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This is Cerf's pseudoisotopy-implies-isotopy theorem. Cerf's result is true in high dimensions, while it's independently known in a low-dimensional range. In dimension $2$ it goes back to the Earle–E …

Yes, I believe there are many. For example, if you think of hyperbolic $2$-space as a geodesic subspace of hyperbolic $3$-space, any group of hyperbolic isometries of hyperbolic $2$-space extends natu …

Two different answers using almost identical techniques! Allen's response got me to think through my response more carefully. Let me edit in a comment to point out my sloppiness, as it points out a …

This was done by Schubert, in 1949 "Die eindeutige Zerlegbarkeit eines Knotens in Primknoten". In his original proof he uses a decomposition of pairs $(𝑆^3,𝐾)$ using embedded spheres with two marked …

The first obstruction is that the normal bundle to $N$ in $M$ must have a $1$-dimensional subbundle. That is the obstruction I used in my example involving $TS^2$ in the above comment. If the norma …

Here is a plot of a grid of lines coming out of the transmitter centred at $(0,-0.5)$ in green, together with the first reflection lines, in yellow. There is 300 emission lines. From the picture you …

Although this does not answer your question, there are partial answers in dimension 3. For example, if you construct an orientable 3-manifold via a random Heegaard splitting (constructing the gluin …

You have all the tools to compute the geometric decomposition of knot and link exteriors in the software Regina. I'm one of the authors, although my hands haven't been over that part of the code very …

I don't know the answer to your question, but I asked Fred Cohen. He had this to say: Most of the computations are in Mahowald's work with the EHP sequence. This gives infinite families at p = 2 wit …

The result they use is Moise's theorem: Moise, Edwin E. (1952), Affine structures in 3-manifolds. V. The triangulation theorem and Hauptvermutung, Annals of Mathematics. Second Series 56: 96–114, Th …

Ben Burton has a paper where he experimentally does Pachner moves to simplify unknot complements. It appears to be very effective. https://arxiv.org/abs/1211.1079 I'd type more but my phone is a litt …

There's a point-set model given in terms of a compactification of the configuration space of points in an interval $[0,1]$. This is popular in the "spaces of knots" literature that uses the Goodwill …

Regarding question 1, yes you can always ensure the image is closed. You prove the strong Whitney by perturbing a generic map $M \to \mathbb R^{2m}$ to an immersion, and then doing a local double-poi …

The theorem you're looking for is proven in Cerf's dissertation. J. Cerf, Topologie de certains espaces de plongements, Bull S.M.F., tome 89 (1961) 227-380. This is for the case you mention, when …

You could consider images such as yours to be a knot in $(S^1)^3$ -- a single-component embedded 1-manifold in the "3-torus" $(S^1)^3$. In your case you get the 3-torus by modding out by your periodi …