Ambient isotopyIn the mathematical subject of topology, an ambient isotopy, also called an h-isotopy, is a kind of continuous distortion of an ambient space, for example a manifold, taking a submanifold to another submanifold. For example in knot theory, one considers two knots the same if one can distort one knot into the other without breaking it. Such a distortion is an example of an ambient isotopy. More precisely, let and be manifolds and and be embeddings of in .
Hyperbolic linkIn mathematics, a hyperbolic link is a link in the 3-sphere with complement that has a complete Riemannian metric of constant negative curvature, i.e. has a hyperbolic geometry. A hyperbolic knot is a hyperbolic link with one component. As a consequence of the work of William Thurston, it is known that every knot is precisely one of the following: hyperbolic, a torus knot, or a satellite knot. As a consequence, hyperbolic knots can be considered plentiful. A similar heuristic applies to hyperbolic links.
Knot polynomialIn the mathematical field of knot theory, a knot polynomial is a knot invariant in the form of a polynomial whose coefficients encode some of the properties of a given knot. The first knot polynomial, the Alexander polynomial, was introduced by James Waddell Alexander II in 1923. Other knot polynomials were not found until almost 60 years later. In the 1960s, John Conway came up with a skein relation for a version of the Alexander polynomial, usually referred to as the Alexander–Conway polynomial.
Prime knotIn knot theory, a prime knot or prime link is a knot that is, in a certain sense, indecomposable. Specifically, it is a non-trivial knot which cannot be written as the knot sum of two non-trivial knots. Knots that are not prime are said to be composite knots or composite links. It can be a nontrivial problem to determine whether a given knot is prime or not. A family of examples of prime knots are the torus knots. These are formed by wrapping a circle around a torus p times in one direction and q times in the other, where p and q are coprime integers.
Chiral knotIn the mathematical field of knot theory, a chiral knot is a knot that is not equivalent to its mirror image (when identical while reversed). An oriented knot that is equivalent to its mirror image is an amphicheiral knot, also called an achiral knot. The chirality of a knot is a knot invariant. A knot's chirality can be further classified depending on whether or not it is invertible. There are only five knot symmetry types, indicated by chirality and invertibility: fully chiral, invertible, positively amphicheiral noninvertible, negatively amphicheiral noninvertible, and fully amphicheiral invertible.
Linking numberIn 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. In Euclidean space, the linking number is always an integer, but may be positive or negative depending on the orientation of the two curves (this is not true for curves in most 3-manifolds, where linking numbers can also be fractions or just not exist at all).
Tangle (mathematics)In mathematics, a tangle is generally one of two related concepts: In John Conway's definition, an n-tangle is a proper embedding of the disjoint union of n arcs into a 3-ball; the embedding must send the endpoints of the arcs to 2n marked points on the ball's boundary. In link theory, a tangle is an embedding of n arcs and m circles into – the difference from the previous definition is that it includes circles as well as arcs, and partitions the boundary into two (isomorphic) pieces, which is algebraically more convenient – it allows one to add tangles by stacking them, for instance.
Hopf linkIn mathematical knot theory, the Hopf link is the simplest nontrivial link with more than one component. It consists of two circles linked together exactly once, and is named after Heinz Hopf. A concrete model consists of two unit circles in perpendicular planes, each passing through the center of the other. This model minimizes the ropelength of the link and until 2002 the Hopf link was the only link whose ropelength was known. The convex hull of these two circles forms a shape called an oloid.
CodimensionIn mathematics, codimension is a basic geometric idea that applies to subspaces in vector spaces, to submanifolds in manifolds, and suitable subsets of algebraic varieties. For affine and projective algebraic varieties, the codimension equals the height of the defining ideal. For this reason, the height of an ideal is often called its codimension. The dual concept is relative dimension. Codimension is a relative concept: it is only defined for one object inside another.
UnlinkIn the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane. An n-component link L ⊂ S3 is an unlink if and only if there exists n disjointly embedded discs Di ⊂ S3 such that L = ∪i∂Di. A link with one component is an unlink if and only if it is the unknot. The link group of an n-component unlink is the free group on n generators, and is used in classifying Brunnian links. The Hopf link is a simple example of a link with two components that is not an unlink.
