In mathematics , quandles are an algebraic structure that is mainly used in knot theory .
definition
A quandle is a set with one operation , so the following applies to all :
![Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/8752c7023b4b3286800fe3238271bbca681219ed)
![\ triangleright](https://wikimedia.org/api/rest_v1/media/math/render/svg/23951eaf8dba0ab46e9884c6c0f717558acc2195)
![x, y, z \ in Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/76c2b24ad81db86df82d6b8f6c01c1af89b8d89d)
- (i)
- (ii) the mapping defined by is a bijection
![{\ displaystyle f_ {y} (x) = x \ triangleright y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8465b6c477193c11bdefafa63b7578eb48bf5f47)
- (iii) .
![{\ displaystyle (x \ triangleright y) \ triangleright z = (x \ triangleright z) \ triangleright (y \ triangleright z)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7830506527c9dea947ccdff5a133963c0c22bb)
Condition (iii) is called self-distributivity .
Because there is a bijection, there is an inverse mapping . The operation is for by
![{\ displaystyle f_ {y} ^ {- 1} \ colon Q \ to Q}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cdc80cea9f8250f45f63b5be00bebba5cb7979bf)
![{\ displaystyle \ triangleright ^ {- 1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8139265be3db618de8df29b36e47fbe10cbf48)
![x, y \ in Q](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef1e7c725895862675aec01c3d43046eb49bab41)
![{\ displaystyle x \ triangleright ^ {- 1} y: = f_ {y} ^ {- 1} (x)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdd7c64f2692aa454284d6246fc8708f10731a29)
Are defined.
Reidemeister movements
The Quandle operations can be interpreted using the Reidemeister movements of node diagrams :
Examples
- Every Abelian group is a Quandle with the operation
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
-
.
- For a group and one defines the quandle as the set with the operation
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![n \ in \ mathbb {N}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d059936e77a2d707e9ee0a1d9575a1d693ce5d0b)
![{\ displaystyle Conj_ {n} (G)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4d1ad646a8f5e40a0f244dff5bf717eead9f4ec)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
-
.
- For a group , the quandle is defined as the set with the operation
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle Core (G)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/696a7543c80dda32911f832ee0efb3d186e8e16b)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
-
.
- Each - module is a Quandle with surgery
![{\ displaystyle \ mathbb {Z} \ left [t ^ {\ pm 1} \ right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1ffa18b92d23581bf13760d359ba59a19b380ab2)
-
.
- These quandles are known as Alexander quandles .
- The fundamental quandle of a knot (or more generally a link ) is defined as follows. Be the complement of a regular environment and . Define
![K \ subset S ^ {3}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4747a42a2fb271b3e3d158d0672230d8421d3dd4)
![{\ displaystyle X_ {K} = S ^ {3} \ setminus int (N (K))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/821840d3b2015a7090ba0f5f9e1d0716ad0d9ee2)
![{\ displaystyle x_ {0} \ in \ partial X_ {K}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da19e689e204480f82c20d4d74059f922a3efabd)
![{\ displaystyle Q (K) = \ pi _ {1} (C (K), \ partial C (K), x_ {0})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fc9a61225fc8dc537a456799df3b6ec2b1ba6a8)
- with the (well-defined) link
-
,
- where the meridian is denoted by.
![{\ displaystyle m _ {\ gamma _ {2} (1)}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8d5720cfc243643702286107136605c8c15c88c)
![{\ displaystyle \ gamma _ {2} (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba119497fad6793fe87f05820784dcb117d9279)
literature
-
David Joyce : A classifying invariant of knots, the knot quandle. J. Pure Appl. Algebra 23 (1982) no. 1, 37-65.
-
Sergei Matwejew : Distributive groupoids in knot theory. (Russian) Mat. Sb. (NS) 119 (161) (1982), no. 1, 78-88, 160.
Web links