QQuandle Cocycle Quivers
Karina Cho ∗ Sam Nelson † Abstract
We incorporate quandle cocycle information into the quandle coloring quivers we defined in [2] todefine weighted directed graph-valued invariants of oriented links we call quandle cocycle quivers . Thisconstruction turns the quandle cocycle invariant into a small category, yielding a categorification of thequandle cocycle invariant. From these graphs we define several new link invariants including a 2-variablepolynomial which specializes to the usual quandle cocycle invariant. Examples and computations areprovided.
Keywords:
Quandles, Enhancements, Quivers, Cocycle Enhancements Introduction
Quandles , algebraic structures whose axioms are derived from the Reidemeister moves, were introduced in[4, 5] and have been studied ever since. Associated to every oriented knot or link there exists a fundamentalquandle , also called the knot quandle of the knot or link. The isomorphism class of the fundamental quandleof a knot determines the group system and hence the knot up to mirror image; in this sense the knot quandleis a complete invariant for knots, though not for split links or other generalizations such as virtual knots.The number of homomorphisms from a knot’s fundamental quandle to a finite coloring quandle, called the quandle counting invariant , is an integer-valued invariant of oriented knots and links.The quandle counting invariant forms the basis for a class of computable knot and link invariants known as enhancements , including in particular quandle cocycle invariants . In [1], a theory of quandle cohomology wasintroduced. Quandle-colored link diagrams can be identified with certain elements of the second homologyof the coloring quandle, and it naturally follows that elements of the second cohomology define invariants ofquandle-colored isotopy. See [1, 3] etc. for more.In [2], we introduced quandle coloring quivers , directed graph-value invariants of oriented knots and linksassociated to pairs (
X, S ) consisting of a finite quandle X and a subset S of the ring of endomorphisms of X . From these quivers we defined a new polynomial knot invariant, using this quiver structure to enhancethe quandle counting invariant, called the in-degree polynomial . Since directed graphs can be interpreted ascategories, this construction yielded a categorification of the quandle counting invariant.In this paper we enhance quandle coloring quivers with quandle cocycles, obtaining quandle cocyclequivers. As in the previous case, this construction yields a categorification of the quandle cocycle invariant.From these quivers we derive new polynomial knot and link invariants, including a two-variable enhancementof the quandle 2-cocycle invariant which has both the quandle counting invariant and the quandle cocycleinvariant as specializations but in general is a stronger invariant than either. The paper is organized asfollows. In Section 2 we review quandles and quandle cohomology. In Section 3 we define quandle cocyclequivers and provide examples demonstrating the computation of the invariant and showing that the newpolynomial invariant is a proper enhancement. We conclude in Section 4 with some questions for futureresearch. ∗ Email: [email protected]. † Email: [email protected]. Partially supported by Simons Foundation collaboration grant 316709 a r X i v : . [ m a t h . G T ] A p r Quandles and Quandle Cohomology
We begin with some preliminary definitions.
Definition 1.
A set X with a binary operation (cid:46) is a quandle if it satisfies(i) for all x ∈ X , x (cid:46) x = x (ii) for all y ∈ X , the map f y : X → X defined by f y ( x ) = x (cid:46) y is a bijection, and(iii) for all x, y, z ∈ X , ( x (cid:46) y ) (cid:46) z = ( x (cid:46) z ) (cid:46) ( y (cid:46) z ) . Definition 2.
For an oriented link L , associate a label to each arc of L . At each crossing of L , assign arelation between arc labels as in the figure below.In words, if we orient a crossing so that the overstrand (labeled y ) points up, and we have arcs labeled z and x on the left and right respectively of the arc labeled y , then we have the crossing relation z = x (cid:46) y .The fundamental quandle Q ( L ) is the quandle generated by the set of arc labels under the equivalencerelations defined by crossing relations for each crossing of L . Remark 1.
The quandle axioms are defined in such a way that the fundamental quandle of an oriented linkis invariant under Reidemeister moves, so Q ( L ) is an invariant of L . Definition 3.
Let X by a finite quandle and L be an oriented link. Then a homomorphism φ : Q ( L ) → X is called an X -coloring of L . Example 1.
Let X be the quandle defined by the operation table (cid:46) L be the figure-8 knot as drawn below, with arc labels a, b, c, d . Then Q ( L ) = (cid:104) a, b, c, d | b = a (cid:46) d = d (cid:46) c, c = a (cid:46) b = d (cid:46) a (cid:105) , and φ : Q ( L ) → X that maps a (cid:55)→ , b (cid:55)→ , c (cid:55)→ , d (cid:55)→ X -coloring of L , which we may visualize withthe picture below. 2he quandle axioms are the conditions required to ensure that for a given quandle coloring of a diagramon one side of a move, there is a unique quandle coloring of the diagram on the other side of the move whichagrees with the original coloring outside the neighborhood of the move. From this, we obtain the followingstandard result: Theorem 1.
Let X be a finite quandle. The number of colorings of an oriented knot or link diagram is aninteger-valued invariant of ambient isotopy. This theorem also follows from the observation that the set of quandle colorings of a knot or link diagramcan be identified with the set Hom( Q ( L ) , X ) of quandle homomorphisms from the fundamental quandle ofthe knot or link L to the coloring quandle X . See [3] for more. Definition 4.
Let X be a finite quandle, A an abelian group and C Rn ( X ; A ) = A [ X n ], the set of A -linearcombinations of ordered n -tuples of X . Let C Dn ( X ; A ) be the subgroup generated by elements ( x , . . . , x n )with x j = x j +1 for some j , and let C Qn ( X ; A ) = C Rn ( X ; A ) /C Dn ( X ; A ). Define ∂ n : C Rn ( X ; A ) → C Rn − ( X ; A )by setting ∂ n ( (cid:126)x ) = n (cid:88) k =1 ( − k (cid:0) ∂ n ( (cid:126)x ) − ∂ n ( (cid:126)x ) (cid:1) where we have ∂ n ( x , . . . , x n ) = ( x , . . . , x k − , x k +1 , . . . , x n ) ∂ n ( x , . . . , x n ) = ( x (cid:46) x k , . . . , x k − (cid:46) x k , x k +1 , . . . , x n )and extending linearly. Since ∂ n ( C Dn ( X ; A )) ⊂ C Dn − ( X ; A ), ∂ n induces ∂ Qn : C Qn ( X ; A ) → C Qn − ( X ; A ).Then the n th quandle homology of X is H Qn ( X ) = Ker( ∂ Qn ) / Im( ∂ Qn +1 ) . Dualizing, we have C n ∗ ( X ; A ) = Hom( C ∗ n ( X ; A ) , A ), δ n : C nR ( X ) → C n +1 R ( X ; A ) defined by( δ n f )( x , . . . , x n ) = f ∂ n +1 ( x , . . . , x n +1 )and n th quandle cohomology of X given by H nQ ( X ) = Ker( δ nQ ) / Im( δ n − Q ) . An element of Ker( δ nQ ) is called a quandle n -cocycle and an element of Im( δ n − Q ) is called a quandle n -coboundary .In this paper, we are particularly interested in quandle 2-cocycles, which are maps φ : A [ X × X ] → A foran abelian group A (usually Z n for us). These can be written as linear combinations of elementary functions χ i,j : X × X → A where χ i,j ( x , x ) = (cid:40) , for i = x , j = x , otherwise. Remark 2.
A quick computation will show that φ is a quandle 2-cocycle if and only if φ satisfies thecondition that φ ( x, y ) + φ ( x (cid:46) y, z ) = φ ( x, z ) + φ ( x (cid:46) z, y (cid:46) z )for all x, y, z ∈ X and φ ( x, x ) = 0 for all x ∈ X . 3 efinition 5. Let L be an oriented link, X be a finite quandle, and φ a quandle 2-cocycle. Let v be an X -coloring of L . For an X -colored crossing c of L , we define φ ( c ) by the following based on the whether thecrossing is positively or negatively oriented:In the illustration above, x and y are the elements of X determined by the coloring v . Then we define φ ( v )to be φ ( v ) = (cid:88) c ∈ C φ ( c ) , where C is the set of crossings in L . Remark 3.
It can be shown that the evaluation of φ on a quandle colored link satisfying the conditiondescribed in Remark 2 is equivalent to φ being invariant under a quandle-colored Reidemeister type IIImove. In fact, φ is an invariant of quandle-colored links. See [1, 3] for more. Definition 6.
Let X be a finite quandle, L an oriented link and φ ∈ H Q ( X ; A ). Then the polynomialΦ φX ( L ) = (cid:88) v ∈C ( L,X ) s φ ( v ) where C ( L, X ) is the set of X -colorings of a diagram of L and φ ( v ) is the Boltzmann weight of the X -coloreddiagram v is the quandle 2-cocycle invariant of L . See [1, 3] for more. Coloring Quivers and Cocycle Quivers
Let L be an oriented link diagram. In [2] we defined the quandle coloring quiver invariant Q SX ( L ) in thefollowing way: given a finite quandle X and set S ⊂ Hom(
X, X ) of quandle endomorphisms, we make adirected graph with a vertex for each X -coloring of L and a directed edge from v j to v k whenever v k = f ( v j )in the sense that each arc color in v k is obtained from the corresponding arc color in v j by applying f forsome f ∈ S . Example 2.
The links L n L n L7n1 L7n2 f (1) = 4 , f (2) = f (3) = f (4) = 3namely . Definition 7.
Let L be an oriented link, X be a finite quandle, S a set of quandle endomorphisms from X to X , and φ a 2-cocycle in C Q ( X ). Then the quandle cocycle quiver Q S,φX ( L ) is the directed graph withvertices corresponding to X -colorings of L , edges from v j to v k whenever v k = f ( v j ) for some f ∈ S , andweights φ ( v j ) at each vertex. When S = { f } is a singleton we will write f instead of { f } for simplicity. Example 3.
The links in example 2 are not distinguished by their coloring quivers with respect to the givenquandle and endomorphism; however, the quandle cocycle quiver with cocycle φ = χ , + 2 χ , + χ , + 2 χ , + 3 χ , + 3 χ , + χ , ∈ C Q ( X ; Z )does distinguish the links. Q f,φX ( L n Q f,φX ( L n Definition 8.
Let L be a link, X a finite quandle, S ⊂ Hom(
X, X ) and φ ∈ C Q ( X ; A ). We define the quiverenhanced cocycle polynomial to be the polynomialΦ S,φX ( L ) = (cid:88) e ∈ E ( Q SX ( L )) s φ ( v j ) t φ ( v k ) where the edge e is directed from vertex v j to vertex v k in the quandle coloring quiver Q SX ( L ).5 emark 4. Since each directed edge contributes its s φ ( v j ) t φ ( v k ) value to the polynomial Φ S,φX ( L ) indepen-dently, if we regard the quandle coloring quiver Q X,S ( L ) as the union of the quivers Q X,f ( L ) for endomor-phisms f ∈ S , then the polynomial can be separated into a sum of the cocycle quiver polynomials for eachindividual endomorphism: Φ S,φX ( L ) = (cid:88) f ∈ S Φ f,φX ( L ) . It follows that evaluating Φ
S,φX ( L ) at t = 1 yields | S | Φ φX ( L ). In particular, when | S | = 1, Φ S,φX ( L ) evaluatesat t = 1 to the classical quandle 2-cocycle invariant as defined in [1] (see e.g. Example 8 in [1] and note thatour s is their t ).Similarly, if f is the identity endomorphism, then Φ f,φX ( L ) is the quandle cocycle invariant evaluated at st . Example 4.
In [2] Example 6 we gave an example of two links L a L a Z X ( L ) = 16 with respect to the quandle X with operation table (cid:46) Q fX ( L ) where f : X → X is the endomorphism given by f (1) = f (3) = 4 and f (2) = f (4) = 2. (The published version incorrectlylists this as “ f (1) = 4 , f (1) = 2 , f (1) = 4 , f (1) = 2”.) We note that for any coboundary φ , the Boltzmannweights are all 0 and the counting invariant Φ Z X ( L ) = 16 is equal to the quiver enhanced polynomial, so thisexample also shows that the Boltzmann weight enhanced quiver is not determined by the quiver enhancedpolynomial.The links in example 3 are distinguished by their quandle cocycle quivers, but they are already distin-guished by their quandle cocycle invariants with respect to the given quandle and cocycle. The next exampleshows that the quandle cocycle quiver can distinguish knots which have the same quandle cocycle invariant. Example 5.
Let X be the quandle defined by operation table (cid:46) . This quandle has endomorphism f (1) = f (6) = 2, f (2) = f (5) = 4 and f (3) = f (4) = 6 and cocycle φ = 2 χ (1 , + 2 χ (1 , + 2 χ (1 , + 2 χ (1 , + χ (2 , + 2 χ (2 , + χ (3 , + 2 χ (3 , +2 χ (4 , + χ (4 , + 2 χ (5 , + χ (5 , + χ (6 , + χ (6 , + χ (6 , + χ (6 , in C Q ( X ; Z ). Then the knots 6 and 7 both have quandle cocycle polynomialΦ φX (6 ) = 6 + 12 s + 12 s = Φ φX (7 )but are distinguished by their quiver enhanced polynomialsΦ f,φX (6 ) = 6 + 12 st + 12 s t (cid:54) = 6 + 12 st + 12 s t = Φ f,φX (7 ) . xample 6. Continuing with the same quandle from example 5, we computed Φ f,φX ( L ) using our python code for the prime knots with up to eight crossings and prime links with up to seven crossings with thecocycle with Z coefficients φ = χ + 3 χ + 2 χ + 3 χ + 3 χ + 2 χ + χ + χ + 3 χ + 3 χ + χ + χ + 2 χ + 3 χ +3 χ + χ + χ + 3 χ + 3 χ + χ and three arbitrarily chosen endomorphisms (where we write f by specifying [ f (1) , . . . , f ( n )]): f = [1 , , , , , f = [1 , , , , , f = [3 , , , , , . The results are collected in the table. For simplicity we list only the nontrivial values; unlisted knots haveΦ f,φX ( L ) = 6. L Φ f ,φX ( L ) Φ f ,φX ( L ) Φ f ,φX ( L )3
30 30 306 s t + 4 s + 4 t + 10 16 s t + 14 10 s t + 6 s + 6 t + 87
30 30 307 s t + 4 s + 4 t + 10 16 s t + 14 10 s t + 6 s + 6 t + 88
54 54 548 s t + 4 s + 4 t + 10 16 s t + 14 10 s t + 6 s + 6 t + 88
54 54 548 s t + 16 s + 16 t + 22 64 s t + 38 40 s t + 24 s + 24 t + 148
54 54 548
54 54 548
54 54 54 L a L a L a L a s t + 4 s + 4 t + 16 16 s t + 20 10 s t + 6 s + 6 t + 14 L a L a L a L a L n L a s t + 8 s + 16 t + 20 24 s t + 36 12 s t + 12 s + 12 t + 24 L a L a L a L a s t + 4 s + 4 t + 16 16 s t + 20 10 s t + 6 s + 6 t + 14 L a L a L n L n Definition 9.
Let X be a quandle, φ ∈ H Q ( X ; A ) a quandle 2-cocycle with values in an abelian group A , and S ⊂ Hom(
X, X ) a set of endomorphisms. Then for a oriented link L and choice of numbering for7ertices and edges in Q fX ( L ), we define a matrix M Q fX ( L ) whose entry in row j column k is − φ ( v j ) v j = Source( e k ) φ ( v j ) v j = Target( e k )0 ElseThe matrix M Q fX ( L ) itself depends on our choice of numbering for vertices and edges, but we can obtainfrom it several link invariants including but not limited to: • The rank of M Q fX ( L ) , • The isomorphism class of the linear transformation between R -modules determined by the matrix, • The Smith normal form of the matrix when R is a PID, • The eigenvalues and characteristic polynomial when M Q fX ( L ) is a square matrix, • The elementary ideals of M Q fX ( L ) and more. Example 7.
Let L be the link L a ,
2) torus link) and consider the quandle, endomorphism andcocycle (cid:46) , f = [1 , , , φ = 2 χ + 3 χ + 4 χ + 4 χ ∈ H Q ( X ; Z ) . Then we compute the matrix M Q fX ( L ) ∈ M ( Z ) for the cocycle quiver: . Then for example, we obtain cocycle quiver characteristic polynomial value ( x + 3) x ∈ Z [ x ]. Example 8.
Using the same quandle and cocycle as in example 7 with endomorphism f = [2 , , L a x + 3) x ∈ Z [ x ]. Questions
We conclude with a few questions and direction for future work.As we saw in example 4 the cocycle polynomial does not determine the cocycle quiver. We are curiousabout which R -colored quivers are obtainable as quandle cocycle quivers of knots and links. For example,the out-degree of every vertex must me the same, namely | S | , so not every quiver is eligible. A link withcocycle quiverwould have the same cocycle quiver polynomial as L a R -colored quiver? What are necessary and sufficient conditions for a R -colored quiver to be the quandlecocycle quiver of a knot or link, and given such a quiver how can we construct the set of all links with thegiven R -colored quiver as Q X,S ( L )? References [1] J. S. Carter, D. Jelsovsky, S. Kamada, L. Langford, and M. Saito. Quandle cohomology and state-suminvariants of knotted curves and surfaces.
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