the map induced by the "building A-modules. is step, recall that ' #i+l a lift of and elementary of modules isomorphic A For the inductive . We now construct We f i r s t while the second is induced by ¢i+l = ~i+l wh°se composition . §13. Homogeneous §9). ~, IB/ I+I B ÷ I A / T i + I A , multiplication T-primary by is certainly .

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Algebraic Structure of Knot Modules by J. P. Levine

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