Algebraic Invariants of Links by Jonathan Hillman

By Jonathan Hillman

This e-book 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 now not typically thought of by means of knot theorists, reminiscent of longitudes, the homological complexity of many-variable Laurent polynomial jewelry, loose coverings of homology boundary hyperlinks, the truth that hyperlinks should not often boundary hyperlinks, the decrease critical sequence as a resource of invariants, nilpotent finishing touch and algebraic closure of the hyperlink staff, and disc hyperlinks. Invariants of the categories thought of right here play a necessary function in lots of functions of knot conception to different components of topology.

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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. 5. THE MAXIMAL ABELIAN COVER 35 leads to natural isomorphisms elM = HorriA(M,Q(t)/A), since Q(t) is injective. If N is a finite A-module there are also natural isomorphisms e2N = Homz(N,Q/Z) (see [Le77]). Therefore the Poincare duality isomorphisms in conjunction with the above exact sequences give rise to pairings: (Blanchfield) (, ) : tHq{X; A) x tHn+1_q(X; A) -> Q(*)/A (Farber-Levine) [, ] : zHq(X;A) x zHn-q(X;A) -» Q / Z The Blanchfield pairings are sesquilinear while the Farber-Levine pairings are Z-linear and isometric ([ta,tft] = [a, ft] for any a € zHq(X;A) and ft € zHn-q(X; A)).

Clearly Ek{M) < Ek+1(M) and Ek(Ms) = Ek(M)s for any multiplicative system S in R. More generally, if / : R —> R' is a ring homomorphism then f(Q) is a presentation matrix for M' = R' ®R M over R', and so Ek(M') is the ideal generated by f(Ek(M)) in R!. For each k > 0 let A^M be the kth exterior power of M, and let a^M = Ann(AkM). This notation is due to Auslander and Buchsbaum, who showed that if R is a local domain and a^M is principal for all k then M is a direct sum of cyclic modules, and used this to give criteria for projectivity [AB62].

Robertello showed that if there is a knot K and a planar surface in S3 x [0,1] with boundary L x {0}U K x {1} then Arf(K) depends only on L, provided that L is proper, and so we may set Arf(L) = Arf(K). (Such knots and planar surfaces may be obtained by iterated fusions of distinct components of the link). It is not yet known whether Arf(L) admits a simple description in terms of other invariants of the link, although Arf(L) may be derived from the second derivative of the Conway polynomial of L and its sublinks 46 2.

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