Abstract:
This paper is an introduction to the subject of virtual knot theory and presents a discussion of some new specific theorems about virtual knots. The new results are as follows:
Using a 3-dimensional topology approach, we prove that if a connected sum of two virtual knots K1 and K2 is trivial, then so are both K1 and K2. We establish an algorithm for recognizing virtual links that is based on the Haken–Matveev technique.
Citation:
L. H. Kaufman, V. O. Manturov, “Virtual Knots and Links”, Geometric topology, discrete geometry, and set theory, Collected papers, Trudy Mat. Inst. Steklova, 252, Nauka, MAIK «Nauka/Inteperiodika», M., 2006, 114–133; Proc. Steklov Inst. Math., 252 (2006), 104–121
\Bibitem{KauMan06}
\by L.~H.~Kaufman, V.~O.~Manturov
\paper Virtual Knots and Links
\inbook Geometric topology, discrete geometry, and set theory
\bookinfo Collected papers
\serial Trudy Mat. Inst. Steklova
\yr 2006
\vol 252
\pages 114--133
\publ Nauka, MAIK «Nauka/Inteperiodika»
\publaddr M.
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\transl
\jour Proc. Steklov Inst. Math.
\yr 2006
\vol 252
\pages 104--121
\crossref{https://doi.org/10.1134/S0081543806010111}
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Linking options:
https://www.mathnet.ru/eng/tm66
https://www.mathnet.ru/eng/tm/v252/p114
This publication is cited in the following 26 articles:
A. A. Kazakov, “The State-Sum Invariants for Virtual Knot”, Lobachevskii J Math, 44:12 (2023), 5286
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Joanna A. Ellis-Monaghan, Iain Moffatt, SpringerBriefs in Mathematics, Graphs on Surfaces, 2013, 101
Joanna A. Ellis-Monaghan, Iain Moffatt, SpringerBriefs in Mathematics, Graphs on Surfaces, 2013, 23
VASSILY OLEGOVICH MANTUROV, “PARITY AND PROJECTION FROM VIRTUAL KNOTS TO CLASSICAL KNOTS”, J. Knot Theory Ramifications, 22:09 (2013), 1350044
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Korablev F.G., Matveev S.V., “Reduction of knots in thickened surfaces and virtual knots”, Dokl. Math., 83:2 (2011), 262–264
Matveev S.V., “Decomposition of homologically trivial knots in F×I”, Dokl. Math., 82:1 (2010), 511–513
Traldi L., “A bracket polynomial for graphs. II. Links, Euler circuits and marked graphs”, J. Knot Theory Ramifications, 19:4 (2010), 547–586