On Alexander polynomial and Grid Homology for knots in the 3-sphere and in lens spaces

In this thesis, we analyzed two powerful knot invariants: Grid Homology and the Alexander polynomial, for knots and links in the 3-sphere and in lens spaces. The former provides a combinatorial description of Knot Floer Homology—originally introduced by Ozsváth, Szabó, and Rasmussen—and counts specific rectangles over a grid diagram representation of the knot. The latter is a topological invariant constructed from the first homology group of the universal cover of the knot complement. In particular, we detailed how to compute this invariant using several different approaches: from its Seifert matrix, from its knot group via Fox calculus, through Skein relations, and directly from the grid diagram of the knot. Furthermore, we demonstrated how to compute these invariants for knots in lens spaces. For knots in lens spaces, Grid Homology can be defined in an analogous way, following the work of Baker, Grigsby, and Hedden. Conversely, in place of the classical Alexander polynomial, we defined the twisted Alexander polynomial, which also accounts for the torsion subgroup of the homology of lens spaces. Finally, we analyzed the relationship between the Alexander polynomial and Grid Homology for knots in the 3-sphere, utilizing the graded Euler characteristic of the grid homology groups.