A Survey of Quandles with some recent developments

Transcription

1 A Survey of Quandles with some recent developments University of South Florida QQQ 2016

2 Knots and their diagrams Overview of knot diagrams and Quandles A knot is the image of a smooth embeding S 1 R 3 or S 3. Two knots K and K are called isotopic if K is obtained from K by continuous deformation with no self-intersection at any time. Technically speaking, if there exists a smooth family of homeomorphisms h t : R 3 R 3 for t [0, 1] such that h 0 = Id and h 1 (K) = K. One approach to study knot theory is combinatorial: using what s called diagrams (projection to the plane showing overand under-crossings).

3 Knots and their diagrams Overview of knot diagrams and Quandles For a well-known set S we call the map I : {knots} S an isotopy invariant of knots, if I (K) = I (K ) for any two isotopic knots K and K. Knot diagram = Image of a knot by a projection R 3 R 2 (finitely many transversal double points (crossings: over- and under-). Reidemeister s theorem: {Knots}/ isotopy of R 3 = {Knot Diagrams} / RI, RII, RIII and isotopy of R 2.

4 History of Quandles 1982 Sergei Matveev Distributive groupoids in knot theory (in Russian) David Joyce A classifying invariant of knots, the knot quandle Egbert V. Brieskorn Automorphic sets and singularities. Around 1990 Dehornoy left Distributive sets.

5 Definition of a Quandle A quandle is a set X with a binary operation (a, b) a b such that: For any a X, a a = a (idempotency) For any a, b X, there exists a unique c X such that c a = b For all a, b, c X (a b) c = (a c) (b c) (Self-distributivity).

6 Examples of Quandles Any set with a b = a (trivial quandle). The set of integers mod n, Z n with a b = 2b a (dihedral quandle). The typical example is a group G with conjugation a b = b 1 ab.

7 Examples of Quandle Any Z[t, t 1 ]-module M is a quandle with a b = ta + (1 t)b. This is called Alexander quandle.

8 Knot Quandle The knot quandle (C est la raison d être) Consider a knot K and label the arcs a 1, a 2,... At each crossing, the relation is given by x y = z. The quandle generated by the labeling of arcs with relations at each crossing is called the knot quandle and denoted Q(K).

9 The fundamental quandle is a complete invariant Theorem ( Matveev and Joyce, independently 1982) Two knots K and L are equivalent if and only if Q(K) and Q(L) are isomorphic as quandles. In other words the knot quandle is a complete invariant.

10 Quandle homomorphisms A map f : (X, ) (Y, ) is a quandle homomorphism if f (x y) = f (x) f (y) for all x, y X. The second condition in the definition of quandle: a, b X, there exists a unique c X such that c a = b can be restated as the Right Multiplication map R a : X X, R a (x) = x a is a bijection.

11 Quandle homomorphisms The third condition in the definition of quandle: (a b) c = (a c) (b c) means that R c (a b) = R c (a) R c (b) Thus x X, R x is an automorphism called symmetry at x. Also R c R b = R b c R c R c R b R c 1 = R b c

12 Automomorphism groups of quandles The set of all automorphisms of X is denoted Aut(X ). The subgroup generated by all symmetries of X is called the Inner automorphism group of X, denoted Inn(X ). The equation R c R b R c 1 = R b c implies that the map X Inn(X ), x R x is a quandle homorphism.

13 Colorings of knot diagrams In general it is difficult to distinguish the isomorphism types of two given presentations of quandles. A convenient method is to use a representation to a finite quandle X : a quandle homomorphism ρ : Q(K) X. We call it a coloring of K by X. The generators of the fundamental quandle Q(K) correspond to the arcs in a diagram. Fenn and Rourke proved that the cardinality #Col X (K) of colorings is a knot invariant.

14 A group from quandle: The Associated Quandle Interpret the operation of a quandle X as a conjugation to get the Associated group G X := F (X )/N where N is the normal subgroup generated by (x y)yx 1 y 1. Universal property: f : X G conj,!f # : G X G s.t. the diagram commutes ι X G X f f # G conj id G. Hom Grp (G X, G) = Hom Qdle (X, G conj )

15 Low-dimensional cocycles Low Dimentional Cocycles: 2-cocycles Let A be an abelian group. A 2-cocycle is a function Φ : X X A such that Φ(x, y) + Φ(x y, z) = Φ(x, z) + Φ(x z, y z).

16 Low-dimensional cocycles Low Dimentional Cocycles: 3-cocycles 3-cocycle: A function Ψ : X X A such that Ψ(x, y, z) + Ψ(x, z, w) + Ψ(x z, y z, w) = Ψ(x y, z, w) + Ψ(x w, y w, z w) + Ψ(x, y, w).

17 Low-dimensional cocycles Quandle homology: The chain complex Let C R n (X ) = free abelian group generated by n-tuples (x 1,..., x n ) X n. Define n : C R n (X ) C R n 1 (X ) by n = 0 for n 1 and for n 2, n (x 1, x 2,..., x n ) = n i=2 ( 1)i [(x 1, x 2,..., x i 1, x i+1,..., x n ) (x 1 x i, x 2 x i,..., x i 1 x i, x i+1,..., x n )] This defines a chain complex {C R n (X ), n } ( n n+1 = 0) which gives rack homology theory ( Fenn-Rourke-Sanderson).

18 The chain complex Low-dimensional cocycles Let C D n (X ) subset of C R n (X ) generated by n-tuples (x 1,..., x n ) with x i = x i+1 for some i {1,..., n 1} when n 2. For X quandle, n (Cn D (X )) Cn 1 D (X ) and define C Q n (X ) = C R n (X )/C D n (X ). For an abelian group A, define the chain and co-chain complexes C Q (X, A) = C Q (X ) A and C Q (X, A) = Hom(C Q (X ), A) H Q n (X, A) := H n (C Q (X, A)) and H n Q (X, A) := Hn (C Q (X, A))

19 Examples Low-dimensional cocycles Some Computations for Dihedral quandles H 2 Q (R 3, A) = 0, for any A and H 3 Q (R 3, Z 3 ) = Z 3 H 3 Q (R 3, A) = 0 for any A without order 3 elements H 2 Q (R 4, Z 2 ) = (Z 2 ) 4 H 2 Q (R 4, A) = A A for any A without order 2 elements H 2 Q (R 5, A) = 0, for any A and H 3 Q (R 5, Z 5 ) = Z 5 H 3 Q (R 5, A) = 0 for any A without order 5 elements

20 H 3 of Dihedral quandles Low-dimensional cocycles In fact Mochizuki, in 2003, proved that for p prime H 3 Q (R p, Z p ) = Z p and gave an explicit expression of generating 3-cocycle θ p (x, y, z) = 4(x y)(y z)z p 1 + (x y) 2 [(2z y) p 1 y p 1 ] +(x y) (2z y)p +y p 2z p p Note that the coefficients of (2z y) p + y p 2z p are divisible by p.

21 H 4 of Dihedral quandles Low-dimensional cocycles Maciej Niebrzydowski and Josef Przytycki proposed, in 2008: For p odd prime H Q 4 (R p) contains Z p. Conjecture: H Q n (R p ) = Z fn p Where f n are Delayed Fibbonacci numbers: f n = f n 1 + f n 3, and f 1 = f 2 = 0, f 3 = 1. This conjecture was solved by Frans Clauwens in

22 Low-dimensional cocycles A relation to group cohomology [ Etingof and Graña, 2003] Let G X =< x X y 1 xy = x y >=Associated group of X and M = G X -module, then H 1 (X, M) = H 1 (G X, M). Let Fun(X, A) = set of functions. If A is trivial G X -module then H 2 (X ; A) = H 1 (G X ; Fun(X, A)).

23 Let X be a finite quandle, A be an abelian group and ψ be a 2-cocycle ψ : X X A. Definition The State Sum invariant of a knot K is Φ(K) = ψ(x, y) ɛ(τ) C τ Where the product is taken over all crossings of the given diagram, the sum is over all possible colorings and ɛ(τ) is the sign of the crossing τ. Φ(K) is a knot invariant.

24 Explicit calculations The torus link T(4,2): Let X = R 4 = {0, 1, 2, 3} where i j = 2j i (mod 4) A = group of integers Z =< t > (multiplicative notation). with the 2-cocycle ψ(x, y) = t if x + y odd and ψ(x, y) = 1 if x + y even. Any pair (a, b) in R 4 R 4 colors K. The 8 pairs (a, b) with a + b being odd, each contributes t. All other pairs each contributes 1, thus Φ(K) = 8 + 8t.

25 Explicit calculations Let X = Z 2 [T, T 1 ]/(T 2 + T + 1), A = Z 2 =< u, u 2 = 1 >, and cocycle Φ(a, b) = u if a, b {0, 1, T + 1} and a b. For knots K (up to nine crossings) the Invariants Φ(K) are: 4(1 + 3u) for 3 1, 4 1, 7 2, 7 3, 8 1, 8 4, 8 11, 8 13, 9 1, 9 6, 9 12, 9 13, 9 14, 9 21, 9 23, 9 35, 9 37, 16(1 + 3u) for 8 18, and for 8 5, 8 10, 8 15, , 9 16, 9 22, 9 24, 9 25, , 9 36, 9 38, 9 39, , otherwise.

26 Andruskiewitsch and Grana s homology, 2003 They developed the theory of quandle cohomology further by defining a general quandle cohomology theory that encompasses all the above: Quandle Module: Let X be a quandle and A an abelian group. Consider a pair of families (η x,y ) x,y X Aut(A) and (τ x,y ) x,y X End(A) such that: η x y,z η x,y = η x y,x z η x,z η x y,z τ x,y = τ x y,x z η y,z τ x y,z = η x y,x z τ x,z + τ x y,x z τ y,z η x,x + τ x,x = 1

27 Andruskiewitsch and Grana s homology The free Z-algebra generated by η x,y, τ x,y modulo these relations is called the quandle algebra over X and denoted Z(X ). Example Let Λ = Z[t, t 1 ], Any Λ-module M is X -module by η x,y (a) = ta and τ x,y (b) = (1 t)b, for a, b M, x, y X.

28 Andruskiewitsch and Grana s homology Let C n (X ) = Z(X )X n = the free left Z(X )-module generated by n-tuples (x 1,..., x n ) X n. Define n : C n (X ) C n 1 (X ) by n (x 1, x 2,..., x n+1 ) = ( 1) n+1 n+1 i=2 ( 1)i η [x1,... ˆx i,...x n+1 ],[x i,...,x n+1 ](x 1, x 2,..., x i 1, x i+1,..., x n ) ( 1) n+1 n+1 i=2 (x 1 x i, x 2 x i,..., x i 1 x i, x i+1,..., x n ) +( 1) n+1 τ [x1,x 3,...x n+1 ],[x 2,x 3,...,x n+1 ] where [x 1, x 2,...x n ] = ((...(x 1 x 2 ) x 3 )...) x n. This gives a generalized quandle homology theory.

29 Twisted Quandles Definition A twisted-quandle is a triple (X,, f ) in which X is a set, is a binary operation on X, and f : X X is a map such that, for any x, y, z X, the identity (x y) f (z) = (x z) (y z), (1) and, for any x, y X, there exists a unique z X such that and for each x X, the identity z y = f (x). (2) x x = f (x). (3)

30 Twisted Quandles Example 1 Given any set X and map f : X X, then the operation x y = f (x) for any x, y X gives a twisted-quandle. 2 For any group G and any group endomorphism f of G, take the operation x y = y 1 xf (y). 3 Consider the Dihedral quandle R n, where n 2, and let f be given by f (x) = ax + b, a Z n, b Z n. The binary operation x y = f (2y x) = 2ay ax + b (mod n) gives a twisted-quandle structure.

31 The Associated Group of Twisted Quandles Definition Let (X,, f ) be a twisted-rack. Then there is a natural map ι mapping X to group, called the enveloping group of twisted-rack of X, and defined as G X = F (X )/ < x y = f (y)xy 1, x, y X >, where F (X ) denotes the free group generated by X.

32 Introduction Functoriality of the Associated group of Twisted Quandles Proposition Let (X,, f ) be a twisted-rack and G be a group. Given any twisted-rack homomorphism ϕ : X G conj, where G conj is a group together with a twisted-rack structure along a homomorphism group g, that is the multiplication is defined as a G b = g(b)ab 1. Then, there exists a unique group homomorphism ϕ : G X G which makes the following diagram commutative (X,, f ) ϕ (G conj, G, g) ι id G X G ϕ

33 Modules over twisted-quandles and Cohomology Definition Let (X,, f ) be a twisted-rack, A be an abelian group and g : X X be a homomorphism. A structure of X -module on A consists of a family of automorphisms (η ij ) i,j X and a family of endmorphisms (τ ij ) i,j X of A satisfying the following conditions: η x y,f (z) η x,y = η x z,y z η x,z (4) η x y,f (z) τ x,y = τ x z,y z η y,z (5) τ x y,f (z) g = η x z,y z τ x,z + τ x z,y z τ y,z (6)

34 Modules over twisted-quandles and Cohomology Remark If X is a twisted-quandle, a twisted-quandle structure of X -module on A is a structure of an X -module further satisfies τ f (x),f (x) g = (η f (x),f (x) + τ f (x),f (x) )τ x,x. Furthermore, if f, g = id maps, then it satisfies η x,x + τ x,x = id.

35 Example Let A be a non-empty set and (X, f ) be a twisted-quandle, and κ be a generalized 2-cocycle. For a, b A, let α x,y (a, b) = η x,y (a) + τ x,y (b) + κ x,y. Then, it can be verified directly that α is a dynamical cocycle and the following relations hold: η x y,f (z) η x,y = η x z,y z η x,z η x y,f (z) τ x,y = τ x z,y z η y,z τ x y,f (z) g = η x z,y z τ x,z + τ x z,y z τ y,z η x y,f (z) κ x,y + κ x y,f (z) = η x z,y z κ x,z + τ x z,y z κ y,z + κ x z,y z.

36 Cohomology of twisted-quandles Let (X,, f ) be a twisted-rack where f : X X is a twisted-rack morphism. We will define the most generalized cohomology theories of twisted-racks as follows: For a sequence of elements (x 1, x 2, x 3, x 4,..., x n ) X n define [x 1, x 2, x 3, x 4,..., x n ] = ((... (x 1 x 2 ) f (x 3 )) f 2 (x 4 ))... ) f n 2 (x n ). Notice that for i < n we have [x 1, x 2, x 3, x 4,..., x n ] = [x 1,..., ˆx i,..., x n ] f i 2 [x i,..., x n ]

37 Cohomology of twisted-quandles Theorem Consider the free left Z(X )-module C n (X ) = Z(X )X n with basis X n. = n : C n+1 (X ) C n (X ), where φ(x 1,..., x n+1 ) n+1 = ( 1) i φη [x1,...,ˆx i,...,x n+1 ],f {i 2} [x i,...,x n+1 ] (x 1,..., ˆx i,..., x n+1 ) i=2 n+1 i=2 ( 1) i φ(x 1 x i, x 2 x i,..., x i 1 x i, f (x i+1 ),..., f (x n+1 )) +( 1) n+1 φτ [x1,x 3,...,x n+1 ],[x 2,...,x n+1 ](x 2,..., x n+1 ).

38 Low dimensional Cocycles Example Let η be the multiplication by T and τ be the multiplication by S as in item 4 of Example 4. The 1-cocycle condition is written for a function φ : X A as T φ(y) + T φ(x) + φ(y) φ(x y) = 0. Note that this means that φ : X A is a quandle homomorphism. For ψ : X X A, the 2-cocycle condition can be written as T ψ(x 1, x 2 ) + ψ(x 1 x 2, f (x 3 )) = T ψ(x 1, x 3 ) + Sψ(x 2, x 3 ) + ψ(x 1 x 3, x 2 x 3 ).

39 Topological Quandles Definition A topological rack is a rack X which is a topological space such that the map X X (x, y) x y X is a continuous. In a topological rack, the right multiplication R x : X y y x X is a homeomorphism, for all x X.

40 Topological Quandles Example (The conjugation quandle) Let G be a topological group. The operation x y = yxy 1 makes G into a topological quandle which is denoted by Conj(G) and is called the conjugation quandle of G. In fact, any conjugacy class of G is a topological quandle with this operation.

41 Topological Quandles Example (The core quandle) Let G be a topological group. The operation x y = yx 1 y defines a topological quandle structure on G. This quandle will be denoted by Core(G) and we call it the core of G. Observe that this operation satisfies (x y) y = x. Any quandle in which this equation is satisfied is called an involutive quandle.

42 Topological Quandles Example (Symmetric manifold) First recall that a symmetric manifold M is a Riemannian manifold such that each point x M is an isolated fixed point of an involtutive isometry i x : M M. Given such manifold, every x M endows M with the structure of topological quandle by setting x y = i y (x).

43 Topological Quandles Example Let S n be the unit sphere of R n+1. Then, with respect to the operation x y = 2(x y)y x, x, y S n, where x y is the usual scalar product in R n+1, and the topology inherited from R n+1, S n is a topological quandle.

44 Topological Quandles Example Following the previous example, let λ and µ be real numbers, and let x, y S n. Then In particular, the operation λx µy = λ[2µ 2 (x y)y x]. ±x ±y = ±(x y) provides a structure of topological quandle on the projective space RP n.

45 Topological Quandles Example Let G be a topological group and σ be a homeomorphism of G. Let H be a closed subgroup of G such that σ(h) = h, for all h H. Then G/H is a quandle with operation [x] [y] := [σ(xy 1 )y], where for x G, [x] denotes the class of x in G/H. For example, one can consider the group G to be the group of rotations G = SO(2n + 1), H = SO(2n) and G/H = S 2n+1.

46 The Space of Colorings of a Knot by a Topological Quandle Topological quandles were considered in 2007 by Rubinsztein. He denoted the space of colorings J X (L) of the knot L by X. The invariant space of the figure eight knot 4 1 is J S 2(4 1 ) = S 2 RP 3 RP 3 while the collapsed (the singly graded homology obtained by collapsing along m = i j ) Khovanov homology for 4 1 is given by, Kh m (4 1 ) = H m (S 2 ){ 1} H m (RP 3 ){ 3} H m (RP 3 ){0}. There are similarities between the space of colorings of knots and Khovanov homology for all prime knots with up to seven crossings and for at least some eight-crossing knots.

47 S. Carter, M. Elhamdadi, and M. Saito. Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol. 2 (2002), R. Churchill, M. Elhamdadi, M. Green and M. Makhlouf. Twisted-Racks, Twisted-Quandles, their Extensions and Cohomology, arxiv: (2016). Elhamdadi, M. and Nelson, S., Quandles an introduction to the algebra of knots, 74,2015, American Mathematical Society, Providence, RI, P. Etingof and M. Graña. On rack cohomology. J. Pure Appl. Algebra 177 (2003), no. 1, R. Fenn and C. Rourke. Racks and links in codimension two. J. Knot Theory Ramifications 1 (1992) T. Mochizuki. Mohamed Some Elhamdadi calculations A Survey of cohomology of Quandles with some groups recent developments of