Knots and Links in Lens Spaces: an Introduction to Their Representation and to Several Geometric Invariants

Knot theory is a very interesting field of mathematics, that takes advantage of results of different areas. Its study produced interesting results not only in topology, but also in biology and physics. Usually knots are studied as subsets of the three dimensional sphere, but recent works pointed out the importance of knots inside different manifolds, such as lens spaces. This book contains the description of several existing possible representations of links in lens spaces, explaining how to transform one into another. From one of this representation, a presentation of the knot group is obtained, and it is shown how to compute from it an interesting family of twisted Alexander polynomials. Beside this, the invariant on which the book is focused on is the lift in the 3-sphere. After producing several possible representations of it, it is shown that the invariant is not complete for links in lens spaces, that is, there exist different links with equivalent lift. Using these examples it is proved that several existing invariants of links in lens spaces are essential, that is to say, they may assume different values on links with equivalent lift.