Keeping the distance, or spans of continua

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Keeping the distance, or spans of continua

• Alex Karassev

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What is a continuum?

• In the plane or space an object which is
• bounded
• closed
• connected

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Bounded
Unbounded
Bounded
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Closed
Non-closed
Closed
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Connected
Not connected
Connected
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Examples

• Graphs
• simple closed curves
• triods
• Locally connected continua
• Non-locally connected continua
• Indecomposable continua

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Simple closed curves
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Triods
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Graphs
CN Railways
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Graphs
"Digital Dandelion" or Map of Internet
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Graphs
Fullerene C60Molecule
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Non-locally connected continua
Topologist's sine curve
And so on
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Decomposable continua
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Indecomposable continua
And so on
Knaster's Buckethandle (1922)
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Span (Andrew Lelek, 1964)

• Two cars are traveling along a road network
  (graph). Each car must travel across the whole
  graph. The largest possible distance the cars can
  keep between them while moving along the graph is
  called the surjective span of the graph
• Similar idea can be used to define span of an
  arbitrary continuum

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Circle
Span diameter
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Triod

• We can start atend points
• However, at some moment one of cars must reach
  the center
• So the largest possible value for span is the
  length of the leg

span
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Line segments (arcs)
Examples of arcs
Span 0
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More complicated cases?
Simple closed curve
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Alternative approach
Continuum C
(X1, X2)
X2
X1
X2
X1
Product (square) C x C
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Alternative approach
Two paths in the continuum one path in the
square!
(0,1)
X1
X2
(1,0)
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Representation of distances
Distance lt 1/2
Diagonal (x1x2, distance 0)
Distance 1/2
In general, the set of points withdistance lt d
is represented by asymmetric neighbourhood of
thediagonal
0
1
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It can be very complicated
1m
C
Distance lt 1.5
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Span of an arc is 0
(0,1)
1
C x C
0
Arc C
(1,0)
Any path from (0,1) to (1,0) must intersect the
diagonal,so the cars will be at the distance 0!
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Square of a simple closed curve
C
C x C Torus
Diagonal
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Visualizing the productsimple closed curve
Torus can be viewed as a squarewith identified
opposite sides
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Variants of spans

• Span both cars traverse the whole graph
• Semispan only one car travels across the whole
  graph
• Symmetric span if, at some moment, Car 1 is at
  the point A and Car 2 is at the point B, then
  there must be another moment when Car 1 is at B
  and Car 2 is at A
• Essential span (for simple closed curves)

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Versions of span coincide for
circle
"standard" triod
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Question of Andrew Lelek (1995)

• Do different types of spans always agree for
  certain classes of continua, particularly for
  triods and simple closed curves?

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Some of our results

• Theorem (Logan Hoehn, Alex Karassev)No two of
  these versions of span agree for all simple
  closed curves or for all triods"Equivalent
  metrics and the spans of graphs", Colloquium
  Math, accepted

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Our idea predetermined distances

• Start with a neighbourhood of diagonal that have
  desired properties, and then define suitable
  metric
• This can be done by specifying closed sets in C x
  C which correspond to points with given distance
  (say 1 or 2)

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Example span is not equal to symmetric span for
a simple closed curve
Span 2 (two blue circles)
X2
Symmetric span 1
0
2
1
X1
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Example span is not equal to semispan for a
simple closed curve
Semispan 2 (two blue arcs)
X2
Span 1
lt1
0
2
1
X1
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Question of Howard Cook (1995)
C2
C1
Is the span of C1less than the span of C2?
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It seems very easy in the case of convex curves
d1lt d2
C2
C1
d2
d1
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but it is very complicated in general!