Combinatorics of embeddings

Melikhov, Sergey A.

doi:10.48550/arxiv.1103.5457

Cited by 4 publications

(5 citation statements)

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“…1.1], Taniyama [15], Melikhov [9,Ex. 4.7] and Melikhov [10,Ex. 4.9] show by various arguments that K n 2n+4 is intrinsically linked, in the sense that every embedding in R 2n+1 contains a pair of disjoint n-spheres that have nonzero linking number.…”

Section: Introductionmentioning

confidence: 99%

Some Ramsey-type results on intrinsic linking ofn–complexes

Tuffley

2013

Algebr. Geom. Topol.

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Define the complete n-complex on N vertices, K n N , to be the n-skeleton of an (N − 1)-simplex. We show that embeddings of sufficiently large complete n-complexes in R 2n+1 necessarily exhibit complicated linking behaviour, thereby extending known results on embeddings of large complete graphs in R 3 (the case n = 1) to higher dimensions. In particular, we prove the existence of links of the following types: r-component links, with the linking pattern of a chain, necklace or keyring; 2-component links with linking number at least λ in absolute value; and 2-component links with linking number a non-zero multiple of a given integer q. For fixed n the number of vertices required for each of our results grows at most polynomially with respect to the parameter r, λ or q.

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Section: Introductionmentioning

confidence: 99%

Some Ramsey-type results on intrinsic linking ofn–complexes

Tuffley

2013

Algebr. Geom. Topol.

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“…In the special case when K i are Van Kampen-Flores complexes ∆ n i 2n i +2 (Example 3.16) this result was proved by B. Grünbaum, so Theorem 4.5 is sometimes referred to as the Van Kampen-Flores-Grünbaum-Schild non-embedding theorem. Sergey Melikhov [Mel11] discovered an interesting connection of this result (and its relatives) with the so called dichotomial cell complexes (dichotomial spheres).…”

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confidence: 82%

Topology of unavoidable complexes

Jojić,

Marzantowicz,

Vrećica

et al. 2016

Preprint

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The partition number π(K) of a simplicial complex K ⊆ 2 [m] is the minimum integer ν such that for each partitionMotivated by the problems of Tverberg-Van Kampen-Flores type we prove several results (Theorems 3.14, 3.18, 4.6) which link together the combinatorics and topology of these two classes of complexes. One of our central observations (Theorem 4.6), summarizing and extending results of G. Schild, B. Grünbaum and many others, is that interesting examples of (almost) r-non-embeddable complexes can be found among the joins K = K 1 * . . . * K s of r-unavoidable complexes.

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“…Consequently its simplicial deleted join U ⊛ U is Z/2homeomorphic to S n [3; Corollary 3.16]. Also there exists a Z/2-map from U ⊛ U to the suspension over the simplicial deleted product U ⊗ U (see [3;Lemma 3.25]). Since U ⊗ U ⊂ Ũ , by the Borsuk-Ulam theorem there exists no Z/2-map Σ Ũ → S n−1 .…”

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confidence: 99%