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«Abstract. We discuss two problems in combinatorial geometry. First, given a geometric polyhedral complex in R3 (a family of 3-polytopes attached ...»

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simply perturb all the vertices. In particular, this explains why we must use non-simplicial polytopes in the proof of Theorem 1.1.

It is perhaps less obvious that all geometric realizations produced in Theorem 6.1 are rational. Although the resulting polyhedral complex must have simplicial interior faces, the boundary faces can be arbitrary. Here rationality is a corollary resulting from the nature of the proof: all steps, in particular all projective transformations can be done over Q.

7.6. Recently, two new explicit examples of simplicial balls with further properties were announced in [BL]. They have 12 and 15 vertices, respectively. This can be contrasted with the 9 vertices of the topological polyhedral ball X ′ we construct in the proof of Theorem 1.3.

Acknowledgements The authors are grateful to Karim Adiprasito, Bruno Benedetti, Jes´s De Loera, J´nos Pach, Rom Pinchasi, Carsten Thomassen, Russ Woodroofe and u a G¨nter Ziegler for helpful comments and conversations. The second named author was u partially supported by the BSF and NSF.

References

[AB] K. Adiprasito and B. Benedetti, Metric Geometry and Collapsibility, arXiv:1107.5789.

[BG] D. W. Barnette and G. Gr¨nbaum, Preassigning the shape of a face, Pacific J. Math. 32 (1970), u 299–302.

[BL] B. Benedetti and F. Lutz, Non-evasiveness, collapsibility and explicit knotted triangulations, Oberwolfach Report, 8 (2011), 1–2.

[BFS] L. J. Billera, P. Filliman and B. Sturmfels, Constructions and complexity of secondary polytopes, Adv. Math. 83 (1990), 155–179.

[BP] L. Billera and J. Provan, Decompositions of simplicial complexes related to diameters of convex polyhedra, Math. Op. Res. 5 (1980), 576–594.

[Br] U. Brehm, Realization spaces, talk slides, 2007.

[Ca] S. S. Cairns, Triangulated manifolds which are not Brouwer manifolds, Ann. Math. 41 (1940), 792–795.

[Ch] D. R. J. Chillingworth, Collapsing three-dimensional convex polyhedra, Proc. Camb. Phil. Soc. 63 (1967), 353–357; Erratum in 88 (1980), 307–310.

GEOMETRIC REALIZATIONS OF POLYHEDRAL COMPLEXES 25

[DRS] J. A. De Loera, J. Rambau and F. Santos, Triangulations: Structures and Algorithms, Springer, 2008.

[F] I. F´ry, On straight line representations of planar graphs, Acta Univ. Szeged. Sect. Sci. Math. 11 a (1948), 229–233.

[FPP] H. de Fraysseix, J. Pach and R.Pollack, Small sets supporting Fary Embeddings of Planar Graphs, in Proc. STOC 1988, ACM, 426–433.

[G] B. Gr¨nbaum, Convex polytopes, Springer, 1995.

u [GS] B. Gr¨nbaum and V.P. Sreedharan, An Enumeration of Simplicial 4-Polytopes with 8 vertices, u J. Comb. Theory 2 (1967), 437–465.

[HZ] M. Hachimori and G. M. Ziegler, Decomposition of simplicial balls and spheres with knots consisting of few edges, Math. Z. 235 (2000), 159–171.

[K] E. R. van Kampen, Remark on the address of S. S. Cairns, in Lectures in Topology, 311–313, University of Michigan Press, Ann Arbor, MI, 1941.

[Mn] N. Mn¨v, The universality theorems..., in Lecture Notes in Math. 1346, Springer, Berlin, 1988, e 527–543.

[MT] B. Mojar and C. Thomassen, Graphs on surfaces, JHU Press, 2001.

[P] I. Pak, Lectures on discrete and polyhedral geometry, monograph draft; available electronically at http://www.math.ucla.edu/~pak/book.htm.

[R] J. Richter-Gebert, Realization spaces of polytopes, Lecture Notes in Math., Springer, 1996.

[RZ] J. Richter-Gebert and G. M. Ziegler, Realization spaces of 4-polytopes are universal, Bull. Amer.

Math. Soc. 32 (1995), 403–412.

[T1] W. T. Tutte, Convex representations of graphs, Proc. Lond. Math. Soc. 10 (1960), 304–320.

[T2] W. T. Tutte, How to draw a graph, Proc. Lond. Math. Soc. 13 (1963), 743–768.

[Wi] N. Witte, A vertex decomposable 3-ball and 3-sphere with a knot consisting of 6 edges, Electronic Geometry Model No. 2001.05.003, available at http://tinyurl.com/7bxdoxp [Wo] R. Woodroofe, Vertex decomposable graphs and obstructions to shellability, Proc. Amer. Math.

Soc. 10 (2009), 3235–3234.

[Z1] G. M. Ziegler, Lectures on polytopes, Springer, New York, 1995.

[Z2] G. M. Ziegler, Non-rational configurations, polytopes, and surfaces, Math. Intell. 30 (2008), 36–42.

26 IGOR PAK, STEDMAN WILSON Appendix A. Belt construction in figures and numbers It follows from Theorem 1.1 that the belts cannot be given by explicit (integer) coordinates. We give the explicit description of the belts by specifying the arcs on which the triangular facets of the prisms lie, as well as a function describing their lateral lengths (see Figure 12). Note that the prism lengths are made small except at the boundary to ensure that the belts do not intersect. Furthermore, we must ensure that the arcs bend sufficiently to avoid each other at the top of the core. To create the arcs we start with a family of circles, and then apply a parametrized family of rotations to stretch them. The Mathematica code describing the explicit details of the construction, and used to generate the complete irrational complex and the 3D graphics in this paper, can be found at http://www.math.ucla.edu/~stedmanw/research/.

1.0 2.0 0.8 1.5 0.6 1.0 0.4 0.5 0.2 0.5 1.0 1.5 2.0 0.0 0.2 0.4 0.6 0.8 1.0

–  –  –

In each belt, our construction uses 318 triangular prisms, exactly 2(80 − 1) + 1 = 159 prisms per semi-belt. The core consists of 5 triangular prisms and 1 pentagonal pyramid.

The complete irrational complex thus consists of a total of 4 · 318 + 5 = 1277 triangular prisms and 1 pentagonal pyramid, as in the theorem.

GEOMETRIC REALIZATIONS OF POLYHEDRAL COMPLEXES 27

Since the belts come close to intersecting near the boundary of the core, some checking is necessary. In Figure 13 we show how the belts near-miss each other due to their shape.

We conclude with a rotated view of the irrational polyhedral complex.

–  –  –

Department of Mathematics, UCLA Los Angeles, CA 90095, USA Email: {pak, stedmanw}@math.ucla.edu



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