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Algorithmic Problems for Curves on Surfaces Daniel Štefankovič University of Rochester

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● 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... outline

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How to represent surfaces?

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

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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)

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

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Simple curves on surfaces closed curve homeomorphic image of circle S 1 simple closed curve = is injective (no self-intersections) (free) homotopy equivalent simple closed curves

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

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Combinatorial description of a (homotopy type of) a simple curve in a surface 1.intersection sequence with a triangulation a b c

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

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

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

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

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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” unique (important for surfaces without boundary)

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● 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... outline

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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

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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 Algorithmic problems - History

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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 explicit compressed representations polynomial in compressed representations, for fixed set of curves any pair of curves

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● 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... outline

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

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

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

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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

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Word equations word equations word equations with given lengths

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

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Word equations word equations - NP-hard word equations with given lengths Plandowski, Rytter ’98 – polynomial time algorithm Diekert, Robson ’98 – linear time for quadratic eqns decidability – Makanin 1977 PSPACE – Plandowski 1999 (quadratic = each variable occurs 2 times) In NP ??? exponential upper bound on the length of a minimal solution MISSING: OPEN:

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● 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... outline

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

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

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

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Shortcut number (g,k) k curves on surface of genus g intersecting another curve smallest n such that n intersections reduced drawing

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Shortcut number (g,1) 2 4 1 3 2 4 1 3 2

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Shortcut number (1,2) 6 4 6 6 1 1 3 3 5 5 2 2 4

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Conjecture: g,k) C k Experimentally: ,2) 7 ,3) 31 (?) Known [Schaefer, Š ‘2000]: (0,k) 2 k

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

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

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

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

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

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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.

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

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Embedding on torus

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● 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... outline

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Geometric intersection number minimum number of intersections achievable by continuous deformations.

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

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

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

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STEP1: Moving between the representations 1.intersection sequence with a triangulation 2. normal coordinates (w.r.t. a triangulation) bc -1 bc -1 ba -1 a)=1 b)=3 c)=2 Can we move between these two representations efficiently? a)=1+2 100 b)=1+3.2 100 c)=2 101

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compressed = straight line program (SLP) X 0 a X 1 b X 2 X 1 X 1 X 3 X 0 X 2 X 4 X 2 X 1 X 5 X 4 X 3 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)) X 5 = bbbabb

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compressed = straight line program (SLP) X 0 a X 1 b X 2 X 1 X 1 X 3 X 0 X 2 X 4 X 2 X 1 X 5 X 4 X 3 X 5 = bbbabb Plandowski, Rytter ’98 – polynomial time algorithm Diekert, Robson ’98 – linear time for quadratic eqns OUTPUT OF: 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(n 4 )) ● get SLP for f(w) where f is a substitution * and w is given by SLP

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

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

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Dehn twist of along

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D()D()

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D()D()

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Geometric intersection numbers n¢ i( , )i( , ) -i( , ) i( ,D n ( )) n¢ i( , )i( , )+i( , ) i( ,D n ( ))/i( , ) ! i( ,

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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 (only for surfaces with S 0)

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● 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... outline

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

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PROBLEM #1: Minimal weight representative INPUT: triangulation + gluing normal coordinates of edge weights OUTPUT: ’ minimizing ’(e) eTeT

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PROBLEM #2: Moving between representations 4. Dehn-Thurston coordinates (Dehn ’38, W.Thurston ’76) unique representation for closed surfaces! PROBLEM normal coordinates Dehn-Thurston coordinates in polynomial time? linear time?

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PROBLEM #3: Word equations PROBLEM: are word equations in NP? are quadratic word equations in NP? NP-hard decidability – Makanin 1977 PSPACE – Plandowski 1999

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PROBLEM #4: Computing Dehn-Twists faster? 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 O(n 3 ) randomized, O(n 9 ) deterministic

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PROBLEM #5: Realizing geometric intersection #? our algorithm is very indirect can compress drawing realizing geometric intersection #? can find the drawing?

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