ways a line can cross itself (topology i believe?)

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ways a line can cross itself (topology i believe?)

Postby digory » Mon Nov 01, 2010 2:41 am UTC

if you draw a line on a piece of paper
(not a perfectly straight one but a line that can curve as it likes)
and if that line bends around and crosses itself once
there are two possible configurations
either it will hide both the start and end of the line inside the loop
or it will leave them outside the loop
if it crosses itself twice there are twelve possible configurations
these twelve form three families
if A is a time that the line passes through the first crossing
and B is a time it passes through the second crossing
these three families are
AABB (which has four members)
ABAB (which has two)
ABBA (which has six)
each letter appears twice because that is what a crossing is, a time the line comes to the same place twice.
there is no such family as BAAB for example since it is just the wrong way to write ABBA
using this notation it would appear that their are 15 possible families of three-crossing figures but two are equivalent to each other (if the start and end of the line are the same) and two have no members because they are impossible to actually draw.
-these are very interesting shapes to play with i encourage you to take a stroll in this little world and draw them yourself (i'll post a picture if i can get my scanner working)
-has this been studied? if so do you know what this set of figures is called? or is it equivalent to some other system that has been studied?
-the closest thing i could find was knot theory but that is quite different, as far as i can tell it has only a cosmetic similarity.

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Re: ways a line can cross itself (topology i believe?)

Postby MartianInvader » Mon Nov 01, 2010 6:11 pm UTC

Well, I can toss some words around - I believe you're describing classification of images of embeddings of certain types of graphs into R^2, up to ambient isotopy. The types of graphs you're interested in are connected graphs with two vertices of degree one (the "start" and "end" of your lines), and some fixed number of remaining vertices of degree 4 (the "crossings"). By "graph" here I mean a 1-dimensional CW-complex, not a combinatorial graph.

I don't really know if it's been studied before, but if you wikipedia some of those terms you may be able to state the question in more formalized language, which will let you look for it more easily.

Is it correct to assume here that you don't allow any sort of weirdness in the crossings? For example, you don't allow the curve brushing up against itself at a point without crossing it, or crossing the same point 20 times?

Edit: After re-reading your post I strongly encourage you to try to very carefully and precisely define when two things are considered the same "configuration". For example, if I go around in a single circle and then finish by drawing a segment into the circle, is that the same as if I finish by drawing a segment out of the circle? If I reflect everything along an axis, did I change the configuration? Or is the only thing that matters the order in which I visit these "crossings"? If that's so, what if I re-label them (ie, is AABB the same as BBAA)? Try to come up with a precise definition of a "configuration", and precise rules as to when two count as the same thing.
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Re: ways a line can cross itself (topology i believe?)

Postby digory » Mon Nov 01, 2010 8:04 pm UTC

you are quite correct, no weirdness is allowed
i believe that the two things you mentioned are the two different one-crossing configurations (tails inside the loop, and tails outside)
there is no such thing as BBAA its the same as AABB the first crossing you come to always gets labeled A
reflection, rotation, skewing, etc... don't change anything
the order that the points are visited in is what determines the family it's in but it's not enough to determine the particular configuration.
for instance family AABB contains the one that has just two identical loops in a row, but that's different from the one that has two loops pointing one up one down,
and then there are two more where the second goes all around the first loop and the start. all four of those are AABB.
i never thought of it in terms of graphs, but now that you mention it it seems to fit, i'll see what that turns up
I'm pretty sure that i found all of the two crossings, my "proof" (which isn't rigorous and it would make me happy if someone could poke it full of holes) is that
1. they must be in one of the three families
2. and that all the ones in one family are linked by flipping loops upside down, or turning loops inside out so that they encompass the whole rest of the figure (without undoing any crossings)

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Re: ways a line can cross itself (topology i believe?)

Postby imatrendytotebag » Tue Nov 02, 2010 3:49 am UTC

If I'm right, a formal way of stating this would be: Two line drawings are the same if you can deform one into the other without changing the number of intersection points or making an intersection point coincide with the endpoint of the line.

Even more formally: [imath]f:I \rightarrow R^2 \sim g:I \rightarrow R^2[/imath] if there exists a homotopy [imath]f_t:I \rightarrow R^2[/imath] taking f to g such that the quotient spaces [imath]I/\sim_t[/imath] are all homeomorphic (where [imath]a \sim_t b[/imath] means [imath]f_t(a) = f_t(b)[/imath]).

Then we just need to say what it means for a path to be admissible in the first place: All intersections are really intersections (not tangencies), don't occur at the endpoints, and each intersection occurs at a different place. (Note that, even after these specifications, we do not need to modify the technical definition in the second paragraph)
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Re: ways a line can cross itself (topology i believe?)

Postby cba » Tue Nov 02, 2010 6:37 pm UTC

allowing some of the "weirdness" could make for even more interesting graphs...
like allowing two sections of the lines to just touch would give you more graphs in the ABBA set... and you could even get the graph to have only 2 different intersection of 'touching' points, and get ABAAB, or one point of intersection and get AAA...
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