Montesinos link

A Montesinos link. In this example, $e=-3$ , $\alpha_1 /\beta_1=-3/2$ and $\alpha_2 /\beta_2=5/2$ .

In the mathematical theory of knots, a Montesinos link is a special kind of link that generalizes pretzel links. A Montesinos link which is also a knot (i.e. a link with one component) is a Montesinos knot.

Notation

A Montesinos link is composed of several rational tangles. One notation for a Montesinos link is $K(e;\alpha_1 /\beta_1,\alpha_2 /\beta_2,\ldots,\alpha_n /\beta_n)$.^[1]

In this notation, $e$ and all the $\alpha_i$ and $\beta_i$ are integers. The Montesinos link given by this notation consists of the sum of the rational tangles given by the integer $e$ and the rational tangles $\alpha_1 /\beta_1,\alpha_2 /\beta_2,\ldots,\alpha_n /\beta_n$

A pretzel link can also be described as a Montesinos link with integer tangles.

References

1. Zieschang, Heiner. "Classification of Montesinos knots." Topology. Springer Berlin Heidelberg, 1984. 378–389

Further reading

• Burde, Gerhard; Zieschang, Heiner (2003). Knots. De Gruyter studies in mathematics 5 (2nd revised and extended ed.). Walter de Gruyter. ISBN 3110170051. ISSN 0179-0986. Zbl 1009.57003.