Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time
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Transcript of Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time
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Computing Dehn Twists and Geometric Intersection Numbers in Polynomial
Time
Marcus Schaefer, Eric SedgwickDePaul University (Chicago)
Daniel ŠtefankovičUniversity of Rochester
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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.
outline
![Page 3: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/3.jpg)
Curves on surfaces
different? same?
closed curve homeomorphic image of circle S1
simple closed curve = is injective (no self-intersections)
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Curves on surfaces
different? same?
homotopy equivalent
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Curves on surfaces
different? same?
homotopy equivalent
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Curves on surfaces
different? same?
homotopy equivalent
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Curves on surfaces
different? same?
homotopy equivalent
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Curves on surfaces
not homotopy equivalent
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Curves on surfaces
curves continuous objects
homotopy classes of curves combinatorial objects
1) how to represent them?2) what/how to compute?
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Geometric intersection numberminimum number of intersections achievable by continuous deformations.
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Geometric intersection numberminimum number of intersections achievable by continuous deformations.
i(,)=2
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EXAMPLE: Geometric intersection numbersare well understood on the torus
(3,5) (2,-1)
3 5 2 -1
det = -13
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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.
outline
![Page 14: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/14.jpg)
Algorithmic problems - HistoryContractibility (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, butonly for fixed set of curves
Algorithmic problems - History
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Algorithmic problems – our resultsGeometric 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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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.
outline
![Page 18: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/18.jpg)
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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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.
outline
![Page 23: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/23.jpg)
How to represent simple curvesin surfaces (up to homotopy)?
Ideally the representation is “unique” (each curve has a unique representation)
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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-1bc-1ba-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!
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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-1bc-1ba-1
a)=1 b)=3 c)=2
geometric intersection number
![Page 29: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/29.jpg)
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.
outline
![Page 30: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/30.jpg)
STEP1: Moving between the representations
1. intersection sequence with a triangulation
2. normal coordinates (w.r.t. a triangulation)
bc-1bc-1ba-1
a)=1 b)=3 c)=2
Can we move between these two representations efficiently?
a)=1+2100 b)=1+3.2100 c)=2101
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STEP1: Moving between the representations
1. intersection sequence with a triangulation
2. normal coordinates (w.r.t. a triangulation)
bc-1bc-1ba-1
a)=1 b)=3 c)=2
Can we move between these two representations efficiently?
a)=1+2100 b)=1+3.2100 c)=2101
YES
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compressed = straight line program (SLP)
X0 a X1 b X2 X1X1
X3 X0X2
X4 X2X1
X5 X4X3
Theorem (SSS’08): normal coordinatescompressed intersection sequence in time O( log (e))
compressed intersection sequencenormal coordinates in time O(|T|.SLP-length(S))
X5 = bbbabb
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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.
outline
![Page 34: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/34.jpg)
Main tool: Word equations
xabx =yxy x,y – variablesa,b - constants
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xabx =yxy x,y – variablesa,b - constants
a solution:
x=ab y=ab
Main tool: Word equations
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Word equations with given lengths
x,y – variablesa,b - constantsxayxb = axbxy
additional constraints: |x|=4, |y|=1
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Word equations with given lengths
x,y – variablesa,b - constantsxayxb = 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 algorithmDiekert, Robson ’98 – linear time for quadratic eqns
decidability – Makanin 1977PSPACE – Plandowski 1999
(quadratic = each variable occurs 2 times)
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Simulating curve using quadratic word equations
X
yz
u v
u=xy...v=u
|u|=|v|=(u)...
Diekert-Robsonnumber ofcomponents
w
z
|x|=(|z|+|u|-|w|)/2
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Moving between the representations1. intersection sequence with a triangulation
2. normal coordinates (w.r.t. a triangulation)bc-1bc-1ba-1
a)=1 b)=3 c)=2
Theorem: normal coordinatescompressed intersection sequence in time O( log (e))
“Proof”:
X
yz
u v
u=xy...av=ua
|u|=|v|=|T| (u)
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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.
outline
![Page 43: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/43.jpg)
Dehn twist of along
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Dehn twist of along
D()
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Dehn twist of along
D()
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Geometric intersection numbers
n¢ i(,)i(,) -i(,) i(,Dn
()) n¢ i(,)i(,)+i(,)
i(,Dn())/i(,) ! i(,
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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.
outline
![Page 48: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/48.jpg)
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
![Page 49: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/49.jpg)
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.
outline
![Page 50: Computing Dehn Twists and Geometric Intersection Numbers in Polynomial Time](https://reader035.fdocuments.in/reader035/viewer/2022062301/56814f12550346895dbca488/html5/thumbnails/50.jpg)
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)
eT
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PROBLEM #2: Moving between representations
3. Dehn-Thurston coordinates(Dehn ’38, W.Thurston ’76)
unique representation for closed surfaces!
PROBLEM normal coordinatesDehn-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 1977PSPACE – 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(n3) randomized, O(n9) deterministic