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Izv. RAN. Ser. Mat., 2007, Volume 71, Issue 5, Pages 111–148 (Mi izv1133)  

This article is cited in 14 scientific papers (total in 14 papers)

Khovanov homology for virtual knots with arbitrary coefficients

V. O. Manturov

Moscow State Region University

Abstract: The Khovanov homology theory over an arbitrary coefficient ring is extended to the case of virtual knots. We introduce a complex which is well-defined in the virtual case and is homotopy equivalent to the original Khovanov complex in the classical case. Unlike Khovanov's original construction, our definition of the complex does not use any additional prescription of signs to the edges of a cube. Moreover, our method enables us to construct a Khovanov homology theory for ‘twisted virtual knots’ in the sense of Bourgoin and Viro (including knots in three-dimensional projective space). We generalize a number of results of Khovanov homology theory (the Wehrli complex, minimality problems, Frobenius extensions) to virtual knots with non-orientable atoms.


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English version:
Izvestiya: Mathematics, 2007, 71:5, 967–999

Bibliographic databases:

UDC: 515
MSC: 57M27, 55N99
Received: 12.07.2006

Citation: V. O. Manturov, “Khovanov homology for virtual knots with arbitrary coefficients”, Izv. RAN. Ser. Mat., 71:5 (2007), 111–148; Izv. Math., 71:5 (2007), 967–999

Citation in format AMSBIB
\by V.~O.~Manturov
\paper Khovanov homology for virtual knots with arbitrary coefficients
\jour Izv. RAN. Ser. Mat.
\yr 2007
\vol 71
\issue 5
\pages 111--148
\jour Izv. Math.
\yr 2007
\vol 71
\issue 5
\pages 967--999

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    This publication is cited in the following articles:
    1. Ilyutko D. P., Manturov V. O., “Introduction to graph-link theory”, J. Knot Theory Ramifications, 18:6 (2009), 791–823  crossref  mathscinet  zmath  isi  elib  scopus
    2. M. V. Zenkina, V. O. Manturov, “An invariant of links in a thickened torus”, J. Math. Sci. (N. Y.), 175:5 (2011), 501–508  mathnet  crossref
    3. Blanchet Ch., “An oriented model for Khovanov homology”, J. Knot Theory Ramifications, 19:2 (2010), 291–312  crossref  mathscinet  zmath  isi  elib  scopus
    4. Ito N., “Chain homotopy maps for Khovanov homology”, J. Knot Theory Ramifications, 20:1 (2011), 127–139  crossref  mathscinet  zmath  isi  elib  scopus
    5. D. P. Ilyutko, V. O. Manturov, I. M. Nikonov, “Parity in knot theory and graph-links”, Journal of Mathematical Sciences, 193:6 (2013), 809–965  mathnet  crossref  mathscinet
    6. Grishanov S.A., Vassiliev V.A., “Invariants of links in 3-manifolds and splitting problem of textile structures”, J. Knot Theory Ramifications, 20:3 (2011), 345–370  crossref  mathscinet  zmath  isi  elib  scopus
    7. V. O. Manturov, “Parity, free knots, groups, and invariants of finite type”, Trans. Moscow Math. Soc., 72 (2011), 157–169  mathnet  crossref  zmath  elib
    8. Ilyutko D.P., “An equivalence between the set of graph-knots and the set of homotopy classes of looped graphs”, J. Knot Theory Ramifications, 21:1 (2012), 1250001, 30 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    9. Chrisman M.W., Manturov V.O., “Parity and exotic combinatorial formulae for finite-type invariants of virtual knots”, J. Knot Theory Ramifications, 21:13 (2012), 1240001, 27 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    10. Kaestner A.M.. Kauffman L.H., “Parity, Skein polynomials and categorification”, J. Knot Theory Ramifications, 21:13 (2012), 1240011, 56 pp.  crossref  mathscinet  zmath  isi  elib  scopus
    11. BOŠTJAN GABROVŠEK, “THE CATEGORIFICATION OF THE KAUFFMAN BRACKET SKEIN MODULE OF”, Bull. Aust. Math. Soc, 2013, 1  crossref  zmath  isi  scopus
    12. Tagami K., “A Khovanov Type Invariant Derived From an Unoriented HQFT for Links in Thickened Surfaces”, Int. J. Math., 24:10 (2013), 1350078  crossref  mathscinet  zmath  isi  scopus
    13. Manturov V.O., “Parity and Projection From Virtual Knots to Classical Knots”, J. Knot Theory Ramifications, 22:9 (2013), 1350044  crossref  mathscinet  zmath  isi  elib  scopus
    14. L.H.irsch Kauffman, V.O.legovich Manturov, “Graphical constructions for the sl(3), C2and G2invariants for virtual knots, virtual braids and free knots”, J. Knot Theory Ramifications, 2015, 1550031  crossref  mathscinet  scopus
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