On diagrams of configurations of 7 skew lines of ℝ³

Borobia, Alberto M.; Mazurovskii, V. F.

doi:10.1090/trans2/173/04

Cited by 3 publications

(7 citation statements)

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“…In this section, we describe some computational results. The number of skew configurations,up to isotopy, having n ≤ 7 lines are known (see the survey of Viro and Drobotukhina [20] and the results of Borobia and Mazurovskii [1], [2], [11]): Lines Isotopy classes 2 1 3 2 4 3 5 7 6 19 7 74…”

Section: Computational Results and Lower Bounds For The Number Of Nonmentioning

confidence: 99%

“…The study and classification of configurations of skew lines (up to isotopy) was started by Viro [19] and continued for example in [1], [2], [3], [6], [10], [13], [14], [15], [16] and [20].…”

Section: Introductionmentioning

confidence: 99%

“…. , n} generated by transformations of the following three types: 1 (1) σ ∼ µ if µ(i) = s + σ(i + t (mod n)) (mod n) for some integers 0 ≤ s, t < n (all sums are modulo n).…”

Section: Introductionmentioning

confidence: 99%

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Spindle-configurations of skew lines

Bacher

Garber

2007

Geom. Topol.

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We prove a conjecture of Crapo and Penne which characterizes isotopy classes of skew configurations with spindlestructure. We use this result in order to define an invariant, spindle-genus, for spindle-configurations.We also slightly simplify 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 which constructs a spindle in a given switching class, or proves non-existence of such a spindle.Corresponding author.

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Section: Computational Results and Lower Bounds For The Number Of Nonmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Spindle-configurations of skew lines

Bacher

Garber

2007

Geom. Topol.

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“…In the same way we can prove condition (6). The theorem is proven by observing that (5) and (7) together imply (2), and that (6) and (8) give (4).…”

Section: R Penilementioning

confidence: 97%

“…Two line configurations s and/21 are said to be (rigidly) isotopic if they can be connected by a continuously parametrized family {Et; 0 < t < 1} of line configurations. Viro classified line configurations up to rigid isotopy for n < 5 in [15] (see also [12]), Mazurovskii for n < 6 in [11], and Borobia and Mazurovskii for n < 7 in [2].…”

Section: Line Diagrams and Line Isotopymentioning

confidence: 99%

Yang-Baxter invariants for line configurations

Penne

1996

Discrete Comput Geom

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We study line configurations in 3-space by means of "line diagrams," projections into a plane with an indication of over and under crossing at the vertices. If we orient such a diagram, we can associate a "contracted tensor" T with it in the same spirit as is done in Knot Theory. We give conditions to make T independent of the orientation, and invariant under isotopy. The Yang-Baxter equation is one such condition. Afterwards we restrict ourselves to Yang-Baxter invariants with a topological state model, and give some new invariants for line isotopy.

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Eulerian partitions for configurations of skew lines

Bacher,

Garber

2010

Preprint

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In this paper, which is a complement of [1], we study a few elementary invariants for configurations of skew lines, as introduced and analyzed first by Viro and his collaborators. We slightly simplify the exposition of some known invariants and use them to define a natural partition of the lines in a skew configuration.We also describe an algorithm which constructs a spindle-permutation for a given switching class, or proves non-existence of such a spindlepermutation.A spindle (or isotopy join or horizontal configuration) is a particularly nice configuration of skew lines in which all lines intersect an oriented additional line A, called the axis of the spindle. Its isotopy class

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On diagrams of configurations of 7 skew lines of ℝ³

Cited by 3 publications

References 0 publications

Spindle-configurations of skew lines

Spindle-configurations of skew lines

Yang-Baxter invariants for line configurations

Eulerian partitions for configurations of skew lines

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