The complete enumeration of 4-polytopes and 3-spheres with nine vertices

Firsching, Moritz

doi:10.1007/s11856-020-2070-4

Cited by 9 publications

(12 citation statements)

References 31 publications

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“…As many combinatorially distinct d ‐polytopes can have the same f ‐vector, an f ‐vector is not inscribable if every polytope with this f ‐vector is not inscribable. Firsching [11] extended previous classifications of 4‐polytopes with few vertices by Altshuler and Steinberg [2] and Brinkmann [4]. For a thorough historical account, we refer to [11, Section 1.4] and the references therein.…”

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

“…Firsching [11] extended previous classifications of 4‐polytopes with few vertices by Altshuler and Steinberg [2] and Brinkmann [4]. For a thorough historical account, we refer to [11, Section 1.4] and the references therein. A complete enumeration of all 4‐polytopes with

f_{0} ⩽ 9

f_{3} ⩽ 9

exists and partial results are known for

f_{0}, f_{3} ⩾ 10

and

20 ⩽ f_{0} + f_{3} ⩽ 23

.…”

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

“…In each of Tables 1 and 2, an entry is surrounded by “⋆” symbols, since they are of particular interest: (14,28,21,7)The dual of the cyclic polytope C 4 (7) has this f ‐vector, we discuss two strategies to find an inscribed realization in Sections 3.2 and 3.3. It is well‐known that the f ‐vector of C 4 (7) and its dual determine combinatorially unique polytopes [11, Table 6; 15, Chapter 6.3]. (20,40,28,8)Besides the cyclic polytope C 4 (8), there are precisely two other neighborly 4‐polytopes on eight vertices [2]. Their associated dual polytopes are the only polytopes that have this f ‐vector.…”

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

“…In combination with Corollary D, the smallest f ‐vector (in terms of the sum

f_{0} + f_{3}

) that is not inscribable, must satisfy

16 ⩽ f_{0} + f_{3} ⩽ 28

. According to the complete classifications given in [4, Table 2.3; 11, Tables 6 and 7], there are ten f ‐vectors that satisfy

16 ⩽ f_{0} + f_{3} ⩽ 19

and that determine a combinatorially unique polytope. These f ‐vectors are: (9,26,26,9)We provide an inscribed realization of this polytope below; (7,18,19,8), (7,17,19,9), (9,19,17,7) and (7,18,22,11)Inscribing coordinates for these f ‐vectors are provided in Appendix A; (8, 19, 20, 9) and (9, 20, 19, 8)If we label the vertices of the first polytope by 1, …, 8, then the facets are five tetrahedra, 1234, 2568, 2578, 2678 and 5678, and four 3‐faces 12 356, 12 457, 134 567 and 23 467.…”

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

“…The f ‐vector

f_{T_{1}}

has a unique associated combinatorial type of 4‐polytope, see [11, Table 7]. If we label the vertices 0, 1, …, 8, then the facets are two tetrahedra 0123 and 5678 and 3‐faces

01 245, 01 346, 02 347, 123 567, 24 578, 14 568 and 34 678 .

To realize this polytope (as a Schlegel projection into the facet 123 567), start with an octahedron 123 567 (with diagonals 17, 26 and 35), and cone over vertex 4, placed in its center.…”

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

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Combinatorial Inscribability Obstructions for Higher Dimensional Polytopes

et al. 2020

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For 3-dimensional convex polytopes, inscribability is a classical property that is relatively well-understood due to its relation with Delaunay subdivisions of the plane and hyperbolic geometry. In particular, inscribability can be tested in polynomial time, and for every f-vector of 3-polytopes, there exists an inscribable polytope with that f-vector. For higher dimensional polytopes, much less is known. Of course, for any inscribable polytope, all of its lower dimensional faces need to be inscribable, but this condition does not appear to be very strong. We observe non-trivial new obstructions to the inscribability of polytopes that arise when imposing that a certain inscribable face be inscribed. Using this obstruction, we show that the duals of the 4-dimensional cyclic polytopes with at least eight vertices-all of whose faces are inscribable-are not inscribable. This result is optimal in the following sense: We prove that the duals of the cyclic 4-polytopes with up to seven vertices are, in fact, inscribable. Moreover, we interpret this obstruction combinatorially as a forbidden subposet of the face lattice of a polytope, show that d-dimensional cyclic polytopes with at least d + 4 vertices are not circumscribable, and that no dual of a neighborly 4-polytope with eight vertices, that is, no polytope with f-vector (20,40,28,8), is inscribable.

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Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

f_{0} ⩽ 9

f_{3} ⩽ 9

exists and partial results are known for

f_{0}, f_{3} ⩾ 10

and

20 ⩽ f_{0} + f_{3} ⩽ 23

.…”

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

“…In combination with Corollary D, the smallest f ‐vector (in terms of the sum

f_{0} + f_{3}

) that is not inscribable, must satisfy

16 ⩽ f_{0} + f_{3} ⩽ 28

. According to the complete classifications given in [4, Table 2.3; 11, Tables 6 and 7], there are ten f ‐vectors that satisfy

16 ⩽ f_{0} + f_{3} ⩽ 19

Section: Inscribability and Small F‐vectorsmentioning

confidence: 99%

“…The f ‐vector

f_{T_{1}}

has a unique associated combinatorial type of 4‐polytope, see [11, Table 7]. If we label the vertices 0, 1, …, 8, then the facets are two tetrahedra 0123 and 5678 and 3‐faces

01 245, 01 346, 02 347, 123 567, 24 578, 14 568 and 34 678 .

To realize this polytope (as a Schlegel projection into the facet 123 567), start with an octahedron 123 567 (with diagonals 17, 26 and 35), and cone over vertex 4, placed in its center.…”