What is a virtual link

By Greg Kuperberg. 2002

261 citations recorded by the source corpus.

Several authors have recently studied virtual knots and links because they admit invariants arising from R-matrices. We prove that every virtual link is uniquely represented by a link L SI in a thickened, compact, oriented surface S such that the link complement (SI)nL has no essential vertical cylinder. AMS Classication 57M25; 57M27 57M15

Subfields: Applied Mathematics, Geometry and Topology, Mechanical Engineering

View Paper: https://www.semanticscholar.org/paper/aed20f732014fbd9d6177a20207519fa9091a57b

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In Applied Mathematics

In Geometry and Topology

In Mechanical Engineering

Citations in the corpus

Virtual Crossing Number and the Arrow Polynomial (2008, background): “can also apply the generalized arrow polynomial to representations of virtual links. For a representation of a virtual link, 22 there is a unique surface with minimum genus in which these links embed [13]. A virtual link with minimal genus g is a link diagram that corresponds to a representation wit”
Virtual knot theory-unsolved problems (2005, unknown): “Might one of these be equivalent via handle stabilization to a classical knot? In [71] Kuperberg shows the uniqueness of the embedding of minimal genus in the stable class for a given virtual link.”
An Affine Index Polynomial Invariant of Virtual Knots (2012, background): “For more information the reader is referred to [17, 10, 19, 20, 21, 24].”
Virtual crossings, convolutions and a categorification of the SO(2N) Kauffman polynomial (2007, background): “ural to extend the deﬁnition (1.18) of the categoriﬁcation complex of a link to virtual links. Virtual links were invented by L. Kauﬀman [5] and their topological meaning was clariﬁed by G. Kuperberg [10] (for the details see V. Manturov’s book [11]). From the combinatorial point of view, they are e”
A self-linking invariant of virtual knots (2004, unknown): “(See also [10].”
A polynomial invariant of virtual links (2013, background): “In [15] Kuperberg showed that for a virtual knot the embedding type of the minimal genus surface is unique.”
Minimal surface representations of virtual knots and links (2004, methodology): “1 (Kuperberg [15]) A minimal representation of a virtual link is unique up to homeomorphism of the thickened surface.”
Khovanov Homology: Torsion and Thickness (2004, methodology): “vanov homology. If we get a higher genus surface we have to use [APS] theory. This can be utilitised also to construct Khovanov homology of virtual links (via Kuperberg minimal genus embedding theory [Ku]). For example, if the graph Gs+ is a loop with adjacent edge(s) ordered e,−e then the diagram i”
Virtual Braids (2004, background): “For reference to previous work on virtual braids the reader should consult [5, 6, 9, 10, 11, 17, 18, 19, 12, 13, 14, 20, 21, 23, 25, 26, 27, 28].”
Cobordism of knots on surfaces (2007, background): “This area has been especially active in the context of the Kauffman theory of virtual knots; see [9, 10, 11].”
Biquandles and virtual links (2004, background): “Two virtual link diagrams are equivalent if and only if their correspond surface embeddings are stably equivalent, [13,15,17,4] .”
VIRTUAL BRAIDS AND THE L-MOVE (2005, background): “For reference to previous work on virtual links and braids the reader should consult [2, 6, 7, 9, 11, 13, 14, 15, 16, 18, 19, 20, 21, 22, 23, 27, 28, 32, 36, 37].”
Virtual knots and links (2005, methodology): “An important Theorem by Kuperberg [ KUP ] says the following.”
Finite-type invariants of w-knotted objects, I: w-knots and the Alexander polynomial (2014, background): “v-knots may be understood either as knots drawn on surfaces modulo the addition or removal of empty handles [59], or as “Gauss diagrams” (see Remark 3.”
The rack space (2003, methodology): “We finish by classifying virtual links (Kauffman [19], see also Kuperberg [21]) up to cobordism, in terms of the 2-dimensional homology of the rack space.”