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

@article{Parsa2018OnTL, title={On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space}, author={Salman Parsa}, journal={Discrete \& Computational Geometry}, year={2018}, volume={59}, pages={663-679} }

We consider d-dimensional simplicial complexes which can be PL embedded in the 2d-dimensional Euclidean space. In short, we show that in any such complex, for any three vertices, the intersection of the link-complexes of the vertices is linklessly embeddable in the $$(2d-1)$$(2d-1)-dimensional Euclidean space. In addition, we use similar considerations on links of vertices to derive a new asymptotic upper bound on the total number of d-simplices in an (continuously) embeddable complex in 2d…

## 11 Citations

### Correction to: On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space

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In the following we refer to the original paper [1]. The main reason for writing this correction is the incorrect statement of Theorem 4. Section 4.1 should be corrected by replacing [3] = {1, 2, 3},…

### On the Smith classes, the van Kampen obstruction and embeddability of $[3]*K$

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In this survey-research paper, we first introduce the theory of Smith classes of complexes with fixed-point free, periodic maps on them. These classes, when defined for the deleted product of a…

### Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

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We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k…

### Combinatorial Lefschetz theorems beyond positivity

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Consider a simplicial complex that allows for an embedding into $\mathbb{R}^d$. How many faces of dimension $\frac{d}{2}$ or higher can it have? How dense can they be?
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### Geometric Embeddability of Complexes is ∃R-complete

- MathematicsArXiv
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We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k…

### On Codimension One Embedding of Simplicial Complexes

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We study d-dimensional simplicial complexes that are PL embeddable in \(\mathbb{R}^{d+1}\). It is shown that such a complex must satisfy a certain homological condition. The existence of this…

### Geometry and the Simplex: Results, Questions and Ideas

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### A short exposition of S. Parsa's theorem on intrinsic linking and non-realizability

- MathematicsArXiv
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A short statement and proof of the following result by S. Parsa, where L is a graph such that any two disjoint cycles have zero linking number.

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

- MathematicsDiscret. Comput. Geom.
- 2021

This work presents a short exposition of the following result by Salman Parsa, where L admits a PL embedding into R 3 such that any two disjoint cycles have zero linking number.

### Branko Gr\"unbaum in many dimensions.

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Even though he was nearly 90 years old, I was still surprised and sad to hear that Branko Grünbaum had passed away. I took courses from Branko as a graduate student at the University of Washington. I…

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