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Minimal surface representations of virtual knots and links. (English) Zbl 1083.57007
The authors study virtual links considered as links in a surface cross an interval \(\Sigma \times I\) (where the genus of \(\Sigma\) is greater than or equal to the virtual genus of the virtual link), using the surface bracket polynomial. They obtain results about virtual links which differ from classical links by only one virtual crossing or one virtualization (i.e., replacing a classical crossing with the opposite crossing sandwiched between two virtual crossings). In particular, some sufficient conditions are found for such links to be non-classical. A Kishino knot and some related virtual knots are analyzed using the surface bracket polynomial, and the paper concludes with a non-classicality result for a class of virtual links which differ from classical links by two virtualizations.

MSC:
57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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