Quandle Cocycle Invariants for Knots, Knotted Surfaces and

Knotted 3-Manifolds

Witold Rosicki (Gdańsk)

Definition A quandle is a set X with a binary operation (a,b)→ab such that:

1) For any aX, aa=a. 2) For any a,b X, there is a unique c X such that cb=a. 3) For any a,b,c X , we have (ab)c=(ac)(bc).

A rack is a set with a binary operation that satisfies 2) and 3).

A kei or (involutory quandle) is a quandle with the additional property: (ab)b=a for any a,b X.

Kei- Takasaki 1942, rack- Conway 1959, quandle Joyce 1978, Matveev 1982

Examples:

1) X={0,1,…,n-1}, ij= 2j-i mod n.2) X=G a group, ab= b-nabn.

Definition Let X be a fixed quandle and let KK be a given diagramof an oriented classical link and let R be the set of over-arcs (bridges). A quandle coloring is a map c:R→X such that:

c(α)=a

c(β)=b

c(γ)=ab

for every crossing.

The Reidemeister moves

I II

III

The Reidemeister moves preserves the quandle coloring.

I II

a aaa a

b a b a

c

cb

III

a

c

ac

bc (ac)(bc)

a b c

ab

c bc

b c

(ab)c

Similarly we can define the quandle coloring for knotted surfaces

Definition Let X be a fixed quandle and let KK be a given diagramof an oriented knotted surface in R4 with a given regular projection p:R4→R3 . Let D be the closure of the set of higher points of the double points of the projection p and let R be the set of regions, which we obtain removing D fom our surface. A quandle coloring is a map c:R→X such that:

ab

ab

Definition Two knotted surfaces in R4 are equvalent if there exist an ambiet isotopy of R4 maping one onto other.

Theorem (Roseman 1998) Two knotted surfaces are equivalent iff one of the brokensurface diagram can be obtained from the other by a finite sequence of moves from the list of the 7 moves, presented belowand ambient isotopy of the diagrams in 3-space.

Another presentation of Roseman moves:

The Roseman moves preserves the quandle coloring, so the quandle coloring is an invariant of an equivalent class.

Similarly we can define the quandle coloring of knotted 3-manifolds in 5-space. There exist 12 Roseman moves such that two knotted 3-manifolds in 5-space are equivalent iff there exist a finite sequence of these moves between their diagrams. The Roseman moves preserve the quandle coloring of3-manifolds in 5-space.

Homology and Cohomology Theories of Quandles.(J.S.Carter, D.Jelsovsky, S.Kamada, L.Langford and M.Saito1999, 2003) Let CR

n(X) be the free abelian group generated by n-tuples (x1,…,xn) of elements of a quandle X. Define a homomorphism∂n: CR

n(X)→CRn-1(X) by

)],...,,,,...,,(),...,,,...,,[(

),...,,(

212

1211121

21

nii

n

iiiiinii

nn

xxxxxxxxxxxxxx

xxx

for n≥2 and ∂n =0 for n≤1.

Then CR(X)= {CR

n (X),∂n } is a chain complex.

Let CDn(X) be the subset of CR

n(X) generated by n-tuples (x1,…,xn ) with xi =xi+1 or some i {1,…,n-1} if n≥2; otherwise CD

n(X)=0. CD

*(X) is a sub-complex of CR*(X)

CQn(X)= CR

n(X)/CDn(X) with ∂’n induced homomorphism.

For an abelian group G, define the chain and cochain complexes:

CW*(X;G)= CW

*(X) G, ∂=∂ id C*W(X;G)= Hom(CW

*(X),G), δ=Hom(∂,id)

where W= D,R,Q.

As usually, ker ∂n = ZWn (X;G) and im ∂n+1= BW

n(X;G)

HWn(X;G)= Hn(CW

*(X;G))= ZWn(X;G)/BW

n(X;G)

ker δn =ZnW(X;G) and im δn-1= Bn

W(X;G)

HnW(X;G)= Hn(C*W(X;G))= Zn

W(X;G)/BnW(X;G)

Example: A function Φ:X×X→G for which the equalities

Φ(x,z)+Φ(xz,yz)=Φ(xy,z)+Φ(x,y) andΦ(x,x)= 0

are satisfied for all x,y,z X is a quandle 2-cocycle Φ Z2

Q(X;G)

The quandle cocycle knot invariant:

C

yxK

)(),()(

xy

xy

(x,y)

C

yxK

),()()(

I II

a aaa a

b a b a

c

cb=a

III

a

c

ac

(ac)(bc)

a b c

ab

c bc

b c

(ab)c

(a,a)=0-(a,b)

(a,b)

(b,c)

bc

(ac,bc)(a,c)

(a,b)

(ab,c)

(b,c)

(a,c)+(b,c)+(ac,bc)=(a,b)+(b,c)+(ab,c)

(a,c)+(ac,bc)=(a,b)+(ab,c)

Example (from picture 14): A function Φ:X×X→G for which the equalitiesΦ(x,z)+Φ(xz,yz)=Φ(xy,z)+Φ(x,y) and Φ(x,x)= 0are satisfied for all x,y,z X is a quandle 2-cocycle Φ Z (X;G)

a

bc

cb

a bcac

(ac)(bc)

III

a

c

ac

(ac)(bc)

a b c

ab

c bc

b c

(ab)c

c

a

ab (ab)cb

bc

III

b

a bcac

c

a

ab (ab)cb

bc

III

(b,c)

(a,c)

(ac,bc)

(a,b)

(ab,c)

(b,c)

(a,c)+(b,c)+(ac,bc)=(a,b)+(b,c)+(ab,c)

(a,c)+(ac,bc)=(a,b)+(ab,c)

Let C is a given coloring of a knotted surface, then for each triple point we have assigned a 3-cocycle .

a

b

c

(a,b,c)

C

zyxK

)(),,()(

C

zyxK

),,()()(

We can define a quandle 3-cocycle invariant of the positionof a surface in a 4-space. The Roseman moves preservethis invariant.

The first sum is taken over all possible colorings of the given diagram K of the surface in 4-space and the second sum (product) is taken over all triple points.

This theory is described in the book of S.Carter, S.Kamada and M.Saito„Surface in 4-Sace”.

For a knotted 3-manifold in 5-space and its projection we can define similar:

C

vzyxK

)(),,,()(

orC

vzyxK

),,,()()(

where the first sum is taken over all possible colorings of the given diagram K of the 3-manifold in 5-space and the second sum (product) is taken over all with multiplicity 4 points.

Φ is an invariant of position if all 12 Roseman moves preserve it.

Points with the multiplicity 4 appear only in 3 Roseman moves:e, f, l .

In „e” two points τ1, τ2 with multiplicity 4 and opposite orientations appear. Therefore ε(τ1)= -ε(τ2) and ε(τ1)Φ-ε(τ2)Φ=0.

In „l” Φ=0 because two colors must be the same.

The calculation in „f” is essential.We will calculate similarly on a picture in 3-space, similarly like we calculated on a line in the case of a classical knot. We will project the 4-space onto „the horizontal” 3-space. „The vertical” 3-spaces will represent as planes. „The diagonal” 3-space will project onto whole 3-space. The red triangle will represent the plane of the intersection of the horizontal and the diagonal 3-spaces.

x1

x2

x3

x5

x4

x1

x2

x 3

x5

x4

x1x2

x1x3x2x3

5

2

3

4

1

1) +Φ(x2,x3,x4,x5)

2) -Φ(x1x2, x3,x4,x5)

3) +Φ(x1x3,x2x3,x4,x5)

4) +Φ(x1,x2,x3,x5)

5) -Φ(x1,x2,x3,x4)

The orientation in the points with multiplicity 4 is given by normal vectors (represented by red arrows) :[1,0,0,0], [0,-1,0,0], [0,0,1,0], [1,-1,1,1], [0,0,0,-1] .

x1

x2

x3

x5

x4

x2

x 3

x5

x4

x1x2

x1x3

5

23

1

4

x4x5

x1

x1x4

x2x3

x2x5

x 3x 5

1) +Φ(x2,x3,x4,x5)2) -Φ(x1, x3,x4,x5)3) +Φ(x1,x2, x4,x5)4) +Φ(x1x4,,x2x4,x3x4,x5)5) -Φ(x1x5,x2x5,x3x5,x4x5)

If Φ is a 4-cocycle thenδ(Φ)(x1,x2,x3,x4,x5)= Φ(∂(x1,x2,x3,x4,x5 ))==Φ(x1,x3,x4,x5) - Φ(x1x2,x3,x4,x5) - Φ(x1,x2,x4,x5)++ Φ(x1x3,x2x3,x4,x5)+Φ(x1,x2,x3,x5) – Φ(x1x4,x2x4,x3x4,x5) –- Φ (x1,x2,x3,x4)+ Φ(x1x5,x2x5,x3x5,x4x5) =0

=

+Φ(x2,x3,x4,x5) -Φ(x1x2, x3,x4,x5)+Φ(x1x3,x2x3,x4,x5)+Φ(x1,x2,x3,x5)-Φ(x1,x2,x3,x4)

+Φ(x2,x3,x4,x5)-Φ(x1, x3,x4,x5)+Φ(x1,x2, x4,x5)+Φ(x1x4,,x2x4,x3x4,x5)-Φ(x1x5,x2x5,x3x5,x4x5) This observation probably will be a part of a paper which we

are going to write with Jozef Przytycki.