Introduction

Quandle theory is a relatively new subject in abstract algebra which has origins in knot theory and new applications to various other areas of mathematics currently being explored. The history of this subject is a story of an idea which keeps getting reinvented and rediscovered. The earliest currently known example dates back to 1940s Japan when Mituhisa Takasaki defined kei, objects which were later known as involutory quandles [T]. Variants on the quandle idea have been studied by Conway (wracks), Brieskorn (automorphic sets), Matveev (distributive groupoids), and Kauffman (crystals), though the current terminology is due to David Joyce, who coined the word "quandle" in his 1980 doctoral dissertation. The term "rack" (canceling the "w" in Conway's "wrack" while dropping the writhe invariance) was restored by Fenn and Rourke and refers to a more general kind of object than the term "quandle"; racks are also known as Crystals and as Automorphic Sets. Readers who are familiar with abstract algebra should think of quandle theory as analogous to group theory, where the quandle axioms are a bit different from the group axioms. Continuing the analogy, racks are to quandles as monoids are to groups.

 

Quandles

Definition: A quandle is a set X with a binary operation that satisfies

  1. xx=x for all elements x ∈X,
  2. for every pair of elements y,z ∈X, there is a unique element x ∈X such that z=xy, and
  3. (xy)z = (xz)(yz) for all x,y,z∈ X.

Axiom 2 is equivalent to the quandle operation having a right inverse, that is, a second operation -1 such that (x y) -1 y = x for all x,y∈ X.

This quandle operation is generally non-commutative and non-associative, i.e., in general x y ≠y x and (x y)zx (yz). The reader is encouraged to check that the following are examples of quandles:

  • Takasaki kei: Let X be the integers mod n with quandle operation xy = 2y - x mod n,
  • Conjugation quandles: Let X be a group with quandle operation given by conjugation, i.e. set xy=y-1xy,
  • Alexander quandles: Let X be a module over the ring Z [t, t-1] with quandle operation given by xy = tx + (1- t)y.

 

Computation with quandles

In Matrices and finite quandles, Benita Ho and I observed that a finite quandle Q={x1,x2,...,xn} may be represented as a matrix M where the entry M[i,j] in row i column j is k where xk = xixj. This quandle matrix notation enables us to do computations with finite quandles without having a algebraic formula for the quandle operation, which is extremely useful and greatly expands the set of quandle structures we can work with.

For example, let Q be the Takasaki kei of order 6, i.e., the set {1,2,3,4,5,6} with ij = 2j - i mod 6. (Recall that "mod n" means "divide by n and keep only the remainder.") Then Q has quandle matrix MQ as listed.

The quandle axioms tell us which matrices represent quandles and which don't. Specifically, if is a quandle matrix then the diagonal entries must be 1,2,...,n, every column must be a permutation of the numbers 1,2,..,n, and for every triple i,j,k of numbers 1,..,n we must have M[M[i,j],k]= M[M[i,k],M[j,k]].

This matrix notation gives us a convenient way of doing computations with finite quandles. You can download my python code or the older maple code for doing computations with finite quandles. Also available are files containing (in Maple format) all finite quandle matrices of order 6, order 7, and order 8. This last file uncompresses to around 800 MB! The C source for the program which generates these matrices, contributed by my friend Richard at Red Hat, is here. An independently generated list of quandles with up to 6 elements and their homology groups can be found in [K]. A Maple program for finding all Alexander presentations of a given finite quandle can be found here.

 

Quandles and Knots

The relationship between quandles and knots was established by David Joyce in [J], where the knot quandle is defined. Specifically, we take a knot diagram and assign a letter to each arc in the diagram, i.e. the quandle elements are labels for the arcs. Then at each crossing, we have the pictured relationship between the arcs.

That is, the arc labeled x-1y is the result of the arc labeled x crossing under the arc labeled y from left to right, while the arc labeled xy is the result of the arc labeled x crossing under the arc labeled y from right to left. That the knot quandle is an invariant of knot type is easy to check; we just verify that Reidemeister moves don't change the quandle. The quandle axioms are just the conditions required for all the labels on the edges of the diagrams to be identical before and after the moves. These pictures are meant to represent two portions of knot diagrams, and the portions of the diagrams outside the pictured parts are identical. In particular, the labels on the arcs at the edges of the pictures have to be the same before and after the move.

           

Thus, if the labels on the arcs come from a quandle and are chosen according to the crossing rule above, then for every labeling of the diagram before the move, there is exactly one corresponding labeling by the same quandle after the move. That is, the total number of labelings of a knot diagram by elements of a fixed finite quandle satisfying the labeling condition is the same for any two diagrams of the knot. Hence, to prove that two knot diagrams represent distinct knots, we can count the number of labelings of the two diagrams by the same finite quandle which satisfy the labeling condition. If the numbers are the same, then the test doesn't tell us anything, but if the numbers are different, then the diagrams must represent different knots. In this way, we get a knot invariant from a finite quandle, known as the counting invariant, denoted |Hom(Q(K),Q)|.

 

Racks

A rack is like a quandle except we do not require the first quandle axiom; the rack axioms are just the second and third quandle axioms. Racks are invariants of framed isotopy in which the first Reidemeister move is replaced with a writhe-preserving doubled version; in particular, this means we no longer need xx=x. Thus, quandles are a type of racks, and any theorems we can prove about racks are automatically true for quandles as well. The reader is encouraged to verify that examples of rack structures include:

  • Constant action racks: Let X={1,2,3,...,n} and choose any permutation σ∈Sn; then define xy=σ(x),
  • (t,s)-racks: Let X be any module over the ring Z[t,t-1,s]/(s2-(1-t)s) with rack operation defined by xy=tx+sy.

The counting invariant for a finite rack changes when we do ordinary type I moves, but it turns out there is a way to sum these framed isotopy invariants to get an invariant of ambient isotopy; see [S] for more.

 

Enhancements

Much of my research involves enhancements of these and other related counting invariants. The idea is quite simple -- instead of counting "1" for each labeling, we define a kind of "signature" for each labeling which is preserved by Reidemeister moves. We then take the multiset of these signatures to get a new enhanced invariant which specializes to the counting invariant but in general contains more information. The first enhancements were the CJKLS quandle 2-cocycle invariants described in [C]. Many of my papers with undergraduate collaborators are about new enhancements of counting invariants.

 

Other Applications

Quandles and racks are also proving useful in other areas of mathematics. For example, the relationship between a Lie group and its associated Lie algebra can be described in a natural way by using the language of quandle theory; another example is that of monodromies, where a description in terms of quandles automatically satisfies requirements which must be included and checked manually when described in terms of groups [B,Y]. The field of quandle theory is still quite young and full of exciting unanswered questions.

 

References

[B] E. Brieskorn. Automorphic Sets and Braids and Singularities. Contemp. Math. 78 (1988) 45-115.
[C] J.S. Carter, D. Jelsovsky, S. Kamada, L. Langford, M. Saito. State-sum invariants of knotted curves and surfaces from quandle cohomology Trans. Amer. Math. Soc. 355 (2003) 3947--3989.
[K] J. S. Carter, S. Kamada, and M. Saito. Surfaces in 4-space. Encyclopaedia of Mathematical Sciences, 142. Low-Dimensional Topology, III. Springer-Verlag, Berlin, 2004.
[F] R. Fenn. and C. Rourke. Racks and links in codimension two, J. Knot Theory Ramifications 1 (1992) 343-406.
[I] A. Inoue. Quandle Homomorphisms of Knot quandles to Alexander quandles. J. Knot Theory Ramifications 10 (2001) 813-821.
[J] D. Joyce. A classifying invariant of knots, the knot quandle. J. Pure Appl. Alg. 23 (1982) 37-65.
[N] M. Niebrzydowski and J. Przytycki. Burnside kei. Fundam. Math. 190 (2006) 211-229.
[S] S. Nelson. Link invariants from finite racks. To appear in Fund. Math.
[R] H. Ryder. The Structure of Racks, PhD Thesis, U. Warwick.
[T] M. Takasaki. Abstractions of symmetric functions. Tohoku Math. J. 49 (1943) 143-207.
[Y] D. Yetter. Quandles and monodromy. J. Knot Theory Ramifications 12 (2003) 523-541.

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