Re: Infinite Knots


Lee Rudolph wrote:
"zuhair" <zaljohar@xxxxxxxxx> writes:

Hi

If I have untwisted knot ( at zero state twisting ), imagin it as a
circle, and if we twist it horizontally in such a mannter that we'll
get two parts like number 8 ,each is half size the original knot, now
if I twist the lower segment in the same manner that the lower segment
will be composed of two circles each is quarter the original Knot. Now
my question is that if I continue twisting in the same manner of the
lower segment can I have infinite number of twists of the original
knot, though the original knot has finite length?

Now If I untwist this infinitelly twisted knot, what I will have? would
it be a knot with a finite number of segments or infinite number or
simply a not without segments ( the orginal untwisted knot ).

The answers to your questions all depend on how the terms involved
particularly the operation of "twisting" and what "knot" means, are
made precise. Certainly there *are* things, which would be recognized
as "knots" by most knot theorists, that have "infinite number of
twists" and yet have finite length; and one way I can imagine making
your sketched procedure precise would create such a thing (it would
be a trivial knot topologically, in the sense that there would be
a homeomorphism of 3-space onto itself carrying a round circle
onto the knot; but any such homeomorphism would have to be
non-differentiable at one or more points).

On the other hand, there is *no* (reasonable) way I can imagine
of making your sketched procedure precise that does not make your
second question trivial: if "untwist" is inverse to "twist",
then certainly you will get "the original untwisted knot" back
if you untwist the twisted knot!

Maybe you can clarify what you meant and then get a more satisfactory
answer.

Lee Rudolph

Oh, yes, right.

I meant if I untwist the infinitely twisted Knot I mentioned beginning
from the upper segments INFINITELLY , then what I would get. My real
intention behind that question is the following: is the rate at which
twisting and untwisting is taking place will make different answeres?

I mean if we twist and untwist at the same rate then probably well get
the original knot.
but if we untwist at a rate that is more rapid than the twisting rate
then we'll get an infinitelly twisted knot that is the opposit of the
original Knot.
If we twist at a speeder rater than we untwist then what well get? a
finitelly twisted Knot,
or an infinitelly twisted not in the direction of the original knot??

I want to discuss all these possibilities.

Zuhair

.

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