2.1 Combinatorics of slopes and lines

The space of slopes $\operatorname {\mathbb {Q}P}^1\subset \operatorname {\mathbb {R}P}^1 \cong S^1$, endowed with the subspace topology, carries a natural cyclic order: given a finite set of slopes $\{s_1,\ldots,s_n\}$ for some $n\geq 3$, we write

\[ s_1\leq s_2\leq \cdots\leq s_n\leq s_1 \]

if the loop $[0,1]\ni t\mapsto s_1 \cdot e^{2\pi i t}\in S^1\subset \mathbb {C}$ based at $s_1$ meets $s_l$ not before $s_k$ if $k< l$; in short, we choose the counter-clockwise order, as illustrated in Figure 3. We call a tuple $(s_1,\ldots,s_n)$ that satisfies this condition increasing. Note that $s_n\neq s_1$ for any such tuple, unless $s_1=s_2=\cdots =s_n$. If the order is opposite, the tuple is called a decreasing tuple. For pairs of distinct slopes the interval notation $(s_1,s_2)$ denotes the set of slopes $s$ for which $(s_1,s,s_2)$ is increasing. As usual, square and round brackets are used to indicate the inclusion and exclusion of the interval boundaries.

Figure 3. The thin interval relative to an increasing sequence of slopes $(s_1,s_2,s_3,\ldots,s_n)$.

Let $\operatorname {\mathfrak {C}}=\operatorname {\mathbb {Q}P}^1\times G\times \{0,1\}$ where $G$ is either $\mathbb {Z}$ or $\mathbb {Z}/2$. When it is necessary to make the distinction between the choice of $G$, we will write $\operatorname {\mathfrak {C}}=\operatorname {\mathfrak {C}_\mathit {G}}$. Elements $c\in \operatorname {\mathfrak {C}}$ will be called lines; one might represent them geometrically as slopes together with decorations in $G\times \{0,1\}$. (We choose the terminology line for distinction with curve, which will have a slightly different meaning in subsequent sections.) Given a triple $c\in \operatorname {\mathfrak {C}}$, denote the first component, the slope of $c$, by $s(c)$; denote the second component, the grading of $c$, by $\operatorname {g}(c)$; the third component is denoted by $\varepsilon (c)$. A line $c$ is rational if $\varepsilon (c)=0$ and special if $\varepsilon (c)=1$. Note that $G$ acts on the set $\operatorname {\mathfrak {C}}$, and we write

\[ n\cdot c=n\cdot (s,g,\varepsilon)=(s,g+n,\varepsilon) \quad \text{for any }n\in G. \]

Let $\operatorname {g}\colon \thinspace \operatorname {\mathfrak {C}}^2\rightarrow G$ be a function satisfying the following identities for all $c,c',c''\in \operatorname {\mathfrak {C}}$:

(symmetry)$$\begin{gather} \operatorname{g}(c,c')+\operatorname{g}(c',c) =\begin{cases} 0 & \text{if } s(c)=s(c')\\ -1 & \text{otherwise;} \end{cases} \end{gather}$$

(transitivity)$$\begin{gather}\operatorname{g}(c,c')+\operatorname{g}(c',c'')=\operatorname{g}(c,c'') \quad\text{ if ($s(c),s(c'),s(c''))$ is increasing;} \end{gather}$$

(linearity)$$\begin{gather}\operatorname{g}(n\cdot c,n'\cdot c')=\operatorname{g}(c,c')+n'-n . \end{gather}$$

A finite non-empty collection of lines $C = \{c_1,\ldots,c_n\}\subset \operatorname {\mathfrak {C}}$ is called a line set. We call $C$ $s$-rational if $\varepsilon (c)=0$ for all $\{c\in C\,|\,s(c)=s\}$, and $s$-special if $\varepsilon (c)=1$ for all $\{ c\in C \,|\, s(c)=s\}$.

It is often useful to consider the underlying slopes realized by a given line set $C$ in the projection $\operatorname {\mathfrak {C}}\to \operatorname {\mathbb {Q}P}^1$. For this purpose, we define the set of supporting slopes as

\[ \operatorname{\mathcal{S}}_C:=\{s(c)\mid c\in C\}\subset\operatorname{\mathbb{Q}P}^1. \]

We call a line set $C$ trivial if all its lines are special and concentrated in a single slope; in other words, if $\operatorname {\mathcal {S}}_C=\{s\}$ for some slope $s\in \operatorname {\mathbb {Q}P}^1$, and $C$ is $s$-special. Otherwise, we call $C$ non-trivial.

Note that the quotient homomorphism $\mathbb {Z}\rightarrow \mathbb {Z}/2$ induces a canonical map $\operatorname {\mathfrak {C}_{\mathbb {Z}}}\rightarrow \operatorname {\mathfrak {C}_{\mathbb {Z}/2}}$, which allows us to relate lines and line sets with respect to the two choices of $G$. Specifically, the image of a line set $C\subset \operatorname {\mathfrak {C}_{\mathbb {Z}}}$ under this map is a multi-set; after removing any duplicate elements, we obtain a line set in $\operatorname {\mathfrak {C}_{\mathbb {Z}/2}}$, which, by abuse of notation, we also denote by $C$.

Remark 2.1 In §§ 4 and 6, we construct the function $\operatorname {g}$ with the desired properties in the Heegaard Floer and the Khovanov setting, respectively. However, it is not hard to see that such a function exists and that it is essentially unique. For this, it is useful to think of $\operatorname {\mathfrak {C}_{\mathbb {Z}}}$ in terms of a covering space of $\operatorname {\mathbb {Q}P}^1$. More precisely, we can identify $\operatorname {\mathfrak {C}}_0:= \operatorname {\mathbb {Q}P}^1\times \mathbb {Z}\times \{0\}\subseteq \operatorname {\mathfrak {C}_{\mathbb {Z}}}$ with the pullback of the universal cover $p\colon \thinspace \mathbb {R}\rightarrow \operatorname {\mathbb {R}P}^1$ along the inclusion $\operatorname {\mathbb {Q}P}^1\hookrightarrow \operatorname {\mathbb {R}P}^1$. This is done as follows: to define a map $\eta \colon \thinspace \operatorname {\mathfrak {C}}_0\rightarrow \mathbb {R}$, fix some $c_\ast \in \operatorname {\mathfrak {C}}_0$ as a basepoint and define $\eta (c_\ast )$ to be some point $x_\ast \in p^{-1}(s(c_\ast ))$. For each $s\in \operatorname {\mathbb {Q}P}^1\smallsetminus \{s(c_\ast )\}$, there is some element $c_s\in \operatorname {\mathfrak {C}}_0$ of slope $s$ such that $\operatorname {g}(c_\ast,c_s)=0$. Let $\gamma _s$ be an injective path from $s(c_\ast )$ to $s$ which goes in counter-clockwise direction. Define $\eta (c_s)$ as the endpoint of the lift of $\gamma _s$ starting at $x_\ast$. Then extend $\eta$ equivariantly using the action of $G=\mathbb {Z}$ on $\operatorname {\mathfrak {C}}_0$ and the action by deck transformations on $\mathbb {R}$, where $+1$ corresponds to a counter-clockwise loop based at $c_\ast$. Under this identification of $\operatorname {\mathfrak {C}}_0$ with a subspace of $\mathbb {R}$, the function $\operatorname {g}$ is simply the floor function of the signed distance:

\[ \operatorname{g}(c,c')=\lfloor\eta(c')-\eta(c)\rfloor \quad\text{for any }c,c'\in\operatorname{\mathfrak{C}}_0. \]

By taking the product with $\{0,1\}$, one can easily extend this construction to $\operatorname {\mathfrak {C}}$. For $\operatorname {\mathfrak {C}_{\mathbb {Z}/2}}$, a similar interpretation is possible: we simply replace the universal cover of $\operatorname {\mathbb {R}P}^1$ by the connected two-fold cover.

Although the expression for the function $\operatorname {g}$ in Remark 2.1 is very concise, we do not make any further use of this perspective. Instead, we only use the properties of the function $\operatorname {g}$, in particular the symmetry and transitivity property.

Note that if $C$ contains a unique line $c$ for which $s(c)=s$ (or, indeed, if $s\notin \operatorname {\mathcal {S}}_C$), then it is $s$-consistent. Thus, in particular, this is a condition that is relevant when multiple lines project to the same slope. In fact, if $C$ is $s$-consistent there are at most two lines of slope $s$ in $C$, since $C\subset \operatorname {\mathfrak {C}}$. We relax this point of view and allow multi-sets when discussing curves in the Heegaard Floer and Khovanov settings in later sections.

Definition 2.3 We call a pair $(C,D)$ of line sets thin, or, more precisely, $\boldsymbol {G}$-thin, if there exists some constant $n\in G$ such that for all $(c,d)\in C\times D$,

\[ \begin{cases} (\varepsilon(c),\varepsilon(d))\in\{(0,1),(1,0)\} & \text{if } s(c)= s(d),\\ \operatorname{g}(c,d)=n & \text{otherwise.} \end{cases} \]

Note that if $(C,D)$ is thin, then so is $(D,C)$. For any line set $C$, we define

\[ \Theta_G(C) = \{s\in\operatorname{\mathbb{Q}P}^1\mid ((s,0,0),C)\text{ is thin}\}. \]

We write $\Theta$ in place of $\Theta _G$ when the statements are true for both $G=\mathbb {Z}$ and $G=\mathbb {Z}/2$ or when this group is clear from the context.

When $\operatorname {\mathcal {S}}_C$ is a singleton, there are four cases that arise for the set $\Theta (C)$ depending on consistency and the values of $\varepsilon$. These are recorded in the following lemma.

Lemma 2.6 Given a line set $C$, suppose $\operatorname {\mathcal {S}}_C=\{s\}$ for some $s\in \operatorname {\mathbb {Q}P}^1$. Then

\[ \Theta(C)=\begin{cases} \{s\} & \text{if C is not s-consistent and s-special,}\\ \operatorname{\mathbb{Q}P}^1 & \text{if C is s-consistent and s-special,}\\ \varnothing & \text{if C is not s-consistent and not s-special,}\\ \operatorname{\mathbb{Q}P}^1\smallsetminus\{s\} & \text{if C is s-consistent and not s-special.} \end{cases} \]

More generally, for a generic line set $C$ the set $\Theta (C)$ is an interval in $\operatorname {\mathbb {Q}P}^1$, whenever it is non-empty. This behaviour can be characterized precisely as follows.

Proof. The implications $(3)\Rightarrow (4)\Rightarrow (1)$ are obvious. Moreover, the implication $(1)\Rightarrow (2)$ follows from (transitivity) of the function $\operatorname {g}$, as in the proof of Lemma 2.5. Thus, it suffices to show $(2)\Rightarrow (3)$. If part (2) holds, then, by (transitivity), $\operatorname {g}((s',0,0),c_i)$ is constant for all $s'\in (s_n,s_1)$, so $(s_n,s_1)\subseteq \Theta (C)$. Moreover, since $s_1$ and $s_n$ differ

\[\operatorname{g}(c_n,c_1)=-1-\operatorname{g}(c_1,c_n)=-1\]

by (symmetry) of the function $\operatorname {g}$. Then, for any $s'\in (s_1,s_n)$,

\[ \operatorname{g}((s',0,0),c_1) = \operatorname{g}((s',0,0),c_n)+\operatorname{g}(c_n,c_1) = \operatorname{g}((s',0,0),c_n)-1 \]

and, hence, $\Theta (C)\cap (s_1,s_n)=\varnothing$. This establishes part (3).

Taken together, Lemmas 2.6 and 2.7 capture nearly all of the behaviour that is possible.

Therefore, continuing with our observation preceding Lemma 2.8, the only additional case that needs special attention is $|\operatorname {\mathcal {S}}_C|=2$. We can now collect all of the forgoing into a clean statement:

In the generic situation, the difference between $G=\mathbb {Z}$ and $G=\mathbb {Z}/2$ vanishes.

2.2 Characterizing thin pairs of line sets

We now turn to a characterization of thinness. Before stating the main theorem of this subsection, we discuss a certain exceptional class of line sets which requires special care, but which in the Heegaard Floer and Khovanov settings is ultimately a pathology that we have not observed in practice.

Note that if $G=\mathbb {Z}/2$, there do not exist exceptional line sets. In particular, we have the following result.

We first prove a technical lemma that will simplify the proof of Theorem 2.14.

Proof of Theorem 2.14 We start with a reformulation of condition (2) on the right-hand side of the asserted equivalence. Suppose for a moment that condition (1) in Theorem 2.14 holds. Then by non-triviality of $C$ and $D$ and Lemma 2.15, $\partial \Theta (C) \cap \partial \Theta (D) = \operatorname {\mathcal {S}}_{C}\cap \operatorname {\mathcal {S}}_{D}$. Suppose further that $C$ is $s$-rational for some slope $s$. Then $s\not \in \Theta (C)$ by Lemma 2.4. Therefore, $s\in \Theta (D)$ and so by the same lemma, $D$ is $s$-special. Similarly, if $D$ is $s$-rational, we can apply the same argument with reversed roles of $C$ and $D$. It therefore suffices to show that $(C,D)$ is thin if and only if:

1. (1) $\Theta (C) \cup \Theta (D) =\operatorname {\mathbb {Q}P}^1$; and

2. (2′) for all $s\in \operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D$, $C$ is $s$-rational and $D$ is $s$-special or vice versa.

Clearly, $(C,D)$ being thin implies condition (2$'$). Thus, let us assume from now on that $C$ and $D$ satisfy condition (2$'$). Write $C=\{c_1,\ldots,c_m\}$ and $D=\{d_1,\ldots,d_n\}$ for some $m,n\geq 1$, and let $s_i=s(c_i)$ for $i=1,\ldots,m$ and $t_j=s(d_j)$ for $j=1,\ldots,n$. We order the components of $C$ and $D$ such that both $(s_1,\ldots,s_m)$ and $(t_1,\ldots,t_n)$ are increasing tuples. The proof proceeds in four cases indexed by $|\operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D|$.

Case 0: $\operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D=\varnothing$. In this case $(C,D)$ is thin if and only if there exists some $M\in G$ such that $\operatorname {g}(c,d)=M$ for all $(c,d)\in C\times D$. By (transitivity), this is the case if and only if after some cyclic permutation of the indices

\[ (s_1,\ldots, s_m,t_1,\ldots,t_n) \]

is an increasing tuple such that $\operatorname {g}(c_i,c_j)=0$ and $\operatorname {g}(d_i,d_j)=0$ for all $i< j$. (Otherwise, if $\operatorname {\mathcal {S}}_C$ and $\operatorname {\mathcal {S}}_D$ intertwine, in the sense that there exist $i,j,k,\ell$ such that $(s_i,t_j,s_k,t_\ell )$ is increasing, (transitivity) implies $M=g(c_i,d_\ell )=g(c_i,d_j)+g(d_j,c_k)+g(c_k,d_\ell )=M+(-1-M)+M$, which is false.) By Lemmas 2.6 and 2.7, the latter condition is equivalent to $\Theta (C)$ and $\Theta (D)$ being two overlapping intervals.

Case 1: $\operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D=\{s\}$. (a) Suppose $\operatorname {\mathcal {S}}_C=\operatorname {\mathcal {S}}_D=\{s\}$. Then, because neither $C$ nor $D$ are trivial, $C$ and $D$ each contain at least one rational line of slope $s$. Thus, $(C,D)$ is not thin. Moreover, $s$ is neither in $\Theta (C)$ nor in $\Theta (D)$, so property (1) does not hold either.

(b) Suppose $\operatorname {\mathcal {S}}_C=\{s\}$ and $\operatorname {\mathcal {S}}_D\supsetneq \{s\}$. If $C$ is not $s$-consistent, condition (1) is false. This is because in this case, $\Theta (C)=\varnothing$ by Lemma 2.6 and non-triviality of $C$, and $\Theta (D)\neq \operatorname {\mathbb {Q}P}^1$ by Lemma 2.7. On the other hand, $C$ not being $s$-consistent, in conjunction with (transitivity), implies that $(C,D)$ is not thin, so the equivalence holds in this case. Suppose now that $C$ is $s$-consistent. Then $\Theta (C)=\operatorname {\mathbb {Q}P}^1\smallsetminus \{s\}$ by non-triviality of $C$ and Lemma 2.6. Therefore, condition (1) is equivalent to $s\in \Theta (D)$. Now observe that since $C$ is non-trivial, it is not $s$-special. Since we are assuming that condition (2$'$) holds, this implies that $C$ is $s$-rational and $D$ is $s$-special. In particular, $C$ consists of a single rational line. Thus, by (linearity), $(C,D)$ is thin if and only if $((s,0,0),D)$ is thin, i.e. $s\in \Theta (D)$.

(c) Suppose $\operatorname {\mathcal {S}}_C\supsetneq \{s\}$ and $\operatorname {\mathcal {S}}_D=\{s\}$. This is the same as Case 1(b) with reversed roles of $C$ and $D$.

(d) Suppose $|\operatorname {\mathcal {S}}_C|,|\operatorname {\mathcal {S}}_D|>1$. Let us reindex the lines such that $s_m\neq s_1=s=t_n\neq t_1$, and $(s_1,\ldots,s_m)$ and $(t_1,\ldots,t_n)$ are increasing. After potentially interchanging $C$ and $D$, we may assume without loss of generality that $(s,s_k,t_\ell )$ is increasing for some $k,\ell$ such that $s_k\neq s \neq t_\ell$. By (transitivity), $(C,D)$ is thin if and only if (i) $(s,s_m,t_1)$ is an increasing tuple, and (ii) $\operatorname {g}(s_i,s_j)=0$ and $\operatorname {g}(t_i,t_j)=0$ for $i< j$, or equivalently, (ii$'$) $(s,t_1)\subseteq \Theta (D)$, and $(s_m,s) \subseteq \Theta (C)$, by Lemma 2.7. Conditions (ii$'$) and (2$'$) imply that $s\in \Theta (D)$ or $s\in \Theta (C)$. Together with condition (i), part (1) follows. Conversely, suppose conditions (1) and (2$'$) hold. Since by Lemma 2.7, $\Theta (C)$ and $\Theta (D)$ are contained in the closures of open intervals disjoint from any supporting slopes of $C$ and $D$, respectively, condition (1) implies (i) and (ii$'$).

Case 2: $\operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D=\{s,t\}$. (a) Suppose $|\operatorname {\mathcal {S}}_C|=2=|\operatorname {\mathcal {S}}_D|$. Suppose further that $C$ is not $s$-consistent. Then $\Theta (C)=\varnothing$ and, hence, $\Theta (C) \cup \Theta (D) \neq \operatorname {\mathbb {Q}P}^1$. Indeed, $(C,D)$ is not thin in this case. Similarly, one can show that the theorem holds whenever $C$ or $D$ are not $s$- and $t$-consistent. Thus, now let us assume that $C$ and $D$ are $s$- and $t$-consistent. By the assumptions that we have already made, we can write $C=\{c,c'\}$ and $D=\{d,d'\}$ where $s(c)=s(d)=s$ and $s(c')=s(d')=t$. Then, by Lemma 2.7,

\begin{align*} (s,t)\subseteq\Theta(C) &\Leftrightarrow \operatorname{g}(c',c)=0 & (t,s)\subseteq\Theta(D) &\Leftrightarrow \operatorname{g}(d,d')=0, \\ (t,s)\subseteq\Theta(C) &\Leftrightarrow \operatorname{g}(c,c')=0 & (s,t)\subseteq\Theta(D) &\Leftrightarrow \operatorname{g}(d',d)=0. \end{align*}

Now, $(C,D)$ being thin is equivalent to $\operatorname {g}(c,d')=\operatorname {g}(c',d)$. By (transitivity) $\operatorname {g}(c,d')=\operatorname {g}(c,d)+\operatorname {g}(d,d')$ and $\operatorname {g}(c',d)=\operatorname {g}(c',c)+\operatorname {g}(c,d)$, and so the condition $\operatorname {g}(c,d')=\operatorname {g}(c',d)$ is equivalent to $\operatorname {g}(c',c)=\operatorname {g}(d,d')$. By (symmetry) of $\operatorname {g}$, this is equivalent to $\operatorname {g}(c,c')=\operatorname {g}(d',d)$. The latter two conditions, in conjunction with the four equivalences above, are equivalent to the condition $\Theta (C) \cup \Theta (D) \supseteq \operatorname {\mathbb {Q}P}^1\smallsetminus \{s,t\}$, since we are assuming that not both $C$ and $D$ are exceptional. This is equivalent to condition (1) since by condition (2$'$), either $C$ or $D$ is $s$-special and either $C$ or $D$ is $t$-special.

(b) Suppose $|\operatorname {\mathcal {S}}_C|>2$. After potentially interchanging $t$ and $s$, we may assume without loss of generality that $(s,s_k,t)$ is increasing for some $k$ such that $s\neq s_k \neq t$. As in Case 1(c), let us reindex the lines such that $s_m\neq s_1=s=t_n\neq t_1$, and $(s_1,\ldots,s_m)$ and $(t_1,\ldots,t_n)$ are increasing. Then, by (transitivity), $(C,D)$ is thin if and only if $s_m=t=t_1$ and $\operatorname {g}(s_i,s_j)=0$ and $\operatorname {g}(t_i,t_j)=0$ for all $i< j$. This, in turn, is equivalent to $\Theta (C) \cup \Theta (D) \supseteq \operatorname {\mathbb {Q}P}^1\smallsetminus \{s,t\}$. Now conclude as in Case 2(a).

(c) Suppose $|\operatorname {\mathcal {S}}_D|>2$. This is the same as Case 2(b) with reversed roles of $C$ and $D$.

Case 3: $|\operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D|>2$. Say $s,s',s''\in \operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D$ are pairwise distinct slopes such that $(s,s',s'')$ is an increasing triple. Then there exist lines $c,c',c''\in C$ and $d,d',d''\in D$ such that $s=s(c)=s(d)$, $s'=s(c')=s(d')$, and $s''=s(c'')=s(d'')$. We claim that in this case $(C,D)$ is not thin. Suppose $(C,D)$ were thin. Then $\operatorname {g}(c, d')=\operatorname {g}(c,d'')$, so $\operatorname {g}(d',d'')=0$. Cyclically permuting the variables gives $\operatorname {g}(d'',d)=\operatorname {g}(d,d')=0$. Applying (transitivity) and (symmetry) of the function $\operatorname {g}$, this leads to a contradiction. Now observe that $\Theta (C)\cup \Theta (D)\neq \operatorname {\mathbb {Q}P}^1$ according to Lemma 2.7.

Given any line set $C$, let $\mathring {\Theta }(C)$ denote the interior of $\Theta (C)$.

This highlights what turns out to be the generic behaviour, in practice, and gives rise to a quick certification of thinness. As the proof of Theorem 2.14 indicates, the main work is in treating the behaviour at the boundaries of the relevant intervals. Indeed, the converse of Corollary 2.16 is not true as the following example illustrates.

Example 2.17 Let $C=\{c,c_\star \}$ and $D=\{d,d_\star \}$ where $c$ and $d$ are rational, $c_\star$ and $d_\star$ are special, $s(c)=0=s(d_\star )$, and $s(c_\star )=\infty =s(d)$. Let $\Delta _c=\operatorname {g}(c,c_\star )$ and $\Delta _d=\operatorname {g}(d,d_\star )$. Then,

\[ \Theta(C)= \begin{cases} {[\infty,0)} & \text{if } \Delta_c=0,\\ (0,\infty] & \text{if } \Delta_c=-1,\\ \{\infty\} & \text{otherwise,} \end{cases} \quad\text{and}\quad \Theta(D)= \begin{cases} {[0,\infty)} & \text{if } \Delta_d=0,\\ (\infty,0] & \text{if } \Delta_d=-1,\\ \{0\} & \text{otherwise.} \end{cases} \]

See Figure 4 for an illustration of one of those cases. Clearly, the hypothesis of Corollary 2.16 is not satisfied for any values of $\Delta _c$ and $\Delta _d$. Moreover,

\[ \operatorname{g}(c,d)-\operatorname{g}(c_\star,d_\star)= \Delta_c+\operatorname{g}(c_\star,d)-(\operatorname{g}(c_\star,d)+\Delta_d)= \Delta_c-\Delta_d, \]

so $(C,D)$ is thin if and only if $\Delta _c=\Delta _d$. If $\Delta _c\in \{0,-1\}$ or $\Delta _d\in \{0,-1\}$, we can verify this independently using Theorem 2.14. Otherwise, both line sets are exceptional.

Figure 4. An illustration of Example 2.17 for the case $\Delta _c=0=\Delta _d$.

3.1 The definition of $\operatorname {HFT}$

Given a Conway tangle $T$ in a three-ball $B^3$, the invariant $\operatorname {HFT}(T)$ takes the form of a multicurve on a four-punctured sphere $S^2_4$, which can be naturally identified with the boundary of $B^3$ minus the four tangle ends $\partial T$. Here, a multicurve is a collection of immersed curves with local systems. To make this precise: an immersed curve in $S^2_4$ is an immersion of $S^1$, considered up to homotopy, that defines a primitive element of $\pi _1(S^2_4)$, and each of these curves is decorated with a local system, i.e. an invertible matrix over $\mathbb {F}$ considered up to matrix similarity. Local systems can be viewed as vector bundles up to isomorphism, where either $\mathbb {F}$ is equipped with the discrete topology or the bundle is equipped with a flat connection. We always drop local systems from our notation when they are trivial, i.e. if they are equal to the unique one-dimensional local system. Immersed curves carry a $\delta$-grading (described in § 4) and multiple parallel immersed curves in the same $\delta$-grading are set to be equivalent to a single curve with a local system that is the direct sum of the individual local systems. We always assume that parallel immersed curves are bundled up this way. Finally, a multicurve is a collection of $\delta$-graded immersed curves.

With this terminology in place, we can sketch the construction of $\operatorname {HFT}(T)$. It is defined in two steps; for details, see [Reference ZibrowiusZib20].

First, one fixes a particular auxiliary parametrization of $\partial B^3\smallsetminus \partial T$ by four embedded arcs connecting the tangle ends. For example, the four gray dotted arcs in Figure 5(a) define such a parametrization for the $(2,-3)$-pretzel tangle. A tangle with such a parametrization can be encoded in a Heegaard diagram $(\Sigma,\boldsymbol {\alpha },\boldsymbol {\beta })$, where $\Sigma$ is some surface with marked points. From this, one defines a relatively $\delta$-graded curvedFootnote ¹ chain complex $\operatorname {CFT^\partial }(T)$ over a certain fixed $\mathbb {F}$-algebra $\operatorname {\mathcal {A}}^\partial$ as the multi-pointed Heegaard Floer theory of the triple $(\Sigma,\boldsymbol {\alpha },\boldsymbol {\beta })$ (see [Reference ZibrowiusZib20, Section 2.3]), similar to Ozsváth and Szabó's link Floer homology [Reference Ozsváth and SzabóOS08]. One can show that the relatively $\delta$-graded chain homotopy type of $\operatorname {CFT^\partial }(T)$ is an invariant of the tangle $T$ with the chosen parametrization [Reference ZibrowiusZib20, Theorem 2.17].

Figure 5. A simple non-rational tangle and its Heegaard Floer tangle invariant.

The second step uses a classification result, which states that the chain homotopy classes of $\delta$-graded curved chain complexes over $\operatorname {\mathcal {A}}^\partial$ are in one-to-one correspondence with free homotopy classes of $\delta$-graded immersed multicurves on the four-punctured sphere $S^2_4$ (see [Reference ZibrowiusZib20, Theorem 0.4]). This correspondence uses a fixed parametrization of $S^2_4$ by four arcs, and we generally assume that the multicurves intersect this parametrization minimally. Roughly speaking, the intersection points of arcs with a multicurve correspond to generators of the according curved chain complexes and paths between those intersection points correspond to the differentials. Now, $\operatorname {HFT}(T)$ is defined as the collection of relatively $\delta$-graded immersed curves on $S^2_4$ corresponding to the curved complex $\operatorname {CFT^\partial }(T)$. In this definition the parametrization of $S^2_4$ (needed for multicurves) is identified with the parametrization of $\partial B^3\smallsetminus \partial T$ (needed for $\operatorname {CFT^\partial }(T)$), and one can show that this identification is natural. Namely, if a tangle $T'$ is obtained from $T$ by adding twists to the tangle ends, the complex $\operatorname {CFT^\partial }(T')$ determines a new set of immersed curves $\operatorname {HFT}(T')$, which agrees with that obtained by twisting the immersed curves $\operatorname {HFT}(T)$ accordingly [Reference ZibrowiusZib23b, Theorem 0.2].

The name rational tangle originated with Conway, who showed that these tangles are in one-to-one correspondence with fractions ${p}/{q}\in \operatorname {\mathbb {Q}P}^1$ (see [Reference ConwayCon70]). We denote the rational tangle corresponding to a slope ${p}/{q}\in \operatorname {\mathbb {Q}P}^1$ by $Q_{p/q}$. The invariant $\operatorname {HFT}(Q_{p/q})$ consists of a single embedded curve which is the boundary of a disk embedded into $B^3$ that separates the two tangle strands of $Q_{p/q}$ (see [Reference ZibrowiusZib20, Example 2.25]). The local system on this curve is one-dimensional. It is known that $\operatorname {HFT}$ detects rational tangles, as follows.

3.2 A gluing theorem for $\operatorname {HFT}$

The invariant $\operatorname {HFT}(T)$ can be also defined via Zarev's bordered sutured Heegaard Floer theory [Reference ZarevZar09]. In this alternate construction, the curved chain complex $\operatorname {CFT^\partial }(T)$ is replaced by an a posteriori equivalent algebraic object, namely the bordered sutured type D structure associated with the tangle complement, which is equipped with a certain bordered sutured structure; see [Reference ZibrowiusZib23b, Section 5]. This perspective gives rise to the following gluing result which relates the invariant $\operatorname {HFT}$ to link Floer homology $\operatorname {\widehat {HFL}}$ via Lagrangian Floer homology $\operatorname {HF}$. We always assume that tangles are glued as in Figure 1, and when such a decomposition exists, we write $L=T_1\cup T_2$ for a link $L$ consisting of tangles $T_1$ and $T_2$. The mirror image of the link $L$ is expressed as $L^*$; this notation extends to tangles so that diagrammatically the mirror of $T_i$, expressed $T_i^*$, is obtained by interchanging over- and under-crossings. Thus, $L^*=T_1^*\cup T_2^*$. (Note that the mirror $T^*$ of a tangle is expressed as $\operatorname {m} T$ in other papers.) Let $V$ be a two-dimensional vector space supported in a single relative $\delta$-grading.

Theorem 3.5 [Reference ZibrowiusZib20, Theorem 5.9]

If $L=T_1\cup T_2$, then

\[ \operatorname{\widehat{HFK}}(L)\otimes V \cong \operatorname{HF}(\operatorname{HFT}(T_1^*),\operatorname{HFT}(T_2)) \]

if the four open components of the tangles become identified to the same component and

\[ \operatorname{\widehat{HFK}}(L) \cong \operatorname{HF}(\operatorname{HFT}(T_1^*),\operatorname{HFT}(T_2)) \]

otherwise.

In this theorem, the knot Floer homology $\operatorname {\widehat {HFK}}(L)$ should be understood as a $\delta$-graded theory. A similar gluing theorem holds in the bigraded setting and also for link Floer homology, using a multivariate Alexander grading on the tangle invariants.

The Lagrangian Floer homology $\operatorname {HF}(\gamma,\gamma ')$ of two immersed curves with local systems $\gamma$ and $\gamma '$ is a vector space generated by intersection points between the two curves [Reference Hanselman, Rasmussen and WatsonHRW24, Section 4] (see also [Reference KotelskiyKot19, Reference Kotelskiy, Watson and ZibrowiusKWZ19]). More precisely, one first arranges that the components are transverse and do not cobound immersed annuli; then, $\operatorname {HF}(\gamma,\gamma ')$ is the homology of the following chain complex: for each intersection point between $\gamma$ and $\gamma '$, there are $n\cdot n'$ corresponding generators of the underlying chain module, where $n$ and $n'$ are the dimensions of the local systems of $\gamma$ and $\gamma '$, respectively. The differential is defined by counting certain bigons between these intersection points. As a consequence, the dimension of $\operatorname {HF}(\gamma,\gamma ')$ is equal to the minimal intersection number between the two curves times the dimensions of their local systems, provided that the curves are not parallel. If the curves are parallel, the dimension of $\operatorname {HF}(\gamma,\gamma ')$ may be greater than the minimal geometric intersection number for certain choices of local systems; for details, see [Reference ZibrowiusZib20, Sections 4.5 and 4.6, in particular Theorem 4.45]. For a more explicit example, suppose $\gamma$ and $\gamma '$ are parallel embedded curves of the same slope equipped with local systems of dimensions $n$ and $n'$, respectively. Then, $\dim \operatorname {HF}(\gamma,\gamma ')$ can realize any even number between 0 and $2 (n \cdot n')$, depending on the local systems, even though the minimal geometric intersection number between these curves is zero. Throughout, we always assume that $\gamma$ and $\gamma '$ intersect minimally without cobounding immersed annuli.

In other words, the mirror is obtained by reflection in a plane containing the four punctures and the parametrizing arcs. This operation is important in relating $\operatorname {HFT}(T_1^*)$ to $\operatorname {HFT}(T_1)$ (see [Reference ZibrowiusZib20, Definition 5.3 and Proposition 5.4]).

For example, because rational tangles satisfy $Q_s^*=Q_{-s}$ we have that $\operatorname {HFT}(Q_{-s})=\operatorname {m}(\operatorname {HFT}(Q_s))$.

3.3 The geography problem for $\operatorname {HFT}$

Often, it is useful to lift immersed curves to a covering space of $S^2_4$, namely the plane $\mathbb {R}^2$ minus the integer lattice $\mathbb {Z}^2$. We may regard $\mathbb {R}^2$ as the universal cover of the torus, and the torus as the two-fold cover of the sphere $S^2$ branched at four marked points; then the integer lattice $\mathbb {Z}^2$ is the preimage of the branch set. This covering space is illustrated in Figure 5(c), where the standard parametrization of $S^2_4$ has been lifted to $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ and the front face and its preimage under the covering map are shaded gray. This picture also includes the lifts of the curves in $\operatorname {HFT}(P_{2,-3})$: The lift of the embedded (dashed) curve is a straight line of slope ${1}/{2}$, whereas the lifts of the two non-embedded components of $\operatorname {HFT}(P_{2,-3})$ look more complicated. Remarkably, however, this example shows almost all the complexity of the immersed curves that can appear as components of $\operatorname {HFT}(T)$ for Conway tangles $T$.

To understand the geography of components of $\operatorname {HFT}(T)$ for general tangles $T$, observe that the linear action on the covering space $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ by $\mathit {SL}(2,\mathbb {Z})$ corresponds to Dehn twisting in $S^2_4$ or, equivalently, adding twists to the tangle ends; for specific conventions see [Reference ZibrowiusZib23b, Observation 3.2].

Definition 3.8 We call a curve in $S^2_4$ rational if its lift to $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ is a straight line of slope ${p}/{q}$. We denote such a curve by $\mathbf {r}({p}/{q})$ if it has a trivial local system, and $\mathbf {r}_{X}({p}/{q})$ if it has a local system $X$.

We call a curve in $S^2_4$ special if, after some twisting, it is equal to the curve $\mathbf {s}_n(0;{\mathsf {i}},{\mathsf {j}})$ whose lift to $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ is shown in Figure 6. The lift of any special curve can be isotoped into an arbitrarily small neighborhood of a straight line of some rational slope ${p}/{q}\in \operatorname {\mathbb {Q}P}^1$ going through some punctures ${\mathsf {i}}$ and ${\mathsf {j}}$, in which case we denote this curve by $\mathbf {s}_n({p}/{q};{\mathsf {i}},{\mathsf {j}})$.

Figure 6. The lift of the curve $\mathbf {s}_n(0;{\mathsf {i}},{\mathsf {j}})$, where $n\in \mathbb {N}$ and $({\mathsf {i}},{\mathsf {j}})=({\mathsf {4}},{\mathsf {1}})$ or $({\mathsf {2}},{\mathsf {3}})$.

The term rational is chosen because for rational tangles $\operatorname {HFT}(Q_{p/q})=\mathbf {r}({p}/{q})$. One can then show the following [Reference ZibrowiusZib23b, Theorem 0.5].

For example, we can now write $\operatorname {HFT}(P_{2,-3})$ as the union of the rational curve $\mathbf {r}({1}/{2})$ and the two special components $\mathbf {s}_1(0;{\mathsf {4}},{\mathsf {1}})$ and $\mathbf {s}_1(0;{\mathsf {2}},{\mathsf {3}})$. Whether rational components with non-trivial local systems appear in $\operatorname {HFT}$ is currently not known. Special components for $n>1$ show up in the invariants of two-stranded pretzel tangles with more twists [Reference ZibrowiusZib20, Theorem 6.9]. Special components always come in pairs according to the following result, which is a simplified version of [Reference ZibrowiusZib23b, Theorem 0.10].

There are also restrictions on rational components. The following is [Reference ZibrowiusZib20, Observation 6.1].

In analogy with § 2, given some slope $s\in \operatorname {\mathbb {Q}P}^1$, we call a multicurve $s$-rational if it does not contain any special components of slope $s$, and $s$-special if it does not contain any rational components of slope $s$.

4.1 The $\delta$-grading of curves in the covering space

We now develop tools that enable us to better understand the $\delta$-grading in terms of the covering space $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ of the four-punctured sphere $S^2_4$. In § 4.3, this allows us to reduce to the situation of § 2 and to apply the theorems from that section to $\operatorname {HFT}$.

Definition 4.1 Recall that given a map $\gamma \colon \thinspace S^1\to S^2_4$, its lift to $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ is a map $\tilde {\gamma }\colon \thinspace [0,1]\to \mathbb {R}^2\smallsetminus \mathbb {Z}^2$ such that the diagram in Figure 8(a) commutes. Given a map $\gamma \colon \thinspace S^1\to S^2_4$, its infinite connected lift to $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ is a map $\bar {\gamma }\colon \thinspace \mathbb {R} \to \mathbb {R}^2\smallsetminus \mathbb {Z}^2$ such that the diagram in Figure 8(b) commutes.

Figure 8. Lifts versus infinite connected lifts, used for studying curves via the planar cover. In this section we mainly use infinite connected lifts; in § 7 we use lifts nearly exclusively, but for illustration purposes the infinite connected lift is sometimes depicted as well. Note that with this nomenclature, the preimage of a curve in the cover may be called the infinite non-connected lift.

For notation, in this section any symbol decorated with a tilde ‘$\sim$’ on top denotes the lift to $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$; likewise, an overbar ‘–’ denotes the infinite connected lift. Infinite connected lifts are sometimes referred to as ‘lifts’ for simplicity, where the difference is clear from the context. In the following, we treat all points in the integer lattice as marked points (as opposed to punctures). Denote by $P$ the union of the integer lattice points with the preimage of the parametrization of $S^2_4$.

The figures in this section follow the same conventions as in [Reference ZibrowiusZib20]: we use the right-hand rule to determine the orientation of domains and the normal vector fields of $S^2_4$ and $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ are pointing into the page. Thus, the boundary of a region of multiplicity $+1$ is oriented clockwise.

Note that the Euler measure is additive in the sense that $e(D+D')=e(D)+e(D')$ for any two domains $D$ and $D'$. In practice, one computes the Euler measure of a domain $D$ using the following formula, which follows from the Gauss–Bonnet theorem:

\[ e(D)=\chi(D)-\tfrac{1}{4}\{\text{acute corners of }D\}+\tfrac{1}{4}\{\text{obtuse corners of }D\}. \]

Definition 4.3 Given an absolutely $\delta$-graded curve $\gamma$, consider two intersection points $\tilde {x}$ and $\tilde {x}'$ of the lift $\bar {\gamma }$ with the integer lattice graph $P$. A connecting domain from $\tilde {x}$ to $\tilde {x}'$ is a domain $\varphi \in H_2(\mathbb {R}^2,P\cup \bar {\gamma })$ with the property

\[ \partial(\partial\varphi\cap \bar{\gamma})=\tilde{x}-\tilde{x}'. \]

Lemma 4.5 For any connecting domain $\varphi$ as in Definition 4.3,

\[ \delta(x')-\delta(x)=2e(\varphi). \]

Definition 4.6 Let $\bullet$ be an intersection point between two absolutely $\delta$-graded curves $\gamma$ and $\gamma '$. Consider the lifts $\bar {\gamma }$ and $\bar {\gamma }'$ of these two curves, such that they intersect at a lift $\tilde {\bullet }$ of the intersection point $\bullet$. A connecting domain for $\tilde {\bullet }$ from $\bar {\gamma }$ to $\bar {\gamma }'$ is a domain $\varphi \in H_2(\mathbb {R}^2,P\cup \bar {\gamma }\cup \bar {\gamma }')$ with the property

\[ \partial(\partial\varphi\cap\bar{\gamma})=\tilde{x}-\tilde{\bullet} \quad \text{and} \quad \partial(\partial\varphi\cap \bar{\gamma}')=\tilde{\bullet}-\tilde{y} \quad \text{for some } \tilde{x}\in\bar{\gamma}\cap P \text{ and } \tilde{y}\in\bar{\gamma}'\cap P. \]

Intersection points between bigraded curves can be endowed with a $\delta$-grading [Reference ZibrowiusZib20, Definition 4.40], and this can be easily calculated according to the following result.

Lemma 4.7 With notation as in Definition 4.6, the $\delta$-grading of $\bullet$ satisfies

\[ \delta(\bullet\colon\thinspace\gamma\to\gamma')=\delta(y)-\delta(x)+\tfrac{1}{2}-2e(\varphi). \]

Observation 4.8 The domain $-\varphi$ is a connecting domain for the same intersection point $\bullet$, but regarded as a generator of $\operatorname {HF}(\gamma ',\gamma )$. Its $\delta$-grading is equal to 1 minus the original $\delta$-grading:

\[ \delta(\bullet\colon\thinspace\gamma\rightarrow\gamma') = 1-\delta(\bullet\colon\thinspace\gamma'\rightarrow\gamma). \]

Proof of Lemma 4.7 First consider the simplest case in which the domain $\varphi$ consists of a single region of multiplicity 1. Up to rotation, there are only four cases, as shown in Figure 9. The lemma then follows directly from [Reference ZibrowiusZib20, Definition 4.40], since in each of those cases, the intersection point corresponds to some algebra element $a\in \operatorname {\mathcal {A}}^\partial$ and its $\delta$-grading $\delta (a)$ is equal to $\tfrac {1}{2}-2e(\varphi )$ (see [Reference ZibrowiusZib20, Definitions 2.10 and 4.5]).

Figure 9. Basic connecting domains satisfying the formula .

Now consider a general connecting domain $\varphi$. Then near $\tilde {\bullet }$, $\varphi$ looks like one of the basic connecting domains $\psi$ that we have just considered (up to adding multiples of square regions). Suppose $\psi$ connects $\tilde {x}'\in \bar {\gamma }\cap P$ to $\tilde {y}'\in \bar {\gamma }'\cap P$. Then, as we have just verified,

\[ \delta(\bullet\colon\thinspace\gamma\rightarrow\gamma') = \delta(y')-\delta(x')+\tfrac{1}{2}-2e(\psi). \]

Let $\psi _x$ and $\psi _y$ be connecting domains from $\tilde {x}$ to $\tilde {x}'$ and from $\tilde {y}'$ to $\tilde {y}$, respectively. Then, by Lemma 4.5,

\[ \delta(x')-\delta(x)=2e(\psi_x) \quad\text{and}\quad \delta(y)-\delta(y')=2e(\psi_y). \]

Combining all three relations, we see that

\[ \delta(\bullet\colon\thinspace\gamma\rightarrow\gamma') = \delta(y)-\delta(x)+\tfrac{1}{2}-2e(\psi_x+\psi+\psi_y). \]

By construction, $\psi _x+\psi +\psi _y-\varphi$ is a sum of square regions and domains bounding closed components of $\bar {\gamma }$ and $\bar {\gamma }'$, so $e(\psi _x+\psi +\psi _y)=e(\varphi )$.

Definition 4.9 Suppose $\bar {\gamma }_i$ is an infinite connected lift to $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ of some absolutely $\delta$-graded curve $\gamma _i$ in $S^2_4$ for $i=1,\ldots,n$, and let $x_i\in \operatorname {HF}(\gamma _i,\gamma _{i+1})$ be an intersection point between $\gamma _{i}$ and $\gamma _{i+1}$, where we take indices modulo $n$. A symmetric domain for the tuples $(\bar {\gamma }_i)_{i=1,\ldots,n}$ and $(\tilde {x}_i)_{i=1,\ldots,n}$ is a domain $\varphi$ satisfying

\[ \partial(\partial\varphi\cap \bar{\gamma}_{i})=\tilde{x}_{i-1}-\tilde{x}_i, \]

where, again, indices are taken modulo $n$.

Proposition 4.10 For any connecting domain $\varphi$ as in Definition 4.9,

\[ \sum_{i=1}^n \delta(x_i)=\frac{n}{2}-2 e(\varphi). \]

Proof. For each $i=1,\ldots, n$, choose some intersection point $\tilde {y}_i$ of $\bar {\gamma }_i$ with $P$. Then we can write $\varphi$ as a sum of $n$ connecting domains $\varphi _i$ for $\tilde {x}_i$ from $\tilde {y}_i$ to $\tilde {y}_{i+1}$. By Lemma 4.7,

\[ \delta(x_i)=\delta(y_{i+1})-\delta(y_i)+\tfrac{1}{2}-2e(\varphi_i) \]

for $i=1,\ldots,n$. Taking the sum over all $n$ equations, we obtain the desired identity.

Definition 4.11 Given two intersection points $x,y\in \operatorname {HF}(\gamma,\gamma ')$ between two curves $\gamma$ and $\gamma '$, a domain (or asymmetric domain) from $\tilde {x}$ to $\tilde {y}$ is a domain $\varphi \in H_2(\mathbb {R}^2,\bar {\gamma }\cup \bar {\gamma }')$ with the property

\[ \partial(\partial\varphi\cap \bar{\gamma})=\tilde{y}-\tilde{x}. \]

Corollary 4.12 For any domain $\varphi$ as in Definition 4.11,

\[ \delta(y)-\delta(x)=2e(\varphi). \]

Example 4.13 If $\varphi$ is a bigon of multiplicity 1 as in Figure 10, Corollary 4.12 implies that $\delta (y)-\delta (x)=2e(\varphi )=1$; see also [Reference ZibrowiusZib20, Lemma 4.41].

Figure 10. A bigon illustrating Example 4.13; compare with Figure 15 and [Reference ZibrowiusZib20, Figure 31].

4.2 Linear curves

By Theorem 3.9, the invariant $\operatorname {HFT}(T)$ of any Conway tangle $T$ consists of rational and special curves, which are linear. Thus, for the remainder of this section we restrict our attention to linear curves.

The slope of the mirror of a linear curve $\gamma$ (see Definition 3.6) is equal to the mirror of the slope of $\gamma$ (see Definition 1.11). Thus, by Lemma 3.7, mirroring operation commutes with taking the curve invariant $\operatorname {HFT}(-)$ and its slopes:

\[ \operatorname{\mathcal{S}}_{\operatorname{HFT}(T^*)}=\operatorname{\mathcal{S}}_{\operatorname{m}(\operatorname{HFT}(T))}=\operatorname{\mathcal{S}}^{\operatorname{m}}_{\operatorname{HFT}(T)}. \]

Lemma 4.15 Let $\gamma$ be a linear curve of slope $s\in \operatorname {\mathbb {Q}P}^1$. Then unless $s=0$, the images of all intersection points of $\bar {\gamma }$ with the horizontal lines of $P$ have the same $\delta$-grading $\delta _{-}:= \delta _{-}(\gamma )$, and unless $s=\infty$, the images of all intersection points of $\bar {\gamma }$ with the vertical lines of $P$ have the same $\delta$-grading $\delta _{|}:= \delta _{|}(\gamma )$. Moreover,

\[ \delta_{|}= \begin{cases} \delta_{-}-\frac{1}{2} & \text{if } 0< s<\infty, \\ \delta_{-}+\frac{1}{2} & \text{if } \infty< s<0. \end{cases} \]

Definition 4.16 Given two linear curves $\gamma$ and $\gamma '$ of the same slope $s\in \operatorname {\mathbb {Q}P}^1$, we define

\[ \delta(\gamma,\gamma') =\begin{cases} \delta_{-}(\gamma')-\delta_{-}(\gamma) & \text{if } s\neq0 ,\\ \delta_{|}(\gamma')-\delta_{|}(\gamma) & \text{if } s\neq\infty. \end{cases} \]

This is well-defined by Lemma 4.15. The two curves are said to have the same $\delta$-grading if $\delta (\gamma,\gamma ')=0$.

Lemma 4.17 Given two linear curves $\gamma$ and $\gamma '$ of different slopes $s,s'\in \operatorname {\mathbb {Q}P}^1$, the Lagrangian intersection theory $\operatorname {HF}(\gamma,\gamma ')$ is supported in a single $\delta$-grading, which is equal to

\[ \delta(\gamma,\gamma') := \begin{cases} \delta_{-}(\gamma')-\delta_{|}(\gamma)+\tfrac{1}{2} & \text{if } s\in(\infty,s') \text{ for } s'\in(0,\infty], \text{ or } s\in(s',\infty) \text{ for } s'\in[\infty,0), \\ \delta_{|}(\gamma')-\delta_{-}(\gamma)+\tfrac{1}{2} & \text{if } s\in(s',0) \text{ for } s'\in[0,\infty), \text{ or } s\in(0,s') \text{ for } s'\in(\infty,0], \end{cases} \]

using the convention that $(\infty,\infty ):= \operatorname {\mathbb {Q}P}^1\smallsetminus \{\infty \}$ and $(0,0):= \operatorname {\mathbb {Q}P}^1\smallsetminus \{0\}$.

Corollary 4.18 (Symmetry of $\delta$)

For any two linear curves $\gamma$ and $\gamma '$,

\[ \delta(\gamma,\gamma')+\delta(\gamma',\gamma)= \begin{cases} 0 & \text{if } s(\gamma)=s(\gamma'),\\ 1 & \text{if } s(\gamma)\neq s(\gamma'). \end{cases} \]

Figure 11. Some illustrations for the proofs of Lemma 4.17 (a–d) and Theorem 4.19 (e).

In the proof of Theorem 4.27, we relate the grading function $\operatorname {g}$ from § 2 to the negative of $\delta$. This explains the difference in sign between the symmetry of $\delta$ compared with the (symmetry) of $\operatorname {g}$. Similarly, whereas the transitivity of $\operatorname {g}$ holds for increasing triples of slopes, we establish transitivity of $\delta$ in terms of decreasing triples.

Theorem 4.19 (Transitivity of $\delta$)

For any triple $(\gamma,\gamma ',\gamma '')$ of linear curves in $S^2_4$ such that $(s(\gamma ),s(\gamma '),s(\gamma ''))$ is decreasing,

\[ \delta(\gamma,\gamma')+\delta(\gamma',\gamma'')=\delta(\gamma,\gamma''). \]

Proof. If $s(\gamma )=s(\gamma ')$ or $s(\gamma ')=s(\gamma '')$, or both, this follows from Definition 4.16 and Lemma 4.17. Thus, let us suppose the curves have pairwise different slopes. Let us consider some infinite connected lifts $\bar {\gamma }$, $\bar {\gamma }'$, and $\bar {\gamma }''$ of $\gamma$, $\gamma '$, and $\gamma ''$, respectively. These lifts intersect in three points $\tilde {A}\in \bar {\gamma }\cap \bar {\gamma }'$, $\tilde {B}\in \bar {\gamma }'\cap \bar {\gamma }''$, and $\tilde {C}\in \bar {\gamma }''\cap \bar {\gamma }$. Consider a connecting domain $\varphi$ for $\tilde {A}$ starting at some point $\tilde {x}\in \bar {\gamma }\cap P$ and ending at some point $\tilde {y}\in \bar {\gamma }'\cap P$. Then choose a connecting domain $\varphi '$ for $\tilde {B}$ which starts at $\tilde {y}$ and ends at some point $\tilde {z}\in \bar {\gamma }''\cap P$. Since the slopes of these curves form a decreasing triple, the triangle $\Delta \tilde {A}\tilde {B}\tilde {C}$ has multiplicity $-1$ when the vertices are ordered counter-clockwise, as illustrated in Figure 11(e); therefore, $e(\Delta \tilde {A}\tilde {B}\tilde {C})=-\frac {1}{4}$. Thus, $\varphi +\varphi '+\Delta \tilde {A}\tilde {B}\tilde {C}$ is a connecting domain for $\tilde {C}$. Hence,

\begin{align*} \delta(\gamma,\gamma')+\delta(\gamma',\gamma'') &= \big(\delta(y)-\delta(x)-2e(\varphi)+\tfrac{1}{2}\big)+ \big(\delta(z)-\delta(y)-2e(\varphi')+\tfrac{1}{2}\big) \\ &= \delta(z)-\delta(x)-2e(\varphi+\varphi'+\Delta\tilde{A}\tilde{B}\tilde{C})+\tfrac{1}{2} = \delta(\gamma,\gamma''). \end{align*}

Proof. Let us consider each combination of rational and special curves separately. Clearly, the Lagrangian Floer homology of a special and a rational curve vanishes. To compute the Lagrangian Floer homology of two rational curves with local systems of the same slope, we can apply [Reference ZibrowiusZib20, Theorem 4.45] to verify the first statement in this case. If the two local systems are trivial, then their Lagrangian Floer homology does not vanish by [Reference ZibrowiusZib20, Lemma 4.51]. Now let us turn to the final case that $\gamma$ and $\gamma '$ are special. Then, if they wrap around different tangle ends, their Lagrangian Floer homology vanishes. Thus, let us consider $\gamma =\mathbf {s}_n(s;{\mathsf {i}},{\mathsf {j}})$ and $\gamma '=\mathbf {s}_m(s;{\mathsf {i}},{\mathsf {j}})$. To justify the support of $\operatorname {HF}(\gamma,\gamma ')$ for $n\neq m$, we can argue as follows. Let us consider a straight line of slope $s$ through two integer lattice points $\tilde {{\mathsf {i}}}$ and $\tilde {{\mathsf {j}}}$ corresponding to the tangle ends ${\mathsf {i}}$ and ${\mathsf {j}}$, respectively. After some homotopy, we may assume that $\gamma$ and $\gamma '$ are contained in a small neighborhood of the embedded arc that is the image of this straight line in $S^2_4$. Moreover, we may assume that the slopes of any lifts of $\gamma$ and $\gamma '$ are contained in $(s-\varepsilon,s+\varepsilon )$ for some small $\varepsilon >0$. Now given some intersection point $\bullet \in \operatorname {HF}(\gamma,\gamma ')$, choose lifts $\bar {\gamma }$ and $\bar {\gamma }'$ that intersect transversely in some lift $\tilde {\bullet }$ of $\bullet$. Let $t$ and $t'$ be the slopes of $\bar {\gamma }$ and $\bar {\gamma }'$ at $\tilde {\bullet }$, respectively. Then

\[ \delta(\bullet)= \begin{cases} \delta(\gamma,\gamma') & \text{if }s-\varepsilon< t'< t< s+\varepsilon,\\ \delta(\gamma,\gamma')+1 & \text{if } s-\varepsilon< t< t'< s+\varepsilon, \end{cases} \]

which can be seen by applying Lemma 4.7 to thin triangular domains enclosed on two sides by $\bar {\gamma }$ and $\bar {\gamma }'$ and on the third side by either only the vertical or only the horizontal arcs in $P$ (see also Figure 9(a)).

There exist local systems $X$ and $Y$ for which $\operatorname {HF}(\mathbf {r}_X(s),\mathbf {r}_Y(s))=0$. For example, take $X$ to be a permutation matrix of order $n$ and let $Y$ be the companion matrix of the polynomial $f(x)=x^n+x+1$. Then $\ker (f(X))=\ker (X)=0$, so by [Reference ZibrowiusZib20, Theorem 4.45 and Lemma 4.51], $\dim \operatorname {HF}(\mathbf {r}_X(s),\mathbf {r}_Y(s))=2\cdot \dim \ker (f(X))=0$. However, we still have some control over local systems.

A tangle that is Heegaard Floer exceptional is described in Example 7.15.

4.3 Heegaard Floer thin fillings

In this subsection, $G$ is either $\mathbb {Z}$ or $\mathbb {Z}/2$. Define

\[ \operatorname{\mathfrak{C}_{\operatorname{HF}}} := \{\operatorname{HFT}(T)\mid\text{Conway tangles }T\}. \]

In the following, we make implicit use of the following properties that the curves in $\operatorname {\mathfrak {C}_{\operatorname {HF}}}$ are known to satisfy: each multicurve $\Gamma \in \operatorname {\mathfrak {C}_{\operatorname {HF}}}$ consists of linear components only (Theorem 3.9). Moreover, special components come in conjugate pairs of curves that are supported in identical $\delta$-gradings (Theorem 3.10). Finally, $\operatorname {HF}(\Gamma _1,\Gamma _2)\neq 0$ for each $\Gamma _1,\Gamma _2\in \operatorname {\mathfrak {C}_{\operatorname {HF}}}$, because of Theorem 3.5 and the fact that knot Floer homology does not vanish. In particular, if $\operatorname {\mathcal {S}}_\Gamma =\{s\}$ for some $\Gamma \in \operatorname {\mathfrak {C}_{\operatorname {HF}}}$, then $\Gamma$ contains some rational component whose local system is not inhibited. Let $\operatorname {\mathfrak {C}^\mathrm {wb}_{\operatorname {HF}}}\subseteq \operatorname {\mathfrak {C}_{\operatorname {HF}}}$ be the subset of well-behaved multicurves defined by

\[ \operatorname{\mathfrak{C}^\mathrm{wb}_{\operatorname{HF}}} := \{\Gamma\in\operatorname{\mathfrak{C}_{\operatorname{HF}}}\mid \Gamma\text{ does not contain any inhibited rational component}\}. \]

Given two multicurves $\Gamma$ and $\Gamma '$ and a slope $s\in \operatorname {\mathcal {S}}_\Gamma \cap \operatorname {\mathcal {S}}_{\Gamma '}$, the following two local conditions will be relevant.

• Condition (R) At least one of $\Gamma$ and $\Gamma '$ is $s$-rational, i.e. it only contains rational components of slope $s$.

• Condition (R _⋆) The local systems of any two rational components of $\Gamma$ and $\Gamma '$ of slope $s$ are complementary.