Framed knot
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In the mathematical area of knot theory, a framed knot is a tame knot with a particular choice of an extension to an embedding of the solid torus D2 x S1. The framing of the knot is the linking number of the image of 1 x S1 with the knot. By considering the image of the annulus, or "ribbon", given by I x S1, a framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists. An analogous definition is made for framed links. Framed links are said to be equivalent if their extensions to solid torii are ambient isotopic.
Framed link diagrams are link diagrams with each component marked, to indicate framing, by an integer representing a slope with respect to the meridian and preferred longitude. A standard way to view a link diagram without markings as representing a framed link is to use the blackboard framing. This framing is obtained by converting each component to a ribbon lying flat on the plane. Type I Reidemeister moves clearly changes the blackboard framing (it changes the number of twists in a ribbon), but the other two moves do not. Replacing the type I move by a modified type I move gives a result for link diagrams with blackboard framing similar to the Reidemeister theorem: link diagrams, with blackboard framing, represent equivalent framed links if and only they are connected by a sequence of (modified) I, II, and III moves.
