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Ordering the Reidemeister moves of a classical knot. (English) Zbl 1095.57005
This paper refines Reidemeister’s classical theorem to show that if two diagrams represent the same link then there is a sequence of Reidemeister moves connecting them in which increasing moves of type I and then type II are done first, followed by type III moves and finally decreasing moves of type II and then type I. Type I moves can be omitted if corresponding components all have the same writhe and winding number. The proof is combinatorially based, using some preliminary lemmas to permit the inductive modification of an original minimal sequence of unsorted moves between the diagrams.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:
[1] J Hass, J C Lagarias, The number of Reidemeister moves needed for unknotting, J. Amer. Math. Soc. 14 (2001) 399 · Zbl 0964.57005
[2] K Reidemeister, Knotten und Gruppen, Abh. Math. Sem. Univ. Hamburg 5 (1927) 7
[3] B Trace, On the Reidemeister moves of a classical knot, Proc. Amer. Math. Soc. 89 (1983) 722 · Zbl 0554.57003
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