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Algorithmic Problems for Curves on Surfaces

Algorithmic Problems for Curves on Surfaces. Daniel Štefankovič University of Rochester. outline. ● Simple curves on surfaces * representing surfaces, simple curves in surfaces * algorithmic questions, history ● TOOL: (Quadratic) word equations

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Algorithmic Problems for Curves on Surfaces

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  1. Algorithmic Problems for Curves on Surfaces Daniel Štefankovič University of Rochester

  2. outline ● Simple curves on surfaces * representing surfaces, simple curves in surfaces * algorithmic questions, history ● TOOL: (Quadratic) word equations ● Regular structures in drawings (?) ● Using word equations (Dehn twist, geometric intersection numbers, ...) ● What I would like...

  3. How to represent surfaces?

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

  5. 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)

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

  7. Simple curves on surfaces closed curve = homeomorphic image of circle S1 simple closed curve =  is injective (no self-intersections) (free) homotopy equivalent simple closed curves

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

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

  10. 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

  11. 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

  12. 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! (compressed)

  13. Combinatorial description of a (homotopy type of) a simple curve in a surface 3. weighted train track 5 10 13 5 10 3

  14. Combinatorial description of a (homotopy type of) a simple curve in a surface 4. Dehn-Thurston coordinates ● number of intersections ● “twisting number” for each “circle” (important for surfaces without boundary) unique

  15. outline ● Simple curves on surfaces * representing surfaces, simple curves in surfaces * algorithmic questions, history ● TOOL: (Quadratic) word equations ● Regular structures in drawings (?) ● Using word equations (Dehn twist, geometric intersection numbers, ...) ● What I would like...

  16. 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

  17. 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

  18. Algorithmic problems – will show 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

  19. outline ● Simple curves on surfaces * representing surfaces, simple curves in surfaces * algorithmic questions, history ● TOOL: (Quadratic) word equations ● Regular structures in drawings (?) ● Using word equations (Dehn twist, geometric intersection numbers, ...) ● What I would like...

  20. Word equations x,y – variables a,b - constants xabx =yxy

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

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

  23. 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

  24. Word equations word equations word equations with given lengths

  25. Word equations In NP ??? 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)

  26. Word equations OPEN: In NP ??? word equations -NP-hard MISSING: decidability – Makanin 1977 PSPACE – Plandowski 1999 exponential upper bound on the length of a minimal solution 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)

  27. outline ● Simple curves on surfaces * representing surfaces, simple curves in surfaces * algorithmic questions, history ● TOOL: (Quadratic) word equations ● Regular structures in drawings (?) ● Using word equations (Dehn twist, geometric intersection numbers, ...) ● What I would like...

  28. Shortcut number (g,k) k curves on surface of genus g intersecting another curve   (the curves do not intersect)

  29. Shortcut number (g,k) k curves on surface of genus g intersecting another curve  4 1  4 8 3 1 6 1

  30. Shortcut number (g,k) k curves on surface of genus g intersecting another curve  4 1  4 8 3 1 6 1 

  31. Shortcut number (g,k) k curves on surface of genus g intersecting another curve  smallest n such that n intersections reduced drawing

  32. Shortcut number (g,1) = 2 2 3 4 1 2 3 4 1

  33. Shortcut number (1,2) > 6 4 2 5 5 2 3 3 6 4 1 6 1

  34. Shortcut number (1,2) > 6 Conjecture: (g,k)  Ck Experimentally: (,2) = 7 (,3) = 31 (?) Known [Schaefer, Š ‘2000]: (0,k)  2k

  35. Directed shortcut number d(g,k) k curves on surface of genus g intersecting another curve  4 1  4 8 3 1 6 1 BAD 

  36. Directed shortcut number d(g,k) upper bound must depend on g,k d(0,2) = 20 Experimentally: finite?

  37. Directed shortcut number d(g,k) finite? interesting? quadratic word equation  drawing problem bound on d(,)  upper bound on word eq. x=yz z=wB x=Aw y=AB A B y A x w z B

  38. Spirals   spiral of depth 1  (spanning arcs, 3 intersections)  interesting for word equations

  39. Unfortunately: Example with no spirals [Schaefer, Sedgwick, Š ’07]

  40. Spirals and folds spiral of depth 1  (spanning arcs, 3 intersections)  fold of width 3 Pach-Tóth’01: In the plane (with puncures) either a large spiral or a large fold must exist.

  41. Unfortunately: Example with no spirals, no folds [Schaefer, Sedgwick, Š ’07]

  42. Embedding on torus

  43. outline ● Simple curves on surfaces * representing surfaces, simple curves in surfaces * algorithmic questions, history ● TOOL: (Quadratic) word equations ● Regular structures in drawings (?) ● Using word equations (Dehn twist, geometric intersection numbers, ...) ● What I would like...

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

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

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

  47. 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

  48. 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

  49. 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

  50. compressed = straight line program (SLP) X0:= a X1:= b X2:= X1X1 X3:= X0X2 X4:= X2X1 X5:= X4X3 X5 = bbbabb OUTPUT OF: Plandowski, Rytter ’98 – polynomial time algorithm Diekert, Robson ’98 – linear time for quadratic eqns CAN DO (in poly-time): ● count the number of occurrences of a symbol ● check equaltity of strings given by two SLP’s (Miyazaki, Shinohara, Takeda’02 – O(n4)) ● get SLP for f(w) where f is a substitution * and w is given by SLP

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