2 bridge knot

In knot theory , a branch of mathematics , 2-bridge knots or 2-bridge links (also: knots or links with 2 bridges ) are a class of knots or links . They were classified under the name four braids in 1956 by Horst Schubert . Because they can be classified by a rational number , they are often referred to as rational knots or rational links . ${\displaystyle {\frac {p}{q}}}$

definition

A 2-bridge node is a node whose number of bridges ${\displaystyle K}$

${\displaystyle br(K)=2}$

is. This means that it can be broken down into intervals in such a way that for a suitable plane there are intervals in both of the half-spaces bounded by the plane . (Equivalently, one can also require that intervals lie in one plane and the other two intervals in one of the bounded half-spaces.) ${\displaystyle 4}$ ${\displaystyle E\subset \mathbb {R} ^{3}}$ ${\displaystyle 2}$ ${\displaystyle 2}$

Similarly, a link with 2 bridges is defined as a link with a number of bridges . ${\displaystyle L}$ ${\displaystyle br(L)=2}$

An equivalent definition says that the knot or the link has exactly 2 maxima with respect to a height function according to a suitable isotopy . ${\displaystyle h\colon S^{3}\to \mathbb {R} }$

Conway normal form

Schematic representation of a 2-bridge link

The classification below shows that each loop can be represented with 2 bridges as in the picture on the right, whereby the number of half-twists is indicated in the respective box and corresponds to positive left- or right-handed half-twists for even or odd . ${\displaystyle a_{i}\in \mathbb {Z} }$ ${\displaystyle i}$ ${\displaystyle a_{i}}$

This representation is called the Conway normal form.

One can always achieve that all have the same sign. In particular, the Conway normal form then gives an alternating knot diagram . ${\displaystyle a_{i}}$

classification

The 2-fold superposition of the 3-sphere, branched over a 2-bridge loop, is a lens space . The 2-bridge entanglements are classified by these lens spaces. One therefore designates the loop for which the lens space is obtained. ${\displaystyle L(p,q)}$ ${\displaystyle K(p,q)}$ ${\displaystyle L(p,q)}$

In particular, two rational numbers and isotopic links correspond if and only if ${\displaystyle {\frac {p_{1}}{q_{1}}}}$ ${\displaystyle {\frac {p_{2}}{q_{2}}}}$

${\displaystyle p_{1}=p_{2}}$and either or is. ${\displaystyle q_{1}\equiv q_{2}\mod p}$ ${\displaystyle q_{1}q_{2}\equiv 1\mod p}$

Modulo of these identities, 2-bridge links are classified by a rational number , where and can be assumed. ${\displaystyle {\frac {p}{q}}}$ ${\displaystyle p>0}$ ${\displaystyle -p<q<p}$

In the above-described Conway normal form corresponds to the continued fraction : ${\displaystyle {\frac {p}{q}}}$ ${\displaystyle \left[a_{0};a_{1},a_{2},a_{3},\ldots \right]}$

${\displaystyle {\frac {p}{q}}={\cfrac {1}{a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\;\,\ddots }}}}}}}}}$

(The continued fraction representation of a rational number is ambiguous, but all continued fraction decompositions result in the same node .) ${\displaystyle {\frac {p}{q}}}$ ${\displaystyle K(p,q)}$

The mirror image of a 2-bridge node is . An orientation-reversing homeomorphism between two different 2-bridge nodes exists if and only if ${\displaystyle K(p,q)}$ ${\displaystyle K(p,-q)}$ ${\displaystyle (S^{3},K(p_{1},q_{1}))\to (S^{3},K(p_{2},q_{2}))}$

${\displaystyle p_{1}=p_{2}}$and either or is. ${\displaystyle q_{1}\equiv -q_{2}\mod p}$ ${\displaystyle q_{1}q_{2}\equiv -1\mod p}$

In particular, a 2-bridge node is amphichiral if and only if is. ${\displaystyle q^{2}\equiv -1\ mod\ p}$

For 2-bridge links (with 2 components) there is an orientation-preserving homeomorphism if and only if

${\displaystyle p_{1}=p_{2}}$and either or is. ${\displaystyle q_{1}\equiv q_{2}\mod 2p}$ ${\displaystyle q_{1}q_{2}\equiv 1\mod 2p}$

Examples

The only torus knots among the 2-bridge knots are the torus knots . ${\displaystyle (\pm 2,n)}$

All 2-bridge nodes that are not torus nodes are hyperbolic nodes .

The clover leaf loop is the 2-bridge knot with Conway normal form , the figure eight is the 2-bridge knot with Conway normal form . ${\displaystyle K(3,1)}$ ${\displaystyle \left[3\right]}$ ${\displaystyle K(5,2)}$ ${\displaystyle \left[2,2\right]}$

KnotInfo gives a list of all 2-bridge nodes with up to 12 crossings and calculates the known node invariants.

${\displaystyle K(p,q)}$is a node if and only if is odd. If is straight, then the 2-bridge link consists of two components . ${\displaystyle p}$ ${\displaystyle p}$

Properties and Invariants

The node group of the 2-bridge link has the presentation ${\displaystyle K(p,q)}$

${\displaystyle \langle a,b\mid ab^{\epsilon _{1}}a^{\epsilon _{2}}\ldots b^{\epsilon _{\alpha _{1}}}a^{-1}b^{-\epsilon _{\alpha _{1}}}\ldots a^{-\epsilon _{2}}b^{-\epsilon _{1}}\rangle }$

with . ${\displaystyle \epsilon _{k}:=\left(-1\right)^{\left[{\frac {kp}{q}}\right]}}$

The incompressible areas in the complements of 2-bridge nodes were classified by Hatcher and Thurston . In particular, they proved that there are no closed incompressible surfaces. If there is no torus knot, then every stretching surgery gives an irreducible 3-manifold and almost all stretching surgeries give manifolds that are neither hook manifolds nor Seifert fibers . ${\displaystyle K(p,q)}$

Schubert already proved that the 2-fold branched superpositions are lens spaces. The classification of all finite branched superpositions was worked out by Minkus.

The complements of hyperbolic 2-bridge nodes (with the exception of the figure eight node) are not commensurate to any other node complements other than themselves .

There are formulas for calculating the HOMFLY polynomial and especially the Jones polynomial of 2-bridge nodes.

See also

• Montesinos knot

literature

• Horst Schubert: Knot with two bridges , Mathematische Zeitschrift 65, 133-170 (1956). doi : 10.1007 / BF01473875
• John Conway: An enumeration of knots and links, and some of their algebraic properties. Computational Problems in Abstract Algebra (Proc. Conf., Oxford, 1967) 329-358, Pergamon, Oxford (1970). PDF
• Laurent Siebenmann: Exercices sur les noeuds rationnels , Université Paris-Sud (1975).
• Louis H. Kauffman, Sofia Lambropoulou: On the classification of rational knots , L 'Enseignement Mathématique, 49, 357-410 (2003). ArXiv
• CC Adams, The Knot Book. Introduction to the mathematical theory of knots , Spektrum Akademischer Verlag (1995) ISBN 3860253387

Web links

Table of rational nodes with up to 16 intersections

Individual evidence

1. ↑ Exercise 9.2.6 in: Kunio Murasugi: Knot theory & its applications. Translated from the 1993 Japanese original by Bohdan Kurpita. Reprint of the 1996 translation, Modern Birkhäuser Classics. Birkhauser Boston, Inc., Boston, MA, 2008. ISBN 978-0-8176-4718-6
2. ^ Carl Bankwitz, Hans Georg Schumann: About four braids. Dep. Math. Sem. Univ. Hamburg 10 (1934), no. 1, 263-284.
3. ↑ Murasugi, op.cit., P. 189.
4. ↑ Schubert, op. Cit.
5. ↑ Jennifer Schultens: Bridge numbers of torus knots. Math. Proc. Cambridge Philos. Soc. 143 (2007), no. 3, 621-625. (The sentence originally goes back to Horst Schubert.)
6. ↑ KnotInfo:
   • Table of 2-bridge nodes
   • Calculation of all invariants
7. ^ Allen Hatcher, William Thurston: Incompressible surfaces in 2-bridge knot complements. Invent. Math. 79 (1985) no. 2, 225-246.
8. ↑ Jerome Minkus: The branched cyclic coverings of 2 bridge knots and links. Mem. Amer. Math. Soc. 35 (1982), no. 255
9. ^ Alan Reid, Genevieve Walsh: Commensurability classes of 2-bridge knot complements. Algebr. Geom. Topol. 8 (2008), no. 2, 1031-1057.
10. ↑ Shigekazu Nakabo: Formulas on the HOMFLY and Jones polynomials of 2-bridge knots and links. Kobe J. Math. 17 (2000), no. 2, 131-144.