Quandle

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 : ${\displaystyle Q}$ ${\displaystyle \triangleright }$ ${\displaystyle x,y,z\in Q}$

(i) ${\displaystyle x\triangleright x=x}$

(ii) the mapping defined by is a bijection ${\displaystyle f_{y}(x)=x\triangleright y}$ ${\displaystyle f_{y}\colon Q\to Q}$

(iii) . ${\displaystyle (x\triangleright y)\triangleright z=(x\triangleright z)\triangleright (y\triangleright z)}$

Condition (iii) is called self-distributivity .

Because there is a bijection, there is an inverse mapping . The operation is for by ${\displaystyle f_{y}}$ ${\displaystyle f_{y}^{-1}\colon Q\to Q}$ ${\displaystyle \triangleright ^{-1}}$ ${\displaystyle x,y\in Q}$

${\displaystyle x\triangleright ^{-1}y:=f_{y}^{-1}(x)}$

Are defined.

Reidemeister movements

The Quandle operations can be interpreted using the Reidemeister movements of node diagrams :

• 

  Reidemeister I.

• 

  Reidemeister II

• 

  Reidemeister III

Examples

• Every Abelian group is a Quandle with the operation ${\displaystyle A}$

${\displaystyle x\triangleright y=2y-x}$.

• For a group and one defines the quandle as the set with the operation ${\displaystyle G}$ ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle Conj_{n}(G)}$ ${\displaystyle G}$

${\displaystyle x\triangleright y=y^{-n}xy^{n}}$.

• For a group , the quandle is defined as the set with the operation ${\displaystyle G}$ ${\displaystyle Core(G)}$ ${\displaystyle G}$

${\displaystyle x\triangleright y=yx^{-1}y}$.

• Each - module is a Quandle with surgery ${\displaystyle \mathbb {Z} \left[t^{\pm 1}\right]}$

${\displaystyle x\triangleright y=tx+(1-t)y}$.

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 ${\displaystyle K\subset S^{3}}$ ${\displaystyle X_{K}=S^{3}\setminus int(N(K))}$ ${\displaystyle x_{0}\in \partial X_{K}}$

${\displaystyle Q(K)=\pi _{1}(C(K),\partial C(K),x_{0})}$

with the (well-defined) link

${\displaystyle \left[\gamma _{1}\right]\triangleright \left[\gamma _{2}\right]=\left[\gamma _{2}\circ m_{\gamma _{2}(1)}\circ \gamma _{2}^{-1}\circ \gamma _{1}\right]}$,

where the meridian is denoted by. ${\displaystyle m_{\gamma _{2}(1)}}$ ${\displaystyle \gamma _{2}(1)}$

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

• Scott Carter : A Survey of Quandle Ideas
• Gavin Wraith : A personal story about knots