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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 …

A standard reference is: F. Sergeraert "Un theoreme de fonctions implicites sur certains espaces de Frechet et quelques applications," Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 599-660. This isn't a st …

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 …

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 …

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 …

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 …

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 …

I don't believe it's written up anywhere. edit: in the comments Charlie Frohman corrects me: MR2128054 (2005m:57041) Roseman, Dennis(1-IA) Elementary moves for higher dimensional knots. (Eng …

A standard reference for this is Hirsch's Differential Topology textbook. If $X$ is compact near all the topologies you'd like to consider are essentially the same. Sometimes they're called the Whit …

Chad Giusti calls these "Plumbers' Knots": https://arxiv.org/abs/0811.2215 https://arxiv.org/abs/1107.4717 In the first paper Giusti gives the space of plumbers knots a natural stratification which is …

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 …