My guess would be that differentiability is irrelevant for knot theory.
You could rather consider closed subsets homeomorphic to S^1. For instance you could restrict your attention to "paths" consisting of injective sequences of integral points
P1, ... , Pi, ..., Pn=P0, where PiPi+1 is parallel to an axis and of length 1. There is probably a corresponding notion of "discrete" isotopy among such paths, so that if two such paths are continuously isotopic, they are also discretely isotopic . As a consequence knots now live in Z^3.
So you can take the higher inductive type H with Z^3 as set of points, and the above (length one) intervals as generating equalities, and study "synthetic knots" in there. The first thing to check is probably that any path in H is homotopic to a path obtained from a sequence of generators.
I am not sure at all that such a synthetic knot theory could be fruitful May-be more interesting is to fill the holes in the H above by adding generating 2-cells one for each elementary square and 3-cells one for each elementary cube. It is not so clear how to do it if you want these cells to be "invertible" More generally given a triangulated variety of dimension 3, you may try to specify as above a higher-inductive type, so that you could try to formulate (and prove...) a synthetic Poincaré conjecture.
Sorry for divagating slightly too much!
ah