Link invariants from finite racks
We define ambient isotopy invariants of oriented knots and links using the counting invariants of framed links defined by finite racks. These invariants reduce to the usual quandle counting invariant when the rack in question is a quandle. We are able to further enhance these counting invariants with 2-cocycles from the coloring rack’s second rack cohomology satisfying a new degeneracy condition which reduces to the usual case for quandles.
A rack is a generally non-associative algebraic structure whose axioms correspond to blackboard-framed isotopy moves on link diagrams. Racks generalize quandles, an algebraic structure whose axioms correspond to the three Reidemeister moves which combinatorially encode ambient isotopy of knot diagrams.
Given a finite quandle , the set of quandle homomorphisms from a knot quandle to gives us an easily computed knot invariant, namely its cardinality . This is the quandle counting invariant, also sometimes called the quandle coloring invariant since each homomorphism can be pictured as a “coloring” of the knot diagram assigning to each arc in a knot diagram the element such that a quandle coloring condition is satisfied at every crossing. Indeed, Fox 3-coloring is the simplest non-trivial example of a quandle coloring invariant for knots.
If is a non-quandle rack, the set of colorings of arcs of a link diagram by elements of satisfying the coloring condition at every crossing is invariant only under blackboard-framed isotopy – type I Reidemeister moves which change the framing of the knot also change the number of colorings. In this paper we will exploit a property of finite coloring racks to define computable invariants of ambient isotopy of knots and links incorporating these framed isotopy coloring invariants. The usual quandle coloring invariants then form a special case of these more general rack coloring invariants.
The paper is organized as follows. In section 2 we review the basics of racks, framed links and virtual links. In section 3 we define finite rack based counting invariants and give some examples. In particular, we show that the writhe-enhanced invariant specializes to the integral invariant and contains more information. In section 4 we enhance the rack counting invariants with 2-cocycles in the style of . We provide an example showing that the cocycle-enhanced invariant contains more information than the writhe-enhanced and integral rack counting invariants. In section 5 we collect questions for future research.
2 Basic definitions
In this section we review the basic definitions we will need for the remainder of the paper.
We begin with a definition from .
A rack is a set with a binary operation satisfying
for all , the map defined by is invertible, with inverse denoted , and
for all , we have .
A rack in which for all is a quandle. The operation is the dual rack operation – it is also self-distributive, and the two operations are mutually distributive. Note that in , is denoted and is denoted .
The rack axioms correspond to Reidemeister moves II and III where we think of rack elements as arcs in an oriented link diagram and means crossing under from right to left when looking in the positive direction of the overcrossing strand. The dual operation can then be interpreted as crossing under from left to right.
Perhaps the simplest example of a rack structure on a finite set is the constant action rack or permutation rack on associated to a permutation . Specifically, set
for all ; then the action of on remains constant as varies. It is easy to verify that this definition gives us a rack structure, since and we have
If a constant action rack is a quandle, then we have and consequentially for all ; such a quandle is called trivial. There is one trivial quandle for each cardinality , denoted . We will denote the constant action rack associated to by .
A simple example of a nontrivial rack structure from  is the -rack structure: let be the ring modulo the ideal generated by . Then any -module is a rack under the operation
For instance, we can take and choose such that and , e.g. with and . If then is a quandle, known as an Alexander quandle.
One useful way to describe a rack operation on a finite set is to encode its operation table as a matrix whose entry in row column is where . Thus, the constant action rack on defined by has matrix
and the -rack with and has rack matrix
Rack axiom (i) requires the columns of a rack matrix to be permutations. See  for more.
2.2 Framed links
Recall that a framed link is a link with a choice of framing curve for every component of , i.e. is a longitude of a regular neighborhood of . Framing curves are determined up to isotopy by their linking numbers with . In terms of diagrams, we can bestow a canonical framing on every component of a link via the blackboard framing, i.e. drawing a framing curve for each parallel to . This gives a framing with linking number given by the writhe where is the set of crossings where crosses itself, and
Combinatorially, blackboard-framed links can be regarded as equivalence classes of link diagrams under the equivalence relation generated by Reidemeister moves II and III together with a doubled type I move which preserves the framing of each component; see [15, 7].
2.3 Virtual links
Virtual knot theory is a combinatorial generalization of ordinary classical knot theory; geometrically, a virtual link is an ordinary link in which the ambient space is not or but for some compact orientable surface , considered up to stabilization (see [11, 4]). More formally, we have:
A virtual link is an equivalence class of link diagrams with an extra crossing type known as a virtual crossing, , under the equivalence relation determined by the usual Reidemeister moves together with the four virtual moves
We can summarize the rules for virtual moves with the detour move, which says that any strand with only virtual crossings can be replaced by any other strand with the same endpoints with only virtual crossings. That is, a strand with only virtual crossings can move past any virtual tangle.
Virtual crossings have no intrinsic over- or under-sense, as they are artifacts of drawing non-planar link diagrams on planar paper. Classical links are then virtual links whose underlying surface is . Replacing the classical Reidemeister I move with the doubled version yields framed virtual links. For the remainder of this paper, we will use “link diagram” to mean “oriented blackboard-framed virtual or classical link diagram.” See  for more.
2.4 The fundamental rack(s) of a link
Associated in  to a framed link is a rack known as the fundamental rack of , which we will denote by .111Does the notation “FR” stand for “fundamental rack” or “Fenn and Rourke”? Perhaps both!
Geometrically, elements of are homotopy classes of paths in the link complement from the framing curves to a fixed base point where the terminal point is fixed but the initial point is permitted to wander along the framing curve during the homotopy. Any such path has an associated element of the fundamental group defined by traveling backwards along then going around the canonical meridian in intersecting , then going back along . The rack operation is then
where is concatenation of paths.
Combinatorially, given a diagram of , the fundamental rack of consists of equivalence classes of rack words in generators corresponding to arcs in the diagram of under the equivalence relation generated by the rack axioms together with the relations imposed at each crossing. If is a virtual link, we simply ignore the virtual crossings.
The pictured blackboard-framed virtual link has fundamental rack with generators and relation :
For each framing of a given link, we have a fundamental rack, generally distinct from the racks of the other framings. All of these racks have a common quotient quandle obtained by setting for all elements , which is the knot quandle of the unframed link . Elements of the knot quandle may be interpreted geometrically as homotopy classes of paths where the initial point is permitted to wander not just along the framing curve but along all of . See  for more.
3 Racks and counting invariants
Let be an unframed link with an ordering on the components. If has components, then the framings of may be indexed by -tuples , each with its own a priori distinct fundamental rack. At the most basic level, then, there are infinitely many rack counting invariants for a given link with respect to any choice of finite target rack . However, we can make a useful observation which enables us to get computable ambient isotopy invariants from the -set of racks of .
Let be a rack. For any , let for be defined recursively by
For each element , the rack rank of , denoted , is the minimal natural number such that , or if for all . The rack rank of , denoted or just if is understood, is the least common multiple of the rack ranks of the elements of ,
To see that is well defined for all , we first need a lemma.
Let be any rack and . Then .
Two elements are operator equivalent if for all . If is a finite rack, then two elements are operator equivalent iff their columns in the matrix of are identical. Lemma 1 says that the -powers of are all operator equivalent. Indeed, the set of operator equivalence classes of a rack forms a quandle under the natural operation .
Let be a rack. If , then .
Suppose We have and Then implies and rack axiom (i) implies . ∎
In terms of rack matrices, corollary 2 says that like the columns of a rack matrix, the diagonal of a rack matrix must be a permutation. Indeed, if we define by then the diagonal of a rack matrix tell us the permuation . It then easily follows that for any where is a finite rack – indeed, is just the exponent of . This fact also follows from proposition 7.3 in .
If and are ambient isotopic link diagrams, we can modify to obtain a diagram which is framed isotopic to by selecting an arc on each component of and adding positive or negative kinks until the framings match.
The proof of theorem 3 involves taking any Reidemeister move sequence starting with and ending with and replacing every type I move with a double I move to adjust the framed isotopy class; at the end, we can then slide the extra crossings along the component until they arrive at the chosen arc. Note that this argument applies to virtual links as well as classical links, since we can slide a classical kink past a virtual crossing using a detour move. Note also that without loss of generality we can assume that all kinks added have positive winding number since we need not preserve the regular isotopy class, only the blackboard-framed isotopy class.
Let . We say two blackboard-framed oriented link diagrams are -phone cord equivalent if one may be obtained from other by a finite sequence of Reidemeister II and III moves and the following -phone cord move, where is the number of loops:
Let be a finite rack with rack rank . If two link diagrams and are -phone cord isotopic then .
From the definition of rack rank, it is easy to see that -phone cord moves induce a bijection on the set of colorings as illustrated.
For two -tuples , let us write if for all components we have .
Let be a finite rack with rack rank . If two link diagrams and are ambient isotopic and have writhe vectors congruent modulo , then .
Note that if is a finite rack with rack rank and is a link, the set of writhes of each component of modulo can be indexed by where is the number of components of . For ease of notation, when and are understood let us denote and a blackboard-framed diagram of with writhe vector by
We can now define computable unframed knot and link invariants using these cardinalities.
Let be a finite rack and a link with components. The integral rack counting invariant of with respect to is
Note that if is a quandle, then we have and is the ordinary quandle counting invariant . Hence the integral rack counting invariant is the natural generalization of the quandle counting invariant to the finite rack case.
If is a constant action rack defined by an -cycle, an undercrossing color becomes if going right-to-left and if going left-to-right, so pushing a color around the knot yields an ending color of where is the writhe of and our starting color was . Thus, there is a rack coloring of a framed knot by if and only if the writhe of is zero mod . Indeed, there are such colorings for the -framing mod and none for the others, and we have for any knot . This generalizes the fact that for a trivial quandle of cardinality and a knot.
Let be the rack with matrix
The integral rack counting invariant with respect to distinguishes the trefoil from the unknot with and , as the reader can verify from the tables of colorings listed in table 1. Here , so we need only consider one diagram each of and with odd writhe and one of each with even writhe.
We can enhance the integer rack counting invariant by keeping track of which framings contribute which colorings. For a writhe vector let us denote the product by . Then we have:
Let be a finite rack and a link with components and writhe vector . The writhe-enhanced rack counting invariant of with respect to is given by
The writhe-enhanced rack counting invariant holds more information than the simple version, as the next example shows.
The constant action rack with rack matrix has rack rank . The Hopf link and the two-component unlink both have integral rack counting invariant value with respect to , but the writhe-enhanced rack counting invariants are distinct, with and as the reader can easily verify from table 2.
Indeed, generalizing the preceeding example we have
Let be a two-component classical link and a constant action rack with an -cycle. Then the writhe-enhanced rack counting invariant has the form
where is the negative of the linking number of mod .
Traveling around a component, to get a valid coloring the end color must match the initial color, so we must go through crossings (counted algebraicially). Since of these are multi-component crossings which do not contribute to the component’s writhe, we must have
The same holds for both components if is classical. There are choices of starting color for each component and every pair produces exactly one coloring, so there are total colorings. ∎
If is a two-component classical link and a constant action rack with an -cycle such that the exponents of of and differ in any term of , then is non-classical.
4 Rack cocycle invariants
In this section we generalize the quandle 2-cocycle invariants defined in  to the finite rack case.
The rack counting invariants described in the last section are cardinalities of sets of homomorphisms which are unchanged by Reidemeister moves. However, a set is more than a cardinality, and we would like to recover as much information from these sets of homomorphisms as possible.
In , the idea is to associate a sum in an abelian group called a Boltzmann weight to a quandle-colored knot diagram in such a way that the sum does not change under Reidemeister moves. Then, instead of counting “1” for each homomorphism, we count its Boltzmann weight, transforming the set of colorings into a multiset of these weights. Such multisets are commonly encoded as polynomials by taking a generating function, i.e. by converting the multiset elements to exponents and multiplicities to coefficients of a dummy variable, e.g. becomes .
The Boltzmann weights are defined as follows: at every crossing in a rack-colored link diagram, we want to count at a positive crossing or at a negative crossing where is the color on the overcrossing strand and is the color on the inbound understrand for positive crossings and the outbound understrand for negative crossings.
This weighting rule has the advantage that the contributions from the two crossings in a Reidemeister type II move cancel, so the sum is automatically invariant under II moves:
We also note that the weighting rule gives invariance under the doubled type I moves required for blackboard-framed isotopy:
The condition for the sum to be unchanged by Reidemeister III moves is pictured below.
This turns out to be the condition that is a cocycle in the second rack cohomology of the rack with coefficients in . Specifically, the -module spanned by -tuples of elements of is the space of rack -chains ; its dual is the space of rack -cochains . Note that has -generating set where
for . Next, we define a coboundary map by
Then for to yield a Boltzmann weight, we need
To get invariance under the Reidemeister I move in the quandle case, we require that for all . This condition also turns out to have a homological interpretation – the cocycles we want to kill live in a subcomplex called the degenerate cochains. In the non-quandle rack case, however, a weaker condition suffices.
Let be a finite rack with rack rank , an abelian group and Say is -reduced if we have
Now we can define an enhanced version of the polynomial rack counting invariant:
Let be an oriented blackboard-framed link, a finite rack and an -reduced rack 2-cocycle. For a rack-colored framed diagram of , , the Boltzmann weight of the coloring is the sum over the crossings in of the crossing weights,
Then the rack cocycle invariant of with respect to is
and the writhe-enhanced rack cocycle invariant is
Note that if is a quandle, then and is -reduced iff for all degenerate cycles ; in this case we also have . Indeed, in the quandle case this rack cocycle invariant becomes the usual CJKLS quandle 2-cocycle invariant from .
Specializing in recovers the writhe-enhanced rack counting invariant, and likewise in the integral case. To see that the integral rack cocycle invariant is stronger than , consider the following simple example.
The rack with rack matrix
has a reduced cocycle with coefficients. Then the -torus link is distinguished from the two-component unlink by :
Our final example illustrates a pair of virtual links with equal writhe-enhanced rack counting invariant values which are distinguished when we include the rack cocycle information.
Again let be the rack with rack matrix
and . Then distinguishes the two pictured virtual links, both of which have . Note that the subscripts on correspond to the component ordering.
For more on rack cocycle invariants, see .
In this section we collect a few questions for future research.
Rack and quandle (co)homology has been generalized in various ways including twisted quandle (co)homology in , quandle (co)homology with coefficients in quandle modules in  and more. How does the rack cocycle invariant change in these cases?
Quandle 3-cocycles have been used to enhance quandle counting invariants of surface knots, i.e. embeddings of compact orientable 2-manifolds in . How do the rack counting and cocycle invariants extend to the surface knot case?
Other ways of enhancing the quandle counting invariants include using quandle polynomials and exploiting any extra structure the quandle may have (symplectic vector space, -module, etc.); generalizing these ideas to the rack case will be the subject of future papers.
Replacing the arcs in the combinatorial motivation for the rack axioms with semiarcs yields biracks, also known as invertible switches or Yang-Baxter sets (see [8, 6]). The birack analogues of the simple and polynomial rack counting invariants will be the subject of another future paper.
Python code for computing rack counting invariants, reduced rack -cocycles with coefficients, and rack cocycle invariants is available for download at www.esotericka.org.
-  J. S. Carter, M. Elhamdadi, M. Graña and M. Saito. Cocycle knot invariants from quandle modules and generalized quandle homology. Osaka J. Math. 42 (2005) 499–541.
-  J. S. Carter, M. Elhamdadi and M. Saito. Twisted quandle homology theory and cocycle knot invariants. Algebr. Geom. Topol. 2 (2002) 95–135.
-  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. 355 (2003) 3947–3989.
-  J. S. Carter, S. Kamada and M. Saito. Stable equivalence of knots on surfaces and virtual knot cobordisms. J. Knot Theory Ramifications 11 (2002) 311–322.
-  M. Elhamdadi and S. Nelson. -degeneracy in rack homology and link invariants. Hiroshima Math. J. 42 (2012) 127–142.
-  R. Fenn, M. Jordan-Santana and L. Kauffman. Biquandles and virtual links. Topology Appl. 145 (2004) 157–175.
-  R. Fenn and C. Rourke. Racks and links in codimension two. J. Knot Theory Ramifications 1 (1992) 343–406.
-  R. Fenn, C. Rourke and B. Sanderson. Trunks and classifying spaces. Appl. Categ. Structures 3 (1995) 321–356.
-  B. Ho and S. Nelson. Matrices and finite quandles. Homology Homotopy Appl. 7 (2005) 197–208.
-  D. Joyce. A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23 (1982) 37–65.
-  L. Kauffman. Virtual Knot Theory. European J. Combin. 20 (1999) 663–690.
-  L. H. Kauffman and D. Radford. Bi-oriented quantum algebras, and a generalized Alexander polynomial for virtual links. Contemp. Math. 318 (2003) 113–140.
-  P. Lopes and D. Roseman. On finite racks and quandles. Comm. Algebra 34 (2006) 371–406.
-  S. V. Matveev. Distributive groupoids in knot theory. Math. USSR, Sb. 47 (1984) 73–83.
-  B. Trace. On the Reidemeister moves of a classical knot. Proc. Amer. Math. Soc. 89 (1983) 722–724.