The question in my mind basically is: can quantum entanglement, at least for the finite-dimensional cases, be expressed in terms of knot theory? Has there been any approach in this direction? If yes, then I suppose it would be a nice way to visualize qubit entanglements in various scenarios.
In this paper, the authors used Knot theory to define what they call 'Path Model Representation'. In a later section they convert this representation to qubits by switching to binary.