Framed knot

Framed knot

In the mathematical theory of knots, a framed knot is the extension of a tame knot to an embedding of the solid torus "D"2 × "S"1 in "S"3.

The framing of the knot is the linking number of the image of the ribbon "I" × "S"1 with the knot. As explained in (Kauffman 1990), a framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists. This definition generalises to an analogous one for framed links. Framed links are said to be "equivalent" if their extensions to solid tori 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. A type I Reidemeister move 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 if they are connected by a sequence of (modified) I, II, and III moves.

References

*L. H. Kauffman: [http://www.math.uic.edu/~kauffman/IRH.pdf "An invariant of regular isotopy"] , "Transactions of the American Mathematical Society" 318(2), 1990, pp. 317–371


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