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By Sylvester J.J.

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Here we show some examples of such colourings and their one–to–one correspondences. The other possibilities can be obtained from these by some permutation of colours (here colours are marked by numbers 1, 2, 3). Thus, each proper colouring of the initial diagram uniquely corresponds to a colouring of the diagram obtained from the initial one by applying a Reidemeister move. Thus, the number of proper colourings is invariant under Reidemeister moves. Let us now call the number of proper colourings the colouring invariant.

2. Show that if both numbers are equal to zero then the knot is trivial. 1. In the sequel, we shall consider only those knots for which at least one of these numbers is not equal to zero. The following fact is left for the reader as an exercise. 3. For a non-self-intersecting curve these numbers are coprime. 1. Knots in Surfaces. 3. Trefoil on the torus So, each torus knot passes p times the longitude of the torus, and q times its meridian, where GCD(p, q) = 1. It is easy to see that for any coprime p and q such a curve exists: one can just take the geodesic line {qφ − pψ = 0(mod 2π)}.

Fix a point a on the circle S 1 . Consider the set of continuous mappings f : S 1 → X such that f (a) = x0 . The set of homotopy classes of such mappings admits a group structure. Indeed, the multiplication of two such mappings can be represented by concatenating their paths. The reverse element is obtained by passing over the initial path in the reverse order. Obviously, these operations are well defined up to homotopy. 1. The obtained group is called the fundamental group of the space X; it is denoted by π1 (X, x0 ).

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Collected mathematical papers, volume 3 by Sylvester J.J.

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