Abstract:
The availability of an infinite number of different knots and the "mysterious" applications that have been and still are applied to them, make knots desirable for scientific research. In this work, mathematical tools will be applied to study common knots of string or rope, which have been used by mankind in the most diverse activities since ancient times.
In mathematical terms, a knot is defined as an embedding of a circumference in the tridimensional space. The required tools to study this mathematical object are being developed since the end of the XIX century, and surprising results have been obtained in the last few decades. Nowadays, knots are studied by different branches of the mathematical science and the results already obtained are being applied to other scientific areas.
Because knots are an infinite family, one of their "mysteries" is associated with how to tabulate them. The prime purpose of Knot Theory, which is not solved yet, is to understand when two knots looking to be the same are equal or not. There are, however, invariants that allow to differentiate between two knots, and some methods have been developed to tabulate knots with certain properties. This thesis will mainly address these issues.
In the first chapter, basic notions that are essential to the understanding of Knot Theory are presented, as well as the first invariants (classical invariants: tricolorability; crossing number; unknotting number) that allowed to differentiate between certain knots. In 1928, in an attempt to easily tell knots apart, Alexander assigned to each knot to a polynomial. Alexander polynomial was not able to solve all problems in tabulation (in 1934, it was discovered that some distinct knots had the same Alexander polynomial) but introduced a new technique in the field. Following his footsteps, other mathematicians (Conway, Jones, Kauffman ...) defined, with different methods, polynomials associated to knots. The second chapter of tins work is dedicated to the study of these various polynomials.
Finally, and because certain knots with particular properties are able to be unequivocaly tabulated, the third chapter of this thesis addresses, using topological methods, one of those types of knots, the torus knots.

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