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 In the mathematical field of topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot ...

, 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. 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 knots. It can be shown that every n-bridge knot can be decomposed into two trivial n- tangles and hence 2-bridge knots are

rational knot In the mathematical field of knot theory, a 2-bridge knot is a knot which can be regular isotoped so that the natural height function given by the ''z''-coordinate has only two maxima and two minima as critical points. Equivalently, these are th ...

s. 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 classific ...

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 * Stick number * Unknotting number

Definition

Properties

Other numerical invariants

References

Further reading