Framed knot
Encyclopedia
In the mathematical theory of knots
, a framed knot is the extension of a tame knot to an embedding of the solid torus
D2 × S1 in S3.
The framing of the knot is the linking number
of the image of the ribbon I × S1 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.
Knot theory
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life in shoelaces and rope, a mathematician's knot differs in that the ends are joined together so that it cannot be undone. In precise mathematical language, a knot is an embedding of a...
, a framed knot is the extension of a tame knot to an embedding of the solid torus
Solid torus
In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
D2 × S1 in S3.
The framing of the knot is the linking number
Linking number
In mathematics, the linking number is a numerical invariant that describes the linking of two closed curves in three-dimensional space. Intuitively, the linking number represents the number of times that each curve winds around the other...
of the image of the ribbon I × S1 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
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
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
Reidemeister move
In the mathematical area of knot theory, a Reidemeister move refers to one of three local moves on a link diagram. In 1926, Kurt Reidemeister and independently, in 1927, J.W. Alexander and G.B...
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.