## An Introduction to Knot Theory

### Definitions

A **Knot** is defined as a continuous 1-1 function from the circle to the
3-sphere.

**Note**: 1-1 implies that a knot is not allowed to have self-intersecting
points. Such "knots", which are usually called **singular knots**, are
however used in the study of the **Vassiliev** invariants.
Two knots are called **ambient isotopic** and considered topologically
equivalent, iff one may continuously pass from one knot to the other.

Let two knots be defined by the 1-1 continuous functions f(s) and f'(s). These
knots are ambient isotopic iff there is a function g(s,t), where t belongs to
[0,1], and g is continuous with respect to both s and t, such that

- g(s,0)=f(s)
- g(s,1)=f'(s)
- g(s,t)=g(s',t) => s=s'

For a discussion of showing knot ** equivalence **
click **here**.
For a discussion of showing knot ** inequivalence ** click
**here**.

*Charilaos Aneziris, charilaos_aneziris@standardandpoors.com*

**Copyright 1995**

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