I will discuss various notions of $k$-width for both manifolds and embeddings of manifolds in Euclidian space, with special emphasis on three dimensional manifolds and their embeddings in $\mathbb{R}^4$. I will largely follow my recent arXiv posting on this subject.
Links:
[1] https://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/node/3444
[3] https://www.mpim-bonn.mpg.de/node/9096