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  "body": "\n## Definition\n\nLet $K$ be a tame [[knot|QYEAZTYGPT]] in $\\R^3$, and let $h \\colon \\R^3 \\to \\R$ be the height function $h(x, y, z) = z$.\nCall a knot $K'$ in $\\R^3$ *generic* if the restriction of $h$ to $K'$ is a Morse function.\nThe *bridge number* $b(K)$ is the least number of local maxima of $h$ on $K'$, over all generic knots $K'$ ambient isotopic to $K$.\n\n## Remarks\n\nA function on a circle has at least one maximum, so $b(K) \\ge 1$.\nExample: a round circle in a vertical plane has exactly one maximum, so the [[unknot|1TQ06R4Q66]] has bridge number $1$.\nConversely $b(K) = 1$ only for the unknot, since a knot with one maximum and one minimum bounds an embedded disk.\nThe [[trefoil knot|VYTYSVHN4A]] has a position with two maxima and is knotted, so its bridge number is $2$.\nNon-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.\nEquivalently, $b(K)$ is the least number of overpasses (bridges) in a diagram of $K$ whose over-strands are disjoint arcs carrying all overcrossings.\nSchubert proved $b(K_1 \\# K_2) = b(K_1) + b(K_2) - 1$ for the [[knot sum|AGZSZW7GYE]].\n",
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