Wirtinger presentation

In the mathematical subfield of knot theory , the Wirtinger presentation (or Wirtinger presentation) is a method for describing ( presenting ) a group of nodes . It was named after the Austrian mathematician Wilhelm Wirtinger .

Problem

One of the most important topological invariants is the fundamental group of a topological space . For a mathematical knot , the knot group is defined as the fundamental group of the knot complement .

The Wirtinger presentation provides a presentation of the node group, i.e. an explicit description using generators and relations.

It is generally a nontrivial problem to read properties of a group from a presentation. In the case of node groups, however, there are algorithms which, for example, use the presentations of two node groups to decide whether the nodes are equivalent.

Procedure

Let be a knot diagram of a knot and let P be a point outside the knot. We choose a direction of travel and designate the route sections one after the other in the node diagram. For each arc we choose a loop beginning and ending in P , which consists of a section from P almost to and a loop around , which runs once positive ('right hand rule'), and then runs the previously selected section back to P. . ${\displaystyle D}$ ${\displaystyle K}$ ${\displaystyle B_{1},\ldots ,B_{k}}$ ${\displaystyle B_{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle B_{i}}$ ${\displaystyle B_{i}}$ ${\displaystyle B_{i}}$

We say an intersection is positive if the lower strand goes from right to left as seen from the upper strand (with the given orientation). Otherwise we call the intersection negative. At the i-th crossing point the arcs and are separated by an arch . Each crossing point gives a relation like in the following picture. ${\displaystyle x_{i}}$ ${\displaystyle x_{i+1}}$ ${\displaystyle x_{s(i)}}$

The presentation thus obtained

${\displaystyle \langle x_{1},\ldots ,x_{k}\mid r_{1},\ldots ,r_{k}\rangle }$

With

${\displaystyle r_{i}=x_{i}^{-1}x_{s(i)}x_{i+1}x_{s(i)}^{-1}\ {\mbox{bei einer positiven Kreuzung}}}$

${\displaystyle r_{i}=x_{i}^{-1}x_{s(i)}^{-1}x_{i+1}x_{s(i)}\ {\mbox{bei einer negativen Kreuzung}}}$

is called the Wirtinger presentation and one can prove that it is a presentation of the fundamental group of the knot complement.

Examples

• 

  Diagram of the shamrock knot

• 

  Figure eight knot

The Wirtinger presentation of the trefoil knot is

${\displaystyle \langle x_{1},x_{2},x_{3}\mid x_{1}x_{2}x_{3}^{-1}x_{2}^{-1},x_{2}x_{3}x_{1}^{-1}x_{3}^{-1},x_{3}x_{1}x_{2}^{-1}x_{1}^{-1}\rangle }$,

you can use and simplify this too ${\displaystyle x=x_{1}x_{2}}$ ${\displaystyle y=x_{2}^{-1}x_{1}^{-1}x_{2}^{-1}}$

${\displaystyle \langle x,y\mid x^{3}y^{2}\rangle }$.

The Wirtinger presentation of the figure eight is

${\displaystyle \langle x_{1},x_{2},x_{3},x_{4}\mid x_{3}x_{4}^{-1}x_{3}^{-1}x_{1},x_{1}x_{2}^{-1}x_{1}^{-1}x_{3},x_{4}x_{2}^{-1}x_{3}^{-1}x_{2}\rangle }$,

you can use and simplify this too ${\displaystyle x=x_{1}}$ ${\displaystyle y=x_{1}^{-1}x_{3}}$

${\displaystyle \langle x,y\mid y^{-1}xyx^{-1}y^{-2}x^{-1}yx\rangle }$.

Individual evidence

1. ↑ Stefan Friedl: Topologie - Sommersemester 2012. (PDF) P. 136 ff , accessed on April 28, 2020 . 
2. ↑ Geoffrey Hemion: The classification of knots and 3-dimensional spaces. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York 1992, ISBN 0-19-859697-9 .
3. ↑ Stefan Friedl: Topologie - Sommersemester 2012. (PDF) p. 136 ff , accessed on July 31, 2015 . 
4. ↑ Gerhard Burde, Heiner Zieschang, Michael Heusener: Knots (=  De Gruyter Studies in Mathematics . Volume 5 ). 3rd, revised. Edition. De Gruyter, Berlin 2014, ISBN 978-3-11-027074-7 .