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.


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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