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Topic: Topological Rings (Read 14662 times) |
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pjay
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Re: Topological Rings
« Reply #25 on: Dec 4th, 2003, 9:05am » |
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The normal rule in knot theory is that these things have to be
ambient isotopies which i'm pretty sure they are not (i'll spare the definition since it requires a few page of explanation, but you can look it up in any knot theory or graduate topology book).
the basic idea though is that at some point you have to pass through a shape which has a point connected to 4 "line segments". At this point, there is no bi-continuous map between this shape and either of the shapes drawn. Maybe the wording of the problem should be changed o specify exactly what kind of moves can be made...
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Icarus
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Re: Topological Rings
« Reply #26 on: Dec 4th, 2003, 4:44pm » |
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I admit that I never studied links other than Martin Gardner - level recreational mathematics, but I was under the impression that the sort of maneuvers described here was exactly what the theory allowed. I've seen a large number of these recreational level problems, and the solutions always worked like this.
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rmsgrey
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Re: Topological Rings
« Reply #27 on: Dec 5th, 2003, 5:14am » |
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I believe (with no research into the matter) that the knot theory assumes line-segments with branching at points. The problem we're dealing with works with a 2-dimensional surface embedded in 3-dimensional space. Among other things, there are infinitely many lines meeting at every point on the surface, and picking any representative line segments or line loops, the transformations we've described transform them continuously without introducing or removing any intersections. If you draw a set of lines "0---0" on the shape in the obvious way, then the two circles are linked both before and after the transformation, and at no point does the connecting line disappear. After the transformation, depending on the details, one of the circles goes round most of one loop, along the connecting rope and then crosses the other loop somewhere before going back along the connecting rope again.
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pjay
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Re: Topological Rings
« Reply #28 on: Dec 25th, 2003, 11:59am » |
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I was under the impression that we were to think of these objects as one dimensional embedded in 3-space (by one-dimensional, I mean a one dimensional manifold for example, any curve). In normal knot theory, knots are always homeomophic to a circle (meaning the circle can be wiggled around and looped through itself to form "knots"). when we are dealing with more than one of these, they are calle braids. But there is no general study of one dimensional objects that branch out- in other words a point where more than 2 segments meet up. But if you are trying to extend the allowable moves from knot theory, a good requirement would be the following:
at any given fixed time t, there is a homeomorphism from the shape at time t to the shape at time 0. This cannot be done in this puzzle since inevitably we must pass through a point at which time the shape will include one point which has 4 line segments coming out of it...
Anyways, I still think this is a good exercise in visualization, i just think we should be more precise about what types of moves are allowed. I for one cannot think of a good definition of allowable moves, so maybe the puzzle as is, is the best it can be...
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Icarus
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Re: Topological Rings
« Reply #29 on: Dec 26th, 2003, 6:52am » |
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on Dec 25th, 2003, 11:59am, pjay wrote:| I was under the impression that we were to think of these objects as one dimensional embedded in 3-space (by one-dimensional, I mean a one dimensional manifold for example, any curve). |
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No - it is common in these problems to consider the object as a two-dimensional surface, just as the picture indicates. (One could also envision it as a solid in this case, but some other problems explicitly require a surface.)
Quote:| In normal knot theory, knots are always homeomophic to a circle (meaning the circle can be wiggled around and looped through itself to form "knots"). |
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Knot theory starts there, but is more broadly applicable. Perhaps you should think of this problem as an application of knot theory (or link theory, more appropriately) rather than as being in that theory.
Quote:| But there is no general study of one dimensional objects that branch out- in other words a point where more than 2 segments meet up. |
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Algebraic geometry includes such objects. But in this case, the objects in question are two-dimensional and a well-known part of surface theory.
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"Pi goes on and on and on ...
And e is just as cursed.
I wonder: Which is larger
When their digits are reversed? " - Anonymous
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tangent
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Re: Topological Rings
« Reply #30 on: Aug 1st, 2006, 10:06pm » |
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a slightly nicer visual representation, just remember this is really in 3D. I threw this together in 15 minutes.
-mike
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