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Journal articles on the topic 'Complexe of oriented matroids'

Author: Grafiati

Published: 25 May 2024

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1

Mücksch, Paul. "Modular flats of oriented matroids and poset quasi-fibrations." Transactions of the American Mathematical Society, Series B 11, no. 9 (January 30, 2024): 306–28. http://dx.doi.org/10.1090/btran/168.

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We study the combinatorics of modular flats of oriented matroids and the topological consequences for their Salvetti complexes. We show that the natural map to the localized Salvetti complex at a modular flat of corank one is what we call a poset quasi-fibration – a notion derived from Quillen’s fundamental Theorem B from algebraic K K -theory. As a direct consequence, the Salvetti complex of an oriented matroid whose geometric lattice is supersolvable is a K ( π , 1 ) K(\pi ,1) -space – a generalization of the classical result for supersolvable hyperplane arrangements due to Falk, Randell and Terao. Furthermore, the fundamental group of the Salvetti complex of a supersolvable oriented matroid is an iterated semidirect product of finitely generated free groups – analogous to the realizable case. Our main tools are discrete Morse theory, the shellability of certain subcomplexes of the covector complex of an oriented matroid, a nice combinatorial decomposition of poset fibers of the localization map, and an isomorphism of covector posets associated to modular elements. We provide a simple construction of supersolvable oriented matroids. This gives many non-realizable supersolvable oriented matroids and by our main result aspherical CW-complexes.

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2

Chepoi, Victor, Kolja Knauer, and Manon Philibert. "Ample Completions of Oriented Matroids and Complexes of Uniform Oriented Matroids." SIAM Journal on Discrete Mathematics 36, no. 1 (February 24, 2022): 509–35. http://dx.doi.org/10.1137/20m1355434.

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3

Bandelt, Hans-Jürgen, Victor Chepoi, and Kolja Knauer. "COMs: Complexes of oriented matroids." Journal of Combinatorial Theory, Series A 156 (May 2018): 195–237. http://dx.doi.org/10.1016/j.jcta.2018.01.002.

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4

Webster, Julian. "Cell complexes, oriented matroids and digital geometry." Theoretical Computer Science 305, no. 1-3 (August 2003): 491–502. http://dx.doi.org/10.1016/s0304-3975(02)00712-0.

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5

Fukuda, Komei, Hiroyuki Miyata, and Sonoko Moriyama. "Complete Enumeration of Small Realizable Oriented Matroids." Discrete & Computational Geometry 49, no. 2 (December 19, 2012): 359–81. http://dx.doi.org/10.1007/s00454-012-9470-0.

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6

Knauer, Kolja, and Tilen Marc. "On Tope Graphs of Complexes of Oriented Matroids." Discrete & Computational Geometry 63, no. 2 (July 11, 2019): 377–417. http://dx.doi.org/10.1007/s00454-019-00111-z.

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7

Bokowski, Jürgen, and Tomaž Pisanski. "Oriented matroids and complete-graph embeddings on surfaces." Journal of Combinatorial Theory, Series A 114, no. 1 (January 2007): 1–19. http://dx.doi.org/10.1016/j.jcta.2006.06.012.

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8

Naimi, Ramin, and Elena Pavelescu. "Linear embeddings of K9 are triple linked." Journal of Knot Theory and Its Ramifications 23, no. 03 (March 2014): 1420001. http://dx.doi.org/10.1142/s0218216514200016.

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We use the theory of oriented matroids to show that any linear embedding of K9, the complete graph on nine vertices, into 3-space contains a non-split link with three components. This shows that Sachs' conjecture on linear, linkless embeddings of graphs, whether true or false, does not extend to 3-links.

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9

Alfonsín, J. L. Ramírez. "On Linked Spatial Representations." Journal of Knot Theory and Its Ramifications 10, no. 01 (February 2001): 143–50. http://dx.doi.org/10.1142/s0218216501000780.

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What is the smallest positive integer m=m(L) such that every linear spatial representation of the complete graph with n vertices, n≥m contain cycles isotopic to link L? In this paper, we show that [Formula: see text]. The proof uses the well-known cyclic polytope and its combinatorial description in terms of oriented matroids.

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10

Welsh, D. J. A. "ORIENTED MATROIDS." Bulletin of the London Mathematical Society 27, no. 5 (September 1995): 499–501. http://dx.doi.org/10.1112/blms/27.5.499.

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11

Booth, Richard F., Alexandre V. Borovik, Israel M. Gelfand, and Neil White. "Oriented Lagrangian Matroids." European Journal of Combinatorics 22, no. 5 (July 2001): 639–56. http://dx.doi.org/10.1006/eujc.2000.0485.

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12

Bokowski, Jürgen, António Guedes de Oliveira, and Jürgen Richter-Gebert. "Algebraic varieties characterizing matroids and oriented matroids." Advances in Mathematics 87, no. 2 (June 1991): 160–85. http://dx.doi.org/10.1016/0001-8708(91)90070-n.

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13

Delucchi, Emanuele. "Modular elimination in matroids and oriented matroids." European Journal of Combinatorics 32, no. 3 (April 2011): 339–43. http://dx.doi.org/10.1016/j.ejc.2010.10.013.

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14

Wenzel, Walter. "Book Review: Oriented matroids." Bulletin of the American Mathematical Society 31, no. 2 (October 1, 1994): 296–98. http://dx.doi.org/10.1090/s0273-0979-1994-00536-1.

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15

Bachem, A., and W. Kern. "Adjoints of oriented matroids." Combinatorica 6, no. 4 (December 1986): 299–308. http://dx.doi.org/10.1007/bf02579255.

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16

Fenton, William E. "Completeness in oriented matroids." Discrete Mathematics 66, no. 1-2 (August 1987): 79–89. http://dx.doi.org/10.1016/0012-365x(87)90120-8.

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17

McLennan, Andrew, and Rabee Tourky. "Games in oriented matroids." Journal of Mathematical Economics 44, no. 7-8 (July 2008): 807–21. http://dx.doi.org/10.1016/j.jmateco.2007.07.003.

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18

Santos, Francisco. "Triangulations of oriented matroids." Memoirs of the American Mathematical Society 156, no. 741 (2002): 0. http://dx.doi.org/10.1090/memo/0741.

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19

Sturmfels, Bernd, Alexander Postnikov, and Isabella Novik. "Syzygies of oriented matroids." Duke Mathematical Journal 111, no. 2 (February 2002): 287–317. http://dx.doi.org/10.1215/s0012-7094-02-11124-7.

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20

Richter-Gebert, Jürgen. "Two interesting oriented matroids." Documenta Mathematica 1 (1996): 137–48. http://dx.doi.org/10.4171/dm/7.

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21

Hochstättler, Winfried, and Robert Nickel. "Joins of oriented matroids." European Journal of Combinatorics 32, no. 6 (August 2011): 841–52. http://dx.doi.org/10.1016/j.ejc.2011.02.005.

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22

Borovik, Alexandre V., Israel Gelfand, and Neil White. "On exchange properties for Coxeter matroids and oriented matroids." Discrete Mathematics 179, no. 1-3 (January 1998): 59–72. http://dx.doi.org/10.1016/s0012-365x(96)00027-1.

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23

Bokowski, J., and A. G. Deoliveira. "Invariant Theory-like Theorems for Matroids and Oriented Matroids." Advances in Mathematics 109, no. 1 (November 1994): 34–44. http://dx.doi.org/10.1006/aima.1994.1078.

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24

Ardila, Federico, Felipe Rincón, and Lauren Williams. "Positively oriented matroids are realizable." Journal of the European Mathematical Society 19, no. 3 (2017): 815–33. http://dx.doi.org/10.4171/jems/680.

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25

Fukuda, Komei, and Tamas Terlaky. "LINEAR COMPLEMENTARITY AND ORIENTED MATROIDS." Journal of the Operations Research Society of Japan 35, no. 1 (1992): 45–61. http://dx.doi.org/10.15807/jorsj.35.45.

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26

Bachem, Achim, and Walter Kern. "Extension Equivalence of Oriented Matroids." European Journal of Combinatorics 7, no. 3 (July 1986): 193–97. http://dx.doi.org/10.1016/s0195-6698(86)80020-8.

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27

Sturmfels, Bernd. "Neighborly Polytopes and Oriented Matroids." European Journal of Combinatorics 9, no. 6 (November 1988): 537–46. http://dx.doi.org/10.1016/s0195-6698(88)80050-7.

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28

Bachem, Achim, and Alfred Wanka. "Separation theorems for oriented matroids." Discrete Mathematics 70, no. 3 (August 1988): 303–10. http://dx.doi.org/10.1016/0012-365x(88)90006-4.

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29

Fukuda, Komei, and Keiichi Handa. "Antipodal graphs and oriented matroids." Discrete Mathematics 111, no. 1-3 (February 1993): 245–56. http://dx.doi.org/10.1016/0012-365x(93)90159-q.

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30

Sturmfels, Bernd, and Günter M. Ziegler. "Extension spaces of oriented matroids." Discrete & Computational Geometry 10, no. 1 (July 1993): 23–45. http://dx.doi.org/10.1007/bf02573961.

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31

Richter-Gebert, Jürgen. "Oriented matroids with few mutations." Discrete & Computational Geometry 10, no. 3 (September 1993): 251–69. http://dx.doi.org/10.1007/bf02573980.

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32

Miller, Douglas A. "Oriented matroids from smooth manifolds." Journal of Combinatorial Theory, Series B 43, no. 2 (October 1987): 173–86. http://dx.doi.org/10.1016/0095-8956(87)90020-7.

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33

Abello and Kumar. "Visibility Graphs and Oriented Matroids." Discrete & Computational Geometry 28, no. 4 (November 2002): 449–65. http://dx.doi.org/10.1007/s00454-002-2881-6.

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34

Lawrence, Jim. "Oriented Matroids and Associated Valuations." Discrete & Computational Geometry 33, no. 3 (November 16, 2004): 445–62. http://dx.doi.org/10.1007/s00454-004-1114-6.

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35

Lawrence, J. "Mutation Polynomials and Oriented Matroids." Discrete & Computational Geometry 24, no. 2 (September 2000): 365–90. http://dx.doi.org/10.1007/s004540010042.

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36

Cordovil, Raul, and Komei Fukuda. "Oriented Matroids and Combinatorial Manifolds." European Journal of Combinatorics 14, no. 1 (January 1993): 9–15. http://dx.doi.org/10.1006/eujc.1993.1002.

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37

Cordovil, Raul, and Pierre Duchet. "Cyclic Polytopes and Oriented Matroids." European Journal of Combinatorics 21, no. 1 (January 2000): 49–64. http://dx.doi.org/10.1006/eujc.1999.0317.

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38

Anderson, Laura, and Rephael Wenger. "Oriented Matroids and Hyperplane Transversals." Advances in Mathematics 119, no. 1 (April 1996): 117–25. http://dx.doi.org/10.1006/aima.1996.0028.

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39

Roudneff, J. P. "Inseparability graphs of oriented matroids." Combinatorica 9, no. 1 (March 1989): 75–84. http://dx.doi.org/10.1007/bf02122686.

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40

Guzmán-Pro, Santiago, and Winfried Hochstättler. "Oriented cobicircular matroids are GSP." Discrete Mathematics 347, no. 1 (January 2024): 113686. http://dx.doi.org/10.1016/j.disc.2023.113686.

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41

Chen, Xiangying. "An Axiomatization of Matroids and Oriented Matroids as Conditional Independence Models." SIAM Journal on Discrete Mathematics 38, no. 2 (May 17, 2024): 1526–36. http://dx.doi.org/10.1137/23m1558653.

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42

Živaljević, Rade T. "Oriented matroids and Ky Fan’s theorem." Combinatorica 30, no. 4 (July 2010): 471–84. http://dx.doi.org/10.1007/s00493-010-2446-x.

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43

Hartmann, Mark, and Michael H. Schneider. "Max-balanced flows in oriented matroids." Discrete Mathematics 137, no. 1-3 (January 1995): 223–40. http://dx.doi.org/10.1016/0012-365x(93)e0168-4.

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44

Cordovil, R., and M. L. Moreira. "A homotopy theorem on oriented matroids." Discrete Mathematics 111, no. 1-3 (February 1993): 131–36. http://dx.doi.org/10.1016/0012-365x(93)90149-n.

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45

Bokowski, Jürgen, and Bernd Sturmfels. "On the coordinatization of oriented matroids." Discrete & Computational Geometry 1, no. 4 (December 1986): 293–306. http://dx.doi.org/10.1007/bf02187702.

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46

Miyata, Hiroyuki, and Arnau Padrol. "Enumerating Neighborly Polytopes and Oriented Matroids." Experimental Mathematics 24, no. 4 (July 24, 2015): 489–505. http://dx.doi.org/10.1080/10586458.2015.1015084.

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47

Cordovil, R. "A Commutative Algebra for Oriented Matroids." Discrete & Computational Geometry 27, no. 1 (January 2002): 73–84. http://dx.doi.org/10.1007/s00454-001-0053-8.

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48

Rambau, J. "Circuit Admissible Triangulations of Oriented Matroids." Discrete & Computational Geometry 27, no. 1 (January 2002): 155–61. http://dx.doi.org/10.1007/s00454-001-0058-3.

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49

Buchi, J. Richard, and William E. Fenton. "Large convex sets in oriented matroids." Journal of Combinatorial Theory, Series B 45, no. 3 (December 1988): 293–304. http://dx.doi.org/10.1016/0095-8956(88)90074-3.

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50

Las Vergnas, Michel. "Acyclic reorientations of weakly oriented matroids." Journal of Combinatorial Theory, Series B 49, no. 2 (August 1990): 195–99. http://dx.doi.org/10.1016/0095-8956(90)90027-w.

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