TITLE: Framing Integer associated with a Framed Knot/Link
QUESTION [7 upvotes]: I have been looking for a clear definition of the n-framing of a knot unsuccessfully. A framing here refers to a choice of homeomorphism between a solid torus neighborhood (a.k.a, tubular neighborhood) of a knot and $S^1 \times D^2$ .I vaguely understand the number (which is an integer ) has to see with the number of times a basis for the normal space to the knot turns when doing a full $2\pi$ turn about the knot. I also know there are framings defined by a vector field V , where $V(k)$, the image of the (nowhere-zero)vector field at a point $k$ in the knot $K$ determines the framing. I assume a 0-framing means we take a normal vector and somehow parallel-displace it along the knot (having chosen a Riemannian metric $g$ to define the displacement), so that it returns exactly to its initial condition.I know that the framing number is an isotopy invariant. But this is not enough for me to know how to assign an integer to a framing.  Any ideas, refs, please?

REPLY [2 votes]: This is my best understanding hopefully will help illustrate, together with other responses:
1) We choose a normal vector field for the core of  $T:=S^1 \times D^2$ . Then the framing number is the winding number of the normal vector field about the core.
2)Say we select l, a longitude , as a basis element for $H^1 (T;\mathbb Z)$. Then the framing number is the linking number of the longitude with the core.
3)Any two n-framings are isotopic.