Montesinos knot

In knot theory , a branch of mathematics , Montesinos knots or Montesinos links are a class of knots or links. Almost 25% of all nodes with up to 11 crossings are Montesinos nodes.

definition

The Montesinos Loop .

{\ displaystyle K (-3, - {\ frac {3} {2}}, {\ frac {5} {2}})}

Be an integer and be rational numbers. The - Montesinos-entanglement is one of rational tangles existing entanglement whose -th Tangle in the Conway notation of the rational number corresponds. ${\ displaystyle e}$ ${\ displaystyle {\ tfrac {p_ {1}} {q_ {1}}}, \ ldots, {\ tfrac {p_ {n}} {q_ {n}}}}$ ${\ displaystyle (e, {\ tfrac {p_ {1}} {q_ {1}}}, \ ldots, {\ tfrac {p_ {n}} {q_ {n}}})}$ ${\ displaystyle n + 1}$ ${\ displaystyle i}$ ${\ displaystyle {\ tfrac {p_ {i}} {q_ {i}}}}$

If the link is connected, it is the - Montesinos knot . ${\ displaystyle (e, {\ tfrac {p_ {1}} {q_ {1}}}, \ ldots, {\ tfrac {p_ {n}} {q_ {n}}})}$

Montesinos knots with integer coefficients (also ) are called pretzel knots . ${\ displaystyle q_ {1} = \ ldots = q_ {n} = 1}$

Branched Overlays

Montesinos nodes are characterized by the following property: The 2-fold branched overlay of the over is a Seifert fiber with the 2-sphere as the base. ${\ displaystyle K \ subset S ^ {3}}$ ${\ displaystyle S ^ {3}}$ ${\ displaystyle K}$

This generalizes Schubert's characterization of rational knots . (In this case, the 2-way branched overlay is a lens space .)

More generally it was shown by Montesinos that the 2-fold branched overlay over an arborescent node is a graph manifold .

classification

A classification of the Montesinos knots and Montesinos tangles was proved by Bonahon in 1979 , other proofs of the classification were given by Zieschang and Turaev .

The result of the classification is: Montesinos interweaving of rational tangles, with and , are classified by the ordered set ${\ displaystyle n + 1}$ ${\ displaystyle n \ geq 3}$ ${\ displaystyle \ sum _ {i = 1} ^ {n} {\ frac {1} {q_ {i}}} \ leq n-2}$

{\ displaystyle \ left ({\ frac {p_ {1}} {q_ {1}}} \ mod \ 1, \ ldots, {\ frac {p_ {n}} {q_ {n}}} \ mod \ 1 \ right)}

(except for cyclic permutations and reversal of the order) together with the rational number

{\ displaystyle e_ {0} = e + \ sum _ {i = 1} ^ {n} {\ frac {p_ {i}} {q_ {i}}}}

.

Alternatively, Montesinos links are classified by the quotients of the node group according to the normal divisor generated by the meridians .

literature

Gerhard Burde, Heiner Zieschang: Knots. (= de Gruyter Studies in Mathematics. 5). 2nd Edition. Walter de Gruyter & Co., Berlin 2003, ISBN 3-11-017005-1 , chapter 12.

Individual evidence

↑ José M. Montesinos: Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas. In: Bol. Soc. Mat. Mexicana. (2) Volume 18, 1973, pp. 1-32.
↑ José M. Montesinos: Revêtements ramifiés de noeuds, espaces fibers de Seifert et scindements de Heegaard. Prépublications Orsay (1979).
^ F. Bonahon: Involutions et fibers de Seifert dans les variétés de dimension 3rd dissertation. Université Paris-Sud XI - Orsay 1979.
^ Heiner Zieschang: Classification of Montesinos knots. In: Ludwig D. Faddeev, Arkadii A. Malcev (Ed.): Topology. (= Lecture Notes in Math. 1060). Topological Conference Leningrad 1982. Springer, Berlin 1984, ISBN 3-540-13337-2 , pp. 378-389.
^ VG Turaev: Classification of oriented Montesinos links via spin structures. In: Topology and geometry — Rohlin Seminar. (= Lecture Notes in Math. 1346). Springer, Berlin 1988, pp. 271-289.

[1] José M. Montesinos: Variedades de Seifert que son recubridores ciclicos ramificados de dos hojas. In: Bol. Soc. Mat. Mexicana. (2) Volume 18, 1973, pp. 1-32.

[2] José M. Montesinos: Revêtements ramifiés de noeuds, espaces fibers de Seifert et scindements de Heegaard. Prépublications Orsay (1979).

[3] F. Bonahon: Involutions et fibers de Seifert dans les variétés de dimension 3rd dissertation. Université Paris-Sud XI - Orsay 1979.

[4] Heiner Zieschang: Classification of Montesinos knots. In: Ludwig D. Faddeev, Arkadii A. Malcev (Ed.): Topology. (= Lecture Notes in Math. 1060). Topological Conference Leningrad 1982. Springer, Berlin 1984, ISBN 3-540-13337-2 , pp. 378-389.

[5] VG Turaev: Classification of oriented Montesinos links via spin structures. In: Topology and geometry — Rohlin Seminar. (= Lecture Notes in Math. 1346). Springer, Berlin 1988, pp. 271-289.