## CONFIGURATIONS OF SKEW LINES JULIA VIRO AND OLEG VIRO (2006)

### BibTeX

@MISC{06configurationsof,

author = {},

title = {CONFIGURATIONS OF SKEW LINES JULIA VIRO AND OLEG VIRO},

year = {2006}

}

### OpenURL

### Abstract

Abstract. This article is a survey of results on projective configurations of subspaces in general position. It is written in the form of introduction to the subject, with much of the material accessible to advanced high school students. However, in the part of the survey concerning configurations of lines in general position in three-dimensional space we give a detailed exposition. The first version of this paper was written as an elementary introductory text for high-school students. It was published [3] in the the journal “Kvant”, the third issue of 1988, but in a shortened form. Then we expanded the article in order to encompass or at least mention some related questions. However we decided to keep the style of [3], in the hope that it would also be appreciated by a professional mathematician. We apologized to a reader, who would find the style irritating, and we mentioned that the material in the first two-thirds of the article (through the section on “sets of five lines”) was announced in the note [2], while the final third of the article is written in a more traditional style. The expanded version [10] of [3] was published in the first volume of a Russian

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Citation Context ... lines, and 11 types of nonsingular plane configurations of seven lines. But when we reach configurations of more than seven lines, the isotopy and rigid isotopy classifications diverge sharply. Mnev =-=[6]-=- proved a surprising theorem, according to which, roughly speaking, a set of nonsingular plane configurations of lines which are isotopic to one another can have the homotopy type of any affine open s... |

18 |
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Citation Context ...sotopic or if we want to determine the isotopy class of a given interlacing of six lines, it is sufficient to use the polynomial invariant of framed links in RP 3 which was introduced by Drobotukhina =-=[12]-=-. This invariant generalizes the Kauffman polynomial of links in R 3 . Return to the classification of interlacings of six lines. Of the 19 types, 15 consist of isotopy join interlacings. The remainin... |

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Citation Context ...zed to a reader, who would find the style irritating, and we mentioned that the material in the first two-thirds of the article (through the section on “sets of five lines”) was announced in the note =-=[2]-=-, while the final third of the article is written in a more traditional style. The expanded version [10] of [3] was published in the first volume of a Russian journal Algebra i Analiz opening a new se... |

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Citation Context ...ut by the action of the group of projective transformations.) The simplest example known of nonsingular plane configurations of lines which are isotopic but not rigid isotopic can be found in Suvorov =-=[8]-=-. The configurations in this example have 14 lines. High-Dimensional Generalizations of Interlacings of Lines Thus, the theory of nonsingular plane configurations of lines is quite different from the ... |

5 |
Chirality and the isotopy classification of skew lines in projective 3space
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Citation Context ...nslation. Furthermore, some of the results appeared in other papers without appropriate references. The tradition of wrong priority references was established in papers by R. Penne and H. Crapo [14], =-=[17]-=-. We expanded the bibliography for the sake of completeness, but arranged it in the chronological order, to facilitate orientation in the history of the subject. Can skew lines be interlaced? The arti... |

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(Show Context)
Citation Context ...2〉, 〈−2〉, 〈−2〉〉 = jc(5,6,3,4,2,1); 〈+〈−2〉, 〈−2〉, 〈−2〉〉 = jc(1,2,5,6,3,4), 〈−〈+2〉, 〈+2〉, 〈+2〉〉 = jc(4,3,6,5,2,1). Seven Lines Interlacings of seven lines have been classified by Borobia and Mazurovskiĭ=-=[24]-=-. There are 74 types of interlacings of seven lines and 48 of these types are isotopy join. As in the case of interlacings of 6 lines, it turns out that Drobotukhina’s polynomial invariant distinguish... |

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Citation Context ...singular plane configurations of lines have both been solved for configurations where the number of lines is ≤ 7, and in these cases the answer to both problems turns out to be the same (see Finashin =-=[5]-=-). If there are ≤ 5 lines, the isotopy type is determined by the number of lines. There are four types of nonsingular plane configurations of six lines, and 11 types of nonsingular plane configuration... |

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Citation Context ...of translation. Furthermore, some of the results appeared in other papers without appropriate references. The tradition of wrong priority references was established in papers by R. Penne and H. Crapo =-=[14]-=-, [17]. We expanded the bibliography for the sake of completeness, but arranged it in the chronological order, to facilitate orientation in the history of the subject. Can skew lines be interlaced? Th... |

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(Show Context)
Citation Context ...lines can be taken into one another by a homeomorphism of RP 3 which is isotopic to the identity (recall that an isotopy of the homeomorphism h: X → Y is a family of homeomorphisms ht: X → Y with t ∈ =-=[0,1]-=-, h0 = h, such that the map X × [0,1] → Y : (x,t) ↦→ ht(x) is continuous). Then what we earlier called isotopies would be called rigid isotopies. Our cavalier attitude about this is permissible only b... |

2 |
Drobotukhina Spleteniia skrescivaiuscikhsia priamykh, Kvant
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(Show Context)
Citation Context ...in general position in three-dimensional space we give a detailed exposition. The first version of this paper was written as an elementary introductory text for high-school students. It was published =-=[3]-=- in the the journal “Kvant”, the third issue of 1988, but in a shortened form. Then we expanded the article in order to encompass or at least mention some related questions. However we decided to keep... |

2 |
Drobotukhina Configurations of skew lines, Algebra i analiz 1:4
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(Show Context)
Citation Context ...wo-thirds of the article (through the section on “sets of five lines”) was announced in the note [2], while the final third of the article is written in a more traditional style. The expanded version =-=[10]-=- of [3] was published in the first volume of a Russian journal Algebra i Analiz opening a new section “Light reading for the professional”. English version of the paper became available in a translati... |

2 | On isotopic weavings - Gaifullin |

1 |
Configurations of six skew lines Zap
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(Show Context)
Citation Context ... the sums for the seven interlacings are all different. Six Lines It is by no means so easy to show that there are in all 19 types of interlacings of six lines (this theorem was proved by Mazurovskiĭ =-=[4]-=-, [13] in 1987). It is no longer possible to distinguish between nonisotopic interlacings using only the linking numbers of the triples of lines in each interlacing. To prove that the isotopy classes ... |

1 |
Kauffman polynomials for nonsingular configurations of projective lines
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(Show Context)
Citation Context ...ucted in this way are said to be isotopy join. Exercise Which of the interlacings encountered above are isotopy joins? Show that all interlacings of five or fewer lines are isotopy joins. Mazurovskiĭ =-=[9]-=- showed that, if we want to prove that two interlacings of six lines are not isotopic or if we want to determine the isotopy class of a given interlacing of six lines, it is sufficient to use the poly... |

1 |
Non-singular configurations of k-dimensional subspaces of (2k + 1)dimensional real projective space, Vestnik Leningrad Univ
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(Show Context)
Citation Context ...nordered configurations. Two configurations of k-dimensional subspaces of RP 2k+1 are said to be stably equivalent if there exists N such that their N-fold suspensions are rigid isotopic. Mazurovskiuı=-=[11]-=- has shown that this stable equivalence shares properties which are common for various stable equivalences mentioned above. Namely, Mazurovskiĭ [11] has proved that for k > 0 any configuration of ≤ k ... |

1 |
Configurations of at most six lines of RP 3 , Real Algebraic Geometry
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(Show Context)
Citation Context ...sums for the seven interlacings are all different. Six Lines It is by no means so easy to show that there are in all 19 types of interlacings of six lines (this theorem was proved by Mazurovskiĭ [4], =-=[13]-=- in 1987). It is no longer possible to distinguish between nonisotopic interlacings using only the linking numbers of the triples of lines in each interlacing. To prove that the isotopy classes are re... |

1 | Configurations of at most six (2n − 1)-dimensional subspaces of - Mazurovskiĭ - 1994 |

1 |
Classification of ordered nonsingular configurations of ≤ 7 lines of RP 3 up to rigid isotopy, Issledovaniya po topologii
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(Show Context)
Citation Context ...take into account any order of lines. We consider unordered interlacings, in which lines are not numerated or labeled. A classification of ordered interlacings is also possible. Mazurovskiĭand Pavlov =-=[18]-=- have classified ordered interlacings of up to seven lines. When the number of lines is less than 4, the result does not differ from the classification in unordered case. Indeed, by an isotopy one can... |

1 |
Amphicheiral configurations of points and lines and algebraic surfaces of degree 4, Issledovaniya po topologii
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(Show Context)
Citation Context ...t circles on the sphere, with respect to isotopies under which the circles remain disjoint great circles on S 3 .CONFIGURATIONS OF SKEW LINES 25 The following complete answer was found by Podkorytov =-=[19]-=-, after the previous version [10] of this paper was written: Podkorytov Theorem. (See [19].) An amphicheiral nonsingular set of q points and p lines in three-dimensional real projective space exists i... |

1 |
MazurovskiĭStable Equivalence of Real Projective Configurations, Topology of Real Algebraic Varieties and Related Topics
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(Show Context)
Citation Context ...fman bracket polynomial. Thus, there is no generalization of the Kauffman bracket to high-dimensional nonsingular configurations which would be preserved under suspension. Then Khashin and Mazurovskiĭ=-=[20]-=- proved that Two nonsingular configurations of k-dimensional subspaces of RP 2k+1 are stably equivalent if and only if they have the same linking numbers of the subspaces. This means that there exists... |

1 | On diagrams of configurations of 7 skew lines - Borobia, Mazurovskiĭ - 1996 |

1 | An infinite sequence of non-realizable weavings, Discrete Applied Mathematics 150 (2005) 256 – 260. E-mail address: oleg@math.uu.se E-mail address: julia@math.uu.se - Repovˇs, Skopenkovb, et al. |