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Consider the projection of a knot on to the plane. Consider following the knot, starting from a crossing, until we get back to that crossing (on the opposite strand). Why must this cycle have odd length?

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definition of length? Is it the number of edges in a graph induced by the crossings? –  alancalvitti Jan 25 '13 at 5:32
    
Hint: how many times do you go through each crossing? –  Carl Jan 25 '13 at 6:36

1 Answer 1

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There's a theorem that the regions cut off by the knot projection can be two-colored. Actually this theorem is about a closed curve in the plane having each intersection with only two curves crossing (no triple or higher crossings at a point). But it is the same as your projected knot onto the plane. Imagine the regions have been colored black or white.

Now as you go around the curve starting at a given crossing, the colors on your left alternate black, white, black, white, etc. When you first rearrive back to your starting point, you'll have just crossed the opposite color (on your left) from the starting color. This makes for the odd length phenomenon you refer to.

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