Georgia Institute of Technology College of Sciences

This thesis adopts the immersed-curve perspective to analyze the knot Floer complexes of (1,1)-satellite knots. The main idea is to encode the chain model construction through what we call a planar (1,1)-pairing. This combinatorial and geometric object captures the interaction between the companion and the pattern via the geometry of immersed and embedded curves on a torus (or its planar lift). By working with explicitly constructed (1,1)-diagrams and their planar analogs, we derive rank inequalities for knot Floer homology and develop a geometric algorithm for computing torsion orders. The latter, based on a depth-search procedure, translates intricate algebraic operations into tangible geometric moves on planar (1,1)-pairings, further yielding results on unknotting numbers and fusion numbers.