Autour du nombre g eom etrique dintersection Francis Lazarus - - PowerPoint PPT Presentation

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Autour du nombre g eom etrique dintersection Francis Lazarus GIPSA-Lab, Grenoble, CNRS Lionel Walden, Les Docks de Cardiff, 1894 1854 1912 What is a surface? What is a surface? Tibor Rad o 1895 1965 1925 S A + F =

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Autour du nombre g´ eom´ etrique d’intersection

Francis Lazarus

GIPSA-Lab, Grenoble, CNRS

Lionel Walden, Les Docks de Cardiff, 1894

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1854 – 1912

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What is a surface?

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What is a surface?

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Tibor Rad´

  • 1895 – 1965

1925

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S − A + F = χ = 2 − 2g − b

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Henri Poincar´

  • e. V-`

eme compl´ ement ` a l’analysis situs. Rendiconti del Circolo Matematico di Palermo, 18(1):45 –110, 1904. Bruce L. Reinhart. Algorithms for jordan curves on compact surfaces.

  • Ann. of Math., p. 209–222, 1962.

Heiner Zieschang. Algorithmen f¨ ur einfache kurven auf fl¨ achen. Mathematica Scandinavica, 17:17–40, 1965. Heiner Zieschang. Algorithmen f¨ ur einfache kurven auf fl¨ achen II. Mathematica scandinavica, 25:49–58, 1969. David R.J. Chillingworth. Simple closed curves on surfaces. Bull. of London Math. Soc., 1(3):310–314, 1969. David R.J. Chillingworth. An algorithm for families of disjoint simple closed curves on surfaces. Bull. of London Math. Soc., 3(1):23–26, 1971. David R.J. Chillingworth. Winding numbers on surfaces. II. Mathematische Annalen, 199(3):131–153, 1972. Joan S. Birman and Caroline Series. An algorithm for simple curves on

  • surfaces. J. London Math. Soc., 29(2):331–342, 1984.
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Marshall Cohen and Martin Lustig. Paths of geodesics and geometric intersection numbers: I. Ann. of Math. Stud. 111:479–500, 1987. Martin Lustig. Paths of geodesics and geometric intersection numbers:

  • II. Ann. of Math. Stud. 111:501–543, 1987.

Joel Hass and Peter Scott. Shortening curves on surfaces. Topology, 33(1):25–43, 1994. Joel Hass and Peter Scott. Configurations of curves and geodesics on

  • surfaces. Geometry and Topology Monographs, 2:201–213, 1999.

Maurits de Graaf and Alexander Schrijver. Making curves minimally crossing by Reidemeister moves. J. Combinatorial Theory, Series B, 70(1):134–156, 1997. Max Neumann-Coto. A characterization of shortest geodesics on

  • surfaces. Algebraic and Geometric Topology, 1:349–368, 2001.

J.M. Paterson. A combinatorial algorithm for immersed loops in

  • surfaces. Topology and its Applications, 123(2):205–234, 2002.

Daciberg L. Gonc ¸alves, Elena Kudryavtseva, and Heiner Zieschang. An algorithm for minimal number of (self-)intersection points of curves on

  • surfaces. Proc. of the Seminar on Vector and Tensor Analysis,

26:139–167, 2005.

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closed free special surface counting homotopy feature Chillingworth winding ’69 number Birman & retraction Series ’84

  • nto a graph

Cohen & retraction Lustig ’87

  • nto a graph

canonical Lustig ’87

  • representative

de Graaf & Reidemeister Schrijver ’97

  • moves

Reidemeister Paterson ’02

  • moves

Gonc ¸alves algebraic et al. ’05

  • approach
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If a curve c is primitive its lifts are uniquely defined by their limit points. If τ is the hyperbolic translation corresponding to a lift ˜ c0 of a primitive c then ı(c) = |{set of pairs of limit points crossing ˜ c0}/τ|

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If a curve c is primitive its lifts are uniquely defined by their limit points. If τ is the hyperbolic translation corresponding to a lift ˜ c0 of a primitive c then ı(c) = |{set of pairs of limit points crossing ˜ c0}/τ| The plan For a given curve c

1

Determine the primitive root of c.

2

Count the number of classes of crossing pairs of limit points (for the root of c).

3

Use adequate formula if c is not primitive.

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A combinatorial framework

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A combinatorial framework

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A combinatorial framework

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A combinatorial framework

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A combinatorial framework: elementary homotopies

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A combinatorial framework: elementary homotopies

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A combinatorial framework: elementary homotopies

Two rules:

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge add/delete a facial walk

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge add/delete a facial walk

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge add/delete a facial walk

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge add/delete a facial walk

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge add/delete a facial walk

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A combinatorial framework: elementary homotopies

Two rules: add/delete a spur along an edge add/delete a facial walk

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Hyp : All faces are quadrilaterals.

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Hyp : All faces are quadrilaterals.

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Hyp : All faces are quadrilaterals.

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Hyp : All faces are quadrilaterals.

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Hyp : All faces are quadrilaterals.

Discrete curvature κs = 1 − ds 2 + cs 4 ds := degree of s and cs := number of incident corners.

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Hyp : All faces are quadrilaterals.

Discrete curvature κs = 1 − ds 2 + cs 4 ds := degree of s and cs := number of incident corners. Combinatorial Gauss-Bonnet Theorem

  • s∈S

κs = χ PROOF.

  • s∈S κs =
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Hyp : All faces are quadrilaterals.

Discrete curvature κs = 1 − ds 2 + cs 4 ds := degree of s and cs := number of incident corners. Combinatorial Gauss-Bonnet Theorem

  • s∈S

κs = χ PROOF.

  • s∈S κs = S −
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Hyp : All faces are quadrilaterals.

Discrete curvature κs = 1 − ds 2 + cs 4 ds := degree of s and cs := number of incident corners. Combinatorial Gauss-Bonnet Theorem

  • s∈S

κs = χ PROOF.

  • s∈S κs = S − A +
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Hyp : All faces are quadrilaterals.

Discrete curvature κs = 1 − ds 2 + cs 4 ds := degree of s and cs := number of incident corners. Combinatorial Gauss-Bonnet Theorem

  • s∈S

κs = χ PROOF.

  • s∈S κs = S − A + 4F/4
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Hyp : All faces are quadrilaterals.

Discrete curvature κs = 1 − ds 2 + cs 4 ds := degree of s and cs := number of incident corners. Combinatorial Gauss-Bonnet Theorem

  • s∈S

κs = χ PROOF.

  • s∈S κs = S − A + F
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Hyp : All faces are quadrilaterals.

Discrete curvature κs = 1 − ds 2 + cs 4 ds := degree of s and cs := number of incident corners. Combinatorial Gauss-Bonnet Theorem

  • s∈S

κs = χ PROOF.

  • s∈S κs = S − A + F = χ
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The 4 brackets Theorem

Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4.

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The 4 brackets Theorem

Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4.

1 2 2 2 1

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The 4 brackets Theorem

Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. 4 brackets Theorem (Gersten et Short ’90) The boundary of a non-singular disk has at least 4 brackets.

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The 4 brackets Theorem

Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. 4 brackets Theorem (Gersten and Short ’90) The boundary of a non-singular disk has at least 4 brackets. PROOF. By Gauss-Bonnet

s∈S κs = 1.

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The 4 brackets Theorem

Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. 4 brackets Theorem (Gersten and Short ’90) The boundary of a non-singular disk has at least 4 brackets. PROOF. By Gauss-Bonnet

s∈S κs = 1.

If s internal: κs = 1 − ds/2 + cs/4 = 1 − cs/4 ≤ 0.

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The 4 brackets Theorem

Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. 4 brackets Theorem (Gersten and Short ’90) The boundary of a non-singular disk has at least 4 brackets. PROOF. By Gauss-Bonnet

s∈S κs = 1.

If s internal: κs = 1 − ds/2 + cs/4 = 1 − cs/4 ≤ 0. If s on the boundary: κs = (2 − cs)/4.

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The 4 brackets Theorem

Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. 4 brackets Theorem (Gersten and Short ’90) The boundary of a non-singular disk has at least 4 brackets. PROOF. By Gauss-Bonnet

s∈S κs = 1.

If s internal: κs = 1 − ds/2 + cs/4 = 1 − cs/4 ≤ 0. If s on the boundary: κs = (2 − cs)/4. Hence, on the boundary: #{s | cs = 1} ≥ #{s | cs ≥ 3} + 4.

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Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. Corollary Every contractible curve (non reduced to a vertex) without spur contains at least 4 brackets.

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Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. Corollary Every contractible curve (non reduced to a vertex) without spur contains at least 4 brackets. van Kampen ’33 Every contractible curve is the label of a reduced disk diagram.

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Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. Corollary Every contractible curve (non reduced to a vertex) without spur contains at least 4 brackets. van Kampen ’33 Every contractible curve is the label of a reduced disk diagram. PROOF.

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Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. Corollary Every contractible curve (non reduced to a vertex) without spur contains at least 4 brackets. van Kampen ’33 Every contractible curve is the label of a reduced disk diagram. PROOF.

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Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. Corollary Every contractible curve (non reduced to a vertex) without spur contains at least 4 brackets. van Kampen ’33 Every contractible curve is the label of a reduced disk diagram. PROOF.

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Hyp : All faces are quadrilaterals and all internal vertices have degree ≥ 4. Corollary Every contractible curve (non reduced to a vertex) without spur contains at least 4 brackets. van Kampen ’33 Every contractible curve is the label of a reduced disk diagram. PROOF. Apply the 4 brackets Theorem to this disk.

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Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. The 5 brackets Theorem The boundary of a non-singular disk with at least one interior vertex has at least 5 brackets.

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Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. The 5 brackets Theorem The boundary of a non-singular disk with at least one interior vertex has at least 5 brackets. PROOF. By Gauss-Bonnet

s∈S κs = 1.

If s internal: κs = 1 − ds/2 + cs/4 = 1 − cs/4 < 0. If s on the boundary: κs = (2 − cs)/4. Hence, on the boundary: #{s | cs = 1} ≥ #{s | cs ≥ 3} + 5.

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. We shorten by removing brackets and spurs

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. We shorten by removing brackets and spurs

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. We shorten by removing brackets and spurs

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. We shorten by removing brackets and spurs

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. We shorten by removing brackets and spurs

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. We shorten by removing brackets and spurs

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. We shorten by removing brackets and spurs

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. 1 2 2 2 We shorten by removing brackets and spurs and push to the right.

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Canonical representative

Hyp : All faces are quadrilaterals and all internal vertices have degree > 4. 1 2 2 2 We shorten by removing brackets and spurs and push to the right.

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Canonical representative

  • L. and Rivaud ’12, Erickson and Whittlesey ’13

After removing all spurs and brackets and pushing to the right as much as possible, we obtain a canonical representative. It can be computed in linear time.

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Canonical representative

  • L. and Rivaud ’12, Erickson and Whittlesey ’13

After removing all spurs and brackets and pushing to the right as much as possible, we obtain a canonical representative. It can be computed in linear time.

Erickson and Whittlesey ’13

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Canonical representative

  • L. and Rivaud ’12, Erickson and Whittlesey ’13

After removing all spurs and brackets and pushing to the right as much as possible, we obtain a canonical representative. It can be computed in linear time.

Erickson and Whittlesey ’13

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Canonical representative

  • L. and Rivaud ’12, Erickson and Whittlesey ’13

After removing all spurs and brackets and pushing to the right as much as possible, we obtain a canonical representative. It can be computed in linear time. Corollary I One can decide if two curves are homotopic in linear time.

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Canonical representative

  • L. and Rivaud ’12, Erickson and Whittlesey ’13

After removing all spurs and brackets and pushing to the right as much as possible, we obtain a canonical representative. It can be computed in linear time. Corollary I One can decide if two curves are homotopic in linear time. Corollary II One can compute the primitive root of a curve in linear time.

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Non-primitive curves

Lemma ı(cp) = p2 × ı(c) + p − 1

c p x p x c

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Non-primitive curves

Lemma ı(cp, dq) = 2pq × ı(c) if c ∼ d or c ∼ d−1, pq × ı(c, d)

  • therwise.

c d c c

p x q x p x p x q x q x

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The plan For a given curve c

1

Determine the primitive root of c.

2

Count the number of classes of crossing pairs of limit points (for the root of c).

3

Use adequate formula if c is not primitive.

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The plan For a given curve c

1

Determine the primitive root of c.

2

Count the number of classes of crossing pairs of limit points (for the root of c).

3

Use adequate formula if c is not primitive.

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Double paths A double path is a pair of homotopic paths.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

L L'= L

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat. Corollary Each class of crossing pairs of limit points are uniquely identified by a maximal double path.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat. Corollary Each class of crossing pairs of limit points are uniquely identified by a maximal double path. Lemma The set of maximal double paths can be computed in quadratic time. PROOF. A pair of indices (i, j) may occur at most twice in the set of maximal double paths.

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Lemma Double paths in a canonical representative (resp. a geodesic) are quasi-flat. Corollary Each class of crossing pairs of limit points are uniquely identified by a maximal double path. Lemma The set of maximal double paths can be computed in quadratic time. Despr´ e and L. ’16 The geometric intersection number of one (two) curve(s) can be computed in quadratic time.

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Computing an actual minimal configuration

Hass and Scott ’85 A curve with excess intersection has either a monogon or a singular bigon.

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Computing an actual minimal configuration

Hass and Scott ’85 A curve with excess intersection has either a monogon or a singular bigon.

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Computing an actual minimal configuration

Hass and Scott ’85 A curve with excess intersection has either a monogon or a singular bigon.

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Bigon swapping

Hass and Scott ’85 A curve with excess intersection has either a monogon or a singular bigon.

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Bigon swapping

Hass and Scott ’85 A curve with excess intersection has either a monogon or a singular bigon. Despr´ e and L. ’16 Given c, an homotopic immersion with a minimal number of intersections can be computed in O(|c|4) time.

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Case of two curves

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Open problems

Propose an algorithm to compute a minimal immersion of two curves.

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Open problems

Propose an algorithm to compute a minimal immersion of two curves. Is quadratic time optimal to just compute the geometric intersection number?

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Open problems

Propose an algorithm to compute a minimal immersion of two curves. Is quadratic time optimal to just compute the geometric intersection number? Find a better algorithm (less than quartic) to compute a minimal immersion of single curve.

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Open problems

Propose an algorithm to compute a minimal immersion of two curves. Is quadratic time optimal to just compute the geometric intersection number? Find a better algorithm (less than quartic) to compute a minimal immersion of single curve. Design a better algorithm to just decide if the geometric intersection number is null.

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Open problems

Propose an algorithm to compute a minimal immersion of two curves. Is quadratic time optimal to just compute the geometric intersection number? Find a better algorithm (less than quartic) to compute a minimal immersion of single curve. Design a better algorithm to just decide if the geometric intersection number is null. Propose an algorithm to compute a minimal immersion in hyperbolic configuration (cf. Hass and Scott ’99).

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