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.

fig8color1.jpg

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.

fig8color2.jpg
Unless otherwise stated, the content of this page is licensed under Creative Commons Attribution-ShareAlike 3.0 License