$(1,1)$ L-space knots

Let $K$ denote a (doubly pointed) knot in a rational homology sphere $Y$. Let $N(K)$ denote an open tubular neighborhood of $K$, $X=Y\smallsetminus N(K)$ the knot exterior, and $\mu\subset{\partial}X$ a meridian of $K$. Let ${\mathrm{Spin}^{c}}(Y)$ denote the set of ${\mathrm{spin^{c}}}$ structures on $Y$ and ${\mathrm{Spin}^{c}}(X,{\partial}X)$ the set of relative ${\mathrm{spin^{c}}}$ structures on $(X,{\partial}X)$. They are torsors over the groups $H_{1}(Y;{\mathbb{Z}})$ and $H_{1}(X;{\mathbb{Z}})$, respectively. Let ${\mathrm{Spin}^{c}}(K)$ denote the set of orbits in ${\mathrm{Spin}^{c}}(X,{\partial}X)$ under the action by $[\mu]$. It forms a torsor over $H_{1}(X;{\mathbb{Z}})/[\mu]\approx H_{1}(Y;{\mathbb{Z}})$, and there exists a pair of torsor isomorphisms $i_{v},i_{h}:{\mathrm{Spin}^{c}}(K)\to{\mathrm{Spin}^{c}}(Y)$.

We work with the hat-version of knot Floer homology with ${\mathbb{Z}}$ coefficients, graded by $s\in{\mathrm{Spin}^{c}}(K)$:

$$\widehat{HFK}(K)=\bigoplus_{s}\widehat{HFK}(K,s).$$

There exists a further pair of gradings

$$a,m:\widehat{HFK}(K,s)\to{\mathbb{Z}}$$

on each summand, the Alexander and Maslov gradings. An element is homogeneous if it is homogeneous with respect to both gradings. The group $\widehat{HFK}(K)$ comes equipped with differentials $\widetilde{d}_{v}$, $\widetilde{d}_{h}$ that preserve the ${\mathrm{Spin}^{c}}(K)$-grading, respectively lower $m$ and $m-2a$ by one, and respectively raise and lower $a$. They are invariants of $K$, and their homology calculates $\widehat{HF}(Y,i_{v}(s))$ and $\widehat{HF}(Y,i_{h}(s))$, respectively. The manifold $Y$ is an L-space if $\widehat{HF}(Y,t)\approx{\mathbb{Z}}$ for all $t\in{\mathrm{Spin}^{c}}(Y)$.

Note that the requirement that the positive chain basis elements are homogeneous does not appear in [RR15, Definition 3.1]. For example, we could alter the positive chain basis displayed above to an inhomogeneous one by replacing $x_{2}$ by $x_{2}+x_{1}$. However, homogeneity is an intended property of a positive chain basis in the literature. Moreover, an inhomogeneous positive chain basis gives rise to a homogeneous one by replacing each basis element by its homogeneous part of highest bigrading. Therefore, our definition is no more restrictive, and its precision is more convenient when we invoke it in the proofs of Lemmas 2.3 and 2.4.

Let $\lambda\subset{\partial}X$ denote the rational longitude of $K$, the unique slope that is rationally null-homologous in $X$. Note that $\mu\neq\lambda$, since $Y$ is a rational homology sphere. Another slope $\alpha\subset{\partial}X$ is positive or negative according to the sign of $(\mu\cdot\lambda)(\lambda\cdot\alpha)(\alpha\cdot\mu)$, for any orientations on these curves. The knot $K$ is a positive or a negative L-space knot if it has an L-space surgery slope of that sign. The following theorem characterizes positive L-space knots in L-spaces in terms of their knot Floer homology. Ozsváth and Szabó originally proved it for the case of knots in $S^{3}$ [OS05, Theorem 1.2]. Boileau, Boyer, Cebanu, and Walsh promoted a significant component of their result to knots in rational homology spheres [BBCW12]. Building on it, J. and S. Rasmussen established the definitive form of the result that we record here: see [RR15, Lemmas 3.2, 3.3, 3.5], including the proofs of these results.

The invariants can be calculated from any doubly-pointed Heegaard diagram $D=(\Sigma,\alpha,\beta,z,w)$ of $K$ after making some additional analytic choices. Here, as usual, $\Sigma$ denotes a closed, oriented surface of some genus $g$; $\alpha$ and $\beta$ are $g$-tuples of homologically linearly independent, disjoint, simple closed curves in $\Sigma$; and the two basepoints $w$ and $z$ lie in the complement of the $\alpha$ and $\beta$ curves on $\Sigma$. The curve collections induce tori ${\mathbb{T}}_{\alpha},{\mathbb{T}}_{\beta}$ in the $g$-fold symmetric product $\mathrm{Sym}^{g}(\Sigma)$. The underlying group of the Floer chain complex $\widehat{CFK}(D)$ is freely generated by ${\mathfrak{T}}(D)={\mathbb{T}}_{\alpha}\cap{\mathbb{T}}_{\beta}$. The elements of ${\mathfrak{T}}(D)$ fall into equivalence classes in 1-1 correspondence with ${\mathrm{Spin}^{c}}(K)$: two elements $x,y$ lie in the same equivalence class iff the set $\pi_{2}(x,y)$ of homotopy classes of Whitney disks from $x$ to $y$ is non-empty. Write ${\mathfrak{T}}(D,s)$ for the equivalence class corresponding to $s\in{\mathrm{Spin}^{c}}(K)$ and $\widehat{CFK}(D,s)$ for the subgroup of $\widehat{CFK}(D)$ generated by the elements in ${\mathfrak{T}}(D,s)$. Each Whitney disk $\phi\in\pi_{2}(x,y)$ has a pair of multiplicities $n_{z}(\phi)$, $n_{w}(\phi)$ and a Maslov index $\mu(\phi)$. The Alexander and Maslov gradings on $\widehat{CFK}(D,s)$ are characterized up to an overall shift by the relations

$$a(x)-a(y)=n_{z}(\phi)-n_{w}(\phi),\quad m(x)-m(y)=\mu(\phi)-2n_{w}(\phi)$$

for all $x,y\in{\mathfrak{T}}(D,s)$ and $\phi\in\pi_{2}(x,y)$. There exist endomorphisms $d_{0},d_{v},d_{h}$ of $\widehat{CFK}(D,s)$ defined on generators $x\in{\mathfrak{T}}(D,s)$ by the same general prescription:

$$d(x)=\sum_{y,\phi}\#\widehat{\mathcal{M}}(\phi)\cdot y.$$

Here $y$ ranges over ${\mathfrak{T}}(D,s)$; $\phi$ ranges over the elements of $\pi_{2}(x,y)$ with Maslov index $\mu(\phi)=1$ and a constraint on the multiplicities $n_{w}(\phi)$, $n_{z}(\phi)$; and $\#\widehat{\mathcal{M}}(\phi)$ is the count of pseudo-holomorphic representatives of $\phi$. The specific constraints on the multiplicities are $n_{w}(\phi)=n_{z}(\phi)=0$ for $d=d_{0}$; $n_{w}(\phi)=0,n_{z}(\phi)>0$ for $d=d_{v}$; and $n_{w}(\phi)>0,n_{z}(\phi)=0$ for $d=d_{h}$. The maps $d_{0},d_{0}+d_{v},d_{0}+d_{h}$ are all differentials. The differential $d_{0}+d_{v}$ lowers $m$ by one and is filtered with respect to $a$. The differential $d_{0}+d_{h}$ lowers $m-2a$ by one and is filtered with respect to $-a$. The differential $d_{0}$ is the $a$-filtration-preserving component of each. The groups $\widehat{HFK}(K,s)$, $\widehat{HF}(Y,i_{v}(s))$, and $\widehat{HF}(Y,i_{h}(s))$ are the homology groups of $\widehat{CFK}(D,s)$ with respect to $d_{0}$, $d_{0}+d_{v}$, and $d_{0}+d_{h}$, respectively, and the Alexander and Maslov gradings on $\widehat{CFK}(D,s)$ descend to the respective gradings on $\widehat{HFK}(K,s)$. The maps $d_{0}+d_{v}$ and $d_{0}+d_{h}$ induce the differentials $\widetilde{d}_{v}$ and $\widetilde{d}_{h}$ on $\widehat{HFK}(K,s)$, respectively.

The Alexander and Maslov gradings enable us to constrain the existence of pseudo-holomorphic disks for L-space knots.

Now assume that $D$ is a $(1,1)$-diagram, so $\Sigma$ has genus one. In this case, the differentials on $\widehat{CFK}(D)$ admit an explicit description that requires no analytic input, owing to the Riemann mapping theorem and the correspondence between holomorphic disks in the Riemann surface $\Sigma=\mathrm{Sym}^{1}(\Sigma)$ and its universal cover ${\mathbb{C}}$. Specifically, the conditions that $\phi\in\pi_{2}(x,y)$ and $\mu(\phi)=1$ imply that $\#\widehat{\mathcal{M}}(\phi)=\pm 1$ for any choice of analytic data. Furthermore, these conditions are met if and only if the image of $\phi$ lifts under the universal covering map $\pi:{\mathbb{R}}^{2}\to\Sigma$ to a bigon cobounded by lifts of $\alpha$ and $\beta$. See [GMM05] and [OS04, pp.89-96] for more details.

Assuming now that $D$ is reduced, we have $d_{0}=0$ and $\widehat{HFK}(K,s)\approx\widehat{CFK}(D,s)$. We proceed to analyze the differentials $\widetilde{d}_{h}$ and $\widetilde{d}_{v}$. Without loss of generality, we henceforth fix $\Sigma={\mathbb{R}}^{2}/{\mathbb{Z}}^{2}$, $z=\pi({\mathbb{Z}}^{2})$, a horizontal line ${\widetilde{\alpha}}\subset{\mathbb{R}}^{2}\smallsetminus{\mathbb{Z}}^{2}$, and $\alpha=\pi({\widetilde{\alpha}})$. After a further homeomorphism, we may assume that $D$ takes the form shown in Figure 2. Observe that the embedded bigons in $D$ are the obvious ones cobounded by the $\alpha$ with the “rainbow” arcs of $\beta$ above $w$ and below $z$. Coherence is the condition that when the $\beta$ curve is oriented, all of the rainbow arcs around a fixed basepoint orient the same way. This makes it easy to check coherence from a diagram in standard form.

Let $H^{\pm}$ denote the upper and lower half-planes bounded by ${\widetilde{\alpha}}$. Choose a lift to ${\widetilde{\alpha}}$ of any point in ${\mathfrak{T}}(D,s)$. There exists a unique lift ${\widetilde{\beta}}_{s}\subset{\mathbb{R}}^{2}$ that passes through it. The points of ${\widetilde{\alpha}}\cap{\widetilde{\beta}}_{s}$ are in 1-1 correspondence with ${\mathfrak{T}}(D,s)$ under $\pi$, and there are an odd number $2n+1$ of them, since ${\widetilde{\alpha}}\cdot{\widetilde{\beta}}_{s}=\chi(\widehat{CFK}(D,s))=1$ with appropriate orientations on the curves. The curve ${\widetilde{\beta}}_{s}$ meets each half-plane in one closed ray and $n$ compact arcs. These compact arcs cobound positive bigons $D_{k}^{+}\subset H^{+}$ and negative bigons $D_{k}^{-}\subset H^{-}$ with ${\widetilde{\alpha}}$ for $k=1,\dots,n$. A positive bigon attains a local maximum, and its image under $\pi$ is the top of one of the rainbow arcs above $w$. Consequently, it contains a lift of the bigon in $D$ cobounded by that rainbow arc with $\alpha$. Thus, $n_{w}(D_{k}^{+})>0$, and similarly $n_{z}(D_{k}^{-})>0$, for all $k$. Lastly, the positive bigons go from their left-most corner to their right-most corner, and vice versa for the negative bigons.

Drawing on the proof of Lemma 2.4, we make a definition and prove a converse.

We prepare by describing a reduced $(1,1)$-diagram of a 1-bridge braid in a rational homology sphere.

Let $\gamma\cup\delta\subset S^{1}\times D^{2}$ denote a 1-bridge braid. Identify ${\partial}(S^{1}\times D^{2})$ with the flat torus $\Sigma$ in such a way that meridians $\theta\times{\partial}D^{2}$ lift to horizontal lines and longitudes $S^{1}\times x$ lift to vertical lines under $\pi:{\mathbb{R}}^{2}\to\Sigma$. Isotope $\gamma$ rel endpoints into a geodesic on $\Sigma$, so that $\pi^{-1}(\gamma)$ consists of parallel line segments in ${\mathbb{R}}^{2}$ of some common slope $s(\gamma)\in P^{1}({\mathbb{R}})$. Orient $\gamma$ and write ${\partial}\gamma=z-w$ so that in each lift of $\gamma$ to ${\mathbb{R}}^{2}$, the lift of $z$ lies below that of $w$.

Form a genus one rational homology sphere $Y$ by gluing a solid torus $V$ to $S^{1}\times D^{2}$ along their boundaries, and let $K$ denote the image of $\gamma\cup\delta$ in $Y$. Let $\alpha$ denote a meridian of $S^{1}\times D^{2}$ and let $\beta_{0}$ denote the curve to which the meridian of $V$ attaches, which we take to be a geodesic. In meridian-longitude coordinates, $\beta_{0}$ has some slope $p/q\neq 0/1$, and $Y$ is homeomorphic to $L(p,q)$ if $p\neq 1$ and $S^{3}$ otherwise. Assume that $\alpha$ and $\beta_{0}$ are disjoint from ${\partial}\gamma$. Isotope $\beta_{0}$ by a finger-move along $\gamma$ into a curve $\beta_{1}$ disjoint from $\gamma$. Observe that $(\Sigma,\alpha,\beta_{1},z,w)$ is a doubly-pointed Heegaard diagram for $K\subset Y$. Put it into reduced form by further isotoping $\beta_{1}$ so as to eliminate all bigons that do not contain basepoints, one by one, resulting in a curve $\beta_{2}$. Let $D=(\Sigma,\alpha,\beta_{2},z,w)$ denote the resulting reduced diagram of $K\subset Y$. Figure 3 exhibits this procedure for the inclusion of the 1-bridge braid $K(7,4,2)$ into $S^{3}$, the pretzel knot $P(-2,3,7)$. The notation derives from the braid presentation of 1-bridge braids; see [Ber91, Gab90, Wu04]. The $1$–bridge braid $K(\omega,b,m)$ denotes the closure of the braid word $(\sigma_{b}\sigma_{b-1}\cdots\sigma_{1})(\sigma_{\omega-1}\sigma_{\omega-2}\cdots\sigma_{1})^{m}$ in $S^{1}\times D^{2}$, where the $\sigma_{i}$ are the generators of the braid group $B_{w}$. Note that $K(\omega,b,m)$ and $K(\omega,b,m+\omega)$ differ by a full twist, so there is a diffeomorphism of the solid torus that carries one to the other. Therefore, we may assume that $0\leq m<w$.

A pair of distinct slopes $s_{1},s_{2}\in P^{1}({\mathbb{Q}})$ determines an interval $[s_{1},s_{2}]\subset P^{1}({\mathbb{Q}})$ oriented from $s_{1}$ to $s_{2}$ in the counterclockwise sense; thus, $[s_{1},s_{2}]\cup[s_{2},s_{1}]=P^{1}({\mathbb{Q}})$ and $[s_{1},s_{2}]\cap[s_{2},s_{1}]=\{s_{1},s_{2}\}$.

We proceed to sharpen the statement of Theorem 1.4. To do so, we introduce the notion of a strict 1-bridge braid and its basic invariants: the winding number and slope interval. Choose the lift of $\gamma$ with one endpoint at $(0,0)$. Its other endpoint takes the form $(t,\omega)\in{\mathbb{R}}\times{\mathbb{Z}}$. We have $\omega>0$ by our convention on the basepoints, and $t$ is not a proper divisor of $\omega$. Moreover, $\omega$ is an invariant of the isotopy type of $K$, since $[K]=\omega\cdot[S^{1}\times x]\in H_{1}(S^{1}\times D^{2};{\mathbb{Z}})$. This is the winding number $\omega(K)$.

Consider the sweep-out of line segments with one endpoint at $(0,0)$ and the other endpoint varying along line $y=\omega$. There exists a maximal slope interval $I(\gamma)\subset P^{1}({\mathbb{Q}})$ containing $s(\gamma)$ and so that the sweep-out of line segments through slopes in $I(\gamma)$ contains no lattice point in its interior. If $\omega=1$, then $I(\gamma)=P^{1}({\mathbb{Q}})\smallsetminus\{0\}$, and otherwise $I(\gamma)=[s_{-}(\gamma),s_{+}(\gamma)]$ with $s_{-}(\gamma)\leq s_{+}(\gamma)$. The sweep-out descends to an isotopy through 1-bridge braids with slopes in the interior of $I(\gamma)$. If a line segment of slope $s_{\pm}(\gamma)$ has endpoint $(q,\omega)\in{\mathbb{Z}}^{2}$, then write $(q,\omega)=(dq^{\prime},d\omega^{\prime})$, where $d=\gcd(q,\omega)$. Let $\mu$ denote the meridian of the torus knot $T(q^{\prime},\omega^{\prime})$ and $\lambda$ the surface framing, oriented so that $\mu\cdot\lambda=+1$. If $d=1$, then $K$ is isotopic to the torus knot $T(q,\omega)$, and if $d>1$, then $K$ is isotopic to the $(d\lambda\pm\mu)$-cable of $T(q^{\prime},\omega^{\prime})$, where the sign is positive or negative if the slope of the line segment is $s_{+}(\gamma)$ or $s_{-}(\gamma)$, respectively. We call such a cable an exceptional cable of a torus knot. Otherwise, if no line segment has endpoint $(q,\omega)\in{\mathbb{Z}}^{2}$, then we call $K$ a strict 1-bridge braid (and justify the terminology in Corollary 3.3). In this case, there exists a unique $m\in{\mathbb{Z}}$ so that $m<t<m+1$, and $s_{-}(\gamma)$ and $s_{+}(\gamma)$ are consecutive terms in the Farey sequence of fractions from $\omega/(m+1)$ to $\omega/m$ whose denominators are bounded by $|m|$ when expressed in lowest terms.