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

Twisting a knot K in S³ along a disjoint unknot c produces a twist family of knots {Kn} indexed by the integers. We prove that if the ratio of the Seifert genus to the slice genus for knots in a twist family limits to 1, then the winding number of K about c equals either zero or the wrapping number. As a key application, if {Kn} or the mirror twist family {Kn} contains infinitely many tight fibered knots, then the latter must occur. This leads to the characterization that c is a braid axis of K if and only if both {Kn} and {Kn} each contain infinitely many tight fibered knots. We also give a necessary and sufficient condition for {Kn} to contain infinitely many L-space knots, and apply the characterization to prove that satellite L-space knots have braided patterns, which answers a question of both Baker-Moore and Hom in the positive. This result also implies an absence of essential Conway spheres for satellite L-space knots, which gives a partial answer to a conjecture of Lidman-Moore. [ABSTRACT FROM AUTHOR]