Mathematics – Geometric Topology
Scientific paper
2006-08-07
Banach Center Publications, Vol. 42, "Knot Theory", Warszawa, 1998, 275-295
Mathematics
Geometric Topology
28 pages, 27 figures
Scientific paper
This paper has two-fold goal: it provides gentle introduction to Knot Theory starting from 3-coloring, the concept introduced by R. Fox to allow undergraduate students to see that the trefoil knot is non-trivial, and ending with statistical mechanics. On the way we prove various (old and new) facts about knots. We relate Fox 3-colorings to Jones and Kauffman polynomials of links and we use this connection to sketch the method of approximating the unknotting number of a knot. We discuss some elementary open problems in knot theory and show that Fox colorings can be useful in trying to solve them. Finally we demonstrate how analysis of Fox colorings can lead us to understand homology and the fundamental group of branched coverings along links.
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