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Classification of genus 1 virtual knots having at most five classical crossings. (English) Zbl 1302.57008

Summary: The goal of this paper is to tabulate all genus one prime virtual knots having diagrams with \(\leq 5\) classical crossings. First, we construct all nonlocal prime knots in the thickened torus \(T\times I\) which have diagrams with \(\leq 5\) crossings and admit no destabilizations. Then we use a generalized version of the Kauffman polynomial to prove that all those knots are different. Finally, we convert the knot diagrams in \(T\) thus obtained into virtual knot diagrams in the plane.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

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