In the

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

subject of

topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing h ...

, an ambient isotopy, also called an ''h-isotopy'', is a kind of

continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous g ...

distortion of an ambient space, for example a

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...

, taking a

submanifold In mathematics, a submanifold of a manifold ''M'' is a subset ''S'' which itself has the structure of a manifold, and for which the inclusion map satisfies certain properties. There are different types of submanifolds depending on exactly which ...

to another submanifold. For example in

knot theory In the mathematical field of 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 ...

, one considers two

knot A knot is an intentional complication in cordage which may be practical or decorative, or both. Practical knots are classified by function, including hitches, bends, loop knots, and splices: a ''hitch'' fastens a rope to another object; a ...

s 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

N

and

M

be manifolds and

g

and

h

embedding In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup. When some object X is said to be embedded in another object Y, the embedding is gi ...

s of

N

M

. A

continuous map In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...

\rightarrow M

is defined to be an ambient isotopy taking

g

h

F_0

is the

identity map Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unch ...

, each map

F_t

is a

homeomorphism In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorph ...

from

M

to itself, and

F_1 \circ g = h

. This implies that the

orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...

must be preserved by ambient isotopies. For example, two knots that are

mirror image A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...

s of each other are, in general, not equivalent.

References

*M. A. Armstrong, ''Basic Topology'',

Springer-Verlag Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, 1983 *Sasho Kalajdzievski, ''An Illustrated Introduction to Topology and Homotopy'', CRC Press, 2010, Chapter 10: Isotopy and Homotopy Topology Maps of manifolds {{topology-stub

See also

References