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From: kelsar777 on 14 Apr 2010 04:06
Let A (in R^2) is an image of circle by continuous nonconstant map on
R^2.
Does exist a knot in R^3 (homeomorpfic with circle) whose projection
onto plane is A?
From: William Elliot on 14 Apr 2010 05:18
On Wed, 14 Apr 2010, kelsar777 wrote:

> Let A (in R^2) is an image of circle by continuous nonconstant map on
> R^2.

> Does exist a knot in R^3 (homeomorpfic with circle) whose projection
> onto plane is A?
>
Likely there is no such non-trivial knot.
From: Dan Cass on 15 Apr 2010 07:19
> Let A (in R^2) is an image of circle by continuous
> nonconstant map on
> R^2.
> Does exist a knot in R^3 (homeomorpfic with circle)
> whose projection
> onto plane is A?

If the image circle in R^2 looks like a scribble,
with say a finite number of self-intersections,
and these intersections are where "two pieces cross":
Then one could "lift up" one of each crossing piece into
the z-direction slightly. This would
produce a knot in R^3 whose projection would be
the original circle image in R^2.

But it would seem just assuming the circle image in R^2
was a "continuous image" might not give something for
which the above construction would work.
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