Knot (mathematics)

In mathematics, a knot is an embedding of a circle S1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies). A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S j in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory.Reidemeister move 1Reidemeister move 2Reidemeister move 3 In mathematics, a knot is an embedding of a circle S1 in 3-dimensional Euclidean space, R3 (also known as E3), considered up to continuous deformations (isotopies). A crucial difference between the standard mathematical and conventional notions of a knot is that mathematical knots are closed—there are no ends to tie or untie on a mathematical knot. Physical properties such as friction and thickness also do not apply, although there are mathematical definitions of a knot that take such properties into account. The term knot is also applied to embeddings of S j in Sn, especially in the case j = n − 2. The branch of mathematics that studies knots is known as knot theory, and has many simple relations to graph theory. A knot is an embedding of the circle (S1) into three-dimensional Euclidean space (R3). or the 3-sphere, S3, since the 3-sphere is compact. Two knots are defined to be equivalent if there is an ambient isotopy between them. A knot in R3 (or alternatively in the 3-sphere, S3), can be projected onto a plane R2 (respectively a sphere S2). This projection is almost always regular, meaning that it is injective everywhere, except at a finite number of crossing points, which are the projections of only two points of the knot, and these points are not collinear. In this case, by choosing a projection side, one can completely encode the isotopy class of the knot by its regular projection by recording a simple over/under information at these crossings. In graph theory terms, a regular projection of a knot, or knot diagram is thus a quadrivalent planar graph with over/under-decorated vertices. The local modifications of this graph which allow to go from one diagram to any other diagram of the same knot (up to ambient isotopy of the plane) are called Reidemeister moves. The simplest knot, called the unknot or trivial knot, is a round circle embedded in R3. In the ordinary sense of the word, the unknot is not 'knotted' at all. The simplest nontrivial knots are the trefoil knot (31 in the table), the figure-eight knot (41) and the cinquefoil knot (51). Several knots, linked or tangled together, are called links. Knots are links with a single component. A polygonal knot is a knot whose image in R3 is the union of a finite set of line segments. A tame knot is any knot equivalent to a polygonal knot. Knots which are not tame are called wild, and can have pathological behavior. In knot theory and 3-manifold theory, often the adjective 'tame' is omitted. Smooth knots, for example, are always tame. 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. A framed knot can be seen as the embedded ribbon and the framing is the (signed) number of twists. This definition generalizes 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) type I, II, and III moves.Given a knot, one can define infinitely many framings on it.Suppose thatwe are given a knot with a fixed framing. One may obtain a new framing from the existing one by cuttinga ribbon and twisting it an integer multiple of 2π around the knot and then glue back again in the placewe did the cut. In this way one obtains a new framing from an old one, up to the equivalence relationfor framed knots„ leaving the knot fixed. The framing in this sense is associated to the number of twiststhe vector field performs around the knot. Knowing how many times the vector field is twisted aroundthe knot allows one to determine the vector field up to diffeomorphism, and the equivalence class of theframing is determined completely by this integer called the framing integer