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Number of homotopically inequivalent loops that pairwise intersect at most once
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This is a follow-up to my previous question,
"Number of homotopically inequivalent loops on a surface":
Q. What is the largest number of
homotopically inequivalent,
simple closed curves,
each pair of which intersects in at most one point,
that can be drawn on a surface $S$ of genus $g$?
The previous question required the loops to be pairwise disjoint.
@LeeMosher answered: $3g - 3$.
For $g=4$, allowing the loops to intersect increases
the count from $3g-3=9$ to at least
$13$:
    Â
    Â
Figure by Jeff Erickson here.
But I have no confidence that $13$ is the max here.
I would be interested to know if there is some general theory that bounds
the loops according to their intersections.
general-topology riemannian-geometry homotopy-theory geodesic
asked Jul 28 at 17:50
Joseph O'Rourke
17.1k248103
add a comment |Â
up vote
1
down vote
favorite
This is a follow-up to my previous question,
"Number of homotopically inequivalent loops on a surface":
Q. What is the largest number of
homotopically inequivalent,
simple closed curves,
each pair of which intersects in at most one point,
that can be drawn on a surface $S$ of genus $g$?
The previous question required the loops to be pairwise disjoint.
@LeeMosher answered: $3g - 3$.
For $g=4$, allowing the loops to intersect increases
the count from $3g-3=9$ to at least
$13$:
    Â
    Â
Figure by Jeff Erickson here.
But I have no confidence that $13$ is the max here.
I would be interested to know if there is some general theory that bounds
the loops according to their intersections.
general-topology riemannian-geometry homotopy-theory geodesic
asked Jul 28 at 17:50
Joseph O'Rourke
17.1k248103
2
Indeed $13$ is not the maximum. There is another one which intersects the following curves exactly once (and the others zero times): the leftmost black curve, the adjacent red curve; the adjacent green curve; and the outer black curve. But I do not know the maximum, and I have a vague memory that this is a hard problem.
– Lee Mosher
Jul 28 at 18:01
1
This article by Josh Greene is probably the state of the art.
– Mike Miller
Jul 29 at 12:03
@MikeMiller: Excellent: $lesssim g^2 log g$.
– Joseph O'Rourke
Jul 29 at 13:02
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
This is a follow-up to my previous question,
"Number of homotopically inequivalent loops on a surface":
Q. What is the largest number of
homotopically inequivalent,
simple closed curves,
each pair of which intersects in at most one point,
that can be drawn on a surface $S$ of genus $g$?
The previous question required the loops to be pairwise disjoint.
@LeeMosher answered: $3g - 3$.
For $g=4$, allowing the loops to intersect increases
the count from $3g-3=9$ to at least
$13$:
    Â
    Â
Figure by Jeff Erickson here.
But I have no confidence that $13$ is the max here.
I would be interested to know if there is some general theory that bounds
the loops according to their intersections.
general-topology riemannian-geometry homotopy-theory geodesic
asked Jul 28 at 17:50
Joseph O'Rourke
17.1k248103
This is a follow-up to my previous question,
"Number of homotopically inequivalent loops on a surface":
Q. What is the largest number of
homotopically inequivalent,
simple closed curves,
each pair of which intersects in at most one point,
that can be drawn on a surface $S$ of genus $g$?
The previous question required the loops to be pairwise disjoint.
@LeeMosher answered: $3g - 3$.
For $g=4$, allowing the loops to intersect increases
the count from $3g-3=9$ to at least
$13$:
    Â
    Â
Figure by Jeff Erickson here.
But I have no confidence that $13$ is the max here.
I would be interested to know if there is some general theory that bounds
the loops according to their intersections.
general-topology riemannian-geometry homotopy-theory geodesic
asked Jul 28 at 17:50
Joseph O'Rourke
17.1k248103
asked Jul 28 at 17:50
Joseph O'Rourke
17.1k248103
asked Jul 28 at 17:50
Joseph O'Rourke
17.1k248103
asked Jul 28 at 17:50
Joseph O'Rourke
17.1k248103
17.1k248103
2
Indeed $13$ is not the maximum. There is another one which intersects the following curves exactly once (and the others zero times): the leftmost black curve, the adjacent red curve; the adjacent green curve; and the outer black curve. But I do not know the maximum, and I have a vague memory that this is a hard problem.
– Lee Mosher
Jul 28 at 18:01
1
This article by Josh Greene is probably the state of the art.
– Mike Miller
Jul 29 at 12:03
@MikeMiller: Excellent: $lesssim g^2 log g$.
– Joseph O'Rourke
Jul 29 at 13:02
add a comment |Â
2
Indeed $13$ is not the maximum. There is another one which intersects the following curves exactly once (and the others zero times): the leftmost black curve, the adjacent red curve; the adjacent green curve; and the outer black curve. But I do not know the maximum, and I have a vague memory that this is a hard problem.
– Lee Mosher
Jul 28 at 18:01
1
This article by Josh Greene is probably the state of the art.
– Mike Miller
Jul 29 at 12:03
@MikeMiller: Excellent: $lesssim g^2 log g$.
– Joseph O'Rourke
Jul 29 at 13:02
2
2
Indeed $13$ is not the maximum. There is another one which intersects the following curves exactly once (and the others zero times): the leftmost black curve, the adjacent red curve; the adjacent green curve; and the outer black curve. But I do not know the maximum, and I have a vague memory that this is a hard problem.
– Lee Mosher
Jul 28 at 18:01
Indeed $13$ is not the maximum. There is another one which intersects the following curves exactly once (and the others zero times): the leftmost black curve, the adjacent red curve; the adjacent green curve; and the outer black curve. But I do not know the maximum, and I have a vague memory that this is a hard problem.
– Lee Mosher
Jul 28 at 18:01
1
1
This article by Josh Greene is probably the state of the art.
– Mike Miller
Jul 29 at 12:03
This article by Josh Greene is probably the state of the art.
– Mike Miller
Jul 29 at 12:03
@MikeMiller: Excellent: $lesssim g^2 log g$.
– Joseph O'Rourke
Jul 29 at 13:02
@MikeMiller: Excellent: $lesssim g^2 log g$.
– Joseph O'Rourke
Jul 29 at 13:02
add a comment |Â
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2
Indeed $13$ is not the maximum. There is another one which intersects the following curves exactly once (and the others zero times): the leftmost black curve, the adjacent red curve; the adjacent green curve; and the outer black curve. But I do not know the maximum, and I have a vague memory that this is a hard problem.
– Lee Mosher
Jul 28 at 18:01
1
This article by Josh Greene is probably the state of the art.
– Mike Miller
Jul 29 at 12:03
@MikeMiller: Excellent: $lesssim g^2 log g$.
– Joseph O'Rourke
Jul 29 at 13:02