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Signed ordered knotlike quandle presentations. (English) Zbl 1085.57006

The author defines generalized quandle presentations, so called signed ordered knotlike quandle presentations (SOKQ presentations) which are defined in such a way that they exactly represent a virtual knot diagram. The information in which order an arc overcrosses several other arcs is encoded in those presentations. The definition of a SOKQ presentation is rather straight forward. The author then studies the effect of Reidemeister moves of a virtual knot diagram on a SOKQ presentation and defines formal Reidemeister moves. The main result is: Two SOKQ presentations present isotopic virtual knots if and only if they are related by a finite sequence of formal Reidemeister moves. This result is also elementary. At last he transmits his result in an obvious way to welded isotopy.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
20F05 Generators, relations, and presentations of groups
05C10 Planar graphs; geometric and topological aspects of graph theory
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