Simple closed curves embedded in 3-space can be entangled even when they are unknotted (and unlinked). This kind of entanglement is geometrical rather than topological. An example from polymer physics is the super-coiling of DNA molecules. One can try to characterise this kind of entanglement by computing the writhe of the space curve. Consider a simple closed curve in 3-space, and attach an orientation to the curve. If the curve is projected onto a plane it will usually cross itself and each crossing can be associated with +1 or -1 according to the relative orientation of the arrows at the crossing, using a right hand rule. Form the sum of these signed crossing numbers. In general this sum will depend on the projection direction, so average over all projection directions. This average is the WRITHE of the curve. If the curve lives on the simple cubic lattice then the computation of the writhe can be reduced to the computation of the average linking number of the curve with pushoffs in four suitably chosen directions. Since linking number is an integer, this means that the writhe of a simple closed curve on the simple cubic lattice is rational. (This is a theorem due to Chris Lacher and De Witt Sumners.) Surprisingly perhaps, the writhe of a simple closed curve on the face-centred cubic lattice is not always rational.

Some useful books on differential geometry are:

D.J. Struik: Lectures on classical differential geometry [Dover], and E. Kreyszig: Differential geometry [Dover].