Some Three and Four-Dimensional Invariants of Satellite Knots with (1,1)-Patterns

Abstract

We use bordered Floer homology, specifically the immersed curve interpretation of the bordered pairing theorem, to compute various three- and four-dimensional invariants of satellite knots with arbitrary companions and patterns from a family of knots in the solid torus that have the knot type of the trefoil in $S^3$. We compute the three-genus, and bound the four-genus of these satellites. We show that all patterns in this family are fibered in the solid torus. This implies that satellites with fibered companions and patterns from this family are also fibered. We also show that satellites with thin fibered companions or companions $K$ with $\tau(K)=\pm g(K)$ formed from these patterns have left or right veering monodromy. We then use this to show that satellites with thin fibered companion knots $K$ so that $|\tau(K)|<g(K)$ formed from these patterns do not have thin knot Floer homology.