Finally, if there is not a distinct maximum, we can choose one of the maxima and add a small twist to the curve at that point. See Figure 4.14. The idea is that a small twist, applied to the leading edge of the tangents we are comparing to get the turning angle, will increase the angle most at the center of the twist. If this corresponds to a nonunique global maximum, we end up with a unique global maximum. Now the correspondence scheme works. That is, we know that the global maximums must match, and because the turning angle curves are monotonic between isolated crit- ical points, we can find the homeomorphisms in s that move the turning angle curves into correspondence. □ 110 Figure 4.14: A twist perturbation. Notice that if the twist is applied precisely at a global max of the turning angle (as measured by the tangent here and the one lagging it in s ), we will increase the turning angle there and will end up with a unique global maximum. Figure 4.13: In this schematic figure, we illustrate the alternating perturbation around the curve, keeping the curve C 1 close to and messing with the second and third deriva- tives to eliminate any critical points other than isolated maxima and minima. Here the perturbation is of course greatly exaggerated. L$_{2}$ < max { d $^{2}$γ/ds$^{2}$, d$^{3}$γ/ds$^{3}$} < L$_{3}$ L$_{1}$ < max { d $^{2}$γ/ds$^{2}$, d$^{3}$γ/ds$^{3}$} < L$_{2}$