# Algebraic Invariants of Links by Jonathan Hillman By Jonathan Hillman

This booklet is meant as a reference on hyperlinks and at the invariants derived through algebraic topology from overlaying areas of hyperlink exteriors. It emphasizes good points of the multicomponent case no longer generally thought of by way of knot theorists, resembling longitudes, the homological complexity of many-variable Laurent polynomial jewelry, loose coverings of homology boundary hyperlinks, the truth that hyperlinks aren't often boundary hyperlinks, the reduce primary sequence as a resource of invariants, nilpotent crowning glory and algebraic closure of the hyperlink workforce, and disc hyperlinks. Invariants of the kinds thought of the following play a vital function in lots of purposes of knot conception to different parts of topology.

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The Witt class is additive for knots. In §7 we consider additivity for links, although there is no natural sum for links or even for link concordance classes. In §8 we show how to compute the Blanchfield pairing of a boundary 1-link using Seifert surfaces. We conclude with a brief 27 28 2. HOMOLOGY AND DUALITY IN COVERS discussion of signature invariants corresponding to homomorphisms from the Witt group of Blanchfield pairings to Z. We shall consider homology and duality for the maximal free covers of homology boundary links in Chapter 9.

Since dXT is a union of copies of Sn x R the natural homomorphism Hq(X;A) —> Hq(X,dX;A) is an isomorphism if q / 0, 1, n or n + 1, a monomorphism if g = 1 or n + 1 and an epimorphism if # = 0 or n. The kernels and cokernels are torsion modules, since H0(dX;A) = Hn(dX;A) = Z». The rest of (4) now follows from (2), the UCSS and Poincare duality, while (5) is a case of Hopf 's Theorem. Since Hq(X; A) is a torsion module for 1 < q < n the UCSS collapses to give the exact sequences of (6). • Let M be a torsion A-module.

These pairings agree with the ones defined by means of the isomorphisms Dq and so are independent of the triangulation used. The geometric properties of intersections of cells in manifolds imply that Ip(a, (3) = (-1^^4(0, a) if ij, : P n _ p x C n _ p - • Z is the transposed pairing. The Blanchfield pairings may be described in such terms as follows. )-cycle of (XjjjdXn) such that ax = du and (3y — dv for some (q + 1)chain u and (n — q»)-chain v and some nonzero a, f3 6 A^, then V(x,y) = a-lIq(u,y) = (f3)-lIq(x,v) = (P)-lHg&z^Iq{x,gv)g mod A^ in 5(AM).