<< Previous ||
by Lee Stemkoski (Adelphi University)
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.
-  Adams, Colin. The Knot Book. American Mathematical Society, 2004.
-  Bogle, Hearst, Jones, and Stoilov. "Lissajous Knots". Journal of Knot Theory and Its Ramifications (3), 2, 1994.
Available online at http://www.cchem.berkeley.edu/jehgrp/pdfs/_272.pdf.
-  Boocher, Adam, Jay Daigle, Jim Hoste, and Wenjing Zheng. "Sampling Lissajous and Fourier Knots".
Experimental Mathematics (18), 2009.
Available online at http://arxiv.org/abs/0707.4210.
-  Buck, Dorothy, and Erica Flapan. Applications of Knot Theory.
American Mathematical Society, 2009.
-  Brown, Ashley, "Examples of Polynomial Knots". (2006). Available online at http://www.mtholyoke.edu/~adurfee/reu/04/reu04.htm.
-  Durfee, Alan and Donal O'Shea. "Polynomial Knots". Preprint (2006). Available online at http://arxiv.org/abs/math/0612803.
-  Hoste, Jim. "Torus Knots are Fourier-(1,1,2) Knots". Preprint (2007). Available online at
-  Kauffman, Louis. "Fourier Knots", in Ideal Knots, World Scientific Publishing Company, 1999.
Available online at http://arxiv.org/abs/q-alg/9711013
-  The Knot Atlas. Available online at http://katlas.math.toronto.edu/wiki/Main_Page.
-  Kraus, Martin. LiveGraphics3D Homepage. (no date). Available online at http://www.vis.uni-stuttgart.de/~kraus/LiveGraphics3D/.
-  Kraus, Martin and Jonathan Rogness. "Constructing Mathlets Quickly using LiveGraphics3D". Journal of Online Mathematics and its Applications (2006). Available online at http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=1143.
-  Madeti, Prabhakar, and Rama Mishra. "Polynomial Representation for Long Knots". Int. Journal of Math. Analysis (3), 7, 2009.
Available online at http://arxiv.org/abs/0803.3195
-  Madeti, Prabhakar, and Rama Mishra. "Minimal Degree Sequence for Torus Knots of Type (p, q)". Journal of Knot Theory and its Ramifications (18), 4, 2009.
-  Murasugi, K. "On the Braid Index of Alternating Links." Trans. Amer. Math. Soc. (326), 1991.
-  Shastri, Anat. "Polynomial Representations of Knots", Tohoku Math. J. (44), 1992.
Available online via Project Euclid: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.tmj/1178227371.
-  Scharein, Rob. KnotPlot Homepage. Available online at http://www.knotplot.com.
-  Schubert, Horst. "Uber eine numerische Knoteninvariante." Math. Z. (61), 1954.
-  Sumners, De Witt. "Lifting the Curtain: Using Topology to Probe the Hidden Action of Enzymes". Notices of the American Mathematical Society (42), 5, May 1995. Available online at http://www.ams.org/notices/199505/sumners.pdf.
-  Trautwein, Aaron. "Harmonic Knots" (Ph.D. Thesis.), University of Iowa, 1995.
-  Trautwein, Aaron. "An Introduction to Harmonic Knots" in Ideal Knots, World Scientific Publishing Company, 1999.
-  Wikipedia contributors. "Reidemeister Moves", Wikipedia, The Free Encyclopedia, accessed 1 Nov. 2010. Available online at: http://en.wikipedia.org/wiki/Reidemeister_move
<< Previous ||