## Virtual Knots and Links (2005)

Citations: | 2 - 0 self |

### BibTeX

@MISC{Kauffman05virtualknots,

author = {Louis H. Kauffman and Vassily O. Manturov},

title = {Virtual Knots and Links},

year = {2005}

}

### OpenURL

### Abstract

This paper is an introduction to the subject of virtual knot theory, combined with a discussion of some specific new theorems about virtual knots. The new results are as follows: We prove, using a 3-dimensional topology approach 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, using Haken-Matveev technique, for recognizing virtual knots. This paper may be read as both an introduction and as a research paper. For more about Haken-Matveev theory and its application to classical knot theory, see [Ha, Hem, Mat, HL]. 1

### Citations

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Citation Context ...esearch paper. For more about Haken-Matveev theory and its application to classical knot theory, see [Ha, Hem, Mat, HL]. 1 Introduction Virtual knot theory was proposed by Louis Kauffman in 1996, see =-=[KaV]-=-. The combinatorial notion of virtual knot 1 is defined as an equivalence class of 4-valent plane diagrams (4-regular plane graphs with extra structure) where a new type of crossing (called virtual) i... |

153 |
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Citation Context ...k from F). Also, a 3-manifold M is boundary irreducible if for any proper disk D ⊂ M, ∂D bounds a disk on ∂M. Given a 3-manifold with boundary. By a boundary pattern (first proposed by Johannson, see =-=[Joh]-=-) we mean a fixed 1–polyhedron (graph) without isolated points on the boundary of the three manifold (we assume this graph be a subpolyhedron of the selected triangulation). The existence of a boundar... |

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Citation Context ...) classical connected sum K1#K2. This lemma follows directly from the proof of 1. The remaining part of Theorem 2 now follows from the non-triviality of connected sum in the classical case, see, e.g. =-=[CF]-=-. 2.1 Two types of connected sums Having two virtual knots K1 and K2 represented by knots in thickened surfaces, there are two natural possibilities to represent their connected sum as a knot in a thi... |

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Citation Context ... uses a recognition techniques for three-manifolds with boundary pattern (see definition below) connected to virtual links in question. We shall use the following facts from Haken-Matveev theory, see =-=[Mat1]-=-. A compressing disk for a surface F in a 3-manifold M is an embedded disk D ⊂ M which meets F along its boundary, i.e. D ∩ F = ∂D. A surface is called incompressible if it does not admit essential co... |

42 | Stable equivalence of knots on surfaces and virtual knot cobordisms - Carter, Kamada, et al. |

37 |
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Citation Context ...prove the following Theorem 3. There is an algorithm to decide whether two virtual links are equivalent or not. This theorem was first proved in [?], see also [Ma12]. We shall use the result by Moise =-=[Moi]-=- that all 3-manifolds admit a triangulation. In the sequel, each 3-manifold is thought to be tringulated. We shall deal with 3-manifolds (possibly, with boundary) and 2-surfaces in these manifolds. A ... |

35 | The number of Reidemeister moves needed for unknotting - Hass, Lagarias |

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Citation Context ...o an invariant of virtual knots. In some cases one has an invariant of virtuals that is an extension of ideas from classical knot theory that vanishes or is otherwise trivial for classical knots. See =-=[Saw]-=-,[SW], [KR], [Ma2, Ma3, Ma5]. Such invariants are valuable for the study of virtual knots, since they promise the possibility of distinguishing classical from virtual knots in key cases. On the other ... |

27 | Minimal surface representations of virtual knots and links. arXiv:math.AT/0401035 v1 - Dye, Kauffman - 2004 |

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Citation Context ...nt of virtual knots. In some cases one has an invariant of virtuals that is an extension of ideas from classical knot theory that vanishes or is otherwise trivial for classical knots. See [Saw],[SW], =-=[KR]-=-, [Ma2, Ma3, Ma5]. Such invariants are valuable for the study of virtual knots, since they promise the possibility of distinguishing classical from virtual knots in key cases. On the other hand, some ... |

26 |
What is a virtual link
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Citation Context ...ves, was used to analyze the structure of Vassiliev invariants for classical and virtual knots. In both [KaV] and [GPV] it is proved that if two classical knots are equivalent in the virtual category =-=[KUP]-=-, then they are equivalent in the classical category. Thus classical knot theory is properly embedded in virtual knot theory. To date, many invariants of classical knots have been generalized for the ... |

25 | Braid presentation of virtual knots and welded knots
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Citation Context ...If we allow only the forbidden move shown in the left part of Fig. 4, we obtain what are called welded knots, developed by Shin Satoh, [Satoh]. Some initial information on this theory can be found in =-=[Kam]-=-, see also [FRR]. Definition 4. By a mirror image of a virtual link diagram we mean a diagram obtained from the initial one by switching all types of classical crossings (all virtual crossings stay on... |

24 | Detecting non-triviality of virtual links - Kadokami |

24 |
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Citation Context ...the definition of a connected sum depends on initial diagrams and the choice of break point. Figure 7 illustrates a non-trivial connected sum of trivial virtual knots. This example is due to Kishino, =-=[KS]-=-. In what follows, we shall prove results for a connected sum K1#K2, with the intent that our statement holds for any connected sum: here the notation K1#K2 will be used for an arbitrary connected sum... |

23 | The classification of knots and 3-dimensional spaces - Hemion - 1992 |

23 |
The Conway Polynomial in S 3 and Thickened Surfaces: A new Determinant Formulation
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Citation Context ...egular plane graphs with extra structure) where a new type of crossing (called virtual) is allowed. This theory can be regarded as a “projection” of knot theory in thickened surfaces Sg ×R studied in =-=[JKS]-=-. Regarded from this point of view, virtual crossings appear as artifacts of the diagram projection from Sg to R 2 . In such a virtual projection diagram, one does not know the genus of the surface fr... |

22 |
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Citation Context ...ly, the forbidden move is a very strong one. Each virtual knot can be transformed to any other one by using all generalized Reidemeister moves and the forbidden move. This was proved by Sam Nelson in =-=[Nel]-=- (first mentioned in [GPV], see also [Kan]) by using Gauss diagrams of virtual links. Therefore, any two closed virtual knots are 1-equivalent and can be transformed to each other by a sequence of the... |

20 | Knot theory - Manturov - 2004 |

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18 |
Forbidden Moves Unknot a Virtual Knot
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Citation Context ...e. Each virtual knot can be transformed to any other one by using all generalized Reidemeister moves and the forbidden move. This was proved by Sam Nelson in [Nel] (first mentioned in [GPV], see also =-=[Kan]-=-) by using Gauss diagrams of virtual links. Therefore, any two closed virtual knots are 1-equivalent and can be transformed to each other by a sequence of the Reidemeister moves and the forbidden move... |

17 |
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(Show Context)
Citation Context ...esponding tubular braids in four-space. In particular, the move (F1) induces such an isotopy, while the forbidden move (F2) does not. For more on this subject, the reader can consult [Satoh] and also =-=[KaV2]-=- and the references therein. The basic idea 21Figure 15: Braiding of circles for this correspondence is due to Satoh in [Satoh] where torus embeddings in four-space are associated with virtual knot d... |

16 |
Detecting virtual knots, Atti
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(Show Context)
Citation Context ...esponding tubular braids in four-space. In particular, the move (F1) induces such an isotopy, while the forbidden move (F2) does not. For more on this subject, the reader can consult [Satoh] and also =-=[DVK]-=- and the references therein. The basic idea for this correspondence is due to Satoh in [Satoh] where torus embeddings in four-space are associated with virtual knot diagrams. Consider now the surjecti... |

15 | Diagrammatic invariants of knotted curves and surfaces, (unpublished manuscript - Carter, Saito - 1992 |

15 | On invariants of virtual links - Manturov |

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13 |
The Braid Permutation
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(Show Context)
Citation Context ... the forbidden move shown in the left part of Fig. 4, we obtain what are called welded knots, developed by Shin Satoh, [Satoh]. Some initial information on this theory can be found in [Kam], see also =-=[FRR]-=-. Definition 4. By a mirror image of a virtual link diagram we mean a diagram obtained from the initial one by switching all types of classical crossings (all virtual crossings stay on the same positi... |

13 |
Virtual Knot Theory, in preparation
- Kauffman, Manturov
(Show Context)
Citation Context ...nly if their corresponding surface embeddings are stably equivalent. 5Figure 5: Generalized Reidemeister moves and thickened surfaces This result was sketched in [KaV]. The complete proof appears in =-=[KaV3]-=-. A hint to this proof is demonstrated in Fig. 5. Here we wish to emphasize the following important circumstance. Definition 5. A virtual link diagram is minimal if no handles can be removed after a s... |

13 | Kauffman–like polynomial and curves in 2–surfaces - Manturov - 2003 |

13 |
Long virtual knots and their invariants
- Manturov
- 2004
(Show Context)
Citation Context ...oup of the complement of the virtual knot in the one-point suspension of a thickened surface where this knot is presented. Another phenomenon that does not appear in the classical case are long knots =-=[Ma11]-=-: if we break a virtual knot diagram at two different points and take them to the infinity, we may obtain two different long knots. 2❡ � ����� ❅ ❅ ❅ ❅ ❅ ❅ Figure 1: Virtual crossing In the present pa... |

12 | Multivariable polynomial invariants for virtual knots and links - Manturov - 2003 |

12 | Two–variable invariant polynomials for virtual links - Manturov - 2002 |

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9 |
Virtual Knot Presentations of Ribbon Torus Knots
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- 2000
(Show Context)
Citation Context ...by a sequence of the Reidemeister moves and the forbidden moves. If we allow only the forbidden move shown in the left part of Fig. 4, we obtain what are called welded knots, developed by Shin Satoh, =-=[Satoh]-=-. Some initial information on this theory can be found in [Kam], see also [FRR]. Definition 4. By a mirror image of a virtual link diagram we mean a diagram obtained from the initial one by switching ... |

9 | groups and virtual links
- Silver, Williams, et al.
(Show Context)
Citation Context ...nvariant of virtual knots. In some cases one has an invariant of virtuals that is an extension of ideas from classical knot theory that vanishes or is otherwise trivial for classical knots. See [Saw],=-=[SW]-=-, [KR], [Ma2, Ma3, Ma5]. Such invariants are valuable for the study of virtual knots, since they promise the possibility of distinguishing classical from virtual knots in key cases. On the other hand,... |

8 | Virtual Knot Groups - Kim - 1999 |

6 |
Viro O.(2000), Finite type invariants of classical and virtual knots, Topology 39
- Goussarov, Polyak
(Show Context)
Citation Context ... surface that do not have any part of the knot or link embedded in them) [KaV2, KaV4, Ma1, Ma8, Ma10, CKS, KUP]. Another approach to Gauss codes for knots and links is the use of Gauss diagrams as in =-=[GPV]-=-). In this paper by Goussarov, Polyak and Viro, the virtual knot theory, taken as all Gauss diagrams up to Reidemeister moves, was used to analyze the structure of Vassiliev invariants for classical a... |

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4 | On Filamentations and Virtual Knots, Topology and its - Hrencecin, Kauffman |

3 | Virtual Braids and the L-move - Kauffman, Lambropoulou |

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2 |
Homotopy equivalences of 3-manifolds with boundaries
- Johansson
(Show Context)
Citation Context ...compressing disks. Also, a 3-manifold M is boundary irreducible if any proper disk D ⊂ M bounds a disk on ∂M. Given a 3-manifold with boundary. By a boundary pattern (first proposed by Johannson, see =-=[Joh]-=-) we mean a fixed 1–polyhedron (graph) without isolated points on the boundary of the three manifolds (we assume this graph be a subpolyhedron of the selected triangulation). 14The existence of a bou... |

2 | math.GT/0405049, A self-linking invariant of virtual knots
- Kauffman
(Show Context)
Citation Context ... mirror image. If K is a virtual knot and J(K) is non-zero, then K is not equivalent to a classical knot. We leave the proof of this Theorem and the proof of the invariance of J(K) to the reader. See =-=[SelfLink]-=- for more about this invariant its generalizations. View Figure . The two virtual knots in this figure illustrate the application of Theorem 2. In the case of the virtual trefoil K, the Gauss code of ... |

2 | Distributive grouppoids in Knot Theory - Matveev - 1984 |