Knotted and knot-free 2

This is part of a collection of examples of abilities of humans to reason in a
mathematical way even if they don't know they are doing it. See the 'knots' page.

For each of the strings shown below, can you tell simply by looking at the
picture, whether it is possible to straighten the string simply by holding the
two ends and pulling them apart?

To check the answer, you could try laying out a string in the configuration
shown, and try pulling the ends apart. Can any current robot look at one of
these pictures and lay out piece of string in a similar configuration?
Does the configuration used to check your answer for one of the pictures have to
match the string layout in the picture exactly? If not why not?

To answer the question about a particular picture without using a piece of
string, do you need to have seen the precise configuration previously? If you
don't need such prior experience, how can you be sure about your answer?

How could a robot be given these abilities?

Which of these will form a knot if ends are pulled apart?

(Note, if you find the crossing points in the images too unclear you may find it
useful to use your browser's "Show Image" option on each image, to make it bigger.
  1. String 1:

  2. String 2:

  3. String 3:

  4. String 4:

  5. String 5:

  6. String 6:

  7. String 7:

  8. String 8:
(Answers will be added later.)

A subsidiary question (Also on the first knots page.)

Suppose you knew that the string had been laid out by two people: one starting
with a straight piece of string initially pulled taut, and holding the two ends
all the time thereafter, while moving the ends closer together to allow portions
between the end to be curved, looped, etc. on the table by someone else.

Could you then be sure that after such a procedure, the ends of the strings
could be pulled apart without forming a knot -- even in a configuration in which
a string ends up with many loops visibly criss-crossing on the table?

If your answer is yes: how can you be sure?

What sort of knowledge about strings, space, motion, and continuity do you need
in order to understand why the potential to be straightened without a knot is
preserved by moving only parts of a string between its two ends while the ends
are constantly held by a person or a machine with two hands constantly resting
on the table, though they may come closer together if necessary, in order to
allow more slack in the string.

Could you prove, using logic, that this will not allow any knots to form? How can
you be sure about the impossibility, i.e. about what the movements of the
string cannot produce under those conditions?

Back to the knots page.

For more examples, see the 'Toddler Theorems' Web page: