Definition:Knot (Knot Theory)
This page is about knot in the context of Knot Theory. For other uses, see knot.
Definition
Let $Y$ be a manifold and $X \subset Y$ a submanifold of $Y$.
Let $i: X \to Y$ be an inclusion, that is a mapping such that $i \sqbrk X = X$.
Then a knotted embedding is an embedding $\phi: X \to Y$ (or the image of such an embedding) such that $\phi \sqbrk X$ is not freely homotopic to $i \sqbrk X$.
Sphere Knot
A knotted $n$-sphere is a knotted embedding:
- $\phi: \Bbb S^n \to \R^{n + 2}$
Circle Knot
The description of the sphere is dropped for $\Bbb S^1$ and the term knot is used without qualification for knotted embeddings $\phi: \Bbb S^1 \to \R^3$.
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Elementary Knot
Circle knots can often be quite wild and unwieldy - most of modern knot theory concerns itself with a specific kind of knot.
These knots are described as a finite set of points in $\R^3$ called $\set {x_1, x_2, \dots, x_n}$, together with line segments from $x_i$ to $x_{i + 1}$ and a line segment from $x_n$ to $x_1$.
The union of all these line segments is clearly a circle knot, or an unknot, an embedding of the circle which is homotopic to a circle.
Examples
Trefoil Knot
These are the planar diagrams of the two types of trefoil knot:
Left-Handed Trefoil
Right-Handed Trefoil
Also see
- Results about knots in the context of knot theory can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): knot
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): knot: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): knot: 1.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): knot (in a curve)