On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space

Parsa, Salman

doi:10.1007/s00454-017-9936-1

Cited by 8 publications

(14 citation statements)

References 24 publications

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“…So let me give reasons for presenting the footnotes. Part (b) describes relations of the current note to [Pa15,Pa18]. Some of these relations might seem to be a matter of opinion.…”

Section: Herementioning

confidence: 99%

“…This paper provides short proofs of Theorems 1, 3 and 4 below. It can be considered as a complement to Zentralblatt review of [Pa15]. Denote [N] := {1, .…”

Section: A Short Exposition Of S Parsa's Theorems On Intrinsic Linkimentioning

confidence: 99%

“…by changing 'the Remark' to '[Pav5, Remark after Theorem 1]'; I replaced references to a version we discussed by references to publicly available version, in which the corresponding passages are the same. S. Parsa's responses and [Pav5] are public results of our extensive private discussion, not his first reaction and first version of an update. So the footnotes in part (b) explain • why at some point I decided to publish this note instead of sticking to my initial suggestion to S. Parsa of adding this text to his update of [Pa15], • why later I decided to end our private discussion and start a discussion to be published, • why at last I decided to end this public discussion, and why I did not start analogous discussion of Theorem 3 (in spite of my wish not to suppress S. Parsa's opinion).…”

Section: Herementioning

confidence: 99%

“…S. Parsa's responses and [Pav5] are public results of our extensive private discussion, not his first reaction and first version of an update. So the footnotes in part (b) explain • why at some point I decided to publish this note instead of sticking to my initial suggestion to S. Parsa of adding this text to his update of [Pa15], • why later I decided to end our private discussion and start a discussion to be published, • why at last I decided to end this public discussion, and why I did not start analogous discussion of Theorem 3 (in spite of my wish not to suppress S. Parsa's opinion). I have no more time resource to comment on S. Parsa's responses, so I ask the reader to find them in updates of [Pa18] and to check them accurately (just as the reader will check accurately the remarks in this note).…”

Section: Herementioning

confidence: 99%

“…See also an exposition of this idea in [Sk14, §3.3, §3.6] and a technical presentation of this idea in [Pa15, the last paragraph of proof of Lemma 1]. Although [Pa15,Pav5] refer to [Sk03], reference to [Sk03, Example 2] is missing from [Pa15] In Theorems 3 and 4 of [Pa15,Pa18] it is not clear and it is not accurately stated whether the constant involved in O depends on d (or on n). In Lemma 2 of [Pa15,Pa18] it is not clear and it is not accurately stated whether the constant involved in O depends on n, d; it is not clear why the bound for triple intersections is denoted by f (n) but m is not denoted by g(n).…”

Section: Herementioning

confidence: 99%

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A Short Exposition of Salman Parsa’s Theorems on Intrinsic Linking and Non-realizability

Skopenkov

2019

Discrete Comput Geom

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We present a short exposition of the following results by S. Parsa. Let L be a graph such that the join L * {1, 2, 3} (i.e. the union of three cones over L along their common bases) piecewise linearly (PL) embeds into R 4 . Then L admits a PL embedding into R 3 such that any two disjoint cycles have zero linking number.There is C such that every 2-dimensional simplicial complex having n vertices and embeddable into R 4 contains less than Cn 5/3 simplices of dimension 2.We also present corrected statement and proof of the analogue of the second result for intrinsic linking.This paper provides short proofs of Theorems 1, 3 and 4 below. It can be considered as a complement to Zentralblatt review of [Pa15]. Denote [N] := {1, . . . , N}.Theorem 1 (S. Parsa). Let L be a graph such that the join L * [3] (i.e. the union of three cones over L along their common bases) piecewise linearly (PL) embeds into R 4 . Then L admits a PL embedding into R 3 such that any two disjoint cycles have zero linking number. This is formally a corollary of [Pa15, Theorem 1] but essentially a restatement of [Pa15, Theorem 1] accessible to non-specialists. Theorem 1 trivially generalizes to a d-dimensional finite simplicial complex L and embeddings L → R 2d+1 , L * [3] → R 2d+2 . See also Remark 7. Proof of Theorem 1. Consider L * [3] as a subset of R 4 . Take a small general position 4-dimensional PL ball ∆ 4 containing the point ∅ * 1 ∈ R 4 . Then the intersection ∂∆ 4 ∩ (L * [3]) is PL homeomorphic to L. Let us prove that this very embedding of L into the 3-dimensional sphere ∂∆ 4 satisfies the required property. Take any two disjoint oriented closed polygonal lines β, γ ⊂ ∂∆ 4 ∩ (L * [3]) ∼ = L. Then (β * {1, 2}) − Int ∆ 4 and (γ * {1, 3}) − Int ∆ 4 are two disjoint 2-dimensional PL disks in R 4 − ∆ 4 whose bound aries are β and γ. Hence β and γ have zero linking number in the 3-dimensional sphere ∂∆ 4 (by the following well-known lemma applied to the 4-dimensional ball S 4 − Int ∆ 4 ).Lemma 2. If two disjoint oriented closed polygonal lines in the 3-dimensional sphere ∂D 4 bound two disjoint 2-dimensional PL disks in the 4-dimensional ball D 4 , then the polygonal lines have zero integer linking number in ∂D 4 . I am grateful to A. Kupavskii, M. Skopenkov, M. Tancer and S. Parsa for useful private discussions, and to S. Parsa for public discussion presented in Remark 7.b.

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“…So let me give reasons for presenting the footnotes. Part (b) describes relations of the current note to [Pa15,Pa18]. Some of these relations might seem to be a matter of opinion.…”

Section: Herementioning

confidence: 99%

“…This paper provides short proofs of Theorems 1, 3 and 4 below. It can be considered as a complement to Zentralblatt review of [Pa15]. Denote [N] := {1, .…”

Section: A Short Exposition Of S Parsa's Theorems On Intrinsic Linkimentioning

confidence: 99%