Invariants of Strongly Invertible Knots: Concordance and Grid Homology

This thesis studies strongly invertible knots, namely knots equipped with a symmetry that reverses their orientation. The work follows two main directions. The first concerns equivariant concordance, a refinement of classical knot concordance that also incorporates the symmetry of the knot. Within this framework, a new invariant called the moth polynomial is introduced. Its main application is to 2-bridge knots, for which it is shown that they always have infinite order in the equivariant concordance group. The algebraic structure of this group is also examined, leading to the proof that it is not solvable. The second part develops an equivariant version of grid homology. To do so, symmetric grid diagrams are introduced as representations of strongly invertible knots, together with the moves that preserve their equivalence class. A homological theory is then constructed on these diagrams, leading to the definition of new topological invariants associated with strongly invertible knots.