Open main menu
Home
Random
Recent changes
Special pages
Community portal
Preferences
About Wikipedia
Disclaimers
Incubator escapee wiki
Search
User menu
Talk
Dark mode
Contributions
Create account
Log in
Editing
Racks and quandles
(section)
Warning:
You are not logged in. Your IP address will be publicly visible if you make any edits. If you
log in
or
create an account
, your edits will be attributed to your username, along with other benefits.
Anti-spam check. Do
not
fill this in!
==Examples and applications== Every group gives a quandle where the operations come from conjugation: : <math>\begin{align} a \triangleleft b &= a b a^{-1} \\ b \triangleright a &= a^{-1} b a \\ &= a^{-1} \triangleleft b \end{align}</math> In fact, every equational law satisfied by [[inner automorphism|conjugation]] in a group follows from the quandle axioms. So, one can think of a quandle as what is left of a group when we forget multiplication, the identity, and inverses, and only remember the operation of conjugation. {{anchor|Fundamental quandle}}Every [[tame knot]] in [[Three-dimensional space|three-dimensional]] [[Euclidean space]] has a 'fundamental quandle'. To define this, one can note that the [[fundamental group]] of the knot complement, or [[knot group]], has a presentation (the [[Wirtinger presentation]]) in which the relations only involve conjugation. So, this presentation can also be used as a presentation of a quandle. The fundamental quandle is a very powerful invariant of knots. In particular, if two knots have [[isomorphic]] fundamental quandles then there is a [[homeomorphism]] of three-dimensional Euclidean space, which may be [[orientation reversing]], taking one knot to the other. Less powerful but more easily computable invariants of knots may be obtained by counting the homomorphisms from the knot quandle to a fixed quandle <math>\mathrm{Q}.</math> Since the Wirtinger presentation has one generator for each strand in a [[knot diagram]], these invariants can be computed by counting ways of labelling each strand by an element of <math>\mathrm{Q},</math> subject to certain constraints. More sophisticated invariants of this sort can be constructed with the help of quandle [[cohomology]]. The {{visible anchor|Alexander quandles}} are also important, since they can be used to compute the [[Alexander polynomial]] of a knot. Let <math>\mathrm{A}</math> be a module over the ring <math>\mathbb{Z}[t, t^{-1}]</math> of [[Laurent polynomial]]s in one variable. Then the '''Alexander quandle''' is <math>\mathrm{A}</math> made into a quandle with the left action given by :<math>a \triangleleft b = tb + (1 - t)a. </math> Racks are a useful generalization of quandles in topology, since while quandles can represent knots on a round linear object (such as rope or a thread), racks can represent ribbons, which may be twisted as well as knotted. {{anchor|Involutory quandles}}A quandle <math>\mathrm{Q}</math> is said to be '''involutory''' if for all <math>a, b \in \mathrm{Q},</math> : <math> a \triangleleft(a \triangleleft b) = b </math> or equivalently, : <math> (b \triangleright a) \triangleright a = b .</math> Any [[Riemannian symmetric space|symmetric space]] gives an involutory quandle, where <math>a \triangleleft b</math> is the result of 'reflecting <math>b</math> through <math>a</math>'.
Edit summary
(Briefly describe your changes)
By publishing changes, you agree to the
Terms of Use
, and you irrevocably agree to release your contribution under the
CC BY-SA 4.0 License
and the
GFDL
. You agree that a hyperlink or URL is sufficient attribution under the Creative Commons license.
Cancel
Editing help
(opens in new window)