PhD Defense Topology and Algorithms on Combinatorial Maps Vincent Despré Gipsa-lab, G-scop, Labex Persyval
18 Octobre 2016
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
2 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
3 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
4 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
V =number of vertices, E=number of edges and F =number of faces
Euler Formula On a surface that can be deformed to a sphere, any polygonal subdivision verifies:
The Problem The Results
χ(S) = V − E + F = 2
Conclusion
5 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem
Euler Formula On a surface S of genus g, any polygonal subdivision verifies: χ(S) = V − E + F = 2 − 2g
The Results
Conclusion
6 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem
Euler Formula On a surface S of genus g with b boundaries, any polygonal subdivision verifies:
The Results
Conclusion
χ(S) = V − E + F = 2 − 2g − b
7 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
8 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
9 / 29
PhD Defense
Splitting Cycles
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
10 / 29
PhD Defense
Splitting Cycles
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
Cabello et al. (2011) Deciding if a combiantorial map admits a splitting cycle is NP-complete.
The Problem The Results
Conclusion
10 / 29
PhD Defense
Splitting Cycles
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Barnette’s Conjecture (1982) Every triangulations of surfaces of genus at least 2 admit a splitting cycle.
Conclusion
10 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations
Conjecture (Mohar and Thomassen, 2001)
Planar Case Torus Case
Geometric Intersection Number of Curves
Every triangulations of surfaces of genus g ≥ 2 admit a splitting cycle of every different type.
The Problem The Results
Conclusion
11 / 29
PhD Defense
Irreducible Triangulations
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
e
Splitting Cycles The Problem Experimental Approach
T
T'
The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
ú There are a finite number of irreducible triangulations of genus g. (Barnette and Edelson, 1988 and Joret and Wood, 2010)
The Problem The Results
Conclusion
ú There are 396784 irreducible triangulations of genus 2. ú Unreachable for genus 3. 12 / 29
PhD Defense
Genus 2 irreducible triangulations
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles
First implementation by Thom Sulanke. Genus 2: Number of triangulations: 396 784 l
The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
n 10 11 12 13 14 15 16 17
3 2 25 710 8130 36794 661 5
4
5
6
7
8
Average
2 58 1516 13004 30555 1395 3
51 2249 20507 50814 12308 3
681 16138 72001 78059 3328 1
130 7818 22877 16609 205 2
1 11 121 9 1
6.09 6.21 6.00 5.61 4.21 3.04 3.01 3.00
Conclusion
13 / 29
PhD Defense
Genus 6
Vincent DESPRE Introduction The Notion of Surface
We consider the 59 non-isomorphic embeddings of K12 . (Altshuler, Bokowski and Schuchert 1996)
Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point
Average: 7.58 Worst-case: 8
The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
Average: 9.41 Worst-case: 10
Average: 10.32 Worst-case: 12 (Hamiltonian cycle!)
14 / 29
PhD Defense
Complete Graphs
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles
χ(S) = v − e + f = n −
n(n − 1) 2 n(n − 1) + · = 2 − 2g 2 3 2
The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations
(n − 3)(n − 4) 12 (n − 3)(n − 4) ≡ 0[12] ⇔ n ≡ 0, 3, 4 or 7[12] g=
Planar Case Torus Case
Geometric Intersection Number of Curves The Problem
Theorem (Ringel and Youngs, ∼1970) Kn can triangulate a surface if and only if n ≡ 0, 3, 4 or 7[12].
The Results
Conclusion
15 / 29
PhD Defense
Computation time
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles
New implementation in C++. The data-structure used for the triangulations is the flag representation.
The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
n basic
12 2 s.
15 1 h.
16 12 h.
19 ∼10 years
This has been computed with an 8 cores computer with 16 Go of RAM. It uses parallel computation.
The Problem The Results
Conclusion
16 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
17 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
n basic final
15 1 h. 2 s.
16 12 h. 3 s.
19 ∼10 years 8 sec.
···
43 1 h. 18 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
Kn Type 1 2 3 4 5 6 7 8 9 10 11 12 13 14 . . . 29 30 max type
K15
K16
K19
K27
K28
K31
K39
K40
K43
8 11 12 13 14
10 12 14 16 16 16
11 14 16 18 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
12 16 19 20 27 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
12 17 18 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
8 13 15 17 20 21 23 24 28 28 29 ⊥ ⊥ ⊥
12 15 20 24 26 30 32 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
10 15 18 19 24 26 28 30 33 35 36 38 40 ⊥
8 11 12 15 18 20 21 23 24 25 27 29 30 31
⊥
⊥
23
25
⊥ ⊥ ⊥ 31
⊥ ⊥ ⊥ 52
⊥ ⊥ ⊥ 55
. . . 42 ⊥ 65
5
6
10
⊥ = No cycle found.
The Problem The Results
Conclusion
Counter-Examples Mohar and Thomassen conjecture is false. 19 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
Kn Type
K15
K16
K19
K27
K28
K31
K39
K40
K43
8 11 12 13 14
10 12 14 16 16 16
11 14 16 18 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
12 16 19 20 27 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
12 17 18 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
8 13 15 17 20 21 23 24 28 28 29 ⊥ ⊥ ⊥
12 15 20 24 26 30 32 ⊥ ⊥ ⊥ ⊥ ⊥ ⊥ ⊥
10 15 18 19 24 26 28 30 33 35 36 38 40 ⊥
8 11 12 15 18 20 21 23 24 25 27 29 30 31
⊥
⊥
23
25
⊥ ⊥ ⊥ 31
⊥ ⊥ ⊥ 52
⊥ ⊥ ⊥ 55
. . . 42 ⊥ 65
1 2 3 4 5 6 7 8 9 10 11 12 13 14 . . . 29 30 max type
5
6
10
⊥ = No cycle found.
The Problem The Results
Conjecture
Conclusion
For every α > 0, there exists a triangulation with no splitting cycles of type larger than α · g2 . 19 / 29
PhD Defense Vincent DESPRE
Encoding Toroidal Triangulations Properties of the planar case:
Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point
1/ We have a notion of 3-orientation for triangulations. 2/ Every 3-orientation admits a unique Schnyder wood coloration. 3/ Each color corresponds to a spanning tree and so there is no monochromatic cycle.
The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
4/ The 3-orientations of a given triangulation have a structure of distributive lattice. 5/ The minimal element of the lattice has no clockwise oriented cycle. 6/ Triangulations are in bijection with a particular type of decorated embedded trees.
Conclusion
20 / 29
PhD Defense
1/ We have a notion of 3-orientation for triangulations.
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
Kampen (1976)
The Problem The Results
Every planar triangulation admits a 3-orientation.
Conclusion
20 / 29
PhD Defense Vincent DESPRE
2/ Every 3-orientation admits a unique Schnyder wood coloration.
Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
20 / 29
PhD Defense Vincent DESPRE
2/ Every 3-orientation admits a unique Schnyder wood coloration.
Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
de Fraisseix and Ossona de Mendez (2001) Each 3-orientation of a plane simple triangulation admits a unique coloring (up to permutation of the colors) leading to a Schnyder wood.
20 / 29
PhD Defense Vincent DESPRE
3/ Each color corresponds to a spanning tree and so there is no monochromatic cycle.
Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
20 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface
4/ The 3-orientations of a given triangulation have a structure of distributive lattice.
Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
20 / 29
PhD Defense Vincent DESPRE
4/ The 3-orientations of a given triangulation have a structure of distributive lattice.
Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
Propp (1993), Ossona de Mendez (1994), Felsner (2004) The set of the 3-orientations of a given triangulation has a structure of distributive lattice for the appropriate ordering.
20 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves
Properties of the planar case: 1/ We have a notion of 3-orientation for triangulations. 2/ Every 3-orientation admits a unique Schnyder wood coloration. 3/ Each color corresponds to a spanning tree and so there is no monochromatic cycle. 4/ The 3-orientations of a given triangulation have a structure of distributive lattice. 5/ The minimal element of the lattice has no clockwise oriented cycle. 6/ Triangulations are in bijection with a particular type of decorated embedded trees.
The Problem The Results
Conclusion
20 / 29
PhD Defense Vincent DESPRE Introduction
6/ Triangulations are in bijection with a particular type of decorated embedded trees (Poulalhon and Schaeffer, 2006).
The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
20 / 29
PhD Defense Vincent DESPRE Introduction
6/ Triangulations are in bijection with a particular type of decorated embedded trees (Poulalhon and Schaeffer, 2006).
The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
20 / 29
PhD Defense Vincent DESPRE Introduction
6/ Triangulations are in bijection with a particular type of decorated embedded trees (Poulalhon and Schaeffer, 2006).
The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
20 / 29
PhD Defense Vincent DESPRE
Properties of the torus case: Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem
1/ We have a notion of 3-orientation for triangulations. 2/ Every 3-orientation admits a unique Schnyder wood coloration. 3/ Each color corresponds to a spanning tree and so There is no monochromatic contractible cycle. 4/ The 3-orientations of a given triangulation have a structure of distributive lattices. 5/ The minimal element of each lattice has no clockwise oriented contractible cycle. 6/ Triangulations are in bijection with a particular type of decorated unicellular toroidal maps.
The Results
Conclusion
21 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface
6/ Triangulations are in bijection with a particular type of decorated unicellular toroidal maps.
Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem
Tree-cotree Decomposition: (T, C, X). T has n − 1 edges, C has f − 1 edges and X the remaining. χ = n − (n − 1 + f − 1 + x) + f ⇔ x = 2 − χ = 2g
The Results
Conclusion
21 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface
6/ Triangulations are in bijection with a particular type of decorated unicellular toroidal maps.
Topology Combinatorial Maps
Splitting Cycles The Problem
3
3
5
The Key Point
7
The Results
4
7 Encoding Toroidal Triangulations
6
2
5
Experimental Approach
1
2
3
2 5
Planar Case Torus Case
Geometric Intersection Number of Curves
4 6
1 2
4 6
3
The Problem The Results
Conclusion
21 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense
Geometric Intersection Number of Curves
Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
22 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations
(g) Number of crossings:
(h) Number of crossings:
too many!
1 → optimal
Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
Three problems: ú Deciding if a curve can be made simple by homotopy. ú Finding the minimum possible number of self-intersections. ú Finding a corresponding minimal representative. 23 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface
Boundaries b>0 b=0
Simple O((g`)2 ) BS (1984) ? L (1987) O(`5 ) GKZ (2005)
Number O((g`)2 ) CL (1987) ? L (1987) O(`5 ) GKZ (2005)
Representative O((g`)4 ) A (2015) ? dGS (1997)
Any
O(` · log2 (`))
O(`2 )
O(`4 )
Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
BS: Birman and Series, An algorithm for simple curves on surfaces. CL: Cohen and Lustig, Paths of geodesics and geometric intersection numbers: I. L: Lustig, Paths of geodesics and geometric intersection numbers: II. A: Arettines, A combinatorial algorithm for visualizing representatives with minimal self-intersection. dGS: de Graaf and Schrijver, Making curves minimally crossing by Reidemeister moves. GKZ: Gonçalves, Kudryavtseva and Zieschang, An algorithm for minimal number of (self-)intersection points of curves on surfaces.
Conclusion
24 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point
Publications: 1/ Some Triangulated Surfaces without Balanced Splitting: Published in Graphs and Combinatorics. 2/ Encoding Toroidal Triangulations: Accepted in Discrete & Computationnal Geometry. 3/ Computing the Geometric Intersection Number of Curves: Will be submitted to the next SoCG.
The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem The Results
Conclusion
Work in progress: 1/ Looking for a proof that does not require a computer. 2/ There are a lot of implications for the bijection in the plane. Is it possible to generalized them. 3/ It remains to look at the construction of a minimal representative for a couple of curves.
25 / 29
PhD Defense Vincent DESPRE Introduction The Notion of Surface Topology Combinatorial Maps
Splitting Cycles The Problem Experimental Approach The Key Point The Results
Encoding Toroidal Triangulations Planar Case Torus Case
Geometric Intersection Number of Curves The Problem
Conjecture
The Results
Conclusion
Deciding if there is a simple closed walk in a given homotopy class is NP-complete and FPT parametrized by the genus of the surface. 26 / 29
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