Here is my main question: what is the difference between "straight" and "piecewise linear" and "continuous" embeddings of graphs/complexes in d-dimensional space?


Moreover I would like to know if any other type of embedding is defined that is in some sense similar to "straight" or "piecewise linear" or "continuous" embeddings for graphs/complexes?



Difference between straight and piecewise linear and continuous embeddings of graphs / complexes in d-dimensional space?
An embedding of an $n$-dimensional simplicial complex $K$ (a graph if $n=1$) into ${\mathbb R}^d$ is *linear* if the restriction to each simplex of $K$ is a linear map. In the stable range, $d>2n+1$, by general position any $K^n$ linearly embeds into ${\mathbb R}^d$.There are various Ramsey-type results, combining combinatorial and topological aspects of such embeddings. For example, a theorem of Negami:
<http://www.ams.org/journals/tran/1991-324-02/S0002-9947-1991-1069741-9/>
asserts that any knot or link type may be realized by a linear embedding of a complete graph (on sufficiently many vertices) into ${\mathbb R}^3$.


An embedding $K^n\hookrightarrow {\mathbb R}^d$ is *piecewise-linear* if there exists a subdivision of $K$ so that the restriction of the embedding to each resulting simplex is linear. (This notion underlies the subject of PL topology, <http://www.springer.com/mathematics/geometry/book/978-3-540-11102-3> )


For $d\leq 2n$ there is an obstruction theory for embeddability of an $n$-complex $K^n$ into ${\mathbb R}^d$, dating back to van Kampen:
<http://link.springer.com/article/10.1007%2FBF02940628>
and extended by Haefliger-Weber:
<http://link.springer.com/article/10.1007%2FBF02564408?LI=true>
This theory and various extensions are described, for example, in the following recent papers:
<http://iopscience.iop.org/0036-0279/54/6/R02> , 
<http://arxiv.org/abs/math/0612082>


Focusing on $d=2n$, $n>2$, it is interesting to note that if there exists a topological (continuous) embedding $K^n \hookrightarrow {\mathbb R}^{2n}$ then the van Kampen obstruction vanishes and so there also exists a piecewise-linear embedding. Then there is a quantitative version of this question: suppose a simplicial complex $K^n$ topologically embeds into ${\mathbb R}^{2n}$, $n>2$. How many subdivisions of $K$ are needed in general for a piecewise-linear embedding? Both upper and lower bounds are given in  

<http://arxiv.org/abs/1311.2667> This paper also discusses the notion of *cell-wise smooth* embeddings, and other notions of complexity of embeddings such as *thickness* and\*distortion\*.


