Just for the record, a trivial but nice observation: the unknot is the
representation of S3 as the join of S1 with itself:

           p2
S1 x S1 ------> S1
   |            |
   | p1         |
   V            V
  S1 ---------> S3

Unfortunately, no-one has studied SL(2,R) and SL(2,Z) in HoTT yet. Although
a trival way to obtain them is through cohesive HoTT would be nice if it
were possible to define these spaces directly as higher inductive types.

Best,
Egbert

On Thu, Jul 19, 2018 at 1:56 PM, Daniel R. Grayson <
danielrichardgrayson@gmail.com> wrote:

> Quillen identified the complement of the trefoil knot with
> SL(2,R)/SL(2,Z), and the proof is
> on page 84 of Milnor's book "Introduction to algebraic K-theory".
>
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