Last Time | Next Time |
That's one way! There is another, which we'll see today.
But we can easily figure out which is which, because the knot is a single continuous piece of material, whereas the link is two separate pieces of material.
It's all about overs and unders!
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Type I | Type II | Type III |
The third images on that page stems from this picture from a recent Science issue:
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In order to consider the picture on the left a knot, we have to know what its ends are doing. In the figure at right, I assumed that they are just connecting to each other in the simplest way.
The succession of steps then go on to show that the knot is actually an unknot! That's good news for the hyperbusy people!
Here's another picture of an unknot, which could trick you -- but knowing the R1 move saves you:
We'll just try a few with a string, to see what we can learn.
Reidemeister Move I is tricolorable. | Reidemeister Move II is tricolorable. | Reidemeister Move III is tricolorable. |
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The unknot is not tricolorable ("At least two colors must be used") | The trefoil knot is tricolorable: | The figure-eight knot is not tricolorable (it requires four colors): "The figure-eight knot is not tricolorable. In the diagram shown, it has four strands with each pair of strands meeting at some crossing. If three of the strands had the same color, then all strands would be forced to be the same color. Otherwise each of these four strands must have a distinct color. Since tricolorability is a knot invariant, none of its other diagrams can be tricolored either." (source) |
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So, for example: if you've got your picture of a knot down to three crossings, and it's not tricolorable, then it's the unknot.
Which of the 6- and 7-knots are tricolorable? (There are three.)