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?
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*.
