Kamada, Naoko; Kamada, Seiichi Abstract link diagrams and virtual knots. (English) Zbl 0997.57018 J. Knot Theory Ramifications 9, No. 1, 93-106 (2000). 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. Cited in 2 ReviewsCited in 144 Documents 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) Keywords:Gauss data; quandle; state-sum invariant PDFBibTeX XMLCite \textit{N. Kamada} and \textit{S. Kamada}, J. Knot Theory Ramifications 9, No. 1, 93--106 (2000; Zbl 0997.57018) Full Text: DOI 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 [5] Kamada N., World Scientific Publishing Co. pp 377– (1997) [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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.