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 protecting areas of hyperlink exteriors. It emphasizes beneficial properties of the multicomponent case now not often thought of by means of knot theorists, equivalent to longitudes, the homological complexity of many-variable Laurent polynomial jewelry, unfastened coverings of homology boundary hyperlinks, the truth that hyperlinks usually are not often boundary hyperlinks, the reduce critical sequence as a resource of invariants, nilpotent final touch and algebraic closure of the hyperlink workforce, and disc hyperlinks. Invariants of the categories thought of the following play a vital function in lots of purposes of knot concept to different parts of topology.

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Chapitre 1. L’outil vectoriel et analytique
    I. Introduction
    II. Le plan vectoriel (rappels)
    III. Les liaisons « plan ponctuel-plan vectoriel »
    IV. L’outil analytique
    V. Compléments
    Exercices

Chapitre 2. L’outil des transformations
    I. Introduction
    II. adjustments usuelles
    III. motion sur les configurations élémentaires
    IV. modifications associant une determine donnée à une determine donnée
    V. Composition de transformations
    VI. Compléments
    Exercices

Chapitre three. Les angles
    I. Introduction
    II. perspective d’un couple de vecteurs
    III. L’addition des angles
    IV. Propriétés géométriques
    V. Angles et cercles
    VI. Compléments
    Exercices

Chapitre four. Le produit scalaire
    I. Introduction
    II. Produit scalaire de deux vecteurs (rappel)
    III. Produit scalaire en géométrie analytique
    IV. Orthogonalité et cocyclicité
    V. Produit scalaire et lignes de niveau
    VI. Compléments
    Exercices

Chapitre five. Trigonométrie et relatives métriques dans le triangle
    I. Introduction
    II. Cosinus et sinus (rappels)
    III. Cosinus et produit scalaire ; sinus et déterminant
    IV. Trigonométrie
    V. kin métriques dans le triangle
    VI. Compléments
    Trigonométrie (formulaire récapitulatif)
    Exercices

Chapitre 6. Rotations et isométries fixant un aspect donné
    I. advent (quart de tour)
    II. Rotation de centre O et d’angle α
    III. Rotation : théorèmes de composition et propriétés géométriques
    IV. Isométries fixant un aspect donné
    V. Compléments
    Exercices

Chapitre 7. Le calcul vectoriel dans l’espace
    I. Introduction
    II. L’espace vectoriel E
    III. Droites et plans : repères et vecteurs directeurs
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Chapitre eight. Le produit scalaire dans l’espace
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Chapitre 10. Statistiques
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Additional resources for Algebraic Invariants of Links

Example text

This is due to [Tr69] for 1-links and [Le69] for odd-dimensional knots). The function GL is piecewise continuous, with jumps only at the roots of A(t) = det(tA - Atr). 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 sum cr(L) + Ti^jiij is invariant under changes of orientation [Mu70']. More generally, if A is the Seifert matrix associated to the homology in degree q of a Seifert hypersurface for a (2q + l)-link L 44 2. HOMOLOGY AND DUALITY IN COVERS and £ is a complex number of modulus 1 then (1 — £)A + (1 — £)Atr is a (—l) 9+1 -hermitean matrix. The signatures (TA(€) S Z depend only on L, and so define a function GL : S1 —* Z. (This is due to [Tr69] for 1-links and [Le69] for odd-dimensional knots). The function GL is piecewise continuous, with jumps only at the roots of A(t) = det(tA - Atr).

Consider the square knot 3i# — 3i). 2), as such operations do not change the pattern of the singularities. Figure 4. Let R be the ribbon disc of Figure 4 and let K = OR. Then H(R) has the presentation (a,b,c,d | aca-1 — d,dad_1 = b,dcd_1 = b), and so H(R) = Z. It can be shown that there is a homomorphism from nK to SL(2, F7) with nonabelian image. The corresponding ribbon 2-knot is trivial, and so if is a nontrivial slice of a trivial 2-knot [Yn70]. (It is in fact the Kinoshita-Terasaka 11-crossing knot with Alexander polynomial 1 [KT57]).

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