Genus, Fiberedness, $\tau$ and $\epsilon$ of Satellite Knots with $n$-Twisted Generalized Mazur patterns

We study a family of $(1,1)$-pattern knots that generalize the Mazur pattern, and compute the concordance invariants $\tau$ and $\epsilon$ of $n$-twisted satellites formed from these patterns. We show that none of the $n$-twisted patterns from this family act surjectively on the smooth or rational concordance group. We also determine when the $n$-twisted generalized Mazur patterns are fibered in the solid torus, compute their genus in $S^1 \times D^2$, and show that $n$-twisted satellites with generalized Mazur patterns and non-trivial companions are not Floer thin.