![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png) simont |
Fri 2002-11-15 01:13 |
I think it's both, in fact.
I'm fairly sure it's possible to continuously deform a torus-with-knotted-hole into a torus-with-unknotted-hole using only the three dimensions you're given and nothing up your sleeve. I've got a Martin Gardner book which shows the transformation if you're interested, but essentially the trick involves moving one end of the hole around to the other end and then further into the hole itself.
However, that doesn't deal with the case of a torus where the substance, rather than the hole in the middle, is tied in a knot. I don't think you can continuously deform that one into an ordinary unknotted torus; so in that situation you'd have to invent an extra dimension in which to untie it. So although as it happens you don't need the extra dimensions for the knotted-hole case, you would have been allowed to use them if you'd needed them.
My understanding of topology, though, is that all this continuous-deformation stuff is actually a red herring. The true definition of topological equivalence is that there exists a bijection from the points of one surface to the points of the other, such that neighbourhoods are preserved (points close together on one surface end up close together on the other). Now clearly a continuous deformation of one surface into the other (using any number of additional dimensions) implies the existence of such a bijection, but it doesn't work the other way round - two surfaces can be shown to be topologically equivalent without the need to exhibit such a continuous deformation. |
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