KNOTS!!

Distinguishing between Mathematical Knots using Three Coloring

Take the pipe cleaner you were given. Loop it around and through itself one or more times and attach the two free ends. Don’t pull it tight. This is a mathematical knot. It is a closed circle that might be tangled up. 1. Without detaching the ends of your pipe cleaner, can you make your knot look like someone else’s knot in your group? In math, we say two knots are equivalent or are the same knot if you can move the strands around to make one look like the other one.

We often want to draw a picture of knots we’ve made. When we draw a picture on paper, we want to be able to include enough information so that someone can know exactly what knot we started with. To do this, we smush the strands around and then flatten it on to the table making sure: • At every point, at most two strings meet up. • When two strings meet up, we draw a break in the strand to designate which strand is the under strand. These points are called crossings. A picture of a knot is called a knot diagram. 2. Draw a knot diagram of the knot you made. Look at the pictures above to give you an idea of what knot diagrams look like. Is it possible to draw a different diagram of the same knot? Pick up your pipe cleaner, move the strands around some more and draw it again.

3. Now, make the knots at the top of the page out of pipe cleaners. Are any of the knots represented by these diagrams equivalent? If so, which ones? How can you tell? Each of these knots has a special name. The first two are unknots. The next two are trefoils. The final one is a figure 8 knot. These aren’t the only knots! In fact, there are infinitely many! Below is a more complicated one. Its name is K11a351.

If you made this out of string, no matter how much you moved the strings around, you would never be able to draw it with less than 11 crossings! In fact, for any number you can think of, there is a knot that needs at least that many crossings to be drawn.

Here are all of the distinct knots up to 7 crossings.

We can use more than one string when making our knots. When we use more than one string, we call them links instead of knots. We are going to look at two different special links that each have two components (meaning you have two separate strings).

The first one is the unlink, the second is the Hopf Link. It’s easy to tell between knots and links because we can just count the number of components. 4. Below are some knots and links. Circle the ones that are knots.

This is just drawn with thicker string.

Here are all the distinct links up to 7 crossings:

Now we want to know a way to tell apart knots from each other and links from each other. To do this, we’ll use three coloring. The rules for three coloring are as follows:

• You can use either 2 or 3 colors.

• Color a strand all one color until you get to an under crossing (break in the strand). When you get to a break, you can either change colors or stay the same color.

• At each crossing, either all three strands coming in are the same color or all three colors are different. (Within your diagram, it’s ok if some crossings are all the same color while other ones have all three colors, but you can’t have two strands the same color and one different.)

5. Below are a few different diagrams of the unknot. Can you three color them using the rules above?

6. Below are a few different diagrams of trefoils. Can you three color them using the rules above?

Theorem 1 It doesn’t matter how you draw the diagram, if you can’t three color one diagram of a knot you can’t 3 color any of its diagrams. And if you can color a knot diagram or a knot, then you can color all of the other diagrams of the same knot.

For example, if we have a diagram and we can three color it, we know it can’t be a diagram of the unknot, since none of them are three colorable. We want to fill out the table below indicating whether or not each knot has three colorable diagrams and how many components each has. Name Picture How many components does it have? Is it 3 colorable?

Unknot 1 No!

Trefoil

Figure 8 Knot

Unlink

Hopf Link

Using just three coloring and counting the number of strands, we can separate knots into 4 different categories, three colorable knots, three colorable links, non three colorable knots and non three colorable links. This helps us identify everything above except we can’t tell the figure 8 knot from the unknot.

7. Can you think of another way we can tell the figure 8 knot from the unknot?

Final activity: We have a bunch of unknots, trefoils, figure 8 knots, unlinks and Hopf links on the front table. Use three coloring and number of components and the way you found above to distinguish the figure 8 knot from the unknot to determine which type of knot each on is. When you study knots, sometimes your friends ask you funny questions. Someone sent me this picture and asked if it was a unknot or not. What do you think?