Discrete Applied Mathematics 150 (2005) 256–260
An inﬁnite sequence of non-realizable weavings
, Arkady Skopenkov
, Fulvia Spaggiari
Institute for Mathematics, Physics and Mechanics,University of Ljubljana, P. O. Box 2964, 1001 Ljubljana,
Department of Differential Geometry, Faculty of Mechanics and Mathematics,Moscow State University,
Moscow 119992, Russia
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
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
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”
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: email@example.com (D. Repovš), firstname.lastname@example.org (A. Skopenkov),
email@example.com (F. Spaggiari).
0166-218X/$-see front matter © 2005 Elsevier B.V. All rights reserved.