In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces modulo an equivalence relation called stabilization/destabilization. Here is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).
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Overview
editIn the theory of classical knots, knots can be considered equivalence classes of knot diagrams under the Reidemeister moves. Likewise a virtual knot can be considered an equivalence of virtual knot diagrams that are equivalent under generalized Reidemeister moves. Virtual knots allow for the existence of, for example, knots whose Gauss codes which could not exist in 3-dimensional Euclidean space. A virtual knot diagram is a 4-valent planar graph, but each vertex is now allowed to be a classical crossing or a new type called virtual. The generalized moves show how to manipulate such diagrams to obtain an equivalent diagram; one move called the semi-virtual move involves both classical and virtual crossings, but all the other moves involve only one variety of crossing.
A classical knot can also be considered an equivalence class of Gauss diagrams under certain moves coming from the Reidemeister moves. Not all Gauss diagrams are realizable as knot diagrams, but by considering all equivalence classes of Gauss diagrams we obtain virtual knots.
A classical knot can be considered an ambient isotopy class of embeddings of the circle into a thickened 2-sphere. This can be generalized by considering such classes of embeddings into thickened higher-genus surfaces. This is not quite what we want since adding a handle to a (thick) surface will create a higher-genus embedding of the original knot. The adding of a handle is called stabilization and the reverse process destabilization. Thus a virtual knot can be considered an ambient isotopy class of embeddings of the circle into thickened surfaces with the equivalence given by (de)stabilization.
Some basic theorems relating classical and virtual knots:
- If two classical knots are equivalent as virtual knots, they are equivalent as classical knots.
- There is an algorithm to determine if a virtual knot is classical.
- There is an algorithm to determine if two virtual knots are equivalent.
There is a relation among the following.
- Virtual equivalence of virtual 1-knot diagrams, which is a set of virtual 1-knots.
- Welded equivalence of virtual 1-knot diagrams
- Rotational welded equivalence of virtual 1-knot diagrams
- Fiberwise equivalence of virtual 1-knot diagrams
See also
editReferences
edit- Boden, Hans; Nagel, Matthias (2017). "Concordance group of virtual knots". Proceedings of the American Mathematical Society. 145 (12): 5451–5461. arXiv:1606.06404. doi:10.1090/proc/13667. S2CID 119139769.
- Carter, J. Scott; Kamada, Seiichi; Saito, Masahico (2002). "Stable equivalence of knots on surfaces and virtual knot cobordisms. Knots 2000 Korea, Vol. 1 (Yongpyong)". J. Knot Theory Ramifications. 11 (3): 311–322.
- Carter, J. Scott; Silver, Daniel; Williams, Susan (2014). "Invariants of links in thickened surfaces". Algebraic & Geometric Topology. 14 (3): 1377–1394. arXiv:1304.4655. doi:10.2140/agt.2014.14.1377. S2CID 53137201.
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- Kamada, Naoko; Kamda, Seiichi (2000). "Abstract link diagrams and virtual knots". Journal of Knot Theory and Its Ramifications. 9 (1): 93–106. doi:10.1142/S0218216500000049.
- Kauffman, Louis H. (1999). "Virtual knot theory" (PDF). European Journal of Combinatorics. 20 (7): 663–690. doi:10.1006/eujc.1999.0314. ISSN 0195-6698. MR 1721925. S2CID 5993431.
- Kauffman, Louis H.; Manturov, Vassily Olegovich (2005). "Virtual Knots and Links". arXiv:math.GT/0502014.
- Kuperberg, Greg (2003). "What is a virtual link?". Algebraic & Geometric Topology. 3: 587–591. arXiv:math/0208039. doi:10.2140/agt.2003.3.587. S2CID 16803280.
- Manturov, Vassily (2004). Knot Theory. CRC Press. ISBN 978-0-415-31001-7.
- Manturov, Vassily Olegovich (2004). "Virtual knots and infinite dimensional Lie algebras". Acta Applicandae Mathematicae. 83 (3): 221–233. doi:10.1023/B:ACAP.0000038944.29820.5e. S2CID 124019548.
- Turaev, Vladimir (2008). "Cobordism of knots on surfaces". Journal of Topology. 1 (2): 285–305. arXiv:math/0703055. doi:10.1112/jtopol/jtn002. S2CID 17888102.