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].