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Abstract link diagrams and virtual knots. (English) Zbl 0997.57018

Summary: The notion of an abstract link diagram is re-introduced with a relationship with Kauffman’s virtual knot theory. It is proved that there is a bijection from the equivalence classes of virtual link diagrams to those of abstract link diagrams. Using abstract link diagrams, we have a geometric interpretation of the group and the quandle of a virtual knot. A generalization to higher dimensional cases is introduced, and the state-sum invariants are treated.

MSC:

57M25 Knots and links in the \(3\)-sphere (MSC2010)
57Q45 Knots and links in high dimensions (PL-topology) (MSC2010)
57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
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References:

[1] DOI: 10.1006/aima.1997.1618 · Zbl 0870.57032 · doi:10.1006/aima.1997.1618
[2] DOI: 10.1142/S0218216593000167 · Zbl 0808.57020 · doi:10.1142/S0218216593000167
[3] DOI: 10.1007/BF00872903 · Zbl 0853.55021 · doi:10.1007/BF00872903
[4] DOI: 10.1016/0022-4049(82)90077-9 · Zbl 0474.57003 · doi:10.1016/0022-4049(82)90077-9
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[6] DOI: 10.1016/0040-9383(87)90009-7 · Zbl 0622.57004 · doi:10.1016/0040-9383(87)90009-7
[7] DOI: 10.1016/0040-9383(87)90058-9 · Zbl 0628.57004 · doi:10.1016/0040-9383(87)90058-9
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