Specifically, a quandle is a set Q with an operation (i.e., a way of combining two quandle elements a and b to get a new quandle element ab) that satisfies

1. aa=a for all elements a of Q,
2. for every pair of elements b,c of Q, there is a unique element a of Q such that c=ab, and
3. (ab)c = (ac)(bc) for all quandle elements a,b,c.

Axiom (2) is equivalent to the quandle operation having a right inverse, that is, a second operation ^-1 such that (a y) ^-1 b = a for all elements of Q.

This quandle operation is generally non-commutative and non-associative, i.e., x y ≠y x and (x y)z ≠ x (yz) in general, so it takes a little getting used to. The reader is encouraged to check that a group G is a quandle with quandle operation given by conjugation, i.e. set

ab = b^-1ab.

Another important example is the class of Alexander quandles: let A be a module over the ring Z [t, t^-1] of Laurent polynomials in one variable. Then A is a quandle with quandle product given by

ab = ta + (1- t)b.

In Matrices and finite quandles, Benita Ho and I observed that a finite quandle Q={x₁,x₂,...,x_n} may be represented as a matrix M where the entry M[i,j] in row i column j is k where

x_k = x_ix_j.

For example, the dihedral quandle of order n, denoted R_n, is the set {1,2,...,n} with quandle operation given by

ij = 2j - i mod n.

(Here, "mod n" means "divide by n and keep only the remainder.") Thus, R₆ has matrix

The quandle axioms tell us which matrices represent quandles and which don't. Specifically, the diagonal entries should be 1,2,...,n (assuming the matrix is in standard form, i.e., the entry M[i,j] in row i column j represents the element ij), every column should be a permutation of the numbers 1,2,..,n, and for every triple i,j,k of numbers 1,..,n we should have

This matrix notation gives us a convenient way of doing computations with finite quandles. Indeed, you can download some programs in the Maple programming language 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 [C2]. A Maple program for finding all Alexander presentations of a given finite quandle can be found here. For those without access to Maple, we also have Python code below.

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. Then at each crossing, we have the following relationship between the arcs:

That is, the left hand undercrossing arc z is the quandle product of the right hand undercrossing arc x with the overcrossing arcy, that is, z=xy. This depends on the orientation of the overcrossing arc, but not the undercrossing. That the knot quandle is an invariant of knot type is easy to check; we just check that Reidemeister moves don't change the quandle. 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.

           

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.

Thus, if the labels on the arcs come from a quandle and are chosen according to the crossing rule above, then for every coloring of the diagram before the move, there is a corresponding coloring by the same quandle after the move. That is, the total number of colorings of a knot diagram by elements of a fixed finite quandle satisfying the coloring 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 colorings of the two diagrams by the same finite quandle which satisfy the coloring 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. Thus, every finite quandle defines a knot invariant, that is, a computable quantity which is the same for all diagrams of a given knot.

Many classical invariants of knots can be derived from the knot quandle, including the knot group, the Alexander (and hence Conway) polynomials and more. Quandles provide a convenient way of devising new knot invariants using invariants of quandles, such as counting homomorphisms of quandles using various cohomology elements as weights, etc. See [C1] for more on various enhancements of this coloring invariant. When the finite coloring quandle is Alexander, the number of colorings is determined by the classical Alexander invariants [I]; the role of other invariants in determining the quandle counting invariant for non-Alexander quandles is currently under investigation.

Quandles 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]

Files

• Python code for semiquandles
• Python code for racks and quandles (needs Python and SymPy)
• quandles-maple.txt
• Alexander quandles maple code
• Quandle difference code
• biquandles-maple.txt
• Yang-Baxter cohomology maple code
• Bilinear biquandles maple code
• Virtual biquandles maple code
• generalized quandle polynomial maple code
• C program for finding finite quandles, by Richard Henderson
• All quandles of order 6
• All quandles of order 7
• All quandles of order 8
• The Structure of Racks, a dissertation by Hayley Ryder