Talk:Markov theorem

Clarify wording

The wording of the “three conditions [that] are necessary and sufficient” doesn't make sense in mathematical English - these don't make sense as a list of criteria, whether jointly or individually.

I am not a German-speaker, and I don't get much clarity out of the given Markov 1936 source, despite how indisputably authoritative it is.

Of the Birman 1974 reference, I found a Google Books listing, but I wasn't able to find within Google's preview the supposed “no. 82” entry, nor any reference to “Markov”. Someone with access to the full text might have more success.

Google Books’ listing for Kauffman 1991 is similarly unhelpful, stopping before the cited page 95.

The next source I found in English was On Markov's Theorem (Joan S. Birman and William W. Menasco, 2002). Birman and Menasco do state the theorem, and the might be enough to work from there. They're more interested in manifolds than I think should be necessary, though, and I have some hope that their other sources will give an interpretation more suitable for an encyclopedia article.

The three additional sources given in Birman and Menasco 2002 for the theorem are • S. Lambropoulou & C. Rourke, Markov’s theorem in 3-manifolds, Topology and its Applications 78, Nos. 1-2 (1997), 95-122.; • H. Morton, Threading knot diagrams, Math Proc. Camb. Phil. Soc.99, 247-260.; and • P. Traczyk, A new proof of Markov’s braid theorem, Knot Theory, Banach Center Publications 42, Polish Acad. of Sciences (1998), 409-419. In addition they suggest their then-upcoming 2006 paper Stabilization in the braid groups I: Markov’s Theorem Without Stabilization.

That's as far as I'm prepared to trace this theorem at the moment. Best wishes to whomever next follows. — Octavo (talk) 16:18, 5 March 2022 (UTC)

Birman's book contains a statement of the theorem and (i assume) a full proof. The typography and the illustrations are very poorly rendered however. Kaufmann's book gives a much more readable treatment of the moves but refers to Birman for a proof.

I tried to give a more precise statement based on Kaufmann's. It is probably slightly unreadable at the moment, the article should probably spend some time repeating basic stuff on braids and braid groups. jraimbau (talk) 08:45, 7 March 2022 (UTC)

Some additional remarks: 1.it would be nice to have pictures for the Markov moves; 2.there was an additional "elementary move" in the previous version (which appears in Kauffman's statement of the theorem), namely "equivalence of braids"; since in the version i rewrote braids are represented by group elements there is no need for it, however the article might read better if there were two versions of the theorem, one "algebraic" (for group elements), and one "geometric" (for braid diagrams, which necessitates the added moves for equivalence of braid diagrams). jraimbau (talk) 08:57, 7 March 2022 (UTC)

Can we clarify "conjugate" here

Can we clarify "conjugate" here, since it's not explained in braid theory ? - Rod57 (talk) 13:14, 14 July 2022 (UTC)