2017
DOI: 10.1142/s0218216517410012
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Khovanov homology, Lee homology and a Rasmussen invariant for virtual knots

Abstract: The paper contains an essentially self-contained treatment of Khovanov homology, Khovanov-Lee homology as well as the Rasmussen invariant for virtual knots and virtual knot cobordisms which directly applies as well to classical knot and classical knot cobordisms. We give an alternate formulation for the Manturov definition [34] of Khovanov homology [25] [26] for virtual knots and links with arbitrary coefficients. This approach uses cut loci on the knot diagram to induce a conjugation operator in the Frobenius… Show more

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Cited by 44 publications
(84 citation statements)
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“…This theorem is a generalization of a corresponding result for classical knots due to Rasmussen [21]. Theorem 3.5 (On Four-Ball Genus for Positive Virtual Knots [2]). Let K be a positive virtual knot (i.e.…”
Section: Spanning Surfaces For Knots and Virtual Knots And The Four-bmentioning
confidence: 71%
See 3 more Smart Citations
“…This theorem is a generalization of a corresponding result for classical knots due to Rasmussen [21]. Theorem 3.5 (On Four-Ball Genus for Positive Virtual Knots [2]). Let K be a positive virtual knot (i.e.…”
Section: Spanning Surfaces For Knots and Virtual Knots And The Four-bmentioning
confidence: 71%
“…Definition 3.2. The four-ball genus g 4 (K) of a virtual knot or link K is the least genus among all virtual surfaces obtained by virtual cobordism that bound K. As we shall see below, there is a simple upper bound on the four-ball genus for any virtual knot or link and a definite result for the four-ball genus of positive virtual knots [2]. Note that in this definition of four-ball genus we have not made reference to an embedding of the surface in the four-ball D 4 .…”
Section: Virtual Knot Cobordism and Concordancementioning
confidence: 98%
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“…The surface S can be interpreted as a topological realization in the 4-manifold M × I of this combinatorial object. Virtual surfaces in the 4-ball were used in the extension of the Rasmussen invariant to virtual knots by Dye-Kaestner-Kauffman [8].…”
Section: Computing With Virtual Seifert Surfacesmentioning
confidence: 99%