Bridge number

Builds on Knot

Definition

Let $K$ be a tame knot in $\R^3$ , and let $h \colon \R^3 \to \R$ be the height function $h(x, y, z) = z$ . Call a knot $K'$ in $\R^3$ generic if the restriction of $h$ to $K'$ is a Morse function. The bridge number $b(K)$ is the least number of local maxima of $h$ on $K'$ , over all generic knots $K'$ ambient isotopic to $K$ .

Remarks

A function on a circle has at least one maximum, so $b(K) \ge 1$ . Example: a round circle in a vertical plane has exactly one maximum, so the unknot has bridge number $1$ . Conversely $b(K) = 1$ only for the unknot, since a knot with one maximum and one minimum bounds an embedded disk. The trefoil knot has a position with two maxima and is knotted, so its bridge number is $2$ . Non-example: a position of the trefoil with three maxima is generic but does not realize the minimum, so the count $3$ is only an upper bound. Equivalently, $b(K)$ is the least number of overpasses (bridges) in a diagram of $K$ whose over-strands are disjoint arcs carrying all overcrossings. Schubert proved $b(K_1 \# K_2) = b(K_1) + b(K_2) - 1$ for the knot sum.

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Knot concept
Bridge number definition

Tag	`WRYP2XCKJB`
URI	`m2://WRYP2XCKJB`
Kind	definition
Writing	W0 stub
MSC	57K10
Defines	Bridge number
Uses	Knot
Changed	2 November 2026

Bridge number

Definition

Remarks

Reading path

Further reading

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