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 3sphere. 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 3manifold theory, often the adjective "tame" is omitted. Smooth knots, for example, are always tame. Wild knots can be found in some Celtic designs.
YouTube Encyclopedic

1/3Views:1 9021 9911 078

What's a knot? Geometry Terms and Definitions

Aaron Lauda: Math with a Twist

Framed knot
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 1dimensional 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...