Notes for AKT-090924-1/0:30:11

Claim 1: There is a bijection between round knots (i.e. knots on a circle) and long knots (i.e. knots on a long line):

${\mathcal {K}}(\bigcirc )={\mathcal {K}}(|)$

Claim 2: ${\mathcal {K}}(|)$ is an abelian monoid.

Remark: I think it's worth checking that the map from circle to line is independent of the choice of point to 'open up' and the path we 'pull out' the two ends after cutting. However it is indeed independent.