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.

34 pages, 12 figures

References in corpus (3)

• Knot Floer homology detects fibred knots
• On knot Floer homology and cabling II
• Berge-Gabai knots and L-space satellite operations

Cited by in corpus (6)

• Branched covers of quasipositive links and L-spaces
• L-space knots have no essential Conway spheres
• Census L-space knots are braid positive, except for one that is not
• Hyperbolic L-space knots not concordant to algebraic knots
• Taut foliations in branched cyclic covers and left-orderable groups
• Asymptotic behavior of unknotting numbers of links in a twist family