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Quantum Topology

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Volume 1, Issue 4, 2010, pp. 399–411
DOI: 10.4171/QT/10

Published online: 2010-10-17

Minimal generating sets of Reidemeister moves

Michael Polyak[1]

(1) Technion, Haifa, Israel

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves Ω, Ω2 and Ω3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the Ω1 and Ω2 moves, and 8 versions of the Ω3 move. We introduce a minimal generating set of 4 oriented Reidemeister moves, which includes two Ω1 moves, one Ω2 move, and one Ω3 move. We then study which other sets of up to 5 oriented moves generate all moves, and show that only few of them do. Some commonly considered sets are shown not to be generating. An unexpected non-equivalence of different Ω3 moves is discussed.

Keywords: Reidemeister moves, knot and link diagrams

Polyak Michael: Minimal generating sets of Reidemeister moves. Quantum Topol. 1 (2010), 399-411. doi: 10.4171/QT/10