Question: At around the one minute mark of this video by Professor Tadashi Tokieda, 
(See Link: <https://www.youtube.com/watch?v=CJBfpvWBmSs&list=PLTBqohhFNBE_09L0i-lf3fYXF5woAbrzJ&index=13>)

Professor Tokieda makes the claim that it is only in $\mathbb{R}^n$ where $n>5$ that one can unknot a knotted torus. He claims that this is because a torus is of dimension 2, and the surface spanned by a moving torus is of dimension 3. Hence, it is only in $\mathbb{R}^n$ where $n>5$ that this torus can be unknotted.

I have quite a few difficulties when trying to understand this claim. Intuitively, if every coordinate of the torus is given by $(x\_1,x\_2,x\_3)$ then simply by adding one more degree of freedom, $(x\_1,x\_2,x\_3,x\_4)$ for example, one can prevent any two arms of the torus from intersecting and thus the torus can be unknotted in $\mathbb{R}^4$.

Is there something that I am missing? Could someone help clarify my misgivings here.

Answer: 
In order to un-knot the toroidal knot presented in the video, we essentially need to make the transformation from the left to the right part of the following immage:[![enter image description here](https://i.stack.imgur.com/PTW0J.jpg)](https://i.stack.imgur.com/PTW0J.jpg)

Once we have managed this, it is merely a matter of unwinding the remaining twist, and that is obviously possible.

Note that the above are only two-dimensional images of a three-dimensional situation. However, illustration "tricks" such as lighting, shadows, and occlusion were used in its production such that you can imagine and mentally visualize the full 3d scene. To add a fourth dimension, we need yet another trick: We assign a real number to each point of the surface and identify it with a wavelength in the visible spectrum. In other words, we use colours to represent the fourth dimension.
So in a first step, we embed our knot into 4d space by putting it into the "red hyperplane":

[![enter image description here](https://i.stack.imgur.com/Hn5Qt.jpg)](https://i.stack.imgur.com/Hn5Qt.jpg)

Next, we deform it so that a part of our choice moves to the "blue hyperplane". Continuity dictates that the colour changes smoothly from red via orange, yellow, and green to blue:
[![enter image description here](https://i.stack.imgur.com/SkJb5.jpg)](https://i.stack.imgur.com/SkJb5.jpg)

Now the blue part can be moved past the red part, even though it looks as if it passes through it. Any points that occupy the same 3d coordinates during that move would only occupy the same 4d coordinates if they had the same colour, but that doesn't happen:
[![enter image description here](https://i.stack.imgur.com/6K4LX.jpg)](https://i.stack.imgur.com/6K4LX.jpg) 

Done.
If Tokieda said that tori do not unknot in $R^5$, he was wrong. 

Theorem 1. If $M^k$ is a smooth $k$-dimensional manifold (possibly disconnected) then any two smooth embeddings $M^k\to R^{2k++2}$ are isotopic. (This is due to Whitney, if I remember it correctly, with a proof similar to the proof of Whitney embedding theorem.) In particular, there are no nontrivial surface links in $R^6$. 

Theorem 2. If $M^k$ is a connected $k$-manifold, $k\ge 2$, then any two smooth embeddings $M^k\to R^{2k+1}$ are isotopic. (Wu, 1958) In particular, every 2-torus unknots in $R^5$, contrary to the cited claim. 

Theorem 3. There are nontrivial 2-component smooth links in $R^5$ whose components are 2-tori. (This is a variation on the Hopf link construction.) 

Theorem 4. There are smooth knotted 2-dimensional tori in $R^4$. (Hudson, 1963)

You can read more in [this survey](https://arxiv.org/pdf/math/0604045.pdf) by Skopenkov:

Embeddings and knottings of manifolds in Euclidean spaces.