Week 1:

1. To create professional mathematical documents, you will need to learn how to use "LaTeX".
   Read a little bit about LaTeX here: http://www.latex-project.org/intro.html
   You can use LaTeX either by downloading a program or using an online service. For now, we will use a (free) website called ScribTeX: http://www.scribtex.com . To guide you through setting up a ScribTeX account and writing your first mathematical document, I have created and uploaded a video to YouTube, watch it at http://www.youtube.com/watch?v=lh7Py16P90k .
   For more introductory information, read "A Quick Introduction to LaTeX", attached below. For more detailed information, read the attached files "LaTeX - Absolute Beginners", "LaTeX - Mathematics", and "LaTeX - Advanced Mathematics".  The file "LaTeX - Reference - Richeson" is a 2-page "cheat sheet" that might be helpful to have handy.  Your goal for this week is to duplicate the file "Mathematics-LaTeX-Test" attached below.  The file "Mathematics-LaTeX-Hints" contains a list of the commands you might need in order to typeset the test document.
   
2. We will be using the program GeoGebra to visualize a lot of the mathematics we will be doing in this course.  Again, this is a free program that can either be downloaded or used online from http://www.geogebra.org . Read and work through the GeoGebra 3.0 QuickStart at the website http://www.geogebra.org/cms/en/help (and bookmark this webpage -- there are lots of useful documents you may want to come back to later). There is also an amazing amount of useful information at http://mathandmultimedia.com/geogebra/ . In particular, try working through the tutorials:
   
   http://mathandmultimedia.com/2009/11/13/geogebra-tutorial-4-graphs-and-sliders/
   http://mathandmultimedia.com/2010/04/19/geogebra-tutorial-11-sliders-andtrigonometric-functions/
   http://mathandmultimedia.com/2010/05/31/geogebra-tutorial-17-functions-derivatives/
   
   Once you're comfortable with the above material, your next goal is to create two sliders for the variables "a" and "b", and then (using the text input), create the parametric curve: Curve[ t^3 + a*t, t^4 + b*t^2, t, -2.5, 2.5 ].  Change the values of "a" and "b" and see how many different types of curves you can obtain, and graph them on an "(a,b)-plane" as discussed in class. 
   If you finish this with time to spare, try some of the other tutorials at
   http://mathandmultimedia.com/2010/12/01/geogebra-intermediate-tutorial-series/ .
   
3. Read the first chapter of "The Knot Book" by Colin Adams.

Week 2:

1. Read "LaTeX - Document Structure" and try to duplicate the file "Document-LaTeX-Test". 
   This is *much* harder, so check out the "Document-LaTeX-Hints" file as well.
2. As you did last week, create a knot-field graph, this time for the parametric curve t^5 + at, t^3 + bt.
3. Questions:
   (a) Why do all the knots occur (for both parametric equations we have explored)
   
   when both the numbers a and b are negative?  ("Why is Quadrant III cool?")
   
   (b) Why can't you get a trefoil knot with the parametric curve above?
   (c) For the parametric equation from last week, explain why multiplying both a and b by a number S
   
   results in a scaled version of the knot, and nothing else. 
   What does this have do to with the speed with which we traverse the knot?
   
4. Read the Biology/Chemistry/Physics chapter of The Knot Book. (It's good to know why all this is useful.)
   

Week 3:

1. Article-style LaTeX summary of previous results, including knot-field diagrams

2. Recommended Reading:
    - Parameterized Knots (Stemkoski 2010)
    - Polynomial Knots (Durfee and O'Shea 2006)
    - Examples of Polynomial Knots (Brown 2004),
        and Identification of Brown's Equations (Toman 2006)
    - Polynomial Representation for Long Knots (Mishra and Prabhakar 2008)
    - http://rigtriv.wordpress.com/2007/08/02/polynomial-knots-i/
    - http://rigtriv.wordpress.com/2007/08/06/polynomial-knots-ii/