Mixing
Decomposing surfaces into pairs of pants
Clash Royale CLAN TAG#URR8PPP
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I apologise if this question is badly worded/doens't make sense - if I knew how to formulate this question well, I'd probably be half-way to answering it myself.
Let $F_g$ be a closed surface of genus $g$, for $g geq 2$. Let K be a pants decomposition of our surface, i.e. a set of closed curves on $F_g$ such that when we cut along the curves, we obtain a number of disjoint pairs of pants. Furthermore, suppose all curves in K are geodesics on $F_g$.
Finally, suppose $F_g$ admits some hyperbolic metric.
Each pair of pants contains three seams, i.e. lines of shortest distance between each pair of boundary components.
Question: Suppose $P_i, P_j$ are two pairs of pants in $F_g$ which are glued along a common boundary component $delta$. Do the end-points of the seams of $P_i$ in $delta$ match up with the end-points of the seams of $P_j$?
general-topology surfaces geometric-topology
asked Jul 25 at 14:01
Monkud
262
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up vote
5
down vote
favorite
I apologise if this question is badly worded/doens't make sense - if I knew how to formulate this question well, I'd probably be half-way to answering it myself.
Let $F_g$ be a closed surface of genus $g$, for $g geq 2$. Let K be a pants decomposition of our surface, i.e. a set of closed curves on $F_g$ such that when we cut along the curves, we obtain a number of disjoint pairs of pants. Furthermore, suppose all curves in K are geodesics on $F_g$.
Finally, suppose $F_g$ admits some hyperbolic metric.
Each pair of pants contains three seams, i.e. lines of shortest distance between each pair of boundary components.
Question: Suppose $P_i, P_j$ are two pairs of pants in $F_g$ which are glued along a common boundary component $delta$. Do the end-points of the seams of $P_i$ in $delta$ match up with the end-points of the seams of $P_j$?
general-topology surfaces geometric-topology
asked Jul 25 at 14:01
Monkud
262
add a comment |Â
up vote
5
down vote
favorite
up vote
5
down vote
favorite
I apologise if this question is badly worded/doens't make sense - if I knew how to formulate this question well, I'd probably be half-way to answering it myself.
Let $F_g$ be a closed surface of genus $g$, for $g geq 2$. Let K be a pants decomposition of our surface, i.e. a set of closed curves on $F_g$ such that when we cut along the curves, we obtain a number of disjoint pairs of pants. Furthermore, suppose all curves in K are geodesics on $F_g$.
Finally, suppose $F_g$ admits some hyperbolic metric.
Each pair of pants contains three seams, i.e. lines of shortest distance between each pair of boundary components.
Question: Suppose $P_i, P_j$ are two pairs of pants in $F_g$ which are glued along a common boundary component $delta$. Do the end-points of the seams of $P_i$ in $delta$ match up with the end-points of the seams of $P_j$?
general-topology surfaces geometric-topology
asked Jul 25 at 14:01
Monkud
262
I apologise if this question is badly worded/doens't make sense - if I knew how to formulate this question well, I'd probably be half-way to answering it myself.
Let $F_g$ be a closed surface of genus $g$, for $g geq 2$. Let K be a pants decomposition of our surface, i.e. a set of closed curves on $F_g$ such that when we cut along the curves, we obtain a number of disjoint pairs of pants. Furthermore, suppose all curves in K are geodesics on $F_g$.
Finally, suppose $F_g$ admits some hyperbolic metric.
Each pair of pants contains three seams, i.e. lines of shortest distance between each pair of boundary components.
Question: Suppose $P_i, P_j$ are two pairs of pants in $F_g$ which are glued along a common boundary component $delta$. Do the end-points of the seams of $P_i$ in $delta$ match up with the end-points of the seams of $P_j$?
general-topology surfaces geometric-topology
asked Jul 25 at 14:01
Monkud
262
asked Jul 25 at 14:01
Monkud
262
asked Jul 25 at 14:01
Monkud
262
asked Jul 25 at 14:01
Monkud
262
262
add a comment |Â
add a comment |Â
1 Answer
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The answer is "almost never" because you can glue two pairs of pants along some circle with an $mathbbR$ worth of twists. Thus, if the seams do match up then doing some small $epsilon$ twist will make it so they don't match anymore.
However, you can also note that for any of the three boundary components of a pair of pants, two of the seams hit it, and ask "If one pair of seams matches up, does the other match up as well?" and for this one I have no idea.
answered Jul 26 at 2:54
Carl
2,0081926
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
0
down vote
The answer is "almost never" because you can glue two pairs of pants along some circle with an $mathbbR$ worth of twists. Thus, if the seams do match up then doing some small $epsilon$ twist will make it so they don't match anymore.
However, you can also note that for any of the three boundary components of a pair of pants, two of the seams hit it, and ask "If one pair of seams matches up, does the other match up as well?" and for this one I have no idea.
answered Jul 26 at 2:54
Carl
2,0081926
add a comment |Â
up vote
0
down vote
The answer is "almost never" because you can glue two pairs of pants along some circle with an $mathbbR$ worth of twists. Thus, if the seams do match up then doing some small $epsilon$ twist will make it so they don't match anymore.
However, you can also note that for any of the three boundary components of a pair of pants, two of the seams hit it, and ask "If one pair of seams matches up, does the other match up as well?" and for this one I have no idea.
answered Jul 26 at 2:54
Carl
2,0081926
add a comment |Â
up vote
0
down vote
up vote
0
down vote
The answer is "almost never" because you can glue two pairs of pants along some circle with an $mathbbR$ worth of twists. Thus, if the seams do match up then doing some small $epsilon$ twist will make it so they don't match anymore.
However, you can also note that for any of the three boundary components of a pair of pants, two of the seams hit it, and ask "If one pair of seams matches up, does the other match up as well?" and for this one I have no idea.
answered Jul 26 at 2:54
Carl
2,0081926
The answer is "almost never" because you can glue two pairs of pants along some circle with an $mathbbR$ worth of twists. Thus, if the seams do match up then doing some small $epsilon$ twist will make it so they don't match anymore.
However, you can also note that for any of the three boundary components of a pair of pants, two of the seams hit it, and ask "If one pair of seams matches up, does the other match up as well?" and for this one I have no idea.
answered Jul 26 at 2:54
Carl
2,0081926
answered Jul 26 at 2:54
Carl
2,0081926
answered Jul 26 at 2:54
Carl
2,0081926
answered Jul 26 at 2:54
Carl
2,0081926
2,0081926
add a comment |Â
add a comment |Â
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