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

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.