# Proof of the Reidemeister theorem

While preparing for my introduction to topology course, I've realized that I don't know where to find a detailed proof of the Reidemeister theorem (two link diagrams give isotopic links, iff they can be connected by a sequence of Reidemeister moves). The students in my class are not really very advanced and I'm not sure all of them would be able to reconstruct all the details of the proof given in Burde and Zieschang's Knot theory or in other sources I know of. And, being a lazy man, I would like to avoid having to type a detailed proof with pictures when I can refer the students to a book. So does anybody know of a source which would have a proof with all details spelled out?

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Kunio Murasugi's Knot Theory and its Applications contains what you are looking for, I believe. I taught the proof of Reidemeister's theorem as part of a short course on knot theory, from that book. Definitely suitable for undergraduates.

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Thanks, Daniel! Will try to borrow this tomorrow from the library. –  algori Feb 14 '10 at 16:57

Reidemeister's proof involves a single move: replacing 2 (or 1) edges of a triangle with the other edge (edges). It is in the English translation of his book.

I don't know if you can ferret out the details of our first movie move theorem in JKTR, or even the CRS version, but when we did those, I felt that we had addressed the general position issues that Dan mentions.

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Scott, thanks! Your second paragraph is a bit cryptic. For the record: I know how to prove Reidemeister's theorem, I'm looking for a detailed proof accessible to undergraduate students with a limited experience of mathematics, let alone topology. –  algori Feb 14 '10 at 5:30
I did not mean to be cryptic at all, but it was late when I wrote it. Masahico and I proved a "movie move theorem" that was published in Journ. of Knot Thy. and its Ram. There were problems with the interpretations that people could make of it. So we proved a more precise result with Joachim Rieger (CRS). A lot of the transversality arguments are worked out in both places. In particular, when your knot diagram has a height function, you have to deal with critical interchange, zig-zag, and $\psi$ moves. I just don't remember if we gave a proof of the R thm, or not. Probably not. –  Scott Carter Feb 14 '10 at 13:48