## Parameterized Knots

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### Activities

The 3D parametric curve grapher can also be used as the basis of a variety of activities to introduce students to knot theory. Some ideas include:

• Click the "Trefoil Polynomial Knot" button to obtain a graph of the trefoil polynomial knot, and press the Home key to place the graph in its default position. Without rotating the graph, try to vary the coefficients to obtain a curve that has a projection with three crossings but is not a trefoil knot. Again, without rotating, try to vary the coefficients to obtain a curve that has a projection with only one crossing.
• Click the "Trefoil Torus Knot" button to obtain a graph of the trefoil, and press the Home key to place the graph in its default position. The crossing number of the trefoil is 3, and indeed, the default view of this trefoil has three crossings. Rotate this graph to obtain different perspectives with different numbers of crossings. What are the different numbers of crossings that can appear in different perspectives of this graph? What is the largest number of crossings that can appear? (It may be easier to see crossings with different graphics properties: try reducing the tube radius to 0.01, turning off the axes option, and turning off the box option.)
• In some mathematical models, knots are modeled as a tube with a given thickness. Naturally, one requires that the tube must not intersect itself. Find the largest radius possible such that the trefoil torus knot does not intersect itself.

### References

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