Ordering the Reidemeister moves of a classical knot

Abstract

We show that any two diagrams of the same knot or link are connected by a sequence of Reidemeister moves which are sorted by type. 57M25; 57M27It is one of the founding theorems of knot theory that any two diagrams of a given link may be changed from one into the other by a sequence of Reidemeister moves. One of the reasons why this result is so crucial to the subject is that it allows one to define a link invariant as an invariant of a diagram which is unchanged under Reidemeister moves. In this paper we shall address the question of whether, given any two diagrams of a knot or link, there exists a sequence of Reidemeister moves between them which is sorted by type. We answer this question in the affirmative with the following theorem:Theorem 1 Given two diagrams D 1 and D 2 for a link L, D 1 may be turned into D 2 by a sequence of " 1 moves, followed by a sequence of " 2 moves, followed by a sequence of 3 moves, followed by sequence of # 2 moves. Furthermore, if D 1 and D 2 are diagrams of a link where the winding number and framing of each component is the same in each diagram, then D 1 may be turned into D 2 by a sequence of " 2 moves, followed by a sequence of 3 moves, followed by a sequence of # 2 moves.

Minimal generating sets of Reidemeister moves

“…Another set which is not generating Remark 1.3. For non-oriented links a sequence of Reidemeister moves can be arranged in such a form that first a number of Ω1 moves are performed, then Ω2 moves are performed, after this Ω3 moves are performed, and finally Ω2 moves have to be performed again, see [2]. It would be interesting to find such a theorem for oriented case.…”

“…[3]) is rarely included in the list of generators, contrary to a more common move 3b which is the standard choice motivated by braid theory. 2 The reason is that, unexpectedly, these moves have different properties, as we discuss in detail in Section 3. Indeed, Theorem 1.2 below implies that any generating set of Reidemeister moves which includes 3b has at least five moves.…”

A bound for orderings of Reidemeister moves

“…In 2006, Alexander Coward showed [1] that given any sequence of Reidemeister moves between link diagrams D 1 and D 2 , it is possible to construct a new sequence ordered in the following way: first Ω ↑ 1 moves, then Ω ↑ 2 moves, then Ω 3 moves, finally Ω ↓ 2 moves. We present, via the following theorem, an upper bound on the number of moves required for an ordered sequence in terms of the number of moves present in any sequence of Reidemeister moves.…”

“…One cannot overstate the importance of this theorem to knot theory. Mathematicians like Alexander Coward [1,2], Marc Lackenby [2], Bruce Trace [4], Joel Hass and Jeffery Lagarias [3] have all explored properties of sequences of Reidemeister moves.…”

Field theories, stable homotopy theory and Khovanov homology

“…Note in particular that the argument there did not depend on the order of the Reidemeister moves chosen. By Coward's theorem [5], we may choose the moves in such a way that all the R1 moves come first, followed by R2-moves, R3-moves and reversed R2moves. Now since, when considering spin, we interpret the R1-moves as R1A-moves, there must be an even number of such moves on each connected component of the link in order for the spin structures on the links corresponding to D and D to be the same.…”