  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.
