<B> An Introduction to Knot Theory

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

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