All Tied Up In Knots September 16, 2002
Knot Theory Basics
"In the old days a love-sick sailor might send the object of his affections a length of fishline loosely tied in a true-lover's knot. If the knot was sent back as it came the relationship was static. If the knot returned home snugly drawn up the passion was reciprocated. But if the knot was capsized — tacit advice to ship out."

The Ashley Book of Knots

As you can tell, love knots have a symbolic significance of tying a couple together. What other meanings can knots offer?

Imagine a piece of rope tied into a simple knot. Then imagine the two free ends of the rope spliced together. This is a mathematical knot: a closed curve. The curve may be a simple loop or it may have a number of twists and crossings. Some of the crossings may be unknottable and some may not. The central question in knot theory is whether two knots can be rearranged — without cutting — to be the same.

Study the following diagram. The unknot is the simplest of all knots with no crossings. Its crossing number is zero. The crossings in knot #1 are unknottable without cutting the rope. Since knot #1 can be rearranged into the unknot, they are really the same knot. Simplifying knot #1 to equate it with the unknot is like simplifying a fraction, e.g., simplifying 3/6 to the equivalent fraction 1/2.

unknot diagram

Trefoil #1 in the diagram below has three crossings. Can you visualize why knot #2 is the same knot? (Hint: Compare the crossings.) Trefoil #2 also has three crossings, but it is not the same knot as trefoil #1 because the corresponding crossings in trefoil #2 go under where in #1 they go over, and vice versa. Trefoil #2 is the mirror image of trefoil #1.

trefoils

Knots can also be added together by pulling a loop from each of two knots, cutting the loops open, and splicing them together:

adding knots

The arithmetic laws of commutativity and associativity also apply to knots. A knot that cannot be unknotted to fewer crossings is called a prime knot.

This section has just been a sampler of some of the basic concepts in knot theory. For more information, visit one of the following knot theory tutorials:

All Knotted Up
The following two exercises help you think about knots:

  • Knots — This activity is for the entire class and is best done in an open space with no furniture.

    All students stand in a circle with their eyes closed, holding their arms out in front of them. They begin walking forward. When they meet in the middle, each student grabs two available hands. The "knot" is now tied. Students open their eyes and try to untangle the knot. Legal operations are stepping over and under, in and out, and twisting. It is illegal to drop hands and break the chain. Can you untangle the knot?

  • Knot Dictation — Students should pair up, each student with a piece of rope. The pairs sit across from one another at a desk, with a standing book or other barrier blocking the work space from one another's view.

    Student X ties a simple knot in her rope. She instructs Student Y how to tie the same knot, without letting him see it. When he is done, they compare knots. Then they switch roles, with Student Y tying the original knot and instructing Student X how to tie the same one. Each pair repeats the exercise for several more knots, noting the vocabulary they use to describe the knots in their instructions.

Knot Theory as Mathematics
Why do mathematicians find knot theory so fascinating? According to the authors of the Mathematics and Knots site: "Mathematicians are not among those who expect that some new theory will come along which will somehow answer all our questions, a kind of 'general unified problem solver'. We do expect to find new ways of looking at and solving old questions, to find new questions, and to find new, surprising and beautiful intricacies of patterns, structures and relationships at which to marvel."

Knot theory enables mathematicians to examine and apply the following concepts:

  • equivalency: When are two knots the same? Answering this question requires mathematicians to define a list of elements that can be compared in order to classify knots into families.

  • invariants (the factors that do not change): When are two knots not the same? Mathematicians must also determine when one knot cannot be rearranged into another. It is not always easy to prove that something cannot be done. Some invariants applicable to knots are crossing number, the unknotting number (the number of changes required in a knot in order to unknot it), coloring number, and bridge number. You can learn more about these invariants at Mathematics and Knots.

  • analogies between knots and numbers: There are many analogies that can be drawn between knots and numbers, e.g., addition, prime numbers and knots, zero (the unknot), and simplification.

  • mathematical proofs: It may be easy to demonstrate whether two knots are the same or not, but writing a mathematical proof of this fact can present a challenge. Mathematicians may translate the problem into a different branch of mathematics in order to discover a proof, leading to creative thought in two areas at once.
Borromean Rings
Links are a special family of knots made up of more than one strand. The picture below represents Borromean Rings — three circles none of which links with any other, but together they are linked and cannot be pulled apart.

Borromean rings

How are the rings put together? Consider how you interlock the flaps on the top of a cardboard carton. In order for each flap to be over the one to its right yet under the one to its left, you must bend one or more of the flaps slightly as you snap them all into place. A mathematical theorem exists which states that flat Borromean rings do not exist — the rings must either be in different planes or have kinks in them.

Borromean rings

























For the Teacher:
Knot Theory and the Curriculum
Dr. Wally Feurzeig, Principal Scientist at BBN Technologies, notes the following benefits for high school students studying knot theory:
  • It integrates geometry, combinatorics, polynomial algebra, number theory, and intuitive topology in a way that is hard to find elsewhere.

  • Working with two-dimensional projections of three-dimensional knots alongside actual three-dimensional models helps students develop a visual sense about the information conveyed in the stylized pictures.

  • Reasoning with visual images is quite difficult for students, and knot theory provides them with gradual ramps to situations that require geometric arguments. For example, many students find it difficult at first to determine whether a given diagram of a knot stands for a structure that is actually knotted, but they become quite adept at this task with a little practice.

  • The study of the polynomial invariants associated with knots provides students with a clear application of polynomial arithmetic to geometric phenomena.
See also the MEGA Mathematics article "NCTM Standards and the Mathematics of Knots." This site also offers knot theory classroom activities.
Strands That Can Knot Replicate
A strand of DNA is only a few molecules wide, but it can be several centimeters long. The strand can easily become tangled inside the cell nucleus, preventing the double-helix from separating in order to replicate itself during cell division.

When the DNA strand has become knotted, special enzymes in the cell nucleus cut the strand to enable it to replicate. The enzymes then splice the loose ends once the DNA is unknotted. Chemical changes occur in the DNA structure during this process. Molecular biologists have discovered that knot theory and geometry help them analyze these changes.

DNA replication

For more information, see "DNA and Knot Theory Today," at the Knot Theory Online site.

Related Activities
DNA & Replication
Learn more about the process of DNA replication.
 
Other Tie-Ins
Physicists have also found relevance in knot theory. They have identified knotted patterns in the fluid flows, such as the atmosphere around Earth.

Knot theory has also found its way into art. See some examples of such sculpture at Symbolic Sculptures and Mathematics. Also visit the Knot Plot Site to see that art and math are sometimes difficult to distinguish from one another.

The definitive reference book on tying knots is Clifford W. Ashley's 1944 classic, The Ashley Book of Knots. In the book Ashley describes close to 4,000 knots, not only explaining how and when to tie them, but also providing a wealth of knot history and anecdotes. Ashley writes: "There are still old knots that are unrecorded, and so long as there are new purposes for rope, there will always be new knots to discover."