Thurston, Knots to Narnia

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Recommends WillO · 2018-03 · Mathematics Stack Exchange · on “Understanding a Bill Thurston popularization of knot complements”

In this video [this video] (which I've only watched once, so maybe I missed something), Bill Thurston illustrates the topology of knot complements in a fanciful way. First he displays a large unknot (i.e. a circle of wire), and asks us to imagine that when you step through the wire, you are transported to a sort of mirror universe (which he calls Narnia). I assumed at first that he wanted us to imagine that we live in the universal cover of the knot complement, so that stepping through to Narnia means traveling to another sheet of that cover. But this is clearly not what he means, first because there is only one Narnia (as opposed to countably many of them) and second because he says you can walk

Cites user48078 · 2014-09 · Mathematics Stack Exchange · on “William Thurston's 'Knots to Narnia'”

[this video] 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

Cites Kyle Miller · 2021-03 · Mathematics Stack Exchange · on “The 3-component hopf link covering the trefoil”

Let's start by fixing some notation. The ordered configuration space of three distinct points in $\mathbb{C}$ is $$\operatorname{OConf}^3(\mathbb{C})=\mathbb{C}^3-\Delta^3\mathbb{C}$$ where $\Delta^3\mathbb{C}=\{(z_1,z_2,z_3)\in\mathbb{C}^3\mid \exists i\neq j, z_i=z_j\}$ , the fat diagonal [LINK] . Another way to write the fat diagonal is as the zero set of a polynomial. Anticipating discriminants, we'll use $(z_1-z_2)^2(z_1-z_3)^2(z_2-z_3)^2$ . The symmetric group $S_3$ acts on $\mathbb{C}^3$ by permutation of coordinates, and furthermore it acts on $\Delta^3\mathbb{C}$ . Hence, $S_3$ acts on $\operatorname{OConf}^3(\mathbb{C})$ . It does so in a properly discontinuous way, and thus the qu

Cites Kyle Miller · 2021-11 · Mathematics Stack Exchange · on “How to tell if a straight line will pass through a closed loop in 3-space?”

Let's say a line is through a knot if it's not possible to disentangle the knot from the line in the following sense: [LINK] For this to not be trivial, the line must be infinitely long, otherwise you can just drag the knot to the end and off the line, then drag it back. (Equivalently, we can consider manipulating the knot diagram itself while having the knot avoid the blue line that goes perpendicularly through the diagram.) There's a topological trick we can use to reimagine this problem. The main observation is that any disentangling will only ever make use of a some bounded region of the infinite space. This means that if it's possible to disentangle the knot, it's also possible to disen