KNOTS 3
Distinguishing simple and complex modifications.
(DRAFT LIKELY TO BE REORGANISED)

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.

Each picture shows a scene, viewed from above, containing a continuous string
looped over and around itself in various ways, resting on a flat surface.

The pictures are displayed with two in each row, merely to reduce waste of screen
space. Adjacent pairs are no more closely related than the second of each pair
and the first of the next pair. However, the order of pictures, indicated by the
labels, String A, String B, ... String R, is significant. (Please ignore the knot
at the visible free end of the string, required for a different use of the string.)

Your task
In some of the pictures, e.g. String B, it is possible to grasp the two ends of
the string and pull them apart without the string forming a knot: the string
will eventually be straight after continuous alterations of position and
curvature of various segments of the string. You may like to test your ability
to tell which strings are and which are not knot-free, but that's not the main
point of this exercises.

Instead, you are asked to decide whether each picture (apart from the first) is
essentially similar to the preceding one, except for small parts that may have
been moved. (Please ignore the knot at the right hand end of the string,
required for a different use of the string.)

In some cases, the transition from one picture to the next involved keeping the
two ends of the string (approximately) fixed on the table while an intervening
portion of the string was moved along the surface of the table, sometimes also
moving over or under another portion of the string. Can that sort of change
affect whether the result of pulling the ends apart will be a knot?

Can you tell which successive pairs of strings involve only such motions,
i.e. motions with ends held on the table.

Can such motions of intermediate segments, while the ends remain fixed, alter
whether the string will form a knot when the ends are pulled apart?
(The transformations under discussion, require two the ends of the string to
remain approximately fixed, and no portion of the string forming a loop to be
pulled to form a double stranded string and then knotted.)

People who are unfamiliar with processes in which portions of a piece of a
flexible string are moved around on a table top, may find it hard to see some of
the possibilities and invariants except in the simplest cases.

(Please report your experiences, and any thoughts arising, to
a.sloman[AT]cs.bham.ac.uk )

    String A:
    Scene 2a
    String B:
    Scene 3a
    Example: is it possible to transform the string from configuration A above to
    configuration B by moving portions of the string horizontally along the surface
    of the table, while the two ends of the string (one out of sight on left) are
    held in place or slid closer together to increase slack in the string?

    What about possible transitions between String B and String C, or between String
    C and String D, or String D and String E, etc. etc.?

    String C:
    Scene 1
    String D:
    Scene 4a
    String E:
    Scene 6a
    String F:
    Scene 5a
    String G:
    Scene 2b
    String H:
    Scene 3b
    String I:
    Scene 4b
    String J:
    Scene 5b
    String K:
    Scene 6b
    String L:
    Scene 5c
    String M:
    Scene 6c
    String N:
    Scene 4c
    String O:
    Scene 6c
    String P:
    Scene 4c
    String Q:
    Scene 6c
    String R:
    Scene 4c

Back to the knots page. For more examples, see the 'Toddler Theorems' Web page:
http://www.cs.bham.ac.uk/research/projects/cogaff/misc/toddler-theorems.html