# Seifert vs slice genera of knots in twist families and a characterization of braid axes

## TL;DR

This paper investigates how twisting knots affects their genus properties and characterizes braid axes, providing conditions for the existence of tight fibered and L-space knots within twist families.

## Contribution

It establishes new criteria relating twist family behaviors to braid axes and classifies when twist families contain infinitely many tight fibered or L-space knots.

## Key findings

- If genus differences are bounded, the winding number is zero or equals the wrapping number.
- Presence of infinitely many tight fibered knots implies the winding number equals the wrapping number.
- Characterizes when twist families contain infinitely many L-space knots.

## Abstract

Twisting a knot $K$ in $S^3$ along a disjoint unknot $c$ produces a twist family of knots $\{K_n\}$ indexed by the integers. Comparing the behaviors of the Seifert genus $g(K_n)$ and the slice genus $g_4(K_n)$ under twistings, we prove that if $g(K_n) - g_4(K_n) < C$ for some constant $C$ for infinitely many integers $n > 0$ or $g(K_n) / g_4(K_n) \to 1$ as $n \to \infty$, then either the winding number of $K$ about $c$ is zero or the winding number equals the wrapping number. As a key application, if $\{K_n\}$ or the mirror twist family $\{\overline{K_n}\}$ contains infinitely many tight fibered knots, then the latter must occur. We further develop this to show that $c$ is a braid axis of $K$ if and only if both $\{K_n\}$ and $\{\overline{K_n}\}$ each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for $\{ K_n \}$ to contain infinitely many L-space knots, and show (modulo a conjecture) that satellite L-space knots are braided satellites.