|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.
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.
Knots can also be added together by pulling a loop from each of two knots, cutting the loops open, and splicing them together:
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
Knot Theory as Mathematics
Knot theory enables mathematicians to examine and apply the following concepts:
For the Teacher:
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.
For more information, see "DNA and Knot Theory Today," at the Knot Theory Online site.
| 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."