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This might be too elementary. I tried to deform the projection but couldn’t be able to find a projection of knot $8_{10}$ with 3 maximal overpasses. Is there any elementary reference on calculating bridge number?

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N. Owad
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Eric
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The bridge number of a knot has the following equivalent definition. The bridge number of a knot is the minimum over all knot diagrams of the number of local maxima of the knot, where the knot is viewed as a smooth closed curve in space. The diagram of $8_{10}$ in your question has $3$ local maxima, and so its bridge number is at most $3$.

Adam Lowrance
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    Could you briefly explain why a local maxima corresponds to a maximal overpass? Or give any reference? – Eric Nov 27 '23 at 01:50
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    @EricLey Roughly, if you have the knot in one of the two forms (in the sense that its projection to the plane is a diagram in one of the two forms), then you can rotate the knot through the horizontal axis by 90 degrees to switch from one form to the other. This is only roughly true since you need the knot to be well-behaved in the out-of-the-diagram direction (the arcs in overpass position should all have a single local maximum -- if that is satisfied, I think the rotation works). – Kyle Miller Nov 27 '23 at 17:57
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    If you want a combinatorial way to go from the local maximum version of bridge number to the overpass version, I once drew an example -- you can ignore the blue surface in the illustrations if you want. The local maximum version tends to have a lot more efficient diagrams! – Kyle Miller Nov 27 '23 at 17:58