Introduction to Knot theoryIntroduction to Knot theory

Minhoon Kim

POSTECH

August 4, 2010

What is the What is the Knot theory?Knot theory?

=

We want to say that they are same !

Knot : circle in R3 or S3 e.g.

=

When two knots are same?When two knots are same?

K0, K1 : Knots i.e. Ki=fi(S1), where fi:S1→S3 is 1-1 (i=0,1)

We say K0=K1 if there is f:S1Χ[0,1]→S3 satisfying (1),(2)

(1) f0(x)=f(x,0) and f1(x)=f(x,1) for all x in S1

(2) ft:S1→S3 : is 1-1 for all t in [0,1], where ft(x) = f(x,t)

ReidemeisterReidemeister moves(1,2,3)moves(1,2,3)

Reidemeister move1

Reidemeister move2

Reidemeister move3

Theorem(Reidemeister) 3 moves are enough !

ExampleExample

i.e. =

Seifert SurfaceSeifert Surface

Definition F in S3 : Seifert surface

if F is (orientable) surface, ∂F = K

Examples

Figure eight knot Trefoil knot

Classification of closed surfacesClassification of closed surfaces

F2 : compact, orientable, connected surface (∂F=0)

⇒ F2 = Sg for some g ≥ 0

S0 S1 S2 S3

g(F) := genus of F

Χ(F) = 2-2g(F), where Χ(F) : Euler characteristic of F

Genus of KnotGenus of Knot

F:Seifert surface of K, K:knot

g(F):=(1-Χ(F))/2 ⇔ Χ(F) = 1-2g(F)

g(K) := min {g(F)|F in S3, ∂F = K}

Note : ∂F is not empty!

g(K) : genus of a knot K

g(K)=0 ⇔ K is trivial knot

Seifert formSeifert formF : Seifert surface of K with orientation (normal direction)

Define Θ:H1(F)ΧH1(F)→ Z (Seifert form) as follows

Θ(x,y)= lk(x,y+), y+:parallel of y along (+) normal direction.

lk(x,y) is linking number of x and y given by

lk(x,y) = 1 lk(x,y) = -1

Calculating Seifert form (Example)Calculating Seifert form (Example)

1. Choose generator of H1(F)=Z4, {x1,x2,x3,x4}

2. Fix orientation of F and calculate!

Matrix representation of Θ with basis {x1,x2,x3,x4}

Slice knot, Alexander polynomialSlice knot, Alexander polynomial

K : slice if ∃D(disk) in B4 with ∂D=K, ∂B4=S3

Let ΔK(t) = det(t1/2Θ-t-1/2ΘT)

ΔK(t) := Alexander polynomial of K

K: slice ΔK(t) = f(t)f(t-1) for some polynomial f

Signature of KnotSignature of Knot

Signature of a matrix is defined by

# of positive eigenvalues - # of negative eigenvalues

Signature of a knot : Signature of Θ+ΘT

Slice knot : zero signature

Slice genus and knot invariantsSlice genus and knot invariants

1. Ozsváth-szabó τ-invariant (from Knot Floer homology)

2. Rasmussen s-invariant (from Khovanov homology)

τ(K) ≤ gs(K), s(K) ≤ 2gs(K) (bound for gs(K))

gs(K) = min {g(F)|F in B4, ∂F = K, ∂B4=S3}

gs(K) :=slice genus.

K:slice ⇔ gs(K)=0

Interplay with 4Interplay with 4--manifold theorymanifold theoryConjecture (Smooth Poincare Conjecture in 4 dimension)

M4 with π*(M4)=π*(S4)⇒ M4=S4(diffeomorphic)?

Theorem (Freedman,Gompf,Morrison,Walker)

If ‘some’ knot K satisfies s(K) ≠0, then SPC4 is false.

4 manifold problemSlice knot problem

By Freedman’s work, this conjecture ⇔

Are there M4 with M4=S4(homeo.) & M4≠S4(diffeo.)?

AddendumAddendumQuestion : Homeomorphic but not Diffeomorphic?

Answer : Possible!

J.milnor’s 1st example : 28 7-spheres using Pontryagin class

Kervaire,Milnor : Differentiable structures on Sn(n≠4)

Donaldson,Freedman : infinitely many R4

Floer : uncountably many R4

ReferencesReferences1. S.K. Donaldson, An application of Gauge theory to four-manifold topology, J. Diff.

Geo. (1983)

2. M.Freedman, The topology of four-manifolds J. Diff. Geo. (1982)

3. M.Freedman, R.Gompf, S.Morrison, K.Walker, Man and machine thinking about

smooth 4 dimensional Poincare conjecture, arxiv:0906.5177v2[math:GT]

4. Kauffman, On Knots(AM-115), Princeton press (1987)

5. M.A.Kervaire, J.Milnor, Groups of homotopy spheres, Ann. of Math. (1963)

6. J.Milnor, Characteristic classes(AM-86), Princeton press (1974)

7. J.Milnor, On manifolds homeomorphic to 7-spheres, Ann. of Math. (1956)

8. J.Milnor, Poincare conjecture and classification of 3-manifolds, Notices of

AMS(2003)

9. Ozsváth-szabó, Knot floer homology and the four-ball genus, Geom. Top. (2003)

10. Rolfsen, Knots and Links, AMS Chelsea publishing (2003)

11. J.Rasmussen, Khovanov homology and slice genus, to appear in Inv. of Math. (2004)

12. C.H. Taubes, Gauge theory on asympotically periodic 4-manifolds J. Diff. Geo. (1987)

Thank Thank You !!!You !!!