Summary
The topology of knots and links can be studied by examining colorings of their diagrams. We explain how to detect knots and links using the method of Fox tricoloring, and we give a new and elementary proof that an infinite family of Brunnian links are each linked. Our proof is based on the remarkable fact (which we prove) that if a link diagram cannot be tricolored then it must be linked. Our paper introduces readers to the Fox coloring generalization of tricoloring and the further algebraic generalization, called a quandle by David Joyce.
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Acknowledgments
First and foremost, Claudia and Devika would like to thank their advisor, Dr. Louis Kauffman for giving us the opportunity to study with him, for guiding us through the world of knot theory, and for introducing us to its mysteries and complexities. We would also like to thank Jonathan Schneider for being there when we first proved that the infinite class of Brunnian rings could not be tricolored. We want to thank Ethan Bian for helping us develop a model of the modular Brunnian rings and for patiently studying knot theory with us. We want to thank the Illinois Mathematics and Science Academy, specifically the Student Inquiry and Research department for providing transportation to our research.
Louis H. Kauffman is supported by the Laboratory of Topology and Dynamics, Novosibirsk State University (contract no. 14.Y26.31.0025 with the Ministry of Education and Science of the Russian Federation).
Additional information
Notes on contributors
Louis H. Kauffman
Louis H. Kauffman is emeritus professor of Mathematics at the University of Illinois at Chicago and is presently visiting Novosibirsk State University in Russia. He is a knot theorist by trade and the Editor in Chief of Journal of Knot Theory and Its Ramifications.
Devika Prasad
Devika Prasad is a senior at the University of Illinois, Urbana-Champaign, where she is majoring in computer science.
Claudia J. Zhu
Claudia J. Zhu is a recent graduate of the University of Pennsylvania, where she majored in computer science.