Discrete Applied Mathematics 150 (2005) 256–260
www.elsevier.com/locate/dam
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An infinite sequence of non-realizable weavings
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Dušan Repovš
a
, Arkady Skopenkov
b
, Fulvia Spaggiari
c
a
Institute for Mathematics, Physics and Mechanics,University of Ljubljana, P. O. Box 2964, 1001 Ljubljana,
Slovenia
b
Department of Differential Geometry, Faculty of Mechanics and Mathematics,Moscow State University,
Moscow 119992, Russia
c
Dipartimento di Matematica, Università degli Studi di Modena e Reggio Emilia, Via Campi 213/B,
Modena 41100, Italy
Received 7 February 2003; received in revised form 20 October 2003; accepted 7 February 2005
Available online 10 May 2005
Abstract
A weaving is a number of lines drawn in the plane so that no three lines intersect at a point, and
the intersections are drawn so as to show which of the two lines is above the other. For each integer
n
4 we construct a weaving of n lines, which is not realizable as a projection of a number of lines
in 3-space, all of whose subfigures are realizable as such projections.
© 2005 Elsevier B.V. All rights reserved.
MSC: Primary: 51M20
Keywords: Algebraic knot theory; Weaving; Realizability; Projection
A weaving is a collection of lines drawn in the plane so that no three lines intersect at
a point, and the intersections are drawn so as to show which of the two lines is “above”
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Repovšwas supportedinpartby the Ministry forHigherEducation, Science andTechnology of theRepublicof
SloveniaResearch Program No. P1-0292-0101-04. Skopenkov was supported in part by the Russian Fundamental
Research Foundation Grant No. 02-01-00014, INTAS Grant No. YSF-2002-393 and Moscow State University
Stipendium for Young Scientists and Teachers. Spaggiari was supported in part by the Ministero per la Ricerca
Scientifica e Tecnologica of Italy within the project Proprietà Geometriche delle Varietà Reali e Complesse. The
authors thank the referee for several comments and suggestions.
E-mail addresses: dusan.repovs@fmf.uni-lj.si (D. Repovš), skopenko@mccme.ru (A. Skopenkov),
spaggiari@unimo.it (F. Spaggiari).
0166-218X/$-see front matter © 2005 Elsevier B.V. All rights reserved.
doi:10.1016/j.dam.2005.02.011