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# On Knots. (AM-115)

LOUIS H. KAUFFMAN
Pages: 498
Stable URL: http://www.jstor.org/stable/j.ctt1b9rzkz

1. Front Matter (pp. i-vi)
3. PREFACE (pp. ix-2)
4. I INTRODUCTION (pp. 3-8)

These notes constitute a leisurely introduction to knot theory that is (we hope!) in the spirit of Fox’s Quick Trip [FI]. We shall also feel free to digress occasionally, sometimes in the direction of applications, sometimes with an analogous structure, sometimes with philosophy or general ideas.

What is knot theory about? One answer is that knot theory studies the placement problem: Given spaces X and Y, classify how X may be placed within Y. Here the how is usually an embedding, and classify often means up to some form of movement of X in Y (isotopy, for example). If X...

5. II LINKING NUMBERS AND REIDEMEISTER MOVES (pp. 9-18)

Our first model for the theory of knots and links in IR³ is combinatorially based. We say that two knot or link diagrams K and K’ are equiva1ent if there exists a sequence of Reidemeister moves (see [R1]) changing K into K’. Equivalence is denoted by the symbol as in K ~ K’. Equivalent diagrams are ambient isotopic. meaning that there is a continuous deformation through embeddings from one to the other. Reidemeister proved the converse—making equivalence and ambient isotopy identical. There are three basic types of Reidemeister moves:

It is understood that these moves are to be performed...

6. III THE CONWAY POLYNOMIAL (pp. 19-41)

We now introduce a more powerful invariant of oriented knots and links. It is the Conway polynomial, a refined variant of the classical Alexander polynomial (see [A1] and [C1], [K1], [K2]). This polynomial invariant is described by three axioms:

AXIOM 1.To each oriented knot or linkKthere is associated a polynomial▼(z) € Z[z].Equivalent knots and links receive identical polynomials: K ~ K’↓▼k=▼k,.

AXIOM 2.IfK ~ 0 (the unknot)then▼ = 1.

AXIOM 3.Suppose that three knots or links differ at the site of one crossing as shown below:then ▼k= ▼k....

7. IV EXAMPLES AND SKEIN THEORY (pp. 42-69)

Here we continue calculating via the ax ioms of Chapter 3. Note that the end of Chapter 3 has provided a proof of the consistency of the axioms. First some recursions, and then some skein theory (see Conway [C1]).

These are alternately knots and links.

If${\nabla _n} = {\nabla _{{k_n}}}$then${\nabla _n} - {\nabla _{n - 2}} = z{\nabla _{n - 1}}$. Thus we have

For z = 1, this yields the Fibonacci Series 1,1,2,3,5,. . ..Here Fibonacci shows that none of these are equivalent (along with component count for the first two).

Thus${\nabla _k} = {\nabla _{\bar k}} = {\left( {1 + {z^2}} \right)^2}$. (Why?) The knots K and K receive the same polynomial, but they are not equivalent. The proof of...

8. V DETECTING SLICES AND RIBBONS— A FIRST PASS (pp. 70-92)

Here is a ribbon knot:

It is called a ribbon knot because it forms the boundary of a “ribbon” that is immersed into three-dimensional space with ribbon singularities:

The ribbon, or ribbon disk, is the image α(D²) of a mapping α : D²→R³ C S³ whose only singularities are of the form illustrated above. Thus each component of the image singular set consists in a pair of closed intervals in D² one with end points on the boundary of D², one entirely interior to D².

Exercise. Show that every knot K C S³ bounds a disk with clasp singularities....

9. VI MISCELLANY (pp. 93-180)

This chapter is a grab bag of bits of mathematical knottery, pictures, tricks, observations. We take this disk and hang him free, one side smile and one side frown. And yet we’ll turn him around and around. Four turns by k return a smile, but quite a tangle up above.

Recall the curl that two twists make:

Then k⁴ is two curls, and thence this isotopy:

So k⁴ is equivalent to 1 , and not a motion of his face!

And now to i and j as well. Each has his own small tail (tale) to tell.

Big curl over....

10. VII SPANNING SURFACES AND THE SEIFERT PAIRING (pp. 181-207)

Let’s begin by determining the genus of the Seifert surface. Recall that the Seifert surface is a surface obtained by Seifert’s algorithm from a knot or link diagram (see Chapter 5.) Also, if F is an orientable surface, then the genus of F, g(F), is given by the formula

2g(F)=ρ(F) - μ(F) + 1

where ρ(F) is the rank of the first homology group H₁(F), and μ(F) is the number of boundary components of F. (We assume that μ(F) > 1.) In other words, the genus of F is the number of handles in the standard form for F’ where...

11. VIII RIBBONS AND SLICES (pp. 208-228)

LEMMA 8.1. Let M³ be a compact, connected orientable 3-manifold with boundary. Denote the boundary by ∂M³ = N². Let j : N →M be the inclusion, and let H*=H*(:Q) denote homology with rational coefficients. Let K = Kernel(j : H₁(N) → H₁(M)). Then, as vector spaces over Q, dim K=[1/2] dim H₁(N).

Proof. Look at the homology exact sequence

We have denoted dimensions by a,b,c,d,1 ,k, and used Poincare-Lefschetz Duality

Thus half the cycles die into the interior of M. The typical case is a handlebody with the meridians representing K:

The lemma is...

12. IX THE ALEXANDER POLYNOMIAL AND BRANCHED COVERINGS (pp. 229-251)

Let K C S³ be an oriented knot or link, and F C S³ a connected oriented spanning surface for K. Let θ : H1(F) x H1(F) → Z be the Seifert pairing.

DEFINITION 9.1. Two polynomials f(t), g(t) € Z[t] are said to be balanced (written f = g) if there is a nonnegative integer n such that ± tnf(t) = g(t) or ± tng(t) = f(t). This definition is also extended to rational functions. Thus$t + \frac{1}{t}$and t² + 1 are balanced and we write$t + \frac{1}{t} = {t^2} + 1$.

DEFINITION 9.2. Let K, F, θ be as above. The Alexander polynomial....

13. X THE ALEXANDER POLYNOMIAL AND THE ARF INVARIANT (pp. 252-261)

Recall that we have defined, for a knot K, the invariant A(K) € Z₂ via A(K) = a₂(K) (modulo-2) where a₂(K) is the second Conway coefficient. And we showed (Chapter V) that A(K) = 0 for ribbon knots. In this chapter we will show that A(K) is identical with the Arf invariant, ARF(K), which is the Arf invariant of a mod-2 quadratic form related to K.

First recall that a mod-2 quadratic form q is a mapping q : V → Z₂ where V is a Z₂-vector space such that V has a bilinear symmetric pairing ( . ) :...

14. XI FREE DIFFERENTIAL CALCULUS (pp. 262-270)

Here is a picture of the figure eight (B) and its universal covering space E. Now π₁(B) = (x,y| ), the free group on two generators. Let G = (x,y| ) and note that G is the group of automorphisms of E over B. Thus E has “generating” 1-simplices X and Y as depicted. X is the lift of x as an element in π₁ starting at *. Y is the lift of y. By regarding E as the set of translates of X and Y under the action of G, we can write the lift of any word w...

15. XII CYCLIC BRANCHED COVERINGS (pp. 271-298)

In Chapter IX we illustrated Seifert’s approach to branched covering spaces. In this chapter we turn to these spaces in a more systematic, and partially four-dimensional manner. Let K be an oriented knot or link in the oriented sphere S³. Then there is a homomorphism φ: π₁ (S³-K) → Z defined by the equation φ(α) = lk(a, K) where this denotes the sum of the linking numbers with individual components of K. Let φb: π₁(S³-K) → Z/nZ denote the composition of φ with the surjection Z → Z/nZ. The n-fold cyclic covering of K is by definition the covering...

16. XIII SIGNATURE THEOREMS (pp. 299-326)

We will return to knots and cyclic branched coverings in the next section. Here we prove general results about the signature of a manifold. (Unless otherwise specified, homology is taken with real coefficients.)

THEOREM 13.1 (Novikov Addition Theorem). Let M⁴nbe a 4n-dimensional manifold that is obtained by gluing two manifolds along a common boundary. Then the signature of M Is the sum of the signatures of these manifolds. Thus if M = Y+U Y_ where Y+ and Y_ are 4n-manifolds, X = Y+∩ Y-is a 4n-1 manifold (X = əY + and X=əy_), then σ(M)= σ(Y+)...

17. XIV G-SIGNATURE THEOREM FOR FOUR MANIFOLDS (pp. 327-331)

The upshot of our work so far is the g-signature theorem for 2-manifolds:

$\sigma \left( {{M^2},g} \right) = \sum - i\cot \left( {{\theta _p}/2} \right)$

where this sum extends over all isolated fixed points p. θpdenotes the rotation angle at p measured in radians.

We now turn to the case of four-manifolds. We will only consider oriented fixed point sets and we make no classification of possible fixed point sets. Thus this is a quick trip through the g-signature theorem. The reader is referred to [G2] and [AS] for more details and other points of view.

We know that the g-signature is a sum of contributions from the fixed point...

18. XV SIGNATURE OF CYCLIC BRANCHED COVERINGS (pp. 332-336)

(Chapters XV to XVII follow the paper [CG].) Let N→N be an m-fold cyclic branched covering of closed (compact, oriented) m-manifolds, with branch set a surface F C N. Let F =π-¹(F) C N denote the inverse image of the branch set.

Exerci se 15.1. Show that [F]² = 1/m[F]² where [F]² denotes the self-intersection of F in N, and [F]² denotes the self-intersection of F in N. Hint: let S be a closed surface, and let L(S) = isomorphism classes of complex line bundles over S (hence D²-bundles). Then

L(S) = [S, BS¹] = [S, CP] = H²(S; Z)...

19. XVI AN INVARIANT FOR COVERINGS (pp. 337-344)

Let M be a closed, oriented, 3-manifold and suppose there is a surjective homomorphism φ : H₁(M;Z) → Zm = Z /mZ. Let M → M be the corresponding covering space, with the generator of covering translations corresponding to 1 € Zm.

Suppose that for some positive integer n there exists an mn-fold cyclic branched covering of 4-manifolds W→W branched along F C Interior(W) such that d(W→W) = n(M→M), and such that the covering translation of W, inducing a 2π/m rotation in fibers normal to F, restricts on each component of əW to the canonical covering transformation determined by Φ....

20. XVII SLICE KNOTS (pp. 345-354)

Let K C S³ be a knot. Let q be a fixed prime, and let be the Mnbe the qn-fold branched cyclic cover of S³, branched along K (n = 1,2,3, . . .). As we shall see, H*(Mn= Q) = H*(S³: Q).

Suppose we have an epimorphism Φ: H₁(M) → Zm.

Since the branched covering projection Mn→ M₁ induces a surjection on π₁, hence on H₁, the composition Фn:Фοπninduces epimorphisms Фn: H₁(Mn)

The main theorem of this section is due to Casson and Gordon [CG]

THEOREM 17.1. Suppose K is a slice knot. Then there is...

21. XVIII CALCULATING σr FOR GENERALIZED STEVEDORE’S KNOTS (pp. 355-365)

We consider the knots Kk(k € Z) as shown below:

Kkhas a Seifert surface of genus 1 with corresponding Seifert matrix. Thus, as we showed in Example 8.4 of Chapter VIII, Kkis algebraically slice exactly when 4k+l = l² for some integer l. The first two values give the unknot and the stevedore's knot which are indeed slice. However,

THEOREM 18.1 (Casson and Gordon). Kkis slice only if k = 0,2.

Proof: If Kkis slice, then it is algebraically slice. Hence, for some fixed k such that 4k+l = l², let Mnbe the 2n-fold...

22. XIX SINGULARITIES, KNOTS AND BRIESKORN VARIETIES (pp. 366-416)

A good reference for this section is Milnor's book [M3],Singular Points of Complex Hyper surfaces;also the original papers of pham [PH] and Brieskorn [BK] and the notes by Hirzebruch and Zagier [HZ]. There is a large and continuing literature on this topic. Our intent here is to give a survey of example and constructions. As we shall see, the subject of the topology of algebraic singularities is intimately related to knot theory and to the structure of branched these ideas come together, so that the link of a Brieskorn singularity may be described completely in terms of knots...

23. APPENDIX: GENERALIZED POLYNOMIALS AND A STATES MODEL FOR THE JONES POLYNOMIAL (pp. 417-443)
24. TABLES: KNOT TABLES AND THE L-POLYNOMIAL (pp. 444-473)
25. REFERENCES (pp. 474-480)