Parameterized Knots

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by Lee Stemkoski (Adelphi University)

Knot Software and LiveGraphics3D

Since visualizing knots is vital to understanding and classifying them, producing galleries of knots with interactive three-dimensional graphics is a valuable endeavor.

Perhaps the most well-known software for this purpose is KnotPlot [16]. The KnotPlot website contains an excellent gallery of interactive three-dimensional graphics. An evaluation version of the KnotPlot program (with some features disabled) may be downloaded free of charge; the full version is available for purchase. The novel feature of the gallery in this article is the inclusion of parametric equations that generate the knot being viewed, and a parametric curve grapher that enables the reader to explore additional parameterizations.

The interactive three-dimensional graphics in this article are displayed using LiveGraphics3D, a Java applet written by Martin Kraus [5]. The applets should run on any Java 1.1-enabled internet browser. You may need to install the Java plug-in for your browser. Bugs in specific versions of certain browsers can occasionally prevent the LiveGraphics3D applet from initializing correctly. Often, reloading the page in your browser may fix this issue.

The basic applet controls are as follows:

A LiveGraphics3D applet is provided in Figure 4 for the convenience of the reader who wishes to familiarize themselves with the basic applet controls. For a complete list of applet controls, see the User Interface section of the LiveGraphics3D Documentation [10].

Figure 4. Shastri's Trefoil Polynomial Knot

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