Knot Theory Project - Discovery and classification of 8 -- crossing knots
The main goal of knot theory is the classification of knot types. In this project we will explore this question through the discovery of knots up to 8 crossings. By the end of the semester we hope, as a class, to have all knots of 8 or fewer crossings listed and fully classified. You will each be assigned some knot diagrams to consider. As we learn more about how to classify knots, you should apply your new skills to your diagrams and keep an eye out for possible equivalences. Your final product will be an info sheet for each knot including all the accumulated information about that knot. This includes the following:

• Conway notation(s)
• diagram of the knot
• 3-colorability
• 5-colorability
• determinant
• Alexander polynomial
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• ...more to come.

We will keep an ongoing tally of information here. At one or more points in the semester we may redistribute the our knots in our list so that each student remains responsible for approximately the same number of knots.


Update:(2/22)
1) What about if we had negatives in our Conway notation?
• Look back at your knots.
• Ignore links, and ignore things that give you a mirror image of one you've already looked at.
• Look for Reidemeister moves to simplify...many of these will be unknots or trefoils.
• Try to figure out as many of these as you can. Our goal is to have at least some description of every knot in the knot table (in the back of the book), up to 8 crossings.

2) Which knots (in the table) do we have?
• We're redistributing our list by determinant.
• Look through the list for all the knots with your determinant.
• Look at knotinfo to see what knots from the table have that determinant.
• Calculate Alexander polynomials to help narrow your choices for comparison.
•Use Reidemeister moves to show that the ones in your list are equivalent to knots from the table.


Update:(3/14)
We have reassigned our knots. From here on out, our knots will be assigned by the names in the knot table. Please see blackboard for the updated list. These will be your knots for the rest of the semester.

Having identified the knots in the knot table, we now hope to build a profile for each knot. Your knots should include the ones that you calculated the determinants for in the last stage. Review your calculations up to now, and be sure that you have the following for each of your knots:
• Conway notation(s)
• diagram of the knot
• determinant and colorability
• Alexander polynomial
• a diagram of a Seifert surface for the knot


Update:(5/3)
As we've progressed in class we've discussed several more items that should be a part of your project. To summarize, your project should include everything in the list above, as well as:
• the Seifert matrix for the knot
• the knot group
• samples of calculation of Alexander polynomials by all three methods (through labeling, through Seifert matrices, and through the knot group)
• an estimate of the genus of the knot (exact value if possible)

An excellent project will also include:
• a braid diagram for the knot
• braid notation for the diagram
• an estimate of the braid number of the knot
• an estimate of the bridge number for the knot
• if the knot is a two-bridge knot, the associated continued fraction
• an estimate of the unknotting number for the knot

BONUS: It is often useful to consider a physical model for a knot. You are encouraged to think creatively about how to make a model of your knots, and include some kind of model in your final product.