Mathematics at Leeds University
See e.g. the book
Introduction to Knot Theory by R H Crowell and R H Fox.
here!
Definition: A subset K of R^3 is a knot if it is a homeomorphic image of the unit circle S^1 in R^3.
Are we clear on the various terms here?
R^d means real d-dimensional space
(in our case real 3d space).
The unit circle is a certain specific subset of R^2.
For `homeomorphism' we need to take a step back...
We assume you know what a metric space (X,d) is.
And a topological space (X,O), where O is the set of all open subsets.
(See e.g. MATH3224 module or MATH3225 module at Leeds.)
A map f:X -> Y between topological spaces is continuous if
the preimage of an open set in Y is open in X.
Map f is a homeomorphism if it is a bijection
and the inverse is also continuous.
...
Implicit in our definition of a knot is that both R^3 and S^1
are topological spaces.
Is this clear?
How does this definition compare with a real knot?
It works quite well (although you certainly have to think about it
-- real knots are tied in open strings, not closed loops;
so the closedness is standing in for some other aspect ...).
One issue is that there is scope for some rather unphysical
`infinite' tanglings in our definition.
We can neatly avoid these by restricting to `polygonal'
knots as follows.
Definition: A polygonal knot is one which is the union of finitely many closed straight segments.
A polygonal knot is the union of at least 3 segments. There are infinitely many such triangular knots, but next we will organise knots into equivalence classes with respect to a natural topological equivalence. And all the triangular knots lie in the same class ...
Two knots are equivalent if there is a homeomorphism of R^3 onto itself that takes one onto the other.
One interesting problem is then to classify knots up to equivalence.
The class of the natural embedding of the unit circle is called the unknot.
Are there any other classes?
How many?
...
References:
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