4_1 is not colorable

To see that the 41 knot is not colorable, we begin with a projection of the knot with the strands labelled 1, 2, 3, and 4.


Consider the crossing where strands 1, 2, and 3 meet. If all three crossings strands have the same color, then strand 4 must also have that same color. This does not lead to a coloring of the knot.

So assume that all three strands have different colors; say 1 is red, 2 is blue, and 3 is green. Then we must color strand 4. However, there is no coloring of strand 4 that would lead to an acceptable coloring of the knot. Thus the knot is not colorable.

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