Preliminaries

If all your system settings are correct, the knot tool should 'pop up' as a separate window from your browser. If it does not, then you probably need to download and install the appropriate Java plugin for your platform and browser from Sun Microsystems ( Sun Java plugin page ).

The tool was originally written as a Java application and recently converted to work as a Java applet. To view any information on the knot from the Info pulldown menu, you must open a Java console from your browser. From Microsoft Internet Explorer (IE), a Java console is usually found from the Tools -> Java Console pulldown menu. I don't normally use IE, but I did test this tool from an IE v6.x browser with a Java plugin installed and noted the following. I needed to enable two options. From the Tools -> Internet Options... menu, click on the Advanced tab. Scroll down, and underneath the heading Java (Sun) check the box for "Use Java 2 v1.xx (applet) (requires restart)". Then check the box under the heading Microsoft VM labeled "Java console enabled (requires restart)". Then shut down and restart your browser.

Note that this is an early version of the tool, and a later version of the applet will allow output in a scroll window below the knot diagram.



Instructions For Use

Initially, a simple link will be displayed on a 4X4 grid. By running the mouse pointer over the grid, you will see little circles appear on the nodes which can control the topology of the knot. Click on one of the nodes and change the node to either a "Horizontal Split" or a "Vertical Split". You should then see that it is simple to create a trefoil knot.

Options menu:

The grid size can be changed from the options menu. For example, the simplest form of a knot, the unknot, can be created from a 2 X 2 grid.





The 6_3 knot can be created from a 6 X 4 grid.





Note that you must have an M X N grid, where M and N are even, in order to create a knot, link, or braid. Currently, if either M or N are odd numbers, an error will occur as you can see in the Java console and odd things may occur down the road. I would suggest restarting the applet by going "Back" in your browser, if this occurs.

View menu:

Click on the different options in the View menu, and see what happens. The "Node Markers" are particularly useful for understanding and visualizing how the horizontal and vertical splits shape the knot.

Info menu:

There are a number of options here to print data about the knot to a Java console. Additionally, this is where you can perform a knot analysis for computing the Kauffman bracket polynomial and the Jones polynomial. To perform a knot analysis, click on Info -> Knot Analysis. Press the Start button to run the analysis. If the Animate box is checked, you will see the knot type change on the screen as it steps through the computation of the Kauffman bracket polynomial. You can manually step through each stage of the computation by pressing the Step button and viewing the data in the Java console. After a successful analysis, you can view the Kauffman and Jones polynomial in the Java console from the Info -> Print Knot Analysis pulldown menu.

Note: I have created many of the first twenty or so prime knots with this tool and verified the correctness of the Jones polynomial calculation. However, the results for links are currently very uncertain.

File menu:

The file menu does absolutely nothing from the applet. If you want to exit the applet, then press "Back" on your browser or close the browser.