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

TL;DR

This paper investigates a family of satellite knots generalizing Mazur patterns, computing their concordance invariants, fiberedness, genus, and Floer properties, revealing limitations on their surjectivity and thinness.

Contribution

It introduces a new class of generalized Mazur patterns, computes their invariants, and analyzes their fiberedness, genus, and Floer properties, extending understanding of satellite knot behavior.

Findings

None of the $n$-twisted patterns act surjectively on the concordance group.

The paper determines when these patterns are fibered in the solid torus.

It shows that certain satellites are not Floer thin.

Abstract

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.