In this activity, students continue their experience with knots and knot
addition by thinking about how the process of knot addition might work
in reverse. Students will try to decompose knots into two simpler parts
that can be added together to make the original knot. Students will be
exposed to the notion of prime knot.
Review the idea of knot addition . Ask
students what will
happen if this is done backwards. In other words, is it possible to look
at a knot and figure out how to take it apart or decompose it into two
knots which could be added together to produce the original knot?
Have each student or group of students choose a knot to
decompose and talk about what happens. It seems that some knots can
be broken down into smaller components and others cannot.
Ask students to work in pairs or small groups to make knot decomposition puzzles by following these steps:
Draw two knots on paper, then draw how it will look when you
add them.
Make the knot that results from the addition in the drawing out of
rope. It is important
to do this carefully, because the slightest error will result result in a knot
that decomposes differently than their drawing. It might even be a knot that won't decompose at all..
Twist and deform the knot you have made so that it is no longer
obvious what the two original knots were.
Give the knotted rope to
another student or group of students.
When you receive a knotted rope from another group of students, try to
figure out what the two knots were that the knot you received was made
from. (This may actually turn out to be quite
difficult!)
When a group succeeds in decomposing a knot, the decomposition should be compared to the
drawing that was made. If the knot is decomposed differently than the drawing shows, both groups will need to get together and talk about the result.
Any knots that cannot be decomposed during the class
period can stand as open problems in the
classroom. Open problems are the lifeblood of mathematics.
NOTE: You may want to place some
restrictions on the size (i.e., number of crossings) of the knots that
are used to make these puzzles so that the knots to be decomposed
aren't outlandishly complex.
The knot decomposition puzzles can be made
even
more challenging if more than two knots can be summed to make the puzzle.
If students try to decompose the knots on the knot diagram into
smaller knots, they will meet with quite a bit of difficulty. All of
these knots are believed to be prime knots--knots that
cannot be decomposed into two simpler knots (unless, of course, one of the
two knots is the zero knot ). Whether
these knots are prime knots, (and the conjecture that none of these knots
are equivalent) has not been
conclusively proved, however.
After much experimentation with the aid of computers, no one has
succeeded in decomposing these knots, but this does not mean that there is not a decomposition waiting to be discovered.
If a student should discover one, contact
Nancy Casey (casey931@cs.uidaho.edu) right away!
What do you look for in a knot to see if it will decompose? Are
there any sure signs that it will? that it won't? If you think it
will decompose, how do you go about trying?
The idea of basic building
blocks--irreducible
units that can be put together in a variety of ways to produce more complex
entities--is one that runs all though western mathematics and science.
Identification of the ultimate building blocks
of the universe is one of the quests of particle physics. Amino acids
are the building blocks of the proteins in all living things. The DNA
molecules of which our genes and chromosomes are made are built up
from 4 basic building blocks. Chemists study molecules made of atoms,
as well as how certain clusters of atoms are intermediate size
building blocks for complex molecules. Students may be familiar with
the concept of prime number, and how prime
numbers are the building blocks of multiplication.
Have students identify other topics of mathematics, science, and other
disciplines where they have experienced the idea of building blocks.
Materials 
