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

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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 good points of the multicomponent case now not generally thought of through knot theorists, comparable to longitudes, the homological complexity of many-variable Laurent polynomial earrings, loose coverings of homology boundary hyperlinks, the truth that hyperlinks are usually not often boundary hyperlinks, the decrease significant sequence as a resource of invariants, nilpotent of completion and algebraic closure of the hyperlink team, and disc hyperlinks. Invariants of the kinds thought of the following play an important position in lots of purposes of knot thought to different parts of topology.

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Capping off the components of U in DA and doubling gives a /i-component 2-link DR. The ribbon group of R is H(R) = TTDR. Each throughcut T determines a conjugacy class g(T) C TTL represented by the oriented boundary of a small disc neighbourhood in R of the corresponding slit. (The standard orientation on D2 induces an orientation on this neighbourhood via the local homeomorphism R). Let TC be the normal subgroup determined by the throughcuts of R. 15. Let L be a ribbon 1-link with group IT = TTL and R a ribbon map extending L.

Levine defined the Arf invariant for odd dimensional knots and showed that Arf(K) — 0 in Z/2Z if and only if Afl-(-l) = ± 1 mod (8), where Afl-(t) is the Alexander polynomial of K [Le66]. (Curiously, this is detected by the image of AK(t) in Rx/{t), where R = F 2 [Z/4Z] = ¥2[t]/(tA - 1)). Since the figure eight knot 4i has Alexander polynomial t2 — 3t + 1 its Arf invariant is nontrivial, and since 4i is -amphicheiral it represents an element of order at most 2 in C\. Hence the Arf invariant homomorphism splits off a ZflZ summand of C\.

If R is an integral domain the rank of M is the dimension of Mo = Ro <8>R M as a vector space over the field of fractions RQ. 47 48 3. DETERMINANTAL INVARIANTS We may assume that there is an epimorphism (j>: Rq —> M whose kernel is generated by p elements. If we view the elements of Rq as row vectors then a set of p generators for Ker() determines a p x q presentation matrix Q for M. This presentation has deficiency q—p. If Q is injective the exact sequence 0 - • RP — ^ Ri^M-*0 is a short free resolution of M.

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