hyperbolic fibered slice knots with right-veering monodromy

The authors [HKK⁺21] observes that many low-crossing slice fibered knots have zero FDTC. Any $(p,1)-$cable of slice fibered knot is still slice fibered, whereas FDTC of $(p,1)-$cable equals to $1/p$. The authors therefore ask the above question about hyperbolic fibered slice knots.

Baldwin, Ni and Sivek [BNS22, Corollary 1.7] prove the following related proposition in terms of the $\tau-$invariant in Heegaard Floer homology:

The $\tau-$invariant vanishes for slice knots. Proposition 1.2 explains the case for low-crossing fibered slice knots because many of those are thin.

We have an immediate corollary:

We follow the recipe of Kazez and Roberts [KR13] to construct hyperbolic fibered knots with positive FDTC. The search for ribbon knot is inspired by the work of Hitt and Silver [HS91]. In section 2 we review Nielson-Thurston classification of surface automorphism and examples from Kazez and Roberts. We construct our example $K^{\prime}$ in section 3.

We first recall Nielson-Thurston classfication of surface automorphisms:

In particular, we only regard $h$ as reducible only if it is not periodic to avoid overlap. Let $\Phi:S\times[0,1]\rightarrow S$ be an isotopy from $h$ to its Thurston representative $\phi$. Considering the restriction of $\Phi$ to the boundary $\partial S$, we have a homeomorphism:

$$\Phi_{\partial}:\partial S\times[0,1]\rightarrow\partial S\times[0,1]$$

defined by $\Phi_{\partial}(x,t)=(\Phi_{t}(x),t)$. The fractional Dehn twist coefficient $c(h)$ can be defined as the winding number of the arc $\Phi_{\partial}(\theta\times[0,1])$. Nielson-Thurston classification guarantees that $c(h)\in\mathbb{Q}$.

Thurston proved that a fibered knot is hyperbolic if and only if its monodromy is pesudo-Anosov. Fractional Dehn twist coefficient is closely related to the following notion of right-veeringness.

If $c(h)=0$, one can find two arcs such that one is moved by $h$ to the right and the other to the left. The significance of right-veeringness is highlighted by the following theorem of Honda, Kazez and Matić:

A large source of examples of reducible right-veering homeomorphism comes from the class of fibered cable knots. Indeed, if $h$ is the monodromy of a fibered $(p,q)-$cable knot $K_{p,q}$ with Seifert surface $S$, then $c(h)=1/pq$ and $h$ is reducible. Let $\{C_{i}\}$ be the collection of curves preserved by $h^{\prime}$. $\{C_{i}\}$ partitions $S$ into subsurfaces $\{S_{j}\}$ permuted by $h^{\prime}$. Let $S_{0}$ be the subsurface containing $\partial S=K_{p,q}$, then $h^{\prime}|_{S_{0}}$ is periodic. Kazez and Roberts characterize the monodromy $h$ of a fibered knot $K$ in $S^{3}$ in the following theorem:

In particular, the $(2,1)-$cable of a fibered knot in $S^{3}$ has its monodromy attaining maximum FDTC. We review hyperbolic case in the next section.

Let $U$ be an unknot properly embedded in a surface $F$. We say $U$ is untwisted relative to $F$ if $U$ bounds a disk transverse to $F$ along $U$. A Stallings’ twist [Sta78] is a surgery along such an untwisted $U$. Kazez and Roberts apply Stallings’ twist on $(2,1)-$cables to produce hyperbolic fibered knots with maximum FDTC $=1/2$.

Let $(S,h)$ be an open book decomposition of $S^{3}$ with connected binding $K$, where $h$ is pseudo-Anosov and $c(h)=0$. Let $K_{2,1}$ be the $(2,1)-$cable of $K$. The fibered surface $\Sigma$ of $K_{2,1}$ can be viewed as the union of two copies $S_{0}$, $S_{1}$ of $S$ connected by a 1-handle. Let $H$ be the monodromy of this new open book.

We choose a simple closed curve $C$ in $\Sigma$ such that $C_{0}=C\cap S_{0}$ and $C_{1}=C\cap S_{1}$ are two essential arcs. Moreover, we require $C_{i}$ to be nonseparating in $S_{i}$. Let $T_{C}$ be the right-handed Dehn twist along $C$ and $H^{\prime}=T_{c}\circ H$.