In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, a transverse knot is a smooth embedding

Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....

 of a circle into a three-dimensional contact manifold such that the tangent vector

Tangent vector

A tangent vector is a vector that is tangent to a curve or surface at a given point.Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold....

 at every point of the knot is transverse

Transversality theorem

In differential topology, the transversality theorem, also known as the Thom Transversality Theorem, is a major result that describes the transversal intersection properties of a smooth family of smooth maps...

 to the contact plane at that point.

Any Legendrian knot

Legendrian knot

In mathematics, a Legendrian knot often refers to a smooth embedding of the circle into \mathbb R^3, which is tangent to the standard contact structure on \mathbb R^3...

 can be C⁰-perturbed in a direction transverse to the contact planes to obtain a transverse knot. This yields a bijection

Bijection

A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...

 between the set of isomorphism

Isomorphism

In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...

 classes of transverse knots and the set of isomorphism classes of Legendrian knots modulo

Modulo

In the mathematical community, the word modulo is often used informally. Generally, to say "A is the same as B modulo C" means, more-or-less, "A and B are the same except for differences accounted for or explained by C"....

negative Legendrian stabilization.