Configurations of six skew lines

Mazurovskii, V. F.

doi:10.1007/bf01099247

Cited by 12 publications

(27 citation statements)

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“…Ya. Viro [1], V. F. Mazurovskii [2,3], and A. Borobia and Mazurovskii [4] obtained the rigid isotopy classification of unordered projective m-configurations for m ~ 5, m = 6, and m = 7 respectively. In the present paper, we describe necessary and sufficient conditions for ordered projective m-configurations with m ~ 7 to be rigidly isotopic.…”

Section: Introductionmentioning

confidence: 99%

Classification of ordered nonsingular configurations of at most seven lines of ℝP 3 up to rigid isotopy

Mazurovskii¹,

Pavlov²

1998

J Math Sci

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Section: Introductionmentioning

confidence: 99%

Classification of ordered nonsingular configurations of at most seven lines of ℝP 3 up to rigid isotopy

Mazurovskii¹,

Pavlov²

1998

J Math Sci

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“…Classification of 2-arrangement groups. The rigid isotopy classification of configurations of n ≤ 7 skew lines in R 3 (and, thereby, of 2-arrangements of n ≤ 7 planes in R 4 ) was established by Viro [42], Mazurovskiȋ [34], and Borobia and Mazurovskiȋ [2]. Clearly, if A is rigidly isotopic to A ′ , or to its mirror image, then G(A) ∼ = G(A ′ ).…”

Section: Arrangements Of Transverse Planes In Rmentioning

confidence: 99%

“…For n = 6, there are 4 nonhorizontal arrangements: L, M, and their mirror images. These arrangements were introduced by Mazurovskiȋ in [34]; further details about them can be found in [32]. For n = 7, there are 13 non-horizontal arrangements, see [2].…”

Section: Arrangements Of Complex Hyperplanesmentioning

confidence: 99%

“…The proof is based on the computations displayed in Table 2. The first column lists the rigid isotopy classes (mirror pairs identified) of 2-arrangements A (with n = |A| ≤ 6, or n = 7 and A horizontal), according to the classification by Viro, Mazurovskiȋ, and Borobia [42,34,2]. The next two columns list the Hall invariants δ S 3 and δ A 4 for the corresponding 2-arrangement groups, G = G(A).…”

Section: Arrangements Of Transverse Planes In Rmentioning

confidence: 99%

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Matei¹,

Suciu²

2002

Internat. Math. Res. Notices

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“…Viro in [16], for n = 6 by V. Mazurovskii in [9], and recently for n = 7 by A. Borobia and V. Mazurovskii in [4]. For a treatment of line isotopy from the view point of projective geometry we refer to [5].…”

Section: Isotopy Of Line Configurationsmentioning

confidence: 99%

The Alexander polynomial of a configuration of skew lines in 3-Space

Penne¹

1998

Pacific J. Math.

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In Penne, 1995, we observed that the multi-variable Alexander polynomial can be computed by means of "multi-variable Burau matrices", the entries of which can be visualized by a "path model". In this article we introduce the Alexander polynomial for line configurations. The correspondence between "adjacency of lines" in a configuration, and the factorization of its Alexander polynomial can be well understood by way of the path model. Introduction.In a previous paper ([12]) we introduced a new machinery to compute the multi-variable Alexander polynomial ∆ L of a link L with k components. Although everything was stated for pure links, the techniques of [12] apply for general links if one makes a small additional observation (Theorem 1).Let us quickly repeat the basic facts of the (non-reduced) Alexander polynomial. If we regard the link L as a closed braid β, β ∈ B n , and if we use Artin's representation of the braid group as free automorphisms,we can present the group of L, G(L) (the fundamental group of its complement), by n generators x 1 , . . . , x n subject to the relationsFor more details, we refer to [2,3]. It is well known that each of the n relations in (2) is implied by the others, allowing us to drop one relation, and we come up with a group presentation for G(L) of "deficiency 1" (one less relation than generators): , and extend it to the corresponding group rings, it can be applied to the entries of the Jacobian, resulting in an Alexander matrix A(L) for L. We refer to [8] where a : Z[F n ] → Z[H] composes the quotient map with the abelianizer. Notice thatWe conclude that the Alexander polynomial of a link L = β is immediately obtained once we have computed A(β). In this article, A(β) will be computed by associating so-called multi-variable Burau matrices with the crossings of β, multiplying them in order of appearance. This generalizes the well-known procedure for knots (k = 1), where A(β) is obtained as the Burau representation τ B (β) of β.The fundamental properties of the multi-variable Burau matrices can be found in [12]. In this article we focus on some of their applications, showing how they can improve our understanding of the Alexander polynomial. As pointed out in [12], the entries of a multi-variable Burau matrix can be easily visualized by considering topological paths on the involved braid diagram, which are evaluated by appropriate weights. This path model will be explained in Section 4. The main goal of this article is to illustrate the elegance of the path model by obtaining fundamental combinatorial proofs for classical theorems of Morton, Torres and Sumners-Woods on the Alexander polynomial (Section 5 and Section 6). THE ALEXANDER POLYNOMIAL 317Most of our material will be stated in terms of a special class of links. More precisely, we are motivated by the isotopy problem of line configurations, that are configurations of a finite number of lines in RP 3 . Two line configurations are called rigidly isotopic if there exists an ambient isotopy that connects them and during which the ...

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Configurations of six skew lines

Cited by 12 publications

References 1 publication

Classification of ordered nonsingular configurations of at most seven lines of ℝP 3 up to rigid isotopy

Classification of ordered nonsingular configurations of at most seven lines of ℝP 3 up to rigid isotopy

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The Alexander polynomial of a configuration of skew lines in 3-Space

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