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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" while dropping the writhe
invariance) was restored by Fenn
and Rourke and refers to a more general kind of object than the term
"quandle." 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
- xx=x for all elements x ∈X,
- for every pair of elements y,z ∈X, there is a unique
element x ∈X such that z=xy, and
- (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)z ≠
x (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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Copyright © 2004-2010 Sam Nelson
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