Countably many knots

Question

I think that there are only countably many knots in $S^3$ up to isotopy. How does one prove a statement like this? Similarly, I think it is true that there are countably many smooth structures on a closed manifold up to diffeomorphism. This should follow from the statement for knots and Morse theory.

Answer 1

Every link in $S^3$ has a diagram, and a diagram can be thought of as a $4$-valent planar graph whose vertices are marked with which pair of opposite incident half edges are the understrand. One can can give the combinatorial data of a diagram (of an unoriented link) by labeling each edge with a distinct integer, then giving a set of $4$-tuples of edge labels, with, say, the first and third labels corresponding to the edges of the understrand. Countability of links follows.

For closed $3$-manifolds, the Lickorish-Wallace theorem says that every such manifold is $\pm1$ Dehn surgery on a framed link in $S^3$. The framing can be encoded via the blackboard framing. Countability of $3$-manifolds follows.

Alternatively, every closed $3$-manifold has a finite PL triangulation, and the data for such is finite.