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)→a  b such that: 1) For any a  X, a  a=a. 2) For any a,b  X, there is a unique c  X such that c  b=a. 3) For any a,b,c  X, we have (a  b)  c=(a  c)  (b  c). 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: (a  b)  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}, i  j= 2j-i mod n. 2)X=G a group, a  b= b -n ab n. Definition K Let X be a fixed quandle and let K be a given diagram of 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)=a  b for every crossing.

The Reidemeister moves I II III

The Reidemeister moves preserves the quandle coloring. I II a aaaa a a b ab a c cbcb III a c acac bcbc (a  c)  (b  c) a bc abab c bcbc b c (a  b)  c

Similarly we can define the quandle coloring for knotted surfaces Definition K Let X be a fixed quandle and let K be a given diagram of an oriented knotted surface in R 4 with a given regular projection p:R 4 →R 3. 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: a b abab

Definition Two knotted surfaces in R 4 are equvalent if there exist an ambiet isotopy of R 4 maping one onto other. Theorem (Roseman 1998) Two knotted surfaces are equivalent iff one of the broken surface diagram can be obtained from the other by a finite sequence of moves from the list of the 7 moves, presented below and 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 of 3-manifolds in 5-space.

Homology and Cohomology Theories of Quandles. (J.S.Carter, D.Jelsovsky, S.Kamada, L.Langford and M.Saito 1999, 2003) Let C R n (X) be the free abelian group generated by n-tuples (x 1,…,x n ) of elements of a quandle X. Define a homomorphism ∂ n : C R n (X)→C R n-1 (X) by for n≥2 and ∂ n =0 for n≤1. Then C R  (X)= {C R n (X),∂ n } is a chain complex.

Let C D n (X) be the subset of C R n (X) generated by n-tuples (x 1,…,x n ) with x i =x i+1 or some i  {1,…,n-1} if n≥2; otherwise C D n (X)=0. C D * (X) is a sub-complex of C R * (X) C Q n (X)= C R n (X)/C D n (X) with ∂’ n induced homomorphism. For an abelian group G, define the chain and cochain complexes: C W * (X;G)= C W * (X) G, ∂=∂ id C* W (X;G)= Hom(C W * (X),G), δ=Hom(∂,id) where W= D,R,Q.

As usually, ker ∂ n = Z W n (X;G) and im ∂ n+1 = B W n (X;G) H W n (X;G)= H n (C W * (X;G))= Z W n (X;G)/B W n (X;G) ker δ n =Z n W (X;G) and im δ n-1 = B n W (X;G) H n W (X;G)= H n (C* W (X;G))= Z n W (X;G)/B n W (X;G) Example: A function Φ:X×X→G for which the equalities Φ(x,z)+Φ(x  z,y  z)=Φ(x  y,z)+Φ(x,y) and Φ(x,x)= 0 are satisfied for all x,y,z  X is a quandle 2-cocycle Φ  Z 2 Q (X;G)

The quandle cocycle knot invariant: x y xyxy  (x,y)

I II a aaaa a a b ab a c c  b=a III a c acac (a  c)  (b  c) a bc abab c bcbc b c (a  b)  c  (a,a)=0 -  (a,b)  (a,b)  (b,c) bcbc  (a  c,b  c)  (a,c)  (a,b)  (a  b,c)  (b,c)

 (a,c)+  (b,c)+  (a  c,b  c)=  (a,b)+  (b,c)+  (a  b,c)  (a,c)+  (a  c,b  c)=  (a,b)+  (a  b,c) Example (from picture 14): A function Φ:X×X→G for which the equalities Φ(x,z)+Φ(x  z,y  z)=Φ(x  y,z)+Φ(x,y) and Φ(x,x)= 0 are satisfied for all x,y,z  X is a quandle 2-cocycle Φ  Z (X;G)

a b c

c b a bcbc acac (a  c)  (b  c) III a c acac (a  c)  (b  c) a bc abab c bcbc b c (a  b)  c c a abab b bcbc III

b a bcbc acac c a abab (a  b)  c b bcbc III  (b,c)  (a,c)  (a  c,b  c)  (a,b)  (a  b,c)  (b,c)  (a,c)+  (b,c)+  (a  c,b  c)=  (a,b)+  (b,c)+  (a  b,c)  (a,c)+  (a  c,b  c)=  (a,b)+  (a  b,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)

We can define a quandle 3-cocycle invariant of the position of a surface in a 4-space. The Roseman moves preserve this 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: or 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 τ 1, τ 2 with multiplicity 4 and opposite orientations appear. Therefore ε(τ 1 )= -ε(τ 2 ) and ε(τ 1 )Φ-ε(τ 2 )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.

x1x1 x2x2 x3x3 x5x5 x4x4

x1x1 x2x2 x3x3 x5x5 x4x4 x1x2x1x2 x1x3x1x3 x2x3x2x

1)+Φ(x 2,x 3,x 4,x 5 ) 2)-Φ(x 1  x 2, x 3,x 4,x 5 ) 3)+Φ(x 1  x 3,x 2  x 3,x 4,x 5 ) 4)+Φ(x 1,x 2,x 3,x 5 ) 5)-Φ(x 1,x 2,x 3, x 4 ) 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].

x1x1 x2x2 x3x3 x5x5 x4x4

x2x2 x3x3 x5x5 x4x4 x1x2x1x2 x1x3x1x x4x5x4x5 x1x1 x1x4x1x4 x2x3x2x3 x2x5x2x5 x3x5x3x5

1)+Φ(x 2,x 3,x 4,x 5 ) 2) -Φ(x 1, x 3,x 4,x 5 ) 3) +Φ(x 1,x 2, x 4,x 5 ) 4) +Φ(x 1  x 4,, x 2  x 4,x 3  x 4,x 5 ) 5) -Φ(x 1  x 5,x 2  x 5,x 3  x 5,x 4  x 5 )

If Φ is a 4-cocycle then δ(Φ)(x 1,x 2,x 3,x 4,x 5 )= Φ(∂(x 1,x 2,x 3,x 4,x 5 ))= =Φ(x 1,x 3,x 4,x 5 ) - Φ(x 1  x 2,x 3,x 4,x 5 ) - Φ(x 1,x 2,x 4,x 5 )+ + Φ(x 1  x 3,x 2  x 3,x 4,x 5 )+Φ(x 1,x 2,x 3,x 5 ) – Φ(x 1  x 4,x 2  x 4,x 3  x 4,x 5 ) – - Φ (x 1,x 2,x 3,x 4 )+ Φ(x 1  x 5,x 2  x 5,x 3  x 5,x 4  x 5 ) =0 = +Φ(x 2,x 3,x 4,x 5 ) -Φ(x 1  x 2, x 3,x 4,x 5 ) +Φ(x 1  x 3,x 2  x 3,x 4,x 5 ) +Φ(x 1,x 2,x 3,x 5 ) -Φ(x 1,x 2,x 3,x 4 ) +Φ(x 2,x 3,x 4,x 5 ) -Φ(x 1, x 3,x 4,x 5 ) +Φ(x 1,x 2, x 4,x 5 ) +Φ(x 1  x 4,,x 2  x 4,x 3  x 4,x 5 ) -Φ(x 1  x 5,x 2  x 5,x 3  x 5,x 4  x 5 ) This observation probably will be a part of a paper which we are going to write with Jozef Przytycki.