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

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 unbroken 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.^[1] Bridge numbers were first studied in the 1950s by Horst Schubert.^{[2] [3]}

The bridge number can equivalently be defined geometrically instead of topologically. 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. In this context, the bridge number is often called the crookedness.

Every non-trivial knot has bridge number at least two,^[1] so the knots that minimize the bridge number (other than the unknot) 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 knots.

If K is the connected sum of K₁ and K₂, then the bridge number of K is one less than the sum of the bridge numbers of K₁ and K₂.^[4]

• Crossing number
• Linking number
• Stick number
• Unknotting number

• Cromwell, Peter (1994). Knots and Links. Cambridge. ISBN 9780521548311.