Enhancements of the rack counting invariant via -reduced dynamical cocycles
We introduce the notion of -reduced dynamical cocycles and use these objects to define enhancements of the rack counting invariant for classical and virtual knots and links. We provide examples to show that the new invariants are not determined by the rack counting invariant, the Jones polynomial or the generalized Alexander polynomial.
Keywords: Dynamical cocycles, enhancements of counting invariants, cocycle invariants
2010 MSC: 57M27, 57M25
Racks were introduced in 1992 in  as an algebraic structure for defining representational and functorial 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 or link to a finite quandle defines a computable integer-valued invariant of unframed oriented knots and links known as the quandle counting invariant.
In , a property of finite racks known as rack rank or rack characteristic was used to define an integer-valued invariant of unframed oriented knots and links using non-quandle racks, known as the integral rack counting invariant; for quandles, this invariant coincides with the quandle counting invariant. An enhancement of a counting invariant uses a Reidemeister-invariant signature for each homomorphism rather than merely counting homomorphisms. In , the first enhancement of the quandle counting invariant was defined using Boltzmann weights determined by elements of the second cohomology of a finite quandle. The resulting quandle 2-cocycle invariants of knots and links have been the subject of much study ever since.
In  an enhancement of the integral rack counting invariant was defined using a modification of the rack module structure from , associating a vector space or module to each homomorphism. In this paper we further generalize the enhancement from  using a modified version of an algebraic structure first defined in  known as a dynamical cocycle. In particular, dynamical cocycles satisfying a condition we call -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 counting invariant, and the rack module enhancement. In Section 3 we define -reduced dynamical cocycles and the -reduced dynamical cocycle invariant. In Section 4 we provide some computations and examples, and we conclude in Section 5 with some questions for future study.
2 Racks, the counting invariant and the rack module enhancement
A rack is a set equipped with a binary operation satisfying the following two conditions:
For each , the map defined by is invertible, with inverse denoted by , and
For each , we have
A quandle is a rack with the added condition:
For all , we have .
Note that axiom (ii) is equivalent to the requirement that each map is a rack homomorphism, i.e.
so we can alternatively define a rack as a set with a bijection for each such that every is an automorphism of the structure on defined by .
Standard examples of racks include:
-racks. Any module over is a rack under
If is invertible, then implies and we have a quandle known as an Alexander quandle.
Conjugation racks. Every group is a rack (indeed, a quandle) under -fold conjugation for any :
The Fundamental Rack of a framed oriented link. Let be a link of components, a regular neighborhood of with set of framing curves giving the framing of , a base point and the set of isotopy classes of paths from to where the terminal point of the path can wander along during the isotopy. For each point there is a meridian in , unique up to isotopy, linking the th component of once. Then for each path representing an isotopy class in , let where is path concatenation reading right-to-left. Then is a rack under the operation
Combinatorially, can be understood as equivalence classes of rack words in a set of generators corresponding one-to-one with the set of arcs in a diagram of under the equivalence r elation generated by the rack axioms and crossing relations in . See  for more details.
Let be a finite set. We can specify a rack structure on by a rack matrix in which the th entry is when . Rack axiom (i) is equivalent to the condition that every column of is a permutation; rack axiom (ii) requires checking each triple for the condition .
The -rack structure on with and has rack matrix
Let be a rack and an oriented link diagram. An -labeling or rack labeling of by is an assignment of an element of to each arc in such that the condition below is satisfied:
Indeed, the rack axioms are algebraic distillations of Reidemeister moves II and III under this labeling scheme; the quandle condition corresponds to the unframed Reidemeister move I, and the framed Reidemeister I moves do not impose any additional conditions. Accordingly, labelings of arcs of oriented framed knot or link diagrams by rack elements (respectively, quandle elements) as shown above are preserved by oriented framed Reidemeister moves (respectively, oriented unframed Reidemeister moves) as illustrated in the figures below.
Let be a rack. We call the map defined by the kink map. The rack rank or rack characteristic of , denoted by , is the order of the permutation considered as an element of the symmetric group . Equivalently, for every element , the rank of , denoted by , is the smallest positive integer such that Thus, is the least common multiple of the ranks for all . In particular, the kink map of a rack structure on a finite set given by a rack matrix is the permutation in which sends to the entry of . That is, the image of is given by the entries along the diagonal of .
The rack in Example 1 has kink map satisfying , , and (or, in cycle notation, ) and hence has rack rank .
The quandle condition implies that the rank of every quandle element is 1, and thus the rack rank of a quandle is always 1. Indeed, quandles are simply racks with rack rank .
Rack rank can be understood geometrically in terms of the Reidemeister type I move: if an arc in a knot diagram is labeled with a rack element , going through a positive kink changes the label to . A natural question is then: how many kinks must we go though to end up again with ? This notion of order is the rank of . We can illustrate the concept of rack rank with the -phone cord move pictured below:
If is the rank of , then labelings of a link diagram by are preserved by -phone cord moves. In particular, if is a rack of rack rank , and and are framed oriented links related by framed Reidemeister moves with framings congruent modulo , then the sets of -labelings of and are in bijective correspondence. It follows that the number of homomorphisms is periodic in on each component of a link .
Let be a rack with rank and let be an oriented link of components. Let be a framing vector specifying a framing modulo for each component of , and let us denote a diagram of with framing vector by . We thus obtain a set of diagrams of framings of mod . For each such diagram , we have a set of -labelings corresponding to homomorphisms . Summing the numbers of -labelings over the set , we obtain an invariant of unframed links known as the integral rack counting invariant, which is denoted by:
Let be the rack with rack matrix . As a labeling rule, the rack structure of says that at a crossing, the understrand switches from 1 to 2 or from 2 to 1 since and for . The kink map is the transposition , so . Thus, to compute on a link of components, we must count -labelings on the set of diagrams with writhe vectors in . The -torus link and the Hopf link both have four -labelings as depicted below, so we have .
An enhancement of is a link invariant defined by associating to each -labeling of a quantity which is unchanged by -labeled framed Reidemeister moves and -phone cord moves. Examples include:
Image Enhanced Invariant. The image of rack homomorphism is closed under and thus is unchanged by -phone cord moves. Hence we have an enhancement:
where is a formal variable.
Writhe Enhanced Invariant. Keeping track of which labelings are contributed by which writhes yields another enhancement:
where is a product of formal variables.
Cocycle Invariants. A finite rack has a cohomology theory analogous to group cohomology. For any , define by
and extend linearly. Let be the subgroup of generated by elements of the form
where is the rack rank of . Then is a subcomplex of ; the quotient complex is the -reduced rack cochain complex (or the quandle cochain complex if ), with cohomology groups denoted by . For every element (such a is called an -reduced 2-cocycle) we have an enhancement
where , the Boltzmann weight of , is the sum over all crossings in of evaluated at the arc labelings of each crossing.
In Example 4, the links and have the same number of -labelings over a complete period of framings mod , but these labelings occur at different framing vectors. In particular, all four labelings of occur with writhe vector while all four labelings of occur with writhe vector . Thus the writhe enhanced invariant distinguishes the links, with .
In  an algebra known as the rack algebra was associated to each finite rack ; in  a modified form of the rack algebra was used to define an enhancement of . The idea is to add a secondary labeling to an -labeled link diagram by putting beads on each arc and defining a -rack style operation on the beads at a crossing with and values indexed by the arc labels in as depicted below:
Let be a finite rack with rack rank . The rack algebra of , denoted by , is the quotient of the polynomial algebra generated by noncommuting variables and for each modulo the ideal generated by the relators
for all . An -module is a representation of , that is, an abelian group with automorphisms and endomorphisms such that the maps defined by the relators of are zero.
Let be a commutative ring. Then any -module becomes an -module with a choice of automorphisms and endomorphisms given by multiplication by invertible elements and generic elements such that the ideal is zero. We can express such a structure conveniently with a block matrix where the entries of and are and respectively.
Let be a rack and let be an -labeled link diagram. The fundamental -module of , denoted by , is the quotient of the free -module generated by the set of arcs in modulo the ideal generated by the crossing relations.
In  an enhancement of was defined using the number of bead labelings of an -labeled diagram of a framed oriented link as a signature as follows:
Let be a finite rack and a commutative ring with an -module structure. The rack module enhanced invariant is given by:
Let be the rack from Example 4 and let . The matrix
defines an -module structure on . To compute for the Hopf link , we must compute for each valid -labeling of . For instance, the following -labeled diagram has fundamental -module with listed presentation matrix:
Replacing each and with its value from and row-reducing over , we have
so the solution space (i.e., the set of bead labelings) is the set and this -labeling contributes to . Repeating for the other labelings, we have .
3 Dynamical cocycles and enhancements of the counting invariant
In this section we generalize the rack module idea to remove the restrictions of the abelian group structure, keeping only those conditions required by the Reidemeister moves. The result is a rack structure on the product defined via a map known as a dynamical cocycle. Dynamical cocycles were defined in  and used to construct extension racks; we will use dynamical cocycles satisfying an extra condition, which we call -reduced dynamical cocycles, to define an enhancement of the rack counting invariant .
Let be a finite rack of rack rank and be a finite set. The elements of will be called beads. A map may be understood as a collection of binary operations indexed by pairs of elements of where where we write . Such a map is a dynamical cocycle on if the maps satisfy:
For all and , the map defined by is a bijection, and
For all and , we have
Let be a rack of rack rank and a dynamical cocycle. Define by . Then if the diagram
commutes for every and , we say the cocycle is -reduced.
The definition of a dynamical cocycle is chosen so that bead labelings of an -labeled diagram are preserved under -labeled framed oriented Reidemeister moves as shown below:
The Reidemeister II and framed type I moves require the operations to be right-invertible; the -reduced condition is required by the -phone cord move:
Let be a finite rack and an -module as defined in Section 2. Then the operations
define an -reduced dynamical cocycle on .
More generally, if is a finite rack of cardinality , we can describe a dynamical cocycle on a finite set with an block matrix, encoding the operations tables for
where the th entry of is when .
Let be a finite rack and an -reduced dynamical cocycle on a set . For an -labeled link diagram , let be the set of -labelings of . Then we define the -reduced dynamical cocycle enhanced invariant or -enhanced invariant by:
By construction, we have
Let be a finite rack and an -reduced dynamical cocycle on a set . If and are ambient isotopic links, then .
The -enhanced invariant is well-defined for virtual knots by the usual convention of ignoring virtual crossings.
4 Computations and Examples
In this section we present example computations of the -reduced dynamical cocycle enhanced invariant.
Let be the rack with rack matrix and let be the dynamical cocycle on given by the block matrix
The virtual knots and the unknot both have Jones polynomial and integral rack counting invariant . Let us compare with . Since has rank , we need to consider diagrams with writhes mod 2. The odd writhe diagrams have no valid -labelings, and there are two valid -labelings of the even writhe diagrams. We collect the valid bead labelings in the tables below.
Hence, we have and is not determined by the Jones polynomial or the integral rack counting invariant .
Similarly, the virtual knots and both have generalized Alexander polynomial
but are distinguished by with for the rack and dynamical cocycle from Example 11.
Hence, is not determined by the generalized Alexander polynomial.
We randomly selected a small dynamical cocycle on the set for the dihedral quandle with matrices below.
We then computed for the list of prime classical knots with up to eight crossings and prime classical links with up to seven crossings as listed at the knot atlas . The results are collected below. In particular, note that the invariant values both specailize to the same rack counting invariant value , and we see that is not determined by .
Our python results indicate that of the 116 prime virtual knots with up to 4 classical crossings listed at the knot atlas, for this is for the virtual knots and , for , and for the other virtual knots in the list.
Our python code for computing -reduced dynamical cocycles and their link invariants is available at www.esotericka.org.
5 Questions for future research
In this section we collect a few questions for future research.
For a given pair of knots or links, how can we choose and to maximize the liklihood of distinguishing the knots or links in question? Is there an algorithm, perhaps starting with presentations of the fundamental racks of the knots, to construct a rack and dynamical cocycle such that always distinguishes inequivalent knots?
A natural direction of generalization is to look at knotted surfaces in , which have an integral quandle counting invariant which should be susceptible to enhancement by beads. What analog of the dynamical cocycle condition arises from the Roseman moves with beads on each sheet?
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