Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction
Computing the Geometric Intersection Number of Curves
Surface Representative Consequences
Representative Algorithm
Vincent Despré and Francis Lazarus
Powers Technical Details
Counting Double paths
gipsa-lab, G-SCOP, Grenoble
[email protected]
Algorithm
System of Curves
16 November 2015
Summary
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Geometric Intersection Number
Problem
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
(a) Number of crossings:
(b) Number of crossings:
too many!
1 → optimal
Counting Double paths Algorithm
System of Curves Summary
Three problems: ë Deciding if a curve can be made simple by homotopy. ë Finding the minimum possible number of self-intersections. ë Finding a corresponding minimal representative. 2 / 18
Geometric Intersection Number
An Old Problem
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
ë Poincaré, 5ème complément analysis situs (1905) ë Reinhart, Algorithms for jordan curves on compact surfaces (1962) ë Chillingworth, Simple closed curves on surfaces (1969) ë Birman and Series, An algorithm for simple curves on surfaces (1984)
Counting Double paths Algorithm
System of Curves Summary
ë Cohen and Lustig, Paths of geodesics and geometric intersection numbers (1987) ë Gonçalves et al., An algorithm for minimal number of (self-)intersection points of curves on surfaces (2005)
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Geometric Intersection Number Vincent DESPRE
INPUT: A combinatorial surface of complexity n. A closed walk of length l.
Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm
A choice of edge orders leads to a generic perturbation of the curve.
Powers Technical Details
Counting Double paths
Discrete vs. Continuous
Algorithm
System of Curves Summary
ë Each equivalence class of curves can be described by a closed walk. ë Each minimal configuration can be realized by a closed walk with appropriate orders on the edges.
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Geometric Intersection Number
de Graaf and Schrijver’s Algorithm
Vincent DESPRE Introduction The Problem
Reidemeister moves:
Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths
de Graaf and Schrijver (1997)
Algorithm
System of Curves Summary
Every curve can be made minimally crossing by Reidemeister moves.
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Geometric Intersection Number
System of Quads
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number Vincent DESPRE Introduction The Problem
Canonical Representative Canonical representative: ë Choose a shortest representative and push it to the right as much as possible.
Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves
Erickson and Whittlesey (2013)
Summary
The canonical representative of a curve is its only representative with no spur, no bracket, no angle -1 and not all angles -2. 7 / 18
Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach
Canonical Representative Canonical representative: ë Choose a shortest representative and push it to the right as much as possible.
Untying
Reduction Surface
Erickson and Whittlesey (2013)
Representative Consequences
Representative Algorithm Powers Technical Details
The canonical representative of a curve is its only representative with no spur, no bracket, no angle -1 and not all angles -2.
Counting Double paths Algorithm
System of Curves Summary
Lazarus and Rivaud (2012) After O(n) precomputation time, one can compute the canonical representative of a curve of length ` in O(`) time. Its length is at most 2`.
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Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach
Properties of the Canonical Form Gersten and Short (1990) A nontrivial contractible closed curve on a system of quads must have either a spur or four brackets.
Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers
Lemma A curve in canonical form has no monogon.
Lemma
Technical Details
Counting Double paths Algorithm
System of Curves
A curve in canonical form has only flat bigons (i.e. the two sides of each bigon correspond to the same path in the system of quads).
Summary
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Geometric Intersection Number
Monogons and Bigons
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface
Hass and Scott (1985)
Representative Consequences
Representative Algorithm
If a curve has excess self-intersections then it has a singular monogon or a singular bigon.
Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number
Monogons and Bigons
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface
Hass and Scott (1985)
Representative Consequences
Representative Algorithm
If a curve has excess self-intersections then it has a singular monogon or a singular bigon.
Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number
Computing a Minimal Representative
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Algorithm for primitive curves: ë Compute a canonical representative. ë Choose a random order on the edges. ë Look for a singular bigon (O(`2 )): • If there is one, exchange its two paths to remove intersections. • Repeat until there is no more singular bigon.
Counting Double paths Algorithm
System of Curves Summary
Theorem This algorithm computes a minimal representative in O(`4 ) time.
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Geometric Intersection Number Vincent DESPRE Introduction
Non-Primitive Curves Formula Let c be a curve, p ∈ N∗ . Then,
The Problem Discrete Approach Untying
i(cp ) = p2 · i(c) + p − 1
Reduction Surface Representative
where i(x) is the geometric intersection number of curve x.
Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number
Bigon Removing
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number
Double Paths
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting
Lemma The maximal double paths form a partition of the pairs of “identical” edges of a given curve. So we can compute the number of crossing double paths in O(`2 ) time.
Double paths Algorithm
System of Curves
Cohen and Lustig (1987)
Summary
Let Σ be a combinatorial surface whose faces are all perforated. Let c be a closed walk without spur in Σ, then i(c) is its number of crossing double paths. 13 / 18
Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction
Main Result Theorem Given a curve c of length ` on a surface of complexity n, one can compute the geometric intersection number of c in O(n + `2 ) time.
Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction
Main Result Theorem Given a curve c of length ` on a surface of complexity n, one can compute the geometric intersection number of c in O(n + `2 ) time.
Surface Representative Consequences
Representative Algorithm
ë We first compute the canonical representative of c in O(n + `) time.
Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction
Main Result Theorem Given a curve c of length ` on a surface of complexity n, one can compute the geometric intersection number of c in O(n + `2 ) time.
Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths
ë We first compute the canonical representative of c in O(n + `) time. ë We compute i(c) in the surface with all its faces perforated in O(`2 ) time.
Algorithm
System of Curves Summary
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Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction
Main Result Theorem Given a curve c of length ` on a surface of complexity n, one can compute the geometric intersection number of c in O(n + `2 ) time.
Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves
ë We first compute the canonical representative of c in O(n + `) time. ë We compute i(c) in the surface with all its faces perforated in O(`2 ) time. ë The minimum i(c) is attained by some orders on the edges. It has no bigon.
Summary
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Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction
Main Result Theorem Given a curve c of length ` on a surface of complexity n, one can compute the geometric intersection number of c in O(n + `2 ) time.
Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
ë We first compute the canonical representative of c in O(n + `) time. ë We compute i(c) in the surface with all its faces perforated in O(`2 ) time. ë The minimum i(c) is attained by some orders on the edges. It has no bigon. ë The corresponding representative has no non-flat bigon, no monogon and no singular flat bigon, so by Hass and Scott it is optimal. 14 / 18
Geometric Intersection Number
System of Curves
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
15 / 18
Geometric Intersection Number
System of Curves
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm
ë Hass and Scott for singular bigons does not hold.
Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
15 / 18
Geometric Intersection Number
System of Curves
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting
ë Hass and Scott for singular bigons does not hold. ë The computation of the minimal representative for a single curve cannot be extended to two curves.
Double paths Algorithm
System of Curves Summary
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Geometric Intersection Number
System of Curves
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
ë Hass and Scott for singular bigons does not hold. ë The computation of the minimal representative for a single curve cannot be extended to two curves. ë However, Cohen and Lustig still works.
System of Curves Summary
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Geometric Intersection Number
System of Curves
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves
ë Hass and Scott for singular bigons does not hold. ë The computation of the minimal representative for a single curve cannot be extended to two curves. ë However, Cohen and Lustig still works. ë Although there might be no singular bigon, we have:
Summary
Lemma If (c, d) have excess intersections then there is a bigon between c and d (not necessarily singular). 15 / 18
Geometric Intersection Number Vincent DESPRE Introduction
Summary k = number of curves b = number of boundaries
The Problem Discrete Approach Untying
Reduction
Before k = 1, b > 0
Surface Representative Consequences
Representative Algorithm Powers Technical Details
k = 2, b > 0
k = 2, b = 0
? L (1987) n.a.
Now k=1 k=2
Simple O(` · log2 (`)) n.a.
k = 1, b = 0
Counting Double paths Algorithm
Simple O((g`)2 ) BS (1984) n.a.
System of Curves Summary
Number O((g`)2 ) CL (1987) O((g`)2 ) CL (1987) ? L (1987) ? L (1987) Number O(`2 ) O(`2 )
Representative O((g`)4 ) A (2009) ? dGS (1997) ? dGS (1997) ? dGS (1997) Representative O(`4 ) ? 16 / 18
Geometric Intersection Number Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
Thank you 17 / 18
Geometric Intersection Number
Easy Cases
Vincent DESPRE Introduction The Problem Discrete Approach Untying
Reduction Surface Representative Consequences
Representative Algorithm Powers Technical Details
Counting Double paths Algorithm
System of Curves Summary
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