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.

## 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.^{[1]} Bridge number was first studied in the 1950s by Horst Schubert.^{[2]}

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.

## Properties

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 K_{1} and K_{2}, then the bridge number of K is one less than the sum of the bridge numbers of K_{1} and K_{2}.^{[3]}

## Other numerical invariants

## References

- ^
^{a}^{b}Adams, Colin C. (1994),*The Knot Book*, American Mathematical Society, p. 65, ISBN 9780821886137. **^**Schultens, Jennifer (2014),*Introduction to 3-manifolds*, Graduate Studies in Mathematics,**151**, American Mathematical Society, Providence, RI, p. 129, ISBN 978-1-4704-1020-9, MR 3203728.**^**Schultens, Jennifer (2003), "Additivity of bridge numbers of knots",*Mathematical Proceedings of the Cambridge Philosophical Society*,**135**(3): 539–544, arXiv:math/0111032 , Bibcode:2003MPCPS.135..539S, doi:10.1017/S0305004103006832, MR 2018265.

## Further reading

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