Bridge number

Schematic picture of a 2-bridge knot.

In the mathematical 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. 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. 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.

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.

Other numerical invariants

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

Further reading

• Cromwell, Peter (1994). Knots and Links. Cambridge. ISBN 9780521548311.
• Schultens, Jennifer. Additivity of bridge numbers of knots