Difference between framing and writhe of a knot

Question

The writhe of an oriented knot is the number of positive crossings minus the number of negative crossings while the framing of a knot is defined to be the linking number of the knot with the curve that is obtained by "pushing" the knot along its normal vector field on the boundary of a closed $ε$-neighborhood. Taking the unknot with one positive Reidemeister $1$ move we obtain a knot diagram of the unknot with $1$ writhe but also with $1$ framing. Is there any connection between those two in general?

Answer 1

If I understand what you mean correctly, your definition of framing is from taking the knot diagram as a knot close to the XY plane, then the normal vector field is the vector field pointing in the Z direction.

If you orient the knot, then you can rotate this vector field counterclockwise 90 degrees around the knot, so then the knot and pushoff are both "in the plane of the diagram." The way you produce the push-off (isotopic to the first method) is by walking along the knot diagram, holding out your right hand, and drawing a parallel knot. The linking number between the knot and this push-off is concentrated at the crossings of the original knot. From this, you get the writhe definition you mention.

Beware that there are many framings of a knot, and the one you describe is known as the blackboard framing. A framing is a trivialization of the normal vector bundle, up to isotopy. Equivalently, it is a choice of normal vector field along the knot, up to isotopy. The framing is completely characterized by a single integer, the linking number between the knot and a push-off along the chosen normal vector field.

Every framing may be represented as the blackboard framing of some diagram. Inserting Reidemeister I loops modifies the blackboard framing by $\pm 1$.

There is a special framing (the $0$-framing) given by a Seifert surface of the knot: the neighborhood of the boundary of the surface gives a normal vector field, and the linking number of the push-off with the knot is $0$.