New PDF release: Algebraic Structure of Knot Modules

By J. P. Levine

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Example text

A-module is A / ( T d) module. if and only if every non-zero is finitely generated The degree of A and all have the is the largest d 32 is induced by multiplication multiplication the unique zero. ~ lift of But #~ ! A/B by It by T i+l , is define is certainly a notion a noetherian If is equivalent to A A = Z[t, to the zero. 1: if A for various then A then the Ad_ 1 ~ 0. values is of a local i, (see sum if domain, isomorphism class of and o n l y i f A A~ all (i). is homogeneous same rank, as R-modules.

We may w r i t e for some - ~ by ~", B" ¢ ~S, + ~ B' = (~ B + q ~ 0 ) ( ~ ~(~ 0 c ~k+ls ~k+is (X, ~) c K. -~) + q(~0' = But We w i l l + p B' c T' A X ~ + ~ ~ ¢ ~S, modulo O c TN ~k+is, @) + + n(~o~ o(x ~0 ~ @ + P0 ~ P0 ~ e) - + ~o~)) - ~ - x + y ~ - ~ ) ) ~)) + n y))e @ B c ~k+lT c ~. c ~. ~ [ c ~[' §15. O c ~ k+l S But + ~k+2s. 1, and so is proved. modules is an e l e m e n t a r y is a free R - m o d u l e . n-primary Then A A-module of is a free A/(~d)-module. ,~k c A be c h o s e n so that the cosets m o d ~A 37 are a basis this, of A 0 = A/~A.

The multiplication a vector Hermitian isomorphism by a suit- space form class of (V, [,]) 17 is d e t e r m i n e d by that of such (W, <,>) lifts to some field and, therefore by W (see When least if (W, <,>), <,> (V, given [,]). is uniquely, r, and, conversely, But Ap/(~) any is a finite up to isomorphism, determined [M-l]). ~ = t + 1 p ~ 2. or t - i, we get a less trivial result, Again one defines (-l)r-~-symmetric, W = V/OV, n o n - s i n g u l a r pairing on but now W, <,> at is a which is only a vector space over Z ; i s o m o r p h i s m classes of (V, [,]), for a given r, P in one-one c o r r e s p o n d e n s e with isomorphism classes of (W, <,>).

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