The Maximal Number of Transverse Self-Intersections of Geodesics ...

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$Math 664 Homework #2: Solutions 1. Let Î© = R, F = {A â R : either A ...$

4 1 D k . . . a 3 2 D 1 C k . . . C 2 C1 b We draw the first part of the curve corresponding to c α 1 1 in [0, 1] × C 1 as pictured above (beginning at point 1). Note this begins at the lower left of D k ∩ C 1 and ends at the upper right corner of C 1 ∩ D 1 .1 starts a horizontal line that begins at some point before the left edge of the rectangle, extends past that edge, and continues on the next horizontal line. This "looping" around the rectangle continues until we reach our d β 1 1 . Thisnextphaseofthe curve is marked by "2" and occurs where the corner of the last a meets with this b or B. Now we continue in this fashion only in the vertical direction. 8
a) b) . .. . . a -> b . . . . . . . . . . . . . . . . . . a -> B . . . . . A -> b . . . . . . . . . . . . . . . . . . . . . A -> B . . . c) . . . . . . d) . . . . . . . . . . . . . . . e) . . . f) . . . . . . . . . g) h) . . . . . . . . . . . . . . . . . . . . . b -> a . . . . . . . . . . .. . . . . . . . b -> A B -> A . . . . . . . . . . . . . . . . . . B -> a . . . . . . . . . . . . . . . . . . . . . . . . The figure above refers to the different types of ways two c i and d i can intersect, excluding the first and last cases. In each case, you see that each d β i i intersects the previous c α i i ,making α i β i number of intersections. Again, from the previous figure, note that all of these new vertical lines pass through all of the horizontal lines, with the exception of one (since 1 and 2 form a line in which there are no intersecting lines from d β 1 1 . This process continues when we stop at point 3. We go back to the horizontal lines as defined by c α 2 2 . Note, once again, that these horizontal lines intersect the previous vertical lines. We can let this process continue until it reaches the final d β k k . We observe that this set of vertical lines starts at point 4 in the figure. Since it is a b, weknowitwilltravelupward,circlingaroundβ k times, and connecting back to point 1. Notice, though, that there are two partial lines in this group. There is one near 4, and one that connects back to 1. Thus, only β k − 1 lines intersect all the other horizontal lines. In short, all horizontal lines, with the exception of the one mentioned, intersect all the other vertical lines, again, with the exception of one in the last group. So if we simply look at the intersection of each b with each prior a (excluding the first a and last b), and the intersection of each subsequent a with each prior b (again excluding the two special cases), we can count the total number of maximum intersections. 9
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The Maximal Number of Transverse Self-Intersections of Geodesics ...

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