Combinatorics of Polytopes

Some (mostly on-line) resources

Books:

• B. Grünbaum, Convex polytopes (library link).
• G. Ziegler, Lectures on polytopes (library link).
• H.S.M. Coxeter, Regular polytopes, New York: Pitman, 1948 ( library link).
• R.R. Thomas, Lectures in Geometric Combinatorics, American Mathematical Soc., 2006 ( library link).
• Geometric combinatorics, edited by E. Miller, V. Reiner, B. Sturmfels. Providence, R.I.: American Mathematical Society; Oxford : Oxford University Press, 2007 ( library link).
• V.M.Buchstaber, T.E.Panov, Toric Topology (see Chapters 1-2).

Lecture notes and unpublished books:

• A. Barvinok, Combinatorics of polytopes Lecture notes, 2010.
• I. Pak, Lectures on Discrete and Polyhedral Geometry.
• Handbook of Discrete and Computational Geometry, Third edition, edited by Jacob E. Goodman, Joseph O'Rourke, and Csaba D. Tóth.
• K. Fukuda, Frequently Asked Questions in Polyhedral Computation.

Some papers:

• V.M. Buchstaber, N.Yu. Erokhovets, Fullerenes, Polytopes and Toric Topology.
• G. Kalai, Some Aspects of the Combinatorial Theory of Convex Polytopes.
• G. Ziegler, Convex Polytopes: Extremal Constructions and f-Vector Shapes.

Some special classes of polytopes:

Associahedra, permutahedra, nestohedra, ...:

• C. Lange, Associahedral Structures in Algebra, Combinatorics and Geometry.
• C. Lange, V. Pilaud, Associahedra via Spines.
• S. Fomin and N. Reading, Root Systems and Generalized Associahedra.
• A. Postnikov, Permutohedra, Associahedra, and beyond.
• A. Postnikov, V. Reiner, L. Williams, Faces of Generalised Permutahedra.
• V. Pilaud, Which Nestohedra are Removahedra?.

• G. Ziegler, Lectures on 0/1 polytopes.

• M. Deza, M. D. Sikiric, Fullerenes: applications and generalizations, slides.
• N. Erokhovets, Construction of fullerenes and Pogorelov polytopes, slides (on joint work with V. M. Buchstaber).

• A. Felikson, Hyperbolic Coxeter polytopes, webpage.

Open problems:

• J. A. De Loera, Easy-to-Explain but Hard-to-Solve Problems About Convex Polytopes , slides.
• J. Erickson, Open problems (webpage on open problems in combinatorics and geometry, including many problems on polytopes).

Webpages:

• George W. Hart, Virtual Polyhedra, The Encyclopedia of Polyhedra .
• Mathieu Dutour Sikirić, Regular, SemiRegular, Regular faced and Archimedean polytopes .
• S. Forcey, Hedra Zoo: Encyclopedia of Combinatorial Polytope Sequences... .

Videos:

• Jos Leys, Étienne Ghys, Aurélien Alvarez, Dimensions (9 short videos, including visualisations of 4-dimensional regular polytopes in videos videos 3 and 4).