In the mathematical theory of knots, the **Thurston–Bennequin number**, or **Bennequin number**, of a front diagram of a Legendrian knot is defined as the writhe of the diagram minus the number of right cusps. It is named after William Thurston and Daniel Bennequin.

The maximum Thurston–Bennequin number over all Legendrian representatives of a knot is a topological knot invariant.

In the mathematical theory of knots, the **unknot**, 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. 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 together so that 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, **R**^{3}. Two mathematical knots are equivalent if one can be transformed into the other via a deformation of **R**^{3} upon itself ; these transformations correspond to manipulations of a knotted string that do not involve cutting the string or passing the string through itself.

In the mathematical field of knot theory, a **knot invariant** is a quantity 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, but invariants can range from the simple, such as a yes/no answer, to those as complex as a homology theory. 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.

In mathematics, a **knot** is an embedding of a circle *S*^{1} in 3-dimensional Euclidean space, **R**^{3}, 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 *S*^{n}, 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.

In mathematics, **contact geometry** is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently, such a distribution may be given as the kernel of a differential one-form, and the non-integrability condition translates into a maximal non-degeneracy condition on the form. These conditions are opposite to two equivalent conditions for 'complete integrability' of a hyperplane distribution, i.e. that it be tangent to a codimension one foliation on the manifold, whose equivalence is the content of the Frobenius theorem.

In the mathematical area of knot theory, a **Reidemeister move** is any of three local moves on a link diagram. Kurt Reidemeister (1927) and, independently, James Waddell Alexander and Garland Baird Briggs (1926), demonstrated that two knot diagrams belonging to the same knot, up to planar isotopy, can be related by a sequence of the three Reidemeister moves.

In knot theory, a knot or link diagram is **alternating** if the crossings alternate under, over, under, over, as one travels along each component of the link. A link is **alternating** if it has an alternating diagram.

In the mathematical field of knot theory, a **mutation** is an operation on a knot that can produce different knots. Suppose *K* is a knot given in the form of a knot diagram. Consider a disc *D* in the projection plane of the diagram whose boundary circle intersects *K* exactly four times. We may suppose that the disc is geometrically round and the four points of intersection on its boundary with *K* are equally spaced. The part of the knot inside the disc is a tangle. There are two reflections that switch pairs of endpoints of the tangle. There is also a rotation that results from composition of the reflections. A mutation replaces the original tangle by a tangle given by any of these operations. The result will always be a knot and is called a **mutant** of *K*.

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.

In mathematics, the **slice genus** of a smooth knot *K* in *S*^{3} 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 *D*^{4} bounded by *S*^{3}.

In the mathematical field of knot theory, the **tricolorability** of a knot is the ability of a knot to be colored with three colors subject to certain rules. Tricolorability is an isotopy invariant, and hence can be used to distinguish between two different (non-isotopic) knots. In particular, since the unknot is not tricolorable, any tricolorable knot is necessarily nontrivial.

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 that is always tangent to the contact hyperplane.

In the mathematical field of knot theory, the **hyperbolic volume** of a hyperbolic link is the volume of the link's complement with respect to its complete hyperbolic metric. The volume is necessarily a finite real number, and is a topological invariant of the link. As a link invariant, it was first studied by William Thurston in connection with his geometrization conjecture.

In the mathematical field of knot theory, the **bridge number** is an invariant of a knot defined as the minimal number of bridges required in all the possible bridge representations of a knot.

**Victor Vladimirovich Goryunov** is a Russian mathematician. He is a leading figure in Singularity theory, whose contributions to the subject are fundamental. He has published several books and a vast variety of papers in singularity theory, Finite type invariants, and Legendrian knots. Many of his papers in Lagrangian and Legendrian geometry are now considered to be classical in the subject.

**Lenhard Ng** is an American mathematician, working primarily on symplectic geometry. Ng is a professor of mathematics at Duke University.

In mathematics, a **transverse knot** is a smooth embedding of a circle into a three-dimensional contact manifold such that the tangent vector at every point of the knot is transverse to the contact plane at that point.

**Daniel Bennequin** is a French mathematician, known for the Thurston–Bennequin number introduced in his doctoral dissertation.

- " Thurston–Bennequin number ",
*The Knot Atlas*. - Lee Rudolph (1997). "The slice genus and the Thurston–Bennequin invariant of a knot".
*Proceedings of the American Mathematical Society*.**125**: 3049–3050. doi: 10.1090/S0002-9939-97-04258-5 . MR 1443854.

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