## Spindle configurations of skew lines (2002)

### BibTeX

@TECHREPORT{Bacher02spindleconfigurations,

author = {Roland Bacher and David Garber},

title = {Spindle configurations of skew lines},

institution = {},

year = {2002}

}

### OpenURL

### Abstract

Abstract. We simplify slightly the exposition of some known invariants for configurations of skew lines and use them to define a natural partition of the lines in a skew configuration. Finally, we describe an algorithm constructing a spindle in a given switching class, provided such a spindle exists. 1.

### Citations

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Citation Context ...onnected Eulerian graphs are good designs for zoological gardens or expositions since they can be visited by walking exactly once along every edge. The following well-known result goes back to Seidel =-=[12]-=-. Proposition 5.1. Eulerian graphs on an odd number 2n−1 of vertices are in bijection with switching classes of order 2n − 1. We recall here the simple proof since it yields an algorithm for computing... |

18 | An analogue of the Jones polynomial for links in RP 3 and a generalization of the Kauffman-Murasugi theorem - Drobotukhina - 1991 |

13 |
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Citation Context ...s classes 6 16 16 7 54 54 8 235 243 9 1824 2038 In fact, one can use representation theory of the symmetric groups in order to derive a formula for the number of switching classes of given order (see =-=[6]-=- and Sequence A2854 in [13]). The map {isotopy classes of configurations of skew lines} −→ {switching classes} is perhaps not surjective for all n (there seems to be an unpublished counterexample of P... |

12 |
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Citation Context ...re exist equivalence classes of skew pseudoline diagrams which are not projections of configurations of skew lines: for instance alternating skew pseudoline diagrams with more than 3 pseudolines (see =-=[7]-=-). There are even generic arrangements of pseudolines which are not stretchable, i.e. cannot be realized as an arrangement of straight lines, see [4] for the smallest possible example having 9 lines. ... |

9 |
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(Show Context)
Citation Context ...C1 to C2 (a continuous deformation of configurations of skew lines starting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro =-=[14]-=- and continued for example in [1], [2], [3], [5], [8], [9], [10], [11] and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration ... |

5 |
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(Show Context)
Citation Context ...rix. Such a matrix can then be encoded by a graph on n − 1 vertices encoding all entries outside the chosen row and the corresponding column. This leads to the notion of graphs which are “cousins” in =-=[3]-=-. This notion is of course equivalent to the notion of switching-equivalence as can be easily checked. 5. Euler partitions In this section we study some properties of switching classes. They lead to i... |

4 |
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Citation Context ...rting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in [1], [2], [3], [5], [8], [9], [10], =-=[11]-=- and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line, called the axis of the spindle. The ... |

3 |
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Citation Context ...f skew lines starting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in [1], [2], [3], [5], =-=[8]-=-, [9], [10], [11] and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line, called the axis of ... |

3 |
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(Show Context)
Citation Context ...w lines starting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in [1], [2], [3], [5], [8], =-=[9]-=-, [10], [11] and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line, called the axis of the s... |

3 |
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(Show Context)
Citation Context ...es starting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in [1], [2], [3], [5], [8], [9], =-=[10]-=-, [11] and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line, called the axis of the spindle... |

2 |
On diagrams of configurations of 7 skew lines of R 3
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(Show Context)
Citation Context ... of configurations of skew lines starting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in =-=[1]-=-, [2], [3], [5], [8], [9], [10], [11] and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line,... |

2 |
Nonsingular configurations of 7 lines
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(Show Context)
Citation Context ...onfigurations of skew lines starting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in [1], =-=[2]-=-, [3], [5], [8], [9], [10], [11] and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line, call... |

2 |
Projective graph theory and configurations of lines
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(Show Context)
Citation Context ...ons of skew lines starting at C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in [1], [2], [3], =-=[5]-=-, [8], [9], [10], [11] and [15]. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line, called the axi... |

2 | Configurations of skew lines
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(Show Context)
Citation Context ...C1 and ending at C2). The study and classification of configurations of skew lines (up to isotopy) was initiated by Viro [14] and continued for example in [1], [2], [3], [5], [8], [9], [10], [11] and =-=[15]-=-. A spindle (or isotopy join) is a particularly nice configuration of skew lines in which all lines of the configuration intersect an oriented extra line, called the axis of the spindle. The isotopy c... |

2 |
Stable equivalence of real projective configurations
- Khashin, Mazurovskii
- 1996
(Show Context)
Citation Context ...ses of spindle permutations into switching classes factorizes through an injective map into isotopy classes of spindles. Remark that the following reformulation of a result of Khashin and Mazurovskii =-=[7]-=- is a simple corollary of Theorem 1.1: Theorem 6.6 (Khashin-Mazurovskii, Theorem 3.2). Two spindle configurations are isotopic if and only if they have the same switching class. 7. Spindlegenus This s... |

1 | Glossary of Signed and Gain Graphs and Allied Areas, The Electronic - Zaslavsky - 1998 |