Enhancements of the rack counting invariant via N-reduced dynamical cocycles
EEnhancements of the rack counting invariant via N -reduceddynamical cocycles Alissa S. Crans ∗ Sam Nelson † Aparna Sarkar ‡ Abstract
We introduce the notion of N -reduced dynamical cocycles and use these objects to defineenhancements of the rack counting invariant for classical and virtual knots and links. Weprovide examples to show that the new invariants are not determined by the rack countinginvariant, the Jones polynomial or the generalized Alexander polynomial. Keywords:
Dynamical cocycles, enhancements of counting invariants, cocycle invariants Introduction
Racks were introduced in 1992 in [6] as an algebraic structure for defining representational andfunctorial invariants of framed oriented knots and links. A rack generalizes the notion of a quandle ,an algebraic structure defined in 1980 in [8, 9] which defines invariants of unframed knots and links.More precisely, the number of quandle homomorphisms from the fundamental quandle of a knot orlink to a finite quandle X defines a computable integer-valued invariant of unframed oriented knotsand links known as the quandle counting invariant .In [10], a property of finite racks known as rack rank or rack characteristic was used to definean integer-valued invariant of unframed oriented knots and links using non-quandle racks, known asthe integral rack counting invariant ; for quandles, this invariant coincides with the quandle countinginvariant. An enhancement of a counting invariant uses a Reidemeister-invariant signature for eachhomomorphism rather than merely counting homomorphisms. In [2], the first enhancement of thequandle counting invariant was defined using Boltzmann weights determined by elements of thesecond cohomology of a finite quandle. The resulting quandle 2-cocycle invariants of knots and linkshave been the subject of much study ever since.In [7] an enhancement of the integral rack counting invariant was defined using a modificationof the rack module structure from [1], associating a vector space or module to each homomorphism.In this paper we further generalize the enhancement from [7] using a modified version of an alge-braic structure first defined in [1] known as a dynamical cocycle . In particular, dynamical cocyclessatisfying a condition we call N -reduced yield an enhancement of the rack counting invariant.The paper is organized as follows. In Section 2 we review the basics of racks, the rack countinginvariant, and the rack module enhancement. In Section 3 we define N -reduced dynamical cocyclesand the N -reduced dynamical cocycle invariant. In Section 4 we provide some computations andexamples, and we conclude in Section 5 with some questions for future study. ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ m a t h . G T ] A ug Racks, the counting invariant and the rack module enhancement
We start by reviewing some basic definitions from [6, 8].
Definition 1 A rack is a set X equipped with a binary operation (cid:46) : X × X → X satisfying thefollowing two conditions:(i) For each x ∈ X , the map f x : X → X defined by f x ( y ) = y (cid:66) x is invertible, with inverse f − x ( y ) denoted by y (cid:66) − x , and(ii) For each x, y, z ∈ X , we have ( x (cid:66) y ) (cid:66) z = ( x (cid:66) z ) (cid:66) ( y (cid:66) z ) . A quandle is a rack with the added condition:(iii) For all x ∈ X , we have x (cid:66) x = x .Note that axiom (ii) is equivalent to the requirement that each map f x : X → X is a rackhomomorphism, i.e. f z ( x (cid:46) y ) = ( x (cid:46) y ) (cid:46) z = ( x (cid:46) z ) (cid:46) ( y (cid:46) z ) = f z ( x ) (cid:46) f z ( y ) , so we can alternatively define a rack as a set X with a bijection f x : X → X for each x ∈ X suchthat every f x is an automorphism of the structure on X defined by x (cid:46) y = f y ( x ).Standard examples of racks include: • ( t, s ) -racks. Any module over ¨Λ = Z [ t ± , s ] / ( s − (1 − t ) s ) is a rack under x (cid:46) y = tx + sy. If s is invertible, then s − (1 − t ) s = 0 implies s = 1 − t and we have a quandle known as an Alexander quandle . • Conjugation racks.
Every group G is a rack (indeed, a quandle) under n -fold conjugation forany n ∈ Z : x (cid:46) y = y − n xy n . • The Fundamental Rack of a framed oriented link.
Let L ⊂ S be a link of c components, n ( L )a regular neighborhood of L with set of framing curves F = { F , . . . , F c } giving the framingof L , x ∈ S \ n ( L ) a base point and F R ( L ) the set of isotopy classes of paths from x to F i where the terminal point of the path can wander along F i during the isotopy. For each point x ∈ F i there is a meridian m ( x ) in n ( L ), unique up to isotopy, linking the i th component of L once. Then for each path y : [0 , → S \ n ( L ) representing an isotopy class in F R ( L ), let p ( y ) = y − ∗ m ( y (1)) ∗ y ∈ π ( S \ n ( L )) where ∗ is path concatenation reading right-to-left.Then F R ( L ) is a rack under the operation[ x ] (cid:46) [ y ] = [ x ∗ p ( y )] . Combinatorially,
F R ( L ) can be understood as equivalence classes of rack words in a set ofgenerators corresponding one-to-one with the set of arcs in a diagram of L under the equivalencer elation generated by the rack axioms and crossing relations in L . See [6] for more details. Definition 2
Let X = { x , . . . , x n } be a finite set. We can specify a rack structure on X by a rackmatrix M X in which the ( i, j )th entry is k when x k = x i (cid:46) x j . Rack axiom (i) is equivalent to thecondition that every column of M X is a permutation; rack axiom (ii) requires checking each triplefor the condition M M i,j ,k = M M i,k ,M j,k . 2 xample 1 The ( t, s )-rack structure on Z = { x = 1 , x = 2 , x = 3 , x = 0 } with t = 1 and s = 2has rack matrix M X = . Definition 3
Let X be a rack and L an oriented link diagram. An X -labeling or rack labeling of L by X is an assignment of an element of X to each arc in L such that the condition below is satisfied:Indeed, the rack axioms are algebraic distillations of Reidemeister moves II and III under thislabeling scheme; the quandle condition corresponds to the unframed Reidemeister move I, and theframed Reidemeister I moves do not impose any additional conditions. Accordingly, labelings ofarcs of oriented framed knot or link diagrams by rack elements (respectively, quandle elements) asshown above are preserved by oriented framed Reidemeister moves (respectively, oriented unframedReidemeister moves) as illustrated in the figures below. respectively , Definition 4
Let X be a rack. We call the map π : X → X defined by π ( x ) = x (cid:66) x the kinkmap . The rack rank or rack characteristic of X , denoted by N ( X ), is the order of the permutation π considered as an element of the symmetric group S | X | . Equivalently, for every element x ∈ X , the rank of x , denoted by N ( x ), is the smallest positive integer N such that π N ( x ) = x. Thus, N ( X ) isthe least common multiple of the ranks N ( x ) for all x ∈ X . In particular, the kink map of a rackstructure on a finite set X = { x , . . . , x n } given by a rack matrix M X is the permutation in S | X | which sends k to the ( k, k ) entry of M X . That is, the image of π is given by the entries along thediagonal of M X . Example 2
The rack in Example 1 has kink map satisfying π (1) = 3, π (2) = 4, π (3) = 1 and π (4) = 2 (or, in cycle notation, π = (13)(24)) and hence has rack rank N = 2.3 emark 3 The quandle condition implies that the rank of every quandle element is 1, and thusthe rack rank of a quandle is always 1. Indeed, quandles are simply racks with rack rank N = 1.Rack rank can be understood geometrically in terms of the Reidemeister type I move: if an arcin a knot diagram is labeled with a rack element x , going through a positive kink changes the labelto π ( x ). A natural question is then: how many kinks must we go though to end up again with x ?This notion of order is the rank of x . We can illustrate the concept of rack rank with the N -phonecord move pictured below:If N is the rank of X , then labelings of a link diagram L by X are preserved by N -phone cordmoves. In particular, if X is a rack of rack rank N , and L and L (cid:48) are framed oriented links relatedby framed Reidemeister moves with framings congruent modulo N , then the sets of X -labelings of L and L (cid:48) are in bijective correspondence. It follows that the number of homomorphisms is periodicin N on each component of a link L . Definition 5
Let X be a rack with rank N and let L be an oriented link of c components. Let w ∈ ( Z N ) c be a framing vector specifying a framing modulo N for each component of L , and let usdenote a diagram of L with framing vector w by ( L, w ). We thus obtain a set of N c diagrams offramings of L mod N . For each such diagram ( L, w ), we have a set of X -labelings corresponding tohomomorphisms f : F R ( L, w ) → X . Summing the numbers of X -labelings over the set { ( L, w ) | w ∈ ( Z N ) c } , we obtain an invariant of unframed links known as the integral rack counting invariant, whichis denoted by: Φ Z X ( L ) = (cid:88) w ∈ ( Z N ) c | Hom(
F R ( L, w ) , X ) | . Example 4
Let X be the rack with rack matrix M X = (cid:20) (cid:21) . As a labeling rule, the rackstructure of X says that at a crossing, the understrand switches from 1 to 2 or from 2 to 1 since1 (cid:46) x = 2 and 2 (cid:46) x = 1 for x = 1 ,
2. The kink map is the transposition (12), so N = 2. Thus, tocompute Φ Z X on a link of c = 2 components, we must count X -labelings on the set of N c = 2 = 4diagrams with writhe vectors in ( Z N ) c . The (4 , L a L a X -labelings as depicted below, so we have Φ Z X ( L a
1) = Φ Z X ( L a
1) = 4.4n enhancement of Φ Z X ( L ) is a link invariant defined by associating to each X -labeling of L aquantity which is unchanged by X -labeled framed Reidemeister moves and N -phone cord moves.Examples include: • Image Enhanced Invariant.
The image of rack homomorphism is closed under (cid:46) and thus isunchanged by N -phone cord moves. Hence we have an enhancement:Φ Im X ( L ) = (cid:88) w ∈ ( Z N ) c (cid:88) f ∈ Hom(
F R ( L, w ) ,X ) u | Im( f ) | where u is a formal variable. • Writhe Enhanced Invariant.
Keeping track of which labelings are contributed by which writhesyields another enhancement:Φ W X ( L ) = (cid:88) w ∈ ( Z N ) c | Hom(
F R ( L, w ) , X ) | q w where q ( w ,...,w c ) = q w . . . q w c c is a product of formal variables. • Cocycle Invariants.
A finite rack X has a cohomology theory analogous to group cohomology.For any f ∈ Hom Z ( Z [ X n ] , Z ), define δ n : Z [ X n ] → Z [ X n +1 ] by( δ n f )( x , . . . , x n +1 ) = n +1 (cid:88) k =2 ( − k ( f ( x , . . . , x k − , x k +1 , . . . , x n +1 ) − f ( x (cid:46) x k , . . . , x k − (cid:46) x k , x k +1 , . . . , x n +1 ))and extend linearly. Let D n be the subgroup of Z [ X n ] generated by elements of the form N (cid:88) k =1 ( x , . . . , π k ( x j ) , π k +1 ( x j ) , . . . , x n ) , j = 1 , . . . , n − N is the rack rank of X . Then ( D n , δ n ) is a subcomplex of ( Z [ X n ] , δ n ); the quotient com-plex ( Z [ X n ] /D n , δ n ) is the N -reduced rack cochain complex (or the quandle cochain complex if N = 1), with cohomology groups denoted by H nR/ND ( X ). For every element φ ∈ H R/ND ( X )(such a φ is called an N -reduced 2-cocycle ) we have an enhancementΦ φX ( L ) = (cid:88) w ∈ ( Z N ) c (cid:88) f ∈ Hom(
F R ( L, w ) ,X ) u BW ( f ) where BW ( f ), the Boltzmann weight of f , is the sum over all crossings in f of φ evaluated atthe arc labelings of each crossing.See [2, 5, 10] for further details. 5 xample 5 In Example 4, the links L a L a X -labelings overa complete period of framings mod N , but these labelings occur at different framing vectors. Inparticular, all four labelings of L a w = (0 ,
0) while all four labelings of L a x = (1 , WX distinguishes thelinks, with Φ WX ( L a
1) = 4 (cid:54) = 4 q q = Φ WX ( L a rack algebra Z [ X ] was associated to each finite rack X ; in [7] amodified form of the rack algebra was used to define an enhancement of Φ Z X . The idea is to adda secondary labeling to an X -labeled link diagram by putting beads on each arc and defining a( t, s )-rack style operation on the beads at a crossing with t and s values indexed by the arc labelsin X as depicted below: c = t x,y a + s x,y b Definition 6
Let X be a finite rack with rack rank N . The rack algebra of X , denoted by Z [ X ],is the quotient of the polynomial algebra Z [ t ± x,y , s x,y ] generated by noncommuting variables t ± x,y and s x,y for each x, y ∈ X modulo the ideal I generated by the relators t x(cid:46)y,z t x,y − t x(cid:46)z,y(cid:46)z t x,z , t x(cid:46)y,z s x,y − s x(cid:46)z,y(cid:46)z t x,z , s x(cid:46)y,z − s x(cid:46)z,y(cid:46)z s y,z − t x(cid:46)z,y(cid:46)z s x,z and 1 − N − (cid:89) k =0 (cid:0) t π k ( x ) ,π k ( x ) + s π k ( x ) ,π k ( x ) (cid:1) for all x, y, z ∈ X . An X -module is a representation of Z [ X ], that is, an abelian group G withautomorphisms t x,y : G → G and endomorphisms s x,y : G → G such that the maps defined by therelators of I are zero. Example 6
Let R be a commutative ring. Then any R -module becomes an X -module with achoice of automorphisms and endomorphisms given by multiplication by invertible elements t x,y ∈ R and generic elements s x,y ∈ R such that the ideal I is zero. We can express such a structureconveniently with a block matrix M R = [ T S ] where the ( i, j ) entries of T and S are t x i ,y j and s x i ,y j respectively. Example 7
Let X be a rack and let f ∈ Hom(
F R ( L ) , X ) be an X -labeled link diagram. The fundamental Z [ X ] -module of f , denoted by Z [ f ], is the quotient of the free Z [ X ]-module generatedby the set of arcs in f modulo the ideal generated by the crossing relations.In [7] an enhancement of Φ Z X was defined using the number of bead labelings of an X -labeleddiagram of a framed oriented link L as a signature as follows: Definition 7
Let X be a finite rack and R a commutative ring with an X -module structure. The rack module enhanced invariant is given by:Φ X,R ( L ) = (cid:88) w ∈ ( Z N ) c (cid:88) f ∈ Hom(
F R ( L, w ) ,X ) u | Hom( Z [ f ] ,R ) | . xample 8 Let X be the rack from Example 4 and let R = Z . The matrix M R = [ T | S ] = (cid:20) (cid:21) defines an X -module structure on R . To compute Φ X,R for the Hopf link L a
1, we must compute | Hom( Z [ f ] , R ) | for each valid X -labeling of L a
1. For instance, the following X -labeled diagram hasfundamental Z [ X ]-module with listed presentation matrix: M Z [ f ] = t , + s , − s , − t , t , − s ,
00 0 − t , + s , Replacing each t x,y and s x,y with its value from M R and row-reducing over Z , we have → , so the solution space (i.e., the set of bead labelings) is the set { (0 , , , , (2 , , , , (1 , , , } and this X -labeling contributes u to Φ X,R ( L a X,R ( L a
1) = 4 u . Dynamical cocycles and enhancements of the counting invariant
In this section we generalize the rack module idea to remove the restrictions of the abelian groupstructure, keeping only those conditions required by the Reidemeister moves. The result is a rackstructure on the product X × S defined via a map α : X × X → Maps( S × S, S ) known as a dynamicalcocycle . Dynamical cocycles were defined in [1] and used to construct extension racks; we will usedynamical cocycles satisfying an extra condition, which we call N -reduced dynamical cocycles , todefine an enhancement of the rack counting invariant Φ Z X . Definition 8
Let X be a finite rack of rack rank N and S be a finite set. The elements of S will be called beads . A map α : X × X → Maps( S × S, S ) may be understood as a collectionof binary operations · x,y : S × S → S indexed by pairs of elements of X where where we write a · x,y b = α ( x, y )( a, b ). Such a map α is a dynamical cocycle on S if the maps satisfy:(i) For all x, y ∈ X and b ∈ S , the map f x,yb : S → S defined by f x,yb ( a ) = a · x,y b is a bijection,and(ii) For all x, y, z ∈ X and a, b, c ∈ S , we have( a · x,y b ) · x(cid:46)y,z c = ( a · x,z c ) · x(cid:46)z,y(cid:46)z ( b · y,z c ) . Definition 9
Let X be a rack of rack rank N and α : X × X → Maps( S × S, S ) a dynamical cocycle.Define ρ x : S → S by ρ x ( a ) = a · x,x a . Then if the diagram7ommutes for every x ∈ X and a ∈ S , we say the cocycle α is N -reduced. The definition of a dynamical cocycle is chosen so that bead labelings of an X -labeled diagramare preserved under X -labeled framed oriented Reidemeister moves as shown below: d = b · y,z c d = b · y,z ce = ( a · x,y b ) · x(cid:46)y,z c e = ( a · x,z c ) · x(cid:46)z,y(cid:46)z ( b · y,z c )The Reidemeister II and framed type I moves require the operations · x,y : S × S → S to beright-invertible; the N -reduced condition is required by the N -phone cord move: b = ρ x ( a ) c = ρ π ( x ) ( b ) = ρ π ( x ) ( ρ x ( a ))... a = ρ π N ( x ) ( ρ π N − ( x ) ( . . . ( ρ x ( a )) . . . ) Example 9
Let X be a finite rack and M an X -module as defined in Section 2. Then the operations a · x,y b = t x,y a + s x,y b define an N -reduced dynamical cocycle on M .More generally, if X is a finite rack of cardinality n , we can describe a dynamical cocycle on afinite set S = { b , . . . , b k } with an ( nk ) × ( nk ) block matrix, M x,y , encoding the operations tablesfor · x,y M x,y = M , M , . . . M ,n M , M , . . . M ,n ... ... . . . ... M n, M n, . . . M n,n where the ( i, j )th entry of M x,y is l when b i · x,y b j = b l . Definition 10
Let X be a finite rack and α an N -reduced dynamical cocycle on a set S . For an X -labeled link diagram f , let L ( f ) be the set of S -labelings of f . Then we define the N -reduceddynamical cocycle enhanced invariant or α -enhanced invariant Φ X,α ( L ) by:Φ X,α ( L ) = (cid:88) w ∈ W (cid:88) f ∈ Hom(
F R ( L, w )) u |L ( f ) | .
8y construction, we have
Theorem 1
Let X be a finite rack and α an N -reduced dynamical cocycle on a set S . If L and L (cid:48) are ambient isotopic links, then Φ X,α ( L ) = Φ X,α ( L (cid:48) ) . Remark 10
The α -enhanced invariant is well-defined for virtual knots by the usual convention ofignoring virtual crossings. Computations and Examples
In this section we present example computations of the N -reduced dynamical cocycle enhancedinvariant. Example 11
Let X be the rack with rack matrix M X = (cid:20) (cid:21) and let α be the dynamicalcocycle on S = { , , } given by the block matrix M α = The virtual knots 3 . Z X = 2. Let us compare Φ X,α (3 .
7) with Φ
X,α (Unknot). Since X has rank N = 2, we needto consider diagrams with writhes mod 2. The odd writhe diagrams have no valid X -labelings, andthere are two valid X -labelings of the even writhe diagrams. We collect the valid bead labelings inthe tables below. x a x a x y z w a b c d x y z w a b c d X,α (3 .
7) = 2 u (cid:54) = 2 u = Φ X,α (Unknot) and Φ
X,α is not determined by the Jonespolynomial or the integral rack counting invariant Φ Z X .9 xample 12 Similarly, the virtual knots 3 . .
85 both have generalized Alexander polynomial∆ = ( t − s − st − X,α with Φ
X,α (3 .
7) = 2 u (cid:54) = 2 u = Φ X,α (4 .
85) for the rack X anddynamical cocycle α from Example 11. x y z w a b c d x y z w a b c d X,α is not determined by the generalized Alexander polynomial.
Example 13
We randomly selected a small dynamical cocycle α on the set S = { , , } for thedihedral quandle X with matrices below. M X = M α = We then computed Φ
X,α for the list of prime classical knots with up to eight crossings and primeclassical links with up to seven crossings as listed at the knot atlas [3]. The results are collectedbelow. In particular, note that the invariant values 6 + 3 u (cid:54) = 9 u both specailize to the same rackcounting invariant value Φ Z X = 9, and we see that Φ X,α is not determined by Φ Z X .Φ X,α ( L ) L u Unknot , , , , , , , , , , , , , , , , , , , , , , , ,L a , L a , L a , L a , L a , L n , L a , L a , L a , L a , L a , L n , L n
26 + 3 u , , , , , , , L a , L a , L a , L a u , , , , L a
524 + 3 u Our python results indicate that of the 116 prime virtual knots with up to 4 classical crossingslisted at the knot atlas, Φ
X,α for this α is 6+3 u for the virtual knots 3 . , . , . , . , . , . , . , . , . , .
68 and 4 .
98, Φ αX,S = 9 u for 4 .
99, and Φ
X,α = 3 u for the other virtual knots in the list.Our python code for computing N -reduced dynamical cocycles and their link invariants is avail-able at . Questions for future research
In this section we collect a few questions for future research.10or a given pair of knots or links, how can we choose X and α to maximize the liklihood ofΦ X,α distinguishing the knots or links in question? Is there an algorithm, perhaps starting withpresentations of the fundamental racks of the knots, to construct a rack X and dynamical cocycle α such that Φ X,α always distinguishes inequivalent knots?A natural direction of generalization is to look at knotted surfaces in R , which have an integralquandle counting invariant which should be susceptible to enhancement by beads. What analog ofthe dynamical cocycle condition arises from the Roseman moves with beads on each sheet? References [1] N. Andruskiewitsch and M. Gra˜na. From racks to pointed Hopf algebras.
Adv. Math. (2003)177-243.[2] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford and M. Saito. Quandle cohomology and state-sum invariants of knotted curves and surfaces.
Trans. Am. Math. Soc. (2003) 3947-3989.[3] D. Bar-Natan (Ed.). The Knot Atlas. http://katlas.math.toronto.edu/wiki/Main Page [4] J. Ceniceros and S. Nelson. ( t, s )-racks and their link invariants. arXiv:1011.5455, to appear in
Int’l. J. Math. [5] M. Elhamdadi and S. Nelson. N -degeneracy in rack homology. arXiv: to appear in HiroshimaMath J. [6] R. Fenn and C. Rourke. Racks and links in codimension two.
J. Knot Theory Ramifications (1992) 343-406.[7] A. Haas, G. Heckel, S. Nelson, J. Yuen, Q. Zhang. Rack Module Enhancements of CountingInvariants. arXiv:1008.0114, to appear in Osaka J. Math. [8] D. Joyce. A classifying invariant of knots, the knot quandle.