If a strip of paper is knotted into an open trefoil, what is the linking number? It is assumed that the paper strip is knotted into an open trefoil (forming a pentangle) that lies flat on the table, and that the two ends of the paper strip are continued up to spatial infinity. (See an image at http://upload.wikimedia.org/wikipedia/commons/4/4f/Overhand-folded-ribbon-pentagon.svg )
For a beginner in knot theory, the question has several issues:
1) Is the linking number defined for open ribbons?
2) Is it a topological invariant of open ribbons?
3) If so, is there an easy way to read it off the drawing?
4) Is the linking number the same as the (planar) writhe of one of the two edges of the paper strip?
 A: *

*The two boundary components of a ribbon are knots themselves, and the linking number between them is well-defined.  It can also be thought of as an isotopy invariant of solid tori knots, with the ribbon being an extruded diameter inside the torus.

*If the ribbon is embedded reasonably enough so that the closure of the ribbon has two knots as boundary components, then it is still well-defined up to isotopy.

*Loosen up the knot so that the two boundary components are in "general position," which means the only way they they intersect in a diagram is at isolated vertices (so the linked picture won't work).  Orient the components and number them, and then use the usual linking number formula using the crossings.

*The writhe of a knot diagram is all of this when you give it the so-called blackboard framing: thicken the knot into a ribbon so that the ribbon remains parallel to the page (no folds).  If you take an arbitrary strip, you can perform Reidemeister I moves to eliminate folds (while stretching the paper), and then the writhe of a boundary edge is the linking number of the boundaries of the original strip.  Writhe is a property of diagrams, not knots; or, writhe is a property of paper strips, not knots.
