In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

field of knot theory, the bridge number is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot.

Definition

Given a knot or link, draw a diagram of the link using the convention that a gap in the line denotes an undercrossing. Call an arc in this diagram a bridge if it includes at least one overcrossing. Then the bridge number of a knot can be found as the minimum number of bridges required for any diagram of the knot.. Bridge number was first studied in the 1950s by

Horst Schubert Horst Schubert (11 June 1919 – 2001) was a German mathematician. Schubert was born in Chemnitz and studied mathematics and physics at the Universities of Frankfurt am Main, Zürich and Heidelberg, where in 1948 he received his PhD under Herb ...

. The bridge number can equivalently be defined geometrically instead of

topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

ly. In bridge representation, a knot lies entirely in the plane apart for a finite number of bridges whose projections onto the plane are straight lines. Equivalently the bridge number is the minimal number of local maxima of the projection of the knot onto a vector, where we minimize over all projections and over all conformations of the knot.

Properties

Every non-trivial knot has bridge number at least two, so the knots that minimize the bridge number (other than the

unknot In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embe ...

) are the

2-bridge knot In the mathematical field of knot theory, a 2-bridge knot is a knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loo ...

s. It can be shown that every n-bridge knot can be decomposed into two trivial n-

tangle Tangle may refer to: Science, Technology, Engineering & Mathematics *''The Tangle'' is the name of the ledger, a directed acyclic graph, used for the cryptocurrency IOTA *Tangle (mathematics), a topological object Natural sciences & medicine ...

s and hence 2-bridge knots are rational knots. If K is the

connected sum In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classifi ...

of K₁ and K₂, then the bridge number of K is one less than the sum of the bridge numbers of K₁ and K₂..

Other numerical invariants

* Crossing number *

Linking number In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other. In E ...

* Stick number * Unknotting number

Definition

Properties

Other numerical invariants

References

Further reading