By Jonathan A. Hillman

This quantity is meant as a reference on hyperlinks and at the invariants derived through algebraic topology from overlaying areas of hyperlink exteriors. It emphasizes positive factors of the multicomponent case no longer often thought of via knot theorists, similar to longitudes, the homological complexity of many-variable Laurent polynomial earrings, loose coverings of homology boundary hyperlinks, the truth that hyperlinks are usually not frequently boundary hyperlinks, the decrease significant sequence as a resource of invariants, nilpotent finishing touch and algebraic closure of the hyperlink team, and disc hyperlinks. Invariants of the kinds thought of right here play a necessary function in lots of functions of knot conception to different parts of topology.

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Then P is self-orthogonal with respect to ft © (—6JV), for if m 6 M and n 6 N-^ are such that b © (-bN)((m, [n]), (p, ]p])) = 0 for all p e Nx, then 6( m — n, p) = 0 for all p 6 N1- and so m — n G N-11. In particular, m is in and (m, [n]) = (m, [m]) is in P . Therefore (M,b) © (N1/N11,bN) is neutral. • Let i? o, and with an involution". If (M, b) is an elinking pairing over R and N is a submodule of M then clearly N < N, and as these submodules have the same length they are in fact equal. If M is annihilated by p" then b takes values in the submodule p~nR/R of RQ/R.

The correct choices of "twisting" homeomorphisms h are important here. The disc D used in such an elementary surgery may be isotoped to avoid a finite set of disjoint discs, and so the surgeries of (2) can be performed simultaneously. Thus the conditions of the lemma imply ADDENDUM. / / L and V are link homotopic then there is an embedding T of mSl x D2 in X(L) with core T\msix{0} a trivial 18 1. 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'.

The jump in aL at 0 € S1 is just of [Ma77], [Li84]. If £ is a pth root of unity for some prime p then CTL(0 is a concordance invariant [Tr69]. Since roots of unity of prime order are dense in 5 1 the function obtained from GL by replacing the value at discontinuities by the average of the limits from either side is also a concordance invariant. The function GL is also a skein invariant, for links with nonzero first Alexander polynomial [Li90]. Viro reinterpreted these signatures for roots of unity in terms of the G-Index Theorem.