Here are two ways to think of knots:
<ol><li>As ambient isotopy clases of smooth embeddings of S<sup>1</sup> in S<sup>3</sup>.</li>
<li>As a planar algebra generated by over-crossings and under-crossings, modulo Reidemeister moves.</li>
</ol>
Quantum topology makes ample use of the second viewpoint. But if you're viewing a knot as an element of a planar algebra, or, well, as an operad, then the more natural operad to work with would be one in which endpoints of crossings get matched up, but not necessarily in the plane. Bar-Natan calls this structure a "circuit algebra"  (a modular operad?). In quantum topology, you're looking for a homomorphic expansion of such an operad to some Lie-algebraic object, which carries a parallel operadic structure, such as the Drinfeld double of a finite group. The point now is that a circuit algebra is algebraically better behaved than a planar algebra, and so it's easier to find homomorphic expansions and to calculate them. This is because the planarity restriction for such operadic structures interacts badly with the Lie algebraic structure, so it's algebraically sensible to kill it by passing to circuit algebras. So you obtain invariants which respect the operadic structure (virtual knot invariants extend to virtual tangles- but knot invariants only extend to "non-associative tangles" or to "q-tangles"- not to tangles- because of the presence of associators).