The perfect analogy to knots would be planar and nonplanar graphs. It is always possible to draw a graph without intersections in 3D, but in 2D some graphs are

The perfect analogy to knots would be planar and nonplanar graphs. It is always possible to draw a graph without intersections in 3D, but in 2D some graphs are impossible to draw without intersections (nonplanar graphs), such as 5-vertices complete graph.