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?
2026-03-25 22:04:36.1774476276
Why the bridge index of $8_{10}$ is 3?
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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 $3$.