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Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time

This presentation covers the concepts of curves on surfaces, representing simple curves, Dehn twists, and geometric intersection numbers. It delves into algorithmic problems like contractibility, transformability, and computing Dehn-twists, with historical context and open questions.

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Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time

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  1. Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time Marcus Schaefer, Eric Sedgwick DePaul University (Chicago) Daniel Štefankovič University of Rochester

  2. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  3. Curves on surfaces different? same? closed curve = homeomorphic image of circle S1 simple closed curve =  is injective (no self-intersections)

  4. Curves on surfaces different? same? homotopy equivalent

  5. Curves on surfaces different? same? homotopy equivalent

  6. Curves on surfaces different? same? homotopy equivalent

  7. Curves on surfaces different? same? homotopy equivalent

  8. Curves on surfaces not homotopy equivalent

  9. Curves on surfaces curves continuous objects homotopy classes of curves combinatorial objects • how to represent them? • what/how to compute?

  10. Geometric intersection number minimum number of intersections achievable by continuous deformations.  

  11. Geometric intersection number minimum number of intersections achievable by continuous deformations.   i(,)=2

  12. EXAMPLE: Geometric intersection numbers are well understood on the torus (2,-1) (3,5) 3 5 det = -13 2 -1

  13. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  14. Algorithmic problems - History Contractibility (Dehn 1912) can shrink curve to point? Transformability (Dehn 1912) are two curves homotopy equivalent? Schipper ’92; Dey ’94; Schipper, Dey ’95 Dey-Guha ’99 (linear-time algorithm) Simple representative (Poincaré 1895) can avoid self-intersections? Reinhart ’62; Ziechang ’65; Chillingworth ’69 Birman, Series ’84

  15. Algorithmic problems - History Geometric intersection number minimal number of intersections of two curves Reinhart ’62; Cohen,Lustig ’87; Lustig ’87; Hamidi-Tehrani ’97 Computing Dehn-twists “wrap” curve along curve Penner ’84; Hamidi-Tehrani, Chen ’96; Hamidi-Tehrani ’01 polynomial only in explicit representations polynomial in compressed representations, but only for fixed set of curves

  16. Algorithmic problems – our results Geometric intersection number minimal number of intersections of two curves Reinhart ’62; Cohen,Lustig ’87; Lustig ’87; Hamidi-Tehrani ’97, Schaefer-Sedgewick-Š ’08 Computing Dehn-twists “wrap” curve along curve Penner ’84; Hamidi-Tehrani, Chen ’96; Hamidi-Tehrani ’01, Schaefer-Sedgewick-Š ’08 polynomial in explicitcompressed representations polynomial in compressed representations, for fixed set of curvesany pair of curves

  17. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  18. How to represent surfaces?

  19. Combinatorial description of a surface 1. (pseudo) triangulation b a c bunch of triangles + description of how to glue them

  20. Combinatorial description of a surface 2. pair-of-pants decomposition bunch of pair-of-pants + description of how to glue them (cannnot be used to represent: ball with 2 holes, torus)

  21. Combinatorial description of a surface 3. polygonal schema b a = a b 2n-gon + pairing of the edges

  22. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  23. How to represent simple curves in surfaces (up to homotopy)? Ideally the representation is “unique” (each curve has a unique representation)

  24. Combinatorial description of a (homotopy type of) a simple curve in a surface • intersection sequence with • a triangulation b c a

  25. Combinatorial description of a (homotopy type of) a simple curve in a surface • intersection sequence with • a triangulation b c a bc-1bc-1ba-1 almost unique if triangulation points on S

  26. Combinatorial description of a (homotopy type of) a simple curve in a surface 2. normal coordinates (w.r.t. a triangulation) (b)=3 (c)=2 (a)=1 (Kneser ’29) unique if triangulation points on S

  27. Combinatorial description of a (homotopy type of) a simple curve in a surface 2. normal coordinates (w.r.t. a triangulation) (b)=300 ? (c)=200 ? (a)=100 a very concise representation!

  28. Recap: • how to represent them? • 2) what/how to compute? • intersection sequence with a triangulation bc-1bc-1ba-1 2. normal coordinates (w.r.t. a triangulation) (a)=1 (b)=3 (c)=2 geometric intersection number

  29. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  30. STEP1: Moving between the representations • intersection sequence with a triangulation bc-1bc-1ba-1 2. normal coordinates (w.r.t. a triangulation) (a)=1 (b)=3 (c)=2 Can we move between these two representations efficiently? (c)=2101 (a)=1+2100 (b)=1+3.2100

  31. STEP1: Moving between the representations • intersection sequence with a triangulation bc-1bc-1ba-1 2. normal coordinates (w.r.t. a triangulation) (a)=1 (b)=3 (c)=2 Can we move between these two representations efficiently? YES (c)=2101 (a)=1+2100 (b)=1+3.2100

  32. Theorem (SSS’08): normal coordinatescompressed intersection sequence in time O( log (e)) compressed intersection sequencenormal coordinates in time O(|T|.SLP-length(S)) compressed = straight line program (SLP) X0:= a X1:= b X2:= X1X1 X3:= X0X2 X4:= X2X1 X5:= X4X3 X5 = bbbabb

  33. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  34. Main tool: Word equations x,y – variables a,b - constants xabx =yxy

  35. Main tool: Word equations x,y – variables a,b - constants xabx =yxy a solution: x=ab y=ab

  36. Word equations with given lengths x,y – variables a,b - constants xayxb = axbxy additional constraints: |x|=4, |y|=1

  37. Word equations with given lengths x,y – variables a,b - constants xayxb = axbxy additional constraints: |x|=4, |y|=1 a solution: x=aaaa y=b

  38. Word equations word equations word equations with given lengths

  39. Word equations word equations -NP-hard decidability – Makanin 1977 PSPACE – Plandowski 1999 word equations with given lengths Plandowski, Rytter ’98 – polynomial time algorithm Diekert, Robson ’98 – linear time for quadratic eqns (quadratic = each variable occurs  2 times)

  40. Simulating curve using quadratic word equations u X z v z y w |u|=|v|=(u) ... |x|=(|z|+|u|-|w|)/2 number of components u=xy ... v=u Diekert-Robson

  41. Moving between the representations • intersection sequence with a triangulation bc-1bc-1ba-1 2. normal coordinates (w.r.t. a triangulation) (a)=1 (b)=3 (c)=2 Theorem: normal coordinatescompressed intersection sequence in time O( log (e)) “Proof”: u=xy ... av=ua u X v z |u|=|v|=|T|(u) y

  42. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  43. Dehn twist of  along   

  44. Dehn twist of  along   D()

  45. Dehn twist of  along    D()

  46. Geometric intersection numbers i(a,Dng(b))/i(a,g) ! i(g,b) n¢ i(a,g)i(g,b) -i(a,b) i(a,Dng(b))  n¢ i(a,g)i(g,b)+i(a,b)

  47. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  48. Computing Dehn-Twists (outline) 1. normal coordinates ! word equations with given lengths 2. solution = compressed intersection sequence with triangulation 3. sequences ! (non-reduced) word for Dehn-twist (substitution in SLPs) 4. Reduce the word ! normal coordinates

  49. outline What is a curve? Some history of algorithmic problems. Representing surfaces. Representing simple curves in surfaces. Transforming between various representations. TOOL: Word equations. What is a Dehn twist and why is it interesting? Computing Dehn twists. Open questions.

  50. PROBLEM #1: Minimal weight representative 2. normal coordinates (w.r.t. a triangulation) (b)=3 (c)=2 (a)=1 unique if triangulation points on S

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