 |
This explainer will present information to acquaint visitors with topological invariants.
1. The visitor will identify knots and unknots.
2. The visitor will classify knots by topological
invariants.
Materials are listed with separate procedures.
 |
 |
|
|
crossing - with rope knots, the place
where rope is lying over or under rope
crossing number - the number of crossings
a rope knot has in its simplest form. A rope knot is in its simplest form
when the crossings all alternate : under, over, under, over...
crossing order - with rope knots,
the order the path of a rope takes as it goes from crossing to crossing
knot theory - a subdiscipline of
topology that focuses on the study of closed knots
left-handed knot - a knot that has
an orientation that is opposite its right-handed version.
mirror image - in general, what an
object looks like reflected in a mirror
right-handed knot - a knot that has
an orientation that is opposite its left-handed version.
topological invariants - properties
that are invariant (that is, they don't change) when deformed by stretching
and bending, without tearing cutting or gluing. In topology, some such
properties are connections, minimum crossings, number of holes.
topology - the study of properties
that are invariant (that is, they don't change) when deformed by stretching
and bending, without tearing cutting or gluing. In topology, size and shape
do not matter.
"unknot" - For purposes of this demonstration,
a tangle of rope which, when simplified so that crossings alternate (over,
under, over...) has no crossings at all.
In addition to discovering what is and is not a mathematical knot, the visitor learns of a few knot invariants. Knot invariants covered in this explainer are crossing number and order of crossings.
Knot theory, which extends far beyond the ideas covered in this explainer, was probably started originally by Lord Kelvin - the same Lord Kelvin for whom the Kelvin of temperature is named. He hypothesized that atoms were knotted vortices in the ether, an invisible fluid thought to fill all space. He hoped that in classifying knots, he could better understand chemicals and arrange an accurate periodic table.
Today, knot theory is making a comeback in the sciences. This has instigated a more collaborative effort between mathematics and the sciences. Previously unrelated areas are now being pursued in order to find new and unexpected uses for and advances in knot theory. For example, one researcher of von Neumann algebras, which are used in quantum mechanics, came up with one of the latest and most profound polynomials used to describe knots.
Molecular biologists use knot theory in describing and predicting the formation of strands of DNA. Chemists are using knot theory to try to create and analyze topologically interesting new compounds. As of 1988, they were still attempting to devise artificially knotted chemicals - knots that would only be seen under powerful microscopes, all in the hopes of better understanding molecular and atomic binding behaviors.
The mirror images of knots with odd crossing numbers are distinct, whereas some knots with even crossing numbers have distinct mirror images and others do not. (This is where the study of knots becomes interesting!)
If a knot A has a different crossing number than knot B, it is a different knot. The converse is not always true. Some different knots have the same crossing number.
There are no knots with crossing numbers one or two. There is one name for knots with a crossing number of 3, the trefoil, which has distinct left-handed and right-handed versions, and one with a crossing number of 4, the figure 8 in which the left-hand and right-hand versions are the same knot.
The topics in this explainer are related to the Beyond Numbers elementary and secondary teacher's manual lessons on knots and knot invariants and the following Beyond Numbers components:
MATERIALS:
board with shadow and one fixed rope
answer sheet
seven untied ropes
1. Lay out the board with the shadow and the example. Also lay out the seven untied ropes.
2.Explain how the example would cast a shadow like the one above and how there are seven other possibilities. Ask the visitors to try to make those possibilities. A layout that is exactly like another one except flipped over, is not a different possibility. Be sure to check that no two layouts are the same.
A good way to examine a layout is to start at one end of the rope and follow it along, keeping track of when the rope goes over or under another piece of the rope. The fixed example will be either: over, over, under, under, under, over or over, under, under, under, over, over.
3.As the visitors are arranging the ropes, inform them that there are eight possibilities in all, consisting of four arrangements and their mirror images. The mirror images can also be labeled as left-handed and right-handed.
Once all the layouts are finished, (and you can check the answer sheet to make sure) ask the visitors which layouts are true knots. Then tell the visitors to hold the two ends of a laid out rope together and lift. There should only be two layouts that have a knot when lifted. The other six possibilities are "unknots" (not knots) and will fall apart.
4.The purposes of all this:
a. Just because it resembles a knot, that
doesn't mean it is a knot.
b. A two-dimensional projection (the shadow)
can represent many three-dimensional figures.
 |
 |
 |
 |
 |
 |
 |
 |
 |
MATERIALS:
3 mounted configurations of string on boards
3 untied ropes
tape/clips
1.Present the three mounted knots and explain that some of them may not be in their simplest forms. The simplest form has the fewest amount of crossings needed to maintain the "knottiness" of the knot. A crossing is just where the string goes over or under itself.
2.Present the untied ropes and suggest to the visitors that they try to reproduce the fixed knots. Or create the knot yourself. When done, clip the ends of the rope together. Then let the visitor try to rearrange the knots to get as few crossings as possible.
3.When the knots have been put into their simplest arrangements, count the number of times the ropes cross themselves. That number is the "crossing number" of each rope. The visitors will have reached the simplest forms when all the crossings alternate, that is,. over, under, over or under, over, under, .. Conversely, this also is how you can tell if what you have is a knot. In its simplest form a knot's crossings will alternate.
4.For a knot in its simplest form, its crossing number is invariant. That is, no matter how you twist and turn it and look at it in different ways, the crossing number will stay the same.
 |
 |
 |
