Comments on: How to untie a trefoil knot?
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot/
Comments on Ask MetaFilter post How to untie a trefoil knot?Tue, 30 Aug 2011 22:58:28 -0800Tue, 30 Aug 2011 23:03:07 -0800en-ushttp://blogs.law.harvard.edu/tech/rss60Question: How to untie a trefoil knot?
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot
Mathematics/Topology: how to untie a trefoil knot in four dimensions? <br /><br /> To my general understanding all 3d knots are not really knots four dimensions; practically I thought this simply involved looking at the process of tying the knot over time, and observing that the process is reversible -- that there is thus no point in 4d at which the simple string "unknot" is fundamentally changed to a knot.<br>
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However when I looked at the <a href="http://en.wikipedia.org/wiki/Trefoil_knot">trefoil knot</a> this understanding fell to pieces. Obviously there is no process to build a trefoil that starts from an untied string without changing its topology fundamentally (by closing the ends together).<br>
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Hence my question: probably the trefoil knot is still no knot in 4d, but how so? Can you point me to any resource which helps visualizing the topology of a trefoil in 4d?post:ask.metafilter.com,2011:site.194807Tue, 30 Aug 2011 22:58:28 -0800knztopologymathematicsknotvisualizationtrefoilresolvedBy: Blazecock Pileon
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2804271
<a href="http://www.rdrop.com/~half/Creations/Puzzles/visualizing.4D/index.html">Unknotting a Knot Using Four Dimensions</a>comment:ask.metafilter.com,2011:site.194807-2804271Tue, 30 Aug 2011 23:03:07 -0800Blazecock PileonBy: knz
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2804281
This is really painful to my brain...comment:ask.metafilter.com,2011:site.194807-2804281Tue, 30 Aug 2011 23:16:10 -0800knzBy: JiBB
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2804282
BP linked to some good pictures that help explain how one would do it, but looking at your question there seems to be a more fundamental confusion.<br>
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<blockquote><i>Obviously there is no process to build a trefoil that starts from an untied string without changing its topology fundamentally (by closing the ends together).</i></blockquote><br>
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But the <a>unknot</a> isn't a piece of string with two ends, it's a closed loop, an embedding of a circle into R<sup>3</sup>.comment:ask.metafilter.com,2011:site.194807-2804282Tue, 30 Aug 2011 23:16:12 -0800JiBBBy: JiBB
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2804284
It might help to use a similar visualization to understand an impossible-in-2D task in 3D.<br>
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Imagine that in the 2D plane you have a square inside of a circle, and you're trying to move the square outside of the circle without the two crossing. Using the same visualization as in BP's link of color representing movement into an additional, orthogonal, dimension, we can visualize things moved up as being more green. Move the square up (color it green). Now you can move it outside of the circle without them crossing. Then, move it back into the original plane (color it white again).comment:ask.metafilter.com,2011:site.194807-2804284Tue, 30 Aug 2011 23:24:06 -0800JiBBBy: Blazecock Pileon
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2804289
<em>This is really painful to my brain...</em><br>
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We can't easily visualize objects in four dimensions, because we exist in a three-dimensional reality (at least within the scale that humans exist in) and we evolved a visual or cognitive language for describing and understanding 3D objects and how to manipulate them. <br>
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The best we can manage is a <em>projection</em> that shows shadows of the object in our three dimensions (or in 2D, for the web) as its fourth-dimensional characteristic is manipulated. This is what the coloring in the trefoil figure symbolizes: the author is pulling on the knot in its fourth dimension, where there are no overlaps to bind it.comment:ask.metafilter.com,2011:site.194807-2804289Tue, 30 Aug 2011 23:31:33 -0800Blazecock PileonBy: baf
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2804321
First of all, forget about the idea that time is the fourth dimension. Time is <em>a</em> fourth dimension, sure, and if we were talking about physics we might say "4D" and mean four-dimensional space-time, but if we're talking about topology, "4D" means four spatial dimensions. And that's an area where we can get the best understanding by using analogies from the relationship of 2D to 3D.<br>
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Consider a closed 2D figure lying in a plane. Within that plane, that figure has a definite inside and a definite outside, and if it's a barrier to motion, then an object on the inside would be confined there, like a ball inside a billiards rack. But if that object is allowed to move through three dimensions, it can move from the inside to the outside by leaving the plane entirely, right? Just pick the ball off the table and place it down again outside the rack. While it's completely outside the plane, it can't collide with a figure completely inside it.<br>
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Similarly, a closed surface in 3D space (say, a sphere) has an inside and an outside, and an object on the inside can't move to the outside without breaching the sphere. But if we had access to a fourth dimension, we could move the object perpendicular to the 3D space and then put it back in outside of the sphere. While it's outside of our 3D space, it's effectively intangible to anything inside it.<br>
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Now, having understood that, the easiest way to understand untying a knot through the fourth dimension is to imagine leaving most of the knot in its 3D space, but stretching a piece of it into the fourth dimension, similar to how you'd pluck up one part of a string lying on a table. While that bit of the knot is outside of the 3D space, it is incapable of touching the bits that are inside it. So we can just shift it over to the other side of some strand of the knot and then plunk it back into the 3D space, just like we shifted the billiard ball to the other side of the rack.<br>
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Now, if you really want to break your brain, consider this: Even though 4D space doesn't admit knotted curves, it does create the possibility of knotted <em>surfaces</em>. Visualizing those is left as an exercise for the reader.comment:ask.metafilter.com,2011:site.194807-2804321Wed, 31 Aug 2011 01:05:35 -0800bafBy: kathrynm
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2804398
I think my brain just exploded. But that was one of the coolest illustrations I've ever seen.comment:ask.metafilter.com,2011:site.194807-2804398Wed, 31 Aug 2011 04:51:15 -0800kathrynmBy: knz
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2805046
I like the idea of knotted surfaces; what would a 3d projection of such a beast look like? Did anyone try this already?comment:ask.metafilter.com,2011:site.194807-2805046Wed, 31 Aug 2011 12:25:31 -0800knzBy: fish tick
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2805486
<a href="http://www.knotplot.com/">Knotplot</a> may be your friend.comment:ask.metafilter.com,2011:site.194807-2805486Wed, 31 Aug 2011 18:20:34 -0800fish tickBy: Blazecock Pileon
http://ask.metafilter.com/194807/How-to-untie-a-trefoil-knot#2805796
<em>I like the idea of knotted surfaces; what would a 3d projection of such a beast look like? Did anyone try this already?</em><br>
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Clifford Stoll sells <a href="http://www.kleinbottle.com/">3D projections of Klein bottles</a>.comment:ask.metafilter.com,2011:site.194807-2805796Thu, 01 Sep 2011 00:00:15 -0800Blazecock Pileon