Vignette 12

You have probably been tying knots from a very early age.  But have you ever thought of a knot as a mathematical object?  In the field of topology, mathematicians study the properties of spatial objects that are the most "durable" -- that is, the properties that are unchanged under certain kinds of distortions of space.  If an object can be deformed continuously into another object, then those two objects are thought of as being topologically equuivalent.  (The technical term is homeomorphic.)  For example, a rubber band in its natural shape looks like an elongated oval -- yet it can be continuously deformed into a circle.  Thus, those two objects -- the elongated oval and the circle -- are homeomorphic.  Similarly, a triangle and a circle are homeomorphic, since one could be continuously deformed into the other.


The technical definition of homeomorphism between two objects requires that it be possible to deform one of the objects in a continuous way so as to produce the other object.  The deformation is also required to be a one-to-one correspondence, and it is required that the reversal of the deformation process be continous also.  Thus, while a piece of string could be joined at the two ends to form a circle, this would not satisfy the definition of a homeomorphism, because the process would not be one-to-one: the two endpoints would be merged to a single point.  (In fact, it can be shown that the string (an interval) is indeed not homeomorphic to the circle.)

As a very elementary example, consider the question of whether the interval [0,1] is homeomorphic to the interval [0,2].  Intuitively, we would expect these intervals to be homeomorphic, because [0,1] can be deformed (by stretching) into the interval [0,2] -- and this stretching appears to be a one-to-one correspondence, and appears to be continuously reversible as well.  On a more formal level, the function  defined by  is a one-to-one correspondence between [0,1] and [0,2] which is a continuous function.  Moreover, its reversal, the inverse function , is also a continuous function.  Thus, [0,1] is homeomorphic to [0,2].

Not Knots

We now consider objects that are homeomorphic to the unit circle.  Such an object is called a simple closed curve.  Examples of simple closed curves can be constructed by laying a rubber band on your desk, and stretching and moving it, always keeping it flat on the desk:

We can also construct simple closed curves by utilizing the third dimension -- lifting part of the rubber band out of the plane of the desk -- as long as any apparent "crossover" points really are not intersections of the curve with itself.  In a 2-dimensional drawing, we usually show such an overlap by drawing a small gap in the lower portion of the curve:

All of the above curves are unknotted.  That is, it is possible to move the rubber band (or rope) through 3-dimensional space so as to result in an ordinary circle.  (Mathematically -- there is a homeomorphism of onto itself which maps the curve onto the unit circle.)


Here are some examples of curves that are knotted -- that is, it is impossible to move the rope around through 3-space to produce an ordinary circle.

Trefoil Knot
Figure-eight Knot

Perhaps you have recognized the graphic on the Vignettes Home Page as a movie of a trefoil knot in the process of being drawn.  Knots can be constructed with enormous amounts of complexity, which leads to the mathematical problem of classifying them.

Knot Theory

The main problem of knot theory is the classification of knots -- that is, the determintation of knot types and how to tell when two knots are equivalent.  Most of the classification work has revolved around the number of crossings in a knot.  For instance, the trefoil knot has three corssings, whereas the figure-eight knot has four.  Some things that are known about knots include the following.

Today, algebra plays a large role in the classification of knots.  It is possible to associate a certain polynomial with a knot, in such a way that if the polynomials are different, then the knots are not equivalent.  In this way, a problem that started out as a topological one becomes an algebraic one!  (This idea of solving problems in one branch of mathematics by using another branch is a fairly common pattern in mathematics today.)  However, a truly simple way to determine whether any given messy tangle of rope is actually unknotted -- or equivalent to a known knot -- has not yet been found.

Are They Good for Anything?

Surprisingly, chemists and molecular biologists have begun to look toward knot theory to provide insight into the nature and structure of various compounds such as DNA, which exhibits not only twisting, but also knotting.  Using the flexible viewpoint of topology rather than the rigid viewpoint of geometry to envision chemical structures may lead to a whole new set of molecules with applications not yet even imagined!

Further Exploration

Copyright © 2000 by Carl R. Spitznagel