In mathematics, the slice genus of a smooth knotK in S3 (sometimes called its Murasugi genus or 4-ball genus) is the least integer g such that K is the boundary of a connected, orientable 2-manifold S of genus g properly embedded in the 4-ball D4 bounded by S3.
More precisely, if S is required to be smoothly embedded, then this integer g is the smooth slice genus of K and is often denoted gs(K) or g4(K), whereas if S is required only to be topologically locally flatly embedded then g is the topologically locally flat slice genus of K. (There is no point considering g if S is required only to be a topological embedding, since the cone on K is a 2-disk with genus0.) There can be an arbitrarily great difference between the smooth and the topologically locally flat slice genus of a knot; a theorem of Michael Freedman says that if the Alexander polynomial of K is1, then the topologically locally flat slice genus of K is 0, but it can be proved in many ways (originally with gauge theory) that for every g there exist knots K such that the Alexander polynomial of K is 1 while the genus and the smooth slice genus of K both equalg.
The (smooth) slice genus of a knot K is bounded below by a quantity involving the Thurston–Bennequin invariant of K:
The (smooth) slice genus is zero if and only if the knot is concordant to the unknot.
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.
In the mathematical theory of knots, the unknot, not knot, or trivial knot, is the least knotted of all knots. Intuitively, the unknot is a closed loop of rope without a knot tied into it, unknotted. To a knot theorist, an unknot is any embedded topological circle in the 3-sphere that is ambient isotopic to a geometrically round circle, the standard unknot.
In topology, knot theory is the study of mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, the simplest knot being a ring. In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, . Two mathematical knots are equivalent if one can be transformed into the other via a deformation of upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing it through itself.
In the mathematical field of knot theory, a knot invariant is a quantity (in a broad sense) defined for each knot which is the same for equivalent knots. The equivalence is often given by ambient isotopy but can be given by homeomorphism. Some invariants are indeed numbers (algebraic), but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory (for example, "a knot invariant is a rule that assigns to any knot K a quantity φ(K) such that if K and K' are equivalent then φ(K) = φ(K')."). Research on invariants is not only motivated by the basic problem of distinguishing one knot from another but also to understand fundamental properties of knots and their relations to other branches of mathematics. Knot invariants are thus used in knot classification, both in "enumeration" and "duplication removal".
A knot invariant is a quantity defined on the set of all knots, which takes the same value for any two equivalent knots. For example, a knot group is a knot invariant.
Typically a knot invariant is a combinatorial quantity defined on knot diagrams. Thus if two knot diagrams differ with respect to some knot invariant, they must represent different knots. However, as is generally the case with topological invariants, if two knot diagrams share the same values with respect to a [single] knot invariant, then we still cannot conclude that the knots are the same.
In mathematics, a knot is an embedding of the circle into three-dimensional Euclidean space, R3. Often two knots are considered equivalent if they are ambient isotopic, that is, if there exists a continuous deformation of R3 which takes one knot to the other.
The Chern–Simons theory is a 3-dimensional topological quantum field theory of Schwarz type developed by Edward Witten. It was discovered first by mathematical physicist Albert Schwarz. It is named after mathematicians Shiing-Shen Chern and James Harris Simons, who introduced the Chern–Simons 3-form. In the Chern–Simons theory, the action is proportional to the integral of the Chern–Simons 3-form.
In knot theory, a branch of mathematics, the trefoil knot is the simplest example of a nontrivial knot. The trefoil can be obtained by joining the two loose ends of a common overhand knot, resulting in a knotted loop. As the simplest knot, the trefoil is fundamental to the study of mathematical knot theory.
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
In mathematics, low-dimensional topology is the branch of topology that studies manifolds, or more generally topological spaces, of four or fewer dimensions. Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups. This can be regarded as a part of geometric topology. It may also be used to refer to the study of topological spaces of dimension 1, though this is more typically considered part of continuum theory.
In the mathematical field of knot theory, the Jones polynomial is a knot polynomial discovered by Vaughan Jones in 1984. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a Laurent polynomial in the variable with integer coefficients.
In mathematics, the Alexander polynomial is a knot invariant which assigns a polynomial with integer coefficients to each knot type. James Waddell Alexander II discovered this, the first knot polynomial, in 1923. In 1969, John Conway showed a version of this polynomial, now called the Alexander–Conway polynomial, could be computed using a skein relation, although its significance was not realized until the discovery of the Jones polynomial in 1984. Soon after Conway's reworking of the Alexander polynomial, it was realized that a similar skein relation was exhibited in Alexander's paper on his polynomial.
In mathematics, a Seifert surface is an orientable surface whose boundary is a given knot or link.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
In mathematics, Khovanov homology is an oriented link invariant that arises as the cohomology of a cochain complex. It may be regarded as a categorification of the Jones polynomial.
In mathematics, in the area of symplectic topology, relative contact homology is an invariant of spaces together with a chosen subspace. Namely, it is associated to a contact manifold and one of its Legendrian submanifolds. It is a part of a more general invariant known as symplectic field theory, and is defined using pseudoholomorphic curves.
A slice knot is a mathematical knot in 3-dimensional space that bounds an embedded disk in 4-dimensional space.
In the mathematical field of knot theory, the Arf invariant of a knot, named after Cahit Arf, is a knot invariant obtained from a quadratic form associated to a Seifert surface. If F is a Seifert surface of a knot, then the homology group H1(F, Z/2Z) has a quadratic form whose value is the number of full twists mod 2 in a neighborhood of an embedded circle representing an element of the homology group. The Arf invariant of this quadratic form is the Arf invariant of the knot.
In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into , which is tangent to the standard contact structure on . It is the lowest-dimensional case of a Legendrian submanifold, which is an embedding of a k-dimensional manifold into a (2k+1)-dimensional contact manifold that is always tangent to the contact hyperplane.
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