A Characterization of Braid Axes

Ken Baker (University of Miami)

Let $\{K_n\}$ be the family of knots obtained by twisting a knot $K$ in $S^3$ along an unknot $c$. When the winding number of $K$ about $c$ is non-zero, we show the ratio $g(K_n)/g_4(K_n)$ limits to $1$ if and only if the winding and wrapping numbers of $K$ about $c$ are equal. When equal, this leads to a description of minimal genus Seifert surfaces of $K_n$ for $|n| \gg 0$ and eventually to a characterization of when $c$ is a braid axis for $K$. This is joint work with Kimihiko Motegi that builds upon joint work with Scott Taylor about the behavior of the Thurston norm under Dehn filling.