Once I get some 3D models of knots created, is there a way to let the user explore them, in-browser? Preferably they would be able to interactively control the rotation.

things that display 3D models from a file

yes, it's minimally possible

THere are LOTS of code that produces semi-3D drawings on top of the 2D canvas element. This is all I really need, I really don't need heavy-duty 3D rendering.

WebGL