By Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

ISBN-10: 0821801244

ISBN-13: 9780821801246

ISBN-10: 0821801309

ISBN-13: 9780821801307

ISBN-10: 0821801368

ISBN-13: 9780821801369

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Lickorish in 1962; see [88] . We will discuss some of the insights that go into the proof of this theorem. 6. If the simple closed curves a, /3 in the compact surface S intersect exactly once, then there is a pair Ji , f2 of Dehn twists such that h · fi(a) is isotopic to /3 . Proof. If we choose Ji to be a left Dehn twist along /3 and f2 to be a right Dehn twist along a, then indeed f2 · fi (a) /3. 27. 25. 26. D a and /3 . fi (a) . 6. 7. If a, /3 are oriented simple closed curves in the orientable surface S, then there is a series of Dehn twists Ji , /2, .

Here we will consider decompositions into pairs of pants. The thrice-punctured sphere, or pair of pants, forms an important building block for orientable surfaces. Of interest here is the fact that a pair of pants contains no essential simple closed curves. You will establish this property in the exercises. 3 . 1 . A pants decomposition of a compact surface S is a col lection of pairwise disjoint simple closed curves { c1 , . . , Cn} such that each component of is a pair of pants. Two pants decompositions are equivalent if they are isotopic.

We say that a, b intersect transversely and write a rh b if \:Ix E a n b, there is a chart (U-p

### A Century of mathematics in America by Peter L Duren; Richard Askey; Uta C Merzbach; Harold M Edwards

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