Positive Plücker tree certificates for non-realizability

Jockusch [J. Combin. Theory Ser. A 72 (1995), pp. 318–321] constructed an infinite family of centrally symmetric (cs, for short) triangulations of 3 3 -spheres that are cs- 2 2 -neighborly. Recently, Novik and Zheng [Adv. Math. 370 (2020), 16 pp.] extended Jockusch’s construction: for all d d and n > d n>d , they constructed a cs triangulation of a d d -sphere with 2 n 2n vertices, Δ n d \Delta ^d_n , that is cs- ⌈ d / 2 ⌉ \lceil d/2\rceil -neighborly. Here, several new cs constructions, related to Δ n d \Delta ^d_n , are provided. It is shown that for all k > 2 k>2 and a sufficiently large n n , there is another cs triangulation of a ( 2 k − 1 ) (2k-1) -sphere with 2 n 2n vertices that is cs- k k -neighborly, while for k = 2 k=2 there are Ω ( 2 n ) \Omega (2^n) such pairwise non-isomorphic triangulations. It is also shown that for all k > 2 k>2 and a sufficiently large n n , there are Ω ( 2 n ) \Omega (2^n) pairwise non-isomorphic cs triangulations of a ( 2 k − 1 ) (2k-1) -sphere with 2 n 2n vertices that are cs- ( k − 1 ) (k-1) -neighborly. The constructions are based on studying facets of Δ n d \Delta ^d_n , and, in particular, on some necessary and some sufficient conditions similar in spirit to Gale’s evenness condition. Along the way, it is proved that Jockusch’s spheres Δ n 3 \Delta ^3_n are shellable and an affirmative answer to Murai–Nevo’s question about 2 2 -stacked shellable balls is given.

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Positive Plücker tree certificates for non-realizability

Cited by 3 publications

References 28 publications

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