Knot diagrams are a special sort of tangle diagrams, so I will reinterpret your question as being about tangle diagrams. Tangle diagrams are a "planar algebra" generated by ${\text{overcrossing},\text{undercrossing}}$, so every tangle can be drawn by taking a finite collection of generators, arranging them in a plane, and connecting each of the four "loose ends" on each generator by "bridge arcs" to another of the "loose ends" (of the same or a different generator), or leaving "the loose end" "loose". The relations are Reidemeister relations. If you allow "bridge arcs" to cross (and allow virtual Reidemeister moves), you get virtual tangles, and if not, you get usual tangles.