Algebraic Invariants of Links (Series on Knots and by Jonathan A. Hillman

By Jonathan A. Hillman

This quantity is meant as a reference on hyperlinks and at the invariants derived through algebraic topology from masking areas of hyperlink exteriors. It emphasizes gains of the multicomponent case now not as a rule thought of through knot theorists, corresponding to longitudes, the homological complexity of many-variable Laurent polynomial jewelry, loose coverings of homology boundary hyperlinks, the truth that hyperlinks will not be often boundary hyperlinks, the decrease imperative sequence as a resource of invariants, nilpotent final touch and algebraic closure of the hyperlink team, and disc hyperlinks. Invariants of the categories thought of right here play a vital function in lots of purposes of knot conception to different components of topology.

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Example text

Figure 6. 9 o Wh2 1 3 Let Wh : 2S -> S be the Whitehead link (5? in the tables of [Rol]), and let 9 : X{Wh\) -+ 5 1 x D 2 be a homeomorphism such that 9(4>i{u,v)) = («,u), for all u,« e 5 1 . If K(j) = 9o Wh2 then L o /C is obtained from L by Whitehead doubling the j ' t ' 1 component. (See Figure 6). When fi = 1 this is an untwisted double of the knot L. Since each component of the Whitehead link bounds a punctured torus in the complement of the other component, Whitehead doubling every component of a link gives a boundary link.

LINKS link and a self homeomorphism h of X(T) = S3 — T(mSl that x D2) such (1) lk(Ti,Lj) = 0 for each 1 < i < m and each component Lj ofL; (2) h(Ti{z, s)) = Ti(sz, s) for all (z, s) € ^ ( S 1 x D2) and 1 < i < m; (3) hoL = L'. a We shall say that two links L and 1/ related by such surgeries are surgery equivalent. The requirement that the core be trivial ensures that the 3-manifold resulting from the surgeries is again 5 3 ; the linking number condition implies that the surgery tori lift to abelian covers of L.

For knots multiplication by t — 1 is invertible on H\(X; Ai), and so no information is lost on localization with respect to S. Localization with respect to E annihilates knot modules. It follows easily from this and from Rolfsen's theorem on isotopies that the E-localized Blanchfield pairing is invariant under isotopy. Let bs(L) and b%(L) be the localized Blanchfield pairings of L. 6. 6. Concordance We need some further terminology in order to describe the concordance invariants corresponding to the Blanchfield pairing.

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