Students will work with a variety of knots and manipulatively explore the
idea of equivalence. They will
understand that as long as knots are not cut, they remain the same or
equivalent regardless of how they are pulled, wrenched or otherwise
deformed.
Explain to the students that they will be asking a question
about knots that is a fundamental question which mathematicians are
always asking about the things that they study. That question is:
How can I tell if two of them are the same or different?
Group students and knots together so that each pair of students
or small group has a batch of knots consisting of pairs of identical
knots. Have the students pick out three pairs of identical knots that
they want to work with.
Have the students begin by getting their knots as mixed up as
they can. Pull the loops out, wrap them around, twist them, tuck them
under--do as many things as they can to their knots to change their
form, being careful, of course, not to break the knots apart.
When the knots have been well-deformed, have each group of
students give their knots to a different group.
When the students receive their new group of knots, they must
sort them back out, grouping the identical knots together. Explain
that the way to show that two knots are the same is to take one of the
knots and move the loops around until it is identical to the other
one.
Students will probably not be short of ideas for continuing and
modifying this game. They may want to make more knots if it becomes
too easy. Some suggestions are:
Find out which
of three knots is the "odd" one
Sort out a collection of pairs
of equivalent knots
Verify if two knots are equivalent
blindfolded
Play 20 questions: A student selects a knot and
hides it from the others. Students ask yes/no questions about its
structure to try and determine which knot it is.
Have students describe the techniques they used for determining
if two knots were the same. List them. Explain the idea of an
invariant--something that will be true
about a knot regardless
of how the knot has been deformed. Do any of their techniques for
determining if two knots are the same rely on the concept of
invariance?
Have students describe the ways that they could be sure that two
knots were not the same. List them.
What techniques were reliable in finding possibilities for
deforming knots? List them.
Two particularly tricky knots are the right- and
left-handed trefoil knots which are not equivalent.
Students should be allowed plenty of time to experiment with these to see
if they believe that this is so.
The right- and left-handed trefoil knots are
called mirror images of one another
because (as students can verify with the aid of a mirror) you can make
either one of them look exactly like the other by looking at it in a
mirror. Study the mirror images of other knots. It is not always the case
that a knot and its mirror image are not equivalent (as is the case with
the trefoil knots). Some knots can be transformed into their mirror images
without untying. Select a group of knots and divide them into two
categories: those which are equivalent to their mirror image and those which are not.
Encourage students to talk about what it takes for them to be
satisfied that two knots aren't equivalent. Help them to make
distinctions between being reasonably sure and absolutely certain.
Explain how this fits into the context of mathematical proof .
Materials 
