Talk:Racks and quandles

message: A quandle should be defined in terms of a rack such that the duplication of axioms can be removed. --MarSch 11:16, 25 June 2007 (UTC)
replies: text: Done! --John Baez (talk) 19:50, 13 June 2010 (UTC)
thread_title: Racks first

message: I added some material on involutory quandles and the 'fundamental quandle' of a knot.  It would be nice to have material on the Alexander quandle.  A more glaring omission is that there's no discussion anywhere of the Wirtinger presentation of the knot group. John Baez 08:08, 28 June 2007 (UTC)
thread_title: New material

message: The concept seems wrong.  For consistency with the rest of the article I'll first reword Axiom 2 to swap a and b so we have x*b = a.  Axiom 2 so reworded then says that there is a function, which I'll call a/b for now, such that (a/b)*b = a.  But the concept mysteriously omits this function from the signature, perhaps in order to satisy the article's statement "In fact, every equation satisfied by conjugation in a group follows from the three quandle axioms."
The sensible thing to do obviously is to name the function.  But this falsifies the statement in a minor way by omitting (a/b)*b = a, minor in that it's implied by Axiom 2 so obviously we put it in.  However it further falsifies it in a major way by omitting (a*b)/b = a, which is also true under the interpretation a*b = b'ab in a group, because a/b in that case is necessarily bab', conjugation of a by b'.
There is a further problem with the notion of an involutary quandle as one where Axiom 2 is strengthened to (a*b)*b = a.  We then get a/b = ((a/b)*b)*b = a*b.  If quandles are supposed to come from conjugation then b'ab = bab' or b'b'abb = a, that is, b*c = b where c = aa (i.e. any square) in the group.  Not many non-abelian groups satisfy this, although all abelian groups do.  However a*b = a in an abelian group, making abelian groups a poor motivation for quandles in terms of conjugation.
A better picture I think would be to have three distinct binary operations corresponding to b'ab, ba'b, and bab'.  The first and third of these should be paired up as the definition of "quandle."
The second stands alone as a binary operation that, unlike conjugation, is not only meaningful but very useful in an abelian group as it expresses the concept of "next", as in "the next point after a and b is ba'b," or 2b - a as it becomes in an abelian group.  But even in nonabelian groups it retains that meaning, and can be considered the algebraic expression of discrete or stepwise motion along a geodesic (perhaps what Gavin Wraith had in mind by his term "sequential").  Slide 6 of http://boole.stanford.edu/pub/consgeom.pdf (for a talk I'll be giving tomorrow at http://euclid.colorado.edu/~kasterma/blast/index.php , slides at http://spot.colorado.edu/~szendrei/BLAST2010/pratt2.pdf , that has evolved from earlier versions) gives an idea of the kinds of surfaces that can be assembled from geodesics on a surface for general (nonabelian) groups.  This incidentally is apropos of a program to reorganize Euclid's postulates 1, 2, and 5 algebraically in a way that recovers linear algebra (more precisely affine spaces over the rationals) without introducing numbers.  --Vaughan Pratt (talk) 14:10, 4 June 2010 (UTC)
replies: text: I've attempted to improve the definition to address your concerns.  I'm pretty sure, now that I think about it, that the original definition was fairly standard, but suboptimal because it includes existential quantifiers that aren't really necessary.  That made the result about equational laws satisfied by conjugation in groups difficult to state correctly.    The definition I gave is a standard purely equational definition.  John Baez (talk) 19:48, 13 June 2010 (UTC)
thread_title: Wrong concept