Minimal generating sets of Reidemeister moves

Details Minimal generating sets of Reidemeister moves Minimal generating sets of Reidemeister moves

2009-08-21

arxiv.org/abs/0908.3127v3

Quantum Topology, vol 1 (2010)

Mathematics

Geometric Topology

14 pages, many figures; v2: significantly improved results; v3: minor changes, publication information

Scientific paper

10.4171/QT

It is well known that any two diagrams representing the same oriented link are related by a finite sequence of Reidemeister moves O1, O2 and O3. Depending on orientations of fragments involved in the moves, one may distinguish 4 different versions of each of the O1 and O2 moves, and 8 versions of the O3 move. We introduce a minimal generating set of four oriented Reidemeister moves, which includes two O1 moves, one O2 move, and one O3 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 O3 moves is discussed.

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