I am reading Dale Rolfsen's book KNOTS AND LINKS, at page 115, I can't figure out why the crookedness of a knot equals its bridge index. Please give me some hints or any references available, much thanks!
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$\begingroup$ What's the definition of crookedness? $\endgroup$– Ryan BudneyMay 4, 2019 at 0:42
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$\begingroup$ @RyanBudney In R^3, if you parameterize the knot, the crookedness of a knot K is the minimal number of relative maxima in the z-coordinate, the minima is taken over all knots of this type. This definition is from Milnor's famous paper On the Total Curvature of Knots link $\endgroup$– FredyMay 4, 2019 at 3:16
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$\begingroup$ If you knew the knot in its "crookedness" position, you could assume the z-coordinate is a morse function on the knot, wouldn't that finish the job? $\endgroup$– Ryan BudneyMay 4, 2019 at 17:20
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$\begingroup$ sorry, I forgot to state the definition of the bridge index in my question. It is defined to be the the minimal number of maximal overpasses in any regular projection of all knots in this type. It is said the two definition(bridge version and the overpass version) are equivalent, but I can't find out why. $\endgroup$– FredyMay 5, 2019 at 3:14
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$\begingroup$ I think this isn't meant to be a difficult problem -- you are likely "psyching yourself out". Assuming the z-coordinate is morse and you're in the "crookedness" position, what you do is consider projections in directions "close to orthogonal to the z-axis". Most such projections are regular knot diagrams, and they will be diagrams for the bridge index. $\endgroup$– Ryan BudneyMay 5, 2019 at 5:26
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