I am reading this artical to study Knot Theory, on exercise two, it asks Show that there are no non-trival, two crossing knot. From the proof, I don't understand what branches that do not represent isometries. And, how does the author know those possible combination can not form a knot?
Show that there are no non-trival, two crossing knot
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1 Answers
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By "branches that do not represent isometries" he means that basically he is just listing all possible combination of the numbers that form a knot. Then, he eliminate the major part of the list given those "isometries", as rotations, reflection, etc; that is, he compares all pairs of branches he numbered, and look if some of them are the same, to eliminate it.
Finally, he finds out that there are only $7$ possible different knots of two crossings, which are drawn there, and he notices they are all, in fact, the trivial knot.