Introduction to Knot theoryIntroduction to Knot theory
Minhoon Kim
POSTECH
August 4, 2010
What is the What is the Knot theory?Knot theory?
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We want to say that they are same !
Knot : circle in R3 or S3 e.g.
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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 !!!