A characterization of simplicial polytopes with g2=1

Eran Nevo; Eyal Novinsky

doi:10.1016/j.jcta.2009.12.011

A characterization of simplicial polytopes with g₂=1

Eran Nevo^*, Eyal Novinsky

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

Abstract

Kalai proved that the simplicial polytopes with g₂=0 are the stacked polytopes. We characterize the g₂=1 case.Specifically, we prove that every simplicial d-polytope (d≥4) which is prime and with g₂=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d≥4) with g₂=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphere (d≥4) with g₂=1, and our proof takes advantage of working with this larger class of complexes.

Original language	English
Pages (from-to)	387-395
Number of pages	9
Journal	Journal of Combinatorial Theory. Series A
Volume	118
Issue number	2
DOIs	https://doi.org/10.1016/j.jcta.2009.12.011
State	Published - Feb 2011
Externally published	Yes

Bibliographical note

Funding Information:
E-mail addresses: [email protected] (E. Nevo), [email protected] (E. Novinsky). 1 Research partially supported by an NSF grant DMS-0757828.

Keywords

Graph rigidity
Homology sphere
Polytope

Access to Document

10.1016/j.jcta.2009.12.011

Cite this

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title = "A characterization of simplicial polytopes with g2=1",

abstract = "Kalai proved that the simplicial polytopes with g2=0 are the stacked polytopes. We characterize the g2=1 case.Specifically, we prove that every simplicial d-polytope (d≥4) which is prime and with g2=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d≥4) with g2=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphere (d≥4) with g2=1, and our proof takes advantage of working with this larger class of complexes.",

keywords = "Graph rigidity, Homology sphere, Polytope",

author = "Eran Nevo and Eyal Novinsky",

note = "Funding Information: E-mail addresses: [email protected] (E. Nevo), [email protected] (E. Novinsky). 1 Research partially supported by an NSF grant DMS-0757828.",

year = "2011",

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doi = "10.1016/j.jcta.2009.12.011",

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AU - Nevo, Eran

AU - Novinsky, Eyal

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N2 - Kalai proved that the simplicial polytopes with g2=0 are the stacked polytopes. We characterize the g2=1 case.Specifically, we prove that every simplicial d-polytope (d≥4) which is prime and with g2=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d≥4) with g2=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphere (d≥4) with g2=1, and our proof takes advantage of working with this larger class of complexes.

AB - Kalai proved that the simplicial polytopes with g2=0 are the stacked polytopes. We characterize the g2=1 case.Specifically, we prove that every simplicial d-polytope (d≥4) which is prime and with g2=1 is combinatorially equivalent either to a free sum of two simplices whose dimensions add up to d (each of dimension at least 2), or to a free sum of a polygon with a (d-2)-simplex. Thus, every simplicial d-polytope (d≥4) with g2=1 is combinatorially equivalent to a polytope obtained by stacking over a polytope as above. Moreover, the above characterization holds for any homology (d-1)-sphere (d≥4) with g2=1, and our proof takes advantage of working with this larger class of complexes.

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KW - Polytope

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