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From Wikipedia, the free encyclopedia

 A wild knot.
A wild knot.

In the mathematical theory of knots, a knot is tame if it can be "thickened up", that is, if there exists an extension to an embedding of the solid torus S 1 × D 2 into the 3-sphere. A knot is tame if and only if it can be represented as a finite closed polygonal chain. Knots that are not tame are called wild and can have pathological behavior. In knot theory and 3-manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame. Wild knots can be found in some Celtic designs.

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  • What's a knot? Geometry Terms and Definitions
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Transcription

Suppose you have a piece of ideal string -- that is, string with a length, but no depth to it at all. What I'm holding isn't perfect string. It's rather thick, so it's not a true 1-dimensional string. But if you use your imagination, which everybody has, then you can think of this as a string. If you take the string and wind and wound and twist and turn and tangle things up, and then connect the two ends together, you get a knot. A mathematical knot should not have any dangly ends. The two ends should fuse together. The simplest knot you can make is called a trefoil. Just take a piece of string... tie a loop... and connect the tips. This is what it looks like when you spread it out on a table. ::look right:: Hmm! There are books and books and books written about knots -- mathematicians call it "knot theory." Sometimes you want to know how much can you untangle a knot. How much can you simplify it? Here's another puzzle: when you are given two knots, can you tell if they're identical to each other? Maybe they should teach knot theory to sailors...

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References

This page was last edited on 4 September 2015, at 22:11.
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