Number of bridges

The bridge number is an invariant from the mathematical field of knot theory .

definition

A node has a display with bridges , if he does so in can be decomposed intervals that a suitable level of each interval in both of the plane bounded half spaces are located. (Equivalently, one can also require that intervals lie in one plane and the other intervals in one of the bounded half-spaces.) ${\displaystyle K\subset \mathbb {R} ^{3}}$ ${\displaystyle b}$ ${\displaystyle 2b}$ ${\displaystyle E\subset \mathbb {R} ^{3}}$ ${\displaystyle b}$ ${\displaystyle b}$ ${\displaystyle b}$

The number of bridges of a node is the smallest number for which there is a representation with bridges. ${\displaystyle br(K)}$ ${\displaystyle K}$ ${\displaystyle b}$ ${\displaystyle b}$

Examples

The clover leaf loop has bridges 2.

• The unknot is the only knot with a number of bridges . ${\displaystyle 1}$
• Knots with a number of bridges ${\displaystyle 2}$ were classified by Horst Schubert in 1956 , they are also called rational knots .
• A classification of the 3-bridge nodes has not yet been successful.
• The bridge number of the torus knot is . ${\displaystyle T_{pq}}$ ${\displaystyle b(T_{pq})=\min(p,q)}$
• The number of bridges in an n-strand braid is at most . ${\displaystyle n}$

properties

• The equation applies to the node sum ${\displaystyle K_{1}\sharp K_{2}}$

${\displaystyle br(K_{1}\sharp K_{2})=br(K_{1})+br(K_{2})-1}$

• Lens spaces are branched overlays of a 2-bridge node as a branch set. ${\displaystyle S^{3}}$
• If a closed 3-manifold has a Heegaard decomposition of the gender , then it is a branched overlay of that with a 3-bridge node as a branching set. ${\displaystyle 2}$ ${\displaystyle S^{3}}$

literature

• Gerhard Burde, Heiner Zieschang, Michael Heusener: Knots. (= De Gruyter Studies in Mathematics. 5). 3rd, completely revised and exp. Edition. De Gruyter, Berlin 2014, ISBN 978-3-11-027074-7 .
• Jennifer Schultens: Additivity of bridge numbers of knots. In: Math. Proc. Cambridge Philos. Soc. 135, no. 3, 2003, pp. 539-544.
• Jennifer Schultens: Bridge numbers of torus knots. In: Math. Proc. Cambridge Philos. Soc. 143, no. 3, 2007, pp. 621-625.