http://www.youtube.com/watch?v=IKSrBt2kFD4
Above is the youtube link to the short video called 'Knots to Narnia'. (9 mins)
While learning knot theory, I found this interesting video. In the video, William Thurston talks about 'universes' and 'branching'. In particular, he describes a way to interpret the regions of a knot projection as certain 'gateways', and he also says that the structure of this 'branched universe' can tell us two different knots apart.
I found his words very interesting, and wanted to follow the discussion further, but I failed to find more information. If anyone could provide a reference with further study along these lines, I will greatly appreciate it.
※P.S. Please feel free to add tags if you find an appropriate one missing, since I'm still not really good at it.
One interpretation is that he is talking about cyclic branched coverings of knots. This is definitely talked about in Rolfsen's "Knots and Links" but is probably also described in any other book about knot theory, Lickorish could well be a better option. In particular, there is a lot of interest in double branched coverings due to the Montesinos trick, which is nicely described here:
https://sketchesoftopology.wordpress.com/2008/01/25/double-branched-covers-of-rational-tangles-and-the-montesinos-trick/
http://ldtopology.wordpress.com/2009/07/27/the-montesinos-trick/