Ambient isotopy

Definition and Overview

In the field of topology, **ambient isotopy** refers to a type of isotopy that occurs within a larger space, typically a manifold. Specifically, it is a continuous deformation of one submanifold into another within the ambient space, such that at each stage of the deformation, the submanifold remains embedded in the ambient space. This concept is crucial in the study of knot theory, differential topology, and geometric topology.

Formal Definition

Let \( M \) be a manifold and \( N \) be a submanifold of \( M \). Two submanifolds \( N_0 \) and \( N_1 \) of \( M \) are said to be ambient isotopic if there exists a continuous map \( H: M \times [0,1] \to M \) such that: 1. \( H(x,0) = x \) for all \( x \in M \) (the identity at \( t=0 \)), 2. \( H(N_0, t) \) is a submanifold of \( M \) for all \( t \in [0,1] \), 3. \( H(N_0, 1) = N_1 \).

This map \( H \) is called an ambient isotopy between \( N_0 \) and \( N_1 \).

Applications in Knot Theory

In knot theory, ambient isotopy is used to determine when two knots are equivalent. Two knots are considered equivalent if one can be transformed into the other via an ambient isotopy in three-dimensional space. This concept is fundamental because it allows mathematicians to classify knots up to deformation, ignoring trivial deformations that do not change the knot's essential properties.

Properties and Examples

Ambient isotopy has several important properties: - **Invariance under Homeomorphisms**: If \( f: M \to M' \) is a homeomorphism and \( N_0 \) and \( N_1 \) are ambient isotopic in \( M \), then \( f(N_0) \) and \( f(N_1) \) are ambient isotopic in \( M' \). - **Transitivity**: If \( N_0 \) is ambient isotopic to \( N_1 \) and \( N_1 \) is ambient isotopic to \( N_2 \), then \( N_0 \) is ambient isotopic to \( N_2 \). - **Symmetry**: If \( N_0 \) is ambient isotopic to \( N_1 \), then \( N_1 \) is ambient isotopic to \( N_0 \).

An example of ambient isotopy is the deformation of a circle into an ellipse within the plane. Another example is the transformation of a trefoil knot into its mirror image in three-dimensional space.

Ambient Isotopy vs. Regular Isotopy

It is important to distinguish between ambient isotopy and regular isotopy. Regular isotopy is a more restrictive concept, typically used in the context of plane curves and link diagrams. While ambient isotopy allows for the entire ambient space to deform, regular isotopy only permits deformations within a fixed plane, without allowing the curves to pass through themselves.

Techniques and Tools

Several mathematical tools and techniques are used to study ambient isotopy: - **Alexander's Theorem**: This theorem states that every knot or link in three-dimensional space can be represented as a closed braid. - **Reidemeister Moves**: These are local moves on knot diagrams that correspond to ambient isotopies in three-dimensional space. There are three types of Reidemeister moves, and any two equivalent knot diagrams can be transformed into each other through a finite sequence of these moves. - **Isotopy Invariants**: These are properties of knots and links that remain unchanged under ambient isotopy. Examples include the Jones polynomial, Alexander polynomial, and knot group.

Advanced Topics

Higher-Dimensional Ambient Isotopy

Ambient isotopy is not limited to three-dimensional space. In higher dimensions, the concept becomes more complex and requires advanced techniques from differential topology and homotopy theory. For instance, in four-dimensional space, the study of surface knots and their ambient isotopies involves intricate interactions between topology and geometry.

Ambient Isotopy in Manifolds

In the context of manifolds, ambient isotopy can be used to study the embedding and classification of submanifolds. For example, in 4-manifolds, the study of ambient isotopy is related to the classification of smooth structures and the h-cobordism theorem.

References