Maybe the connection between knots and primes can be stated in terms of SL(2,R) and SL(2,Z), because these groups are essential in number theory, e.g., modular forms, quadratic forms.

José M


On Jul 19, 2018, at 2:38 PM, Egbert Rijke <e.m.rijke@gmail.com> wrote:

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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