## 1. Introduction

Fox famously asked

*What is an alternating knot?*

He was interested in knowing if this property could be characterized without reference to knot diagrams; see Lickorish [Reference LickorishLic97, Chapter 4]. A satisfying answer to Fox's question was provided by Greene [Reference GreeneGre17] and Howie [Reference HowieHow17]: both works prove that a non-split link is alternating if and only if it admits a pair of special spanning surfaces.

Bar-Natan conjectured [Reference Bar-NatanBN02] and Lee proved [Reference LeeLee05] that alternating links have thin Khovanov homology. Subsequently, Ozsváth and Szabó proved that alternating links have thin knot Floer homology [Reference Ozsváth and SzabóOS03a, Reference Ozsváth and SzabóOS08]. That is, the relevant bigraded homology theory in each case is supported along a single diagonal (taking the reduced version in the case of Khovanov's invariant). These diagonals give rise to the integer-valued $\delta$-grading in each theory, so that thinness is defined, algebraically, as follows.

Definition 1.1 A $\delta$-graded vector space is called *thin* if it is supported in at most one $\delta$-grading.

A link is called thin if its associated invariant is thin. Bar-Natan's calculations showed that non-alternating thin links exist in Khovanov homology, suggesting a broader class of links that appears harder to pin down. Restricting coefficients to the rational numbers for the moment, Dowlin's spectral sequence from Khovanov homology to knot Floer homology [Reference DowlinDow18] implies that if a link is thin as measured by Khovanov homology then it must be thin as measured by knot Floer homology. In fact, computations suggest that these notions of thinness coincide. Thus, the question

*What is a thin link?*

is a natural one. In particular, is there a characterization of thinness that does not depend on the bigraded link homology theory used? For example, quasi-alternating links were proved to be thin by Manolescu and Ozsváth [Reference Manolescu and OzsváthMO08]. Interestingly, thin links that are not quasi-alternating exist [Reference GreeneGre10] and indeed arise in infinite families [Reference Greene and WatsonGW13]. A larger class has been proposed, two-fold quasi-alternating [Reference Scaduto and StoffregenSS18], and one might ask whether this exactly captures the property of being thin.

Beyond the homology theory in question, thinness may also depend on the coefficient system. Indeed, Shumakovitch found a knot whose Khovanov homology is thin when computed over $\mathbb {Q}$, but not over the two-element field $\mathbb {F}$ (see [Reference ShumakovitchShu21]); see Example 7.1 for further discussion. The authors are unaware of any such example for knot Floer homology.

### A change of perspective

The question *‘What is a thin link?’* may be placed in a broader context: Given any homology theory $\operatorname {\mathbf {H}_\ast }$ (of CW-complexes, manifolds, links, etc.), a basic observation is that its dimension is bounded below by the absolute value of its Euler characteristic $\chi$. Thus, the following is a natural problem.

Problem 1.2 Characterize the objects $Y$ for which $\dim \operatorname {\mathbf {H}_\ast }(Y)=|\chi \operatorname {\mathbf {H}_\ast }(Y)|$.

Equivalently, the problem is to classify all objects whose homology is supported in gradings of the same parity. Even for singular homology of manifolds, this appears to be a hard question, although some basic facts can be easily established: for oriented two-dimensional manifolds, for example, the identity $\dim \operatorname {\mathbf {H}_\ast }(Y)=|\chi \operatorname {\mathbf {H}_\ast }(Y)|$ characterizes the two-sphere. For unoriented two-dimensional manifolds, the situation already becomes more subtle, because the answer depends on the field of coefficients. For $n$ odd, the Euler characteristic of any $n$-dimensional closed manifold vanishes, so there are no solutions to this identity. A naïve guess for even integers $n\geq 4$ would be that solutions should admit a handle decomposition with no $i$-handles for odd $i$. But such a characterization seems to be difficult to establish even for closed, simply connected four-manifolds; see [Reference KirbyKir95, Problem 4.18].

In the context of Ozsváth and Szabó's Heegaard Floer homology $\operatorname {\widehat {HF}}$ for closed oriented three-manifolds, solutions to $\dim \operatorname {\widehat {HF}}(Y)=|\chi \operatorname {\widehat {HF}}(Y)|$ are known under the name *L-spaces*; see § 7.5 for a detailed discussion of this definition. In this context, Problem 1.2 relates to the question

*What is an L-space?*

(see [Reference Ozsváth and SzabóOS05a, Question 11]), which continues to drive research. Ozsváth and Szabó proved that L-spaces cannot carry taut foliations [Reference Ozsváth and SzabóOS04a] (see also [Reference BowdenBow16, Reference Kazez and RobertsKR17]). At present, the conditions $Y$ not being an L-space, $\pi _1(Y)$ being left-orderable, and $Y$ admitting a taut foliation are known to be equivalent for all graph manifolds [Reference Boyer and ClayBC17, Reference RasmussenRas17, Reference Hanselman, Rasmussen, Rasmussen and WatsonHRRW20]. The equivalence of these three conditions is conjectured in general; see [Reference Boyer, Gordon and WatsonBGW13] or [Reference DunfieldDun20] for further discussion.

Turning now to link homology theories: the reduced Khovanov homology $\widetilde {\operatorname {Kh}}(L;{\mathbf {k}})$ of an $\ell$-component link $L$ in $S^3$ categorifies the Jones polynomial $V_L(t)$, in the sense that

where $h$ denotes the homological grading, $q$ the quantum grading, and ${\mathbf {k}}$ is some field. By setting $t=1$, we see that the ungraded Euler characteristic with respect to the homological grading is equal to $V_L(1)=2^{\ell -1}$. Problem 1.2 in this setting was recently solved by Xie and Zhang [Reference Xie and ZhangXZ18], who showed that the identity $\dim \widetilde {\operatorname {Kh}}(L;{\mathbf {k}})=|\chi _h\widetilde {\operatorname {Kh}}(L;{\mathbf {k}})|=2^{\ell -1}$ characterizes so-called forests of unknots (at least if ${\mathbf {k}}=\mathbb {F}$). Similarly, the knot Floer homology $\operatorname {\widehat {HFK}}(L;{\mathbf {k}})$ categorifies the Alexander polynomial $\Delta _L(t)$:

where $h$ denotes the homological grading (often called the Maslov grading) and $A$ denotes the Alexander grading (or, more precisely, twice the Alexander grading from [Reference Ozsváth and SzabóOS04b]). The ungraded Euler characteristic with respect to the homological grading is equal to $0$ if $\ell >1$ and $\Delta _L(1)=1$ if $\ell =1$. Thus, in the first case, there are no solutions to the identity $\dim \operatorname {\widehat {HFK}}(L;{\mathbf {k}})=|\chi _h\operatorname {\widehat {HFK}}(L;{\mathbf {k}})|$; in the second case, Problem 1.2 reduces to the question about unknot detection for $\operatorname {\widehat {HFK}}$, which was settled by Ozsváth and Szabó [Reference Ozsváth and SzabóOS04a].

Since both $\widetilde {\operatorname {Kh}}$ and $\operatorname {\widehat {HFK}}$ are bigraded homology theories, one is not restricted to taking Euler characteristics with respect to the homological grading. Another choice is the $\delta$-grading, which is defined by $\delta =\frac {1}{2}q-h$ and $\delta =\frac {1}{2}A-h$, respectively. This corresponds to setting $t=-1$ in the polynomial invariants:

This choice seems to be particularly natural, since

where $\det (L)$ is the determinant of $L$, a classical link invariant. It leads us to consider the following.

Definition 1.3 Given a link homology theory $\operatorname {\mathbf {H}_\ast }$, an *A-link* is a link $L$ satisfying

In the following, $\operatorname {\mathbf {H}_\ast }$ will be either $\widetilde {\operatorname {Kh}}$ or $\operatorname {\widehat {HFK}}$ with coefficients in some field ${\mathbf {k}}$. Again, there is a dependence on the homology theory $\mathbf {H}_*$ as well as on ${\mathbf {k}}$, and we will be adding the relevant modifiers where needed. Problem 1.2 relates to the question

*What is an A-link?*

By this question we are interested in knowing if this property can be described (geometrically or topologically) without reference to a link homology theory. Clearly, every thin link is an A-link. For $\operatorname {\mathbf {H}_\ast }=\operatorname {\widehat {HFK}}$, the converse is false, as the family of twisted Whitehead doubles of the trefoil knot that Hedden and Ording consider in [Reference Hedden and OrdingHO08] illustrates; see Example 7.15. For $\operatorname {\mathbf {H}_\ast }=\widetilde {\operatorname {Kh}}$, we expect that all A-links are thin. This is closely related to the question of full support.

Definition 1.4 We say that a link homology theory $\operatorname {\mathbf {H}_\ast }$ has *full support* if for all links $L$ and all $\delta$-gradings $i< j< k$,

Proposition 1.5 Given a link homology theory $\operatorname {\mathbf {H}_\ast }$ with full support, a link is thin if and only if it is an A-link.

Proof. A link $L$ is an A-link if and only if $\mathbf {H}_*(L)$ is supported in gradings of the same parity. Assuming $\operatorname {\mathbf {H}_\ast }$ has full support, the latter is equivalent to $\mathbf {H}_*(L)$ being supported in a single grading, i.e. $L$ being thin.

To the best of the authors’ knowledge, there is no known example of a link violating full support for Khovanov homology: indeed, that this invariant has full support appears to be a folklore conjecture.

Our shift in perspective from thin links to A-links is primarily motivated by the observation that the latter are better behaved with respect to tangle decompositions along Conway spheres, which is the focus of this article. Another reason is the interplay between L-spaces and A-links in the context of two-fold branched covers: there is a spectral sequence due to Ozsváth and Szabó relating the reduced Khovanov homology of a link and the Heegaard Floer homology of the mirror of the two-fold branched cover of the link [Reference Ozsváth and SzabóOS05b]. In particular, given an A-link, the associated two-fold branched cover is an L-space. However, the converse is not true: the Poincaré homology sphere is an L-space that may be obtained as the two-fold branched cover of the torus knot $10_{124}$, which is not an A-knot. Nonetheless, there is a sense in which A-link branch sets might be characterized by sufficiently large L-space surgeries on strongly invertible knots; see the discussion in § 7, as well as [Reference WatsonWat17, Conjecture 30] and [Reference WatsonWat11] for related examples.

### Thin links and Conway spheres

For simplicity, we now restrict to coefficients in the field of two elements $\mathbb {F}$. We focus on characterizing thin links and A-links from the perspective of Conway spheres. This is motivated, in part, by results characterizing L-spaces in the presence of an essential torus. Given a three-manifold with torus boundary $M$ and a parametrization of $\partial M$ by a meridian $\mu$ and a longitude $\lambda$, the space of L-space fillings of $M$ is defined by

where $M({p}/{q})$ is the closed three-manifold obtained by Dehn filling along the slope $p\mu +q\lambda \in H_1(\partial M)$. Rasmussen and Rasmussen showed [Reference Rasmussen and RasmussenRR17, Proposition 1.3 and Theorem 1.6] the following.

Theorem 1.6 For any three-manifold with torus boundary $M$, $\mathcal {L}(M)$ is either empty, a single point, a closed interval or $\operatorname {\mathbb {Q}P}^1$ minus a single point.

Denote the interior of $\mathcal {L}(M)$ by $\mathring {\mathcal {L}}(M)$. Hanselman, Rasmussen, and the second author established the following result [Reference Hanselman, Rasmussen and WatsonHRW24, Theorem 13].

#### Theorem 1.7 (L-space gluing theorem)

Let $Y = M_0 \cup _h M_1$ be a three-manifold where the $M_i$ are boundary incompressible manifolds and $h\colon \thinspace \partial M_1 \rightarrow \partial M_0$ is an orientation-reversing homeomorphism between the torus boundaries. Then $Y$ is an L-space if and only if

A similar result holds without the assumption that $M_i$ be boundary incompressible; see Remark 7.16.

A Conway tangle is a proper embedding of two intervals and a finite (possibly empty) set of circles into a closed 3-dimensional ball. We consider Conway tangles up to isotopy fixing the boundary sphere pointwise. Given a Conway tangle $T$ and a $\delta$-graded link homology theory, we make analogous definitions:

Here $T({p}/{q})$ is the ${p}/{q}$-rational filling of $T$, that is, the link obtained by closing the tangle $T$ with a ${-p}/{q}$-rational tangle, using the tangle gluing convention shown in Figure 1. We call these the *A-link filling space* and the *thin filling space* of the tangle $T$, respectively. Strictly speaking, we should decorate each of these with the homology theory in question; we use subscripts to do so where needed. Often, however, the four spaces $\mathrm {A}_{\operatorname {Kh}}(T)$, $\mathrm {A}_{\operatorname {HF}}(T)$, $\Theta _{\operatorname {HF}}(T)$, and $\Theta _{\operatorname {Kh}}(T)$ coincide. Moreover, the central statements in this paper hold in both the Khovanov and Heegaard Floer setting. Therefore, we do not specify the homology theory in the remainder of this introduction.

#### Theorem 1.8 (Characterization of A-link filling spaces)

For any Conway tangle $T$, $\mathrm {A}(T)$ is either empty, a single point or an interval in $\operatorname {\mathbb {Q}P}^1$.

#### Theorem 1.9 (Characterization of thin filling spaces)

For any Conway tangle $T$, $\Theta (T)$ is either empty, a single point, two distinct points, or an interval in $\operatorname {\mathbb {Q}P}^1$.

Theorems 1.8 and 1.9 illustrate the difference between A-links and thin links. However, whereas Heegaard Floer A-links need not be Heegaard Floer thin, see Example 7.15, we do not know any tangle for which $\Theta (T)$ consists of two distinct points, neither in Heegaard Floer nor Khovanov theory. If such a tangle exists in Khovanov homology, then the rational fillings of this tangle include Khovanov A-links that are not Khovanov thin. In particular, this would establish that Khovanov homology does not have full support. Despite the (potential) existence of such pathological examples, we know that thin and A-link filling spaces coincide generically in the following sense.

Proposition 1.10 If $\Theta (T)$ is an interval, then $\Theta (T)=\mathrm {A}(T)$.

In contrast with $\mathcal {L}(M)$, when $\mathrm {A}(T)$ or $\Theta (T)$ is an interval with two distinct boundary points the interval need not necessarily be closed. This suggests that analogues of the L-space gluing theorem have to be slightly more subtle. The proofs of all results in this paper rely on the homological invariants $\operatorname {HFT}(T)$ and $\widetilde {\operatorname {Kh}}(T)$, which are generalizations of Heegaard Floer and Khovanov homology of links to Conway tangles [Reference Kotelskiy, Watson and ZibrowiusKWZ19, Reference ZibrowiusZib19, Reference ZibrowiusZib20, Reference ZibrowiusZib23b]; these invariants are reviewed in §§ 3 and 5, respectively. Given two Conway tangles $T_1$ and $T_2$, let $\Gamma _1$ denote the Heegaard Floer/Khovanov tangle invariant of $T_1^*$, the mirror of $T_1$, and let $\Gamma _2$ be the corresponding invariant of $T_2$. The link $T_1\cup T_2$ is obtained by identifying the two tangles according to the prescription in Figure 1.

Definition 1.11 For a subset of slopes $X \in \mathbb {Q} P^1$, define its *mirror* as $X^{\operatorname {m}}=\{-s\,|\, s\in X\}$.

#### Theorem 1.12 (A-link gluing theorem)

The link $T_1\cup T_2$ is an A-link if and only if:

(1) $\mathrm {A}^{\operatorname {m}}(T_1) \cup \mathrm {A}(T_2) = \operatorname {\mathbb {Q}P}^1$; and

(2) certain conditions indexed by $\partial \mathrm {A}^{\operatorname {m}}(T_1) \cap \partial \mathrm {A}(T_2)$ hold for $\Gamma _1$ and $\Gamma _2$.

Condition (2) is easy to describe, once the relevant tangle invariants have been reviewed. Note that this condition is vacuously satisfied if $\partial \mathrm {A}^{\operatorname {m}}(T_1) \cap \partial \mathrm {A}(T_2) = \varnothing$, which is true generically. This allows us to obtain the following.

Corollary 1.13 Let $\mathring {\mathrm {A}}(T_i)$ denote the interior of $\mathrm {A}(T_i)$ for $i=1,2$. Then

There is also an analogue of the A-link gluing theorem for thinness. However, due to the characterization results of A-link versus thin filling spaces, this analogue requires an additional hypothesis about the tangle invariants $\operatorname {HFT}(T)$ and $\widetilde {\operatorname {Kh}}(T)$. For this, we introduce the notion of Heegaard Floer/Khovanov exceptionality for tangles (Definitions 4.24 and 6.11). Heegaard Floer exceptional tangles do exist, see Example 7.15. We conjecture that Khovanov exceptional tangles do not exist. That such a conjecture is reasonable is supported by the following.

Proposition 1.14 If a Khovanov exceptional tangle exists, then there exists a link whose Khovanov homology is supported in precisely two non-adjacent $\delta$-gradings.

Once more, the question of full support is brought to the foreground.

#### Theorem 1.15 (Thin gluing theorem)

Suppose at most one of $T_1$ and $T_2$ is exceptional. Then $T_1\cup T_2$ is thin if and only if:

(1) $\Theta ^{\operatorname {m}}(T_1) \cup \Theta (T_2) = \operatorname {\mathbb {Q}P}^1$; and

(2) certain conditions indexed by $\partial \Theta ^{\operatorname {m}}(T_1) \cap \partial \Theta (T_2)$ hold for $\Gamma _1$ and $\Gamma _2$.

Corollary 1.16 Let $\mathring {\Theta }(T_i)$ denote the interior of $\Theta (T_i)$ for $i=1,2$. Then

Corollary 1.13 and Corollary 1.16 provide the condition one checks in practice. Some examples are discussed in § 7.

### How to read this paper

The similarities between Heegaard Floer and Khovanov homology, highlighted by the main results of this paper, extend to the arguments that go into the proofs of these results. In fact, the arguments are so similar that they can be presented without reference to either link homology theory. This is done in § 2, which requires no specialized knowledge. We then show that both the Heegaard Floer invariant $\operatorname {HFT}(T)$ (§§ 3 and 4) and the Khovanov invariant $\operatorname {Kh}(T)$ (§§ 5 and 6) fit into this general framework. Section 7 discusses examples and applications of our main results, focussing primarily on thinness in Khovanov homology.

The sections of this paper need not be read in order, and depending on the interests of the reader certain sections can be skimmed or even skipped. A flow chart of dependencies is given in Figure 2. For instance, having read this introduction, the reader may wish to turn immediately to the Examples in § 7 in order to get a sense of what one observes in nature. Section 2 is entirely combinatorial and makes no reference to any link homology theory. Sections 3 and 4 focus on knot Floer homology whereas §§ 5 and 6 focus on Khovanov homology following a similar structure: in both cases, we review the relevant tangle invariant in the first section and establish our new results in the second.

## 2. Abstracting the main argument

This section lays the combinatorial foundation on which the main results of this paper rely. Towards characterizing thin links and A-links without reference to a given homology theory, we find it compelling that, relative to tangle decompositions, thinness is amenable to the elementary combinatorial abstraction described in the following.

### 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

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.

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

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

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

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:

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.

Definition 2.2 Given $s\in \operatorname {\mathbb {Q}P}^1$, a line set $C$ is called $s$*-consistent* if $\operatorname {g}(c,c')=0$ for all $c,c'\in C$ with $s(c)=s=s(c')$.

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 $(c,d)\in C\times D$,

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

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.

Lemma 2.4 Given a line set $C$, $s(c)\not \in \Theta (C)$ for every rational $c\in C$.

Proof. This is an immediate consequence of the definitions.

Lemma 2.5 Given a line set $C$, suppose $s_0\in \Theta (C)$. Then $C$ is $s$-consistent for all slopes $s\in \operatorname {\mathbb {Q}P}^1\smallsetminus \{s_0\}$.

Proof. Let us write $c_0=(s_0,0,0)$. Since $s_0\in \Theta (C)$, $\operatorname {g}(c_0,c)=\operatorname {g}(c_0,c')$ for any lines $c,c'\in C$ of slopes different from $s_0$. In particular, this holds for lines $c,c'$ of the same slope $s\neq s_0$. In this case, the triple $(c_0,c,c')$ is increasing, so by (transitivity) of the function $\operatorname {g}$, $\operatorname {g}(c,c')=0$.

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

Proof. Suppose $C$ is not $s$-consistent so that there exist $c,c'\in C$ such that $\operatorname {g}(c,c')\neq 0$. Now consider some ‘test’ slope $s_0\neq s$ and let $c_0=(s_0,0,0)$. The triple $(c_0,c,c')$ is increasing, so by (transitivity) $\operatorname {g}(c_0,c)\ne \operatorname {g}(c_0,c')$. Thus, $s_0\notin \Theta (C)$ and $\Theta (C)\subseteq \{s\}$. Similarly, if $C$ is $s$-consistent, (transitivity) implies $\operatorname {\mathbb {Q}P}^1\smallsetminus \{s\}\subseteq \Theta (T)$. Finally, appealing to Lemma 2.4, $s\in \Theta (C)$ if and only if all lines $c\in C$ are special.

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.

Lemma 2.7 Given a line set $C=\{c_1,\ldots,c_n\}$ write $s_i=s(c_i)$ and suppose $(s_1,\ldots,s_n)$ is increasing with $s_1\neq s_n$; see Figure 3. Then the following conditions are equivalent:

(1) there exists some $s\in \Theta (C)$ with $s\in (s_n,s_1)$;

(2) $\operatorname {g}(c_i,c_j)=0$ for all $i< j$;

(3) $(s_n,s_1)\subseteq \Theta (C)\subseteq [s_n,s_1]$;

(4) $(s_n,s_1)\subseteq \Theta (C)$.

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

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

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.

Lemma 2.8 With the same notation as in Lemma 2.7, suppose $|\operatorname {\mathcal {S}}_C|>2$ and $\Theta (C)\subseteq \operatorname {\mathcal {S}}_C$. Then $\Theta (C)\subseteq \{s_i\}$ for some $i$.

Proof. Suppose there exist two distinct slopes $s,s'\in \Theta (C)$. Then by Lemma 2.5, $C$ is $t$-consistent for all $t\in \operatorname {\mathcal {S}}_C$. Since $|\operatorname {\mathcal {S}}_C|>2$, we may assume that, after potentially reindexing the lines, the slopes $s_1=s$, $s_i=s'$, and $s_n$ are pairwise distinct, that $(s_1,\ldots,s_n)$ is increasing, and that $s_{i-1}\neq s_i$. Let $j$ be minimal such that $s_j\neq s$. Then, $\operatorname {g}(c_k,c_\ell )=0$ for all $j\leq k<\ell \leq n$, since $s\in \Theta (C)$. In particular, $\operatorname {g}(c_j,c_n)=0$. Since also $s_i\in \Theta (C)$, we get, in addition, that $\operatorname {g}(c_n,c_k)=0$ for all $1\leq k< i$. This contradicts (symmetry) of the function $\operatorname {g}$ unless $i=j$. However, if $i=j$, then $(s,s')\subset \Theta (C)$ by the direction $(2)\Rightarrow (4)$ of Lemma 2.7, contradicting our initial assumption about $\Theta (C)$.

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:

#### Theorem 2.9 (Characterization of $G$-thin filling spaces)

Let $C$ be a non-trivial line set. Then $\Theta (C)$ is either empty, a single point, two distinct points, or an interval in $\operatorname {\mathbb {Q}P}^1$. For $\Theta _{\mathbb {Z}/2}(C)$, the third case does not arise.

Observation 2.10 We have $\partial \Theta (C)\subseteq \operatorname {\mathcal {S}}_C$ for any line set $C$ by Lemmas 2.6 and 2.7. Moreover, if $C$ is non-trivial, $\operatorname {\mathcal {S}}_C$ is disjoint from the interior of $\Theta (C)$.

Proof of Theorem 2.9 If $|\operatorname {\mathcal {S}}_C|=1$, both statements follow from Lemma 2.6. Thus, we can assume in the following that $|\operatorname {\mathcal {S}}_C|\geq 2$. Let us also assume that $\Theta (C)$ contains some slope $s$. If $s\not \in \operatorname {\mathcal {S}}_C$ then $\Theta (C)$ is an interval by Lemma 2.7. If $\Theta (C)\subseteq \operatorname {\mathcal {S}}_C$ and $|\operatorname {\mathcal {S}}_C|>2$, the set $\Theta (C)$ contains at most one slope by Lemma 2.8. This concludes the proof of the first statement. Suppose $|\Theta _{\mathbb {Z}/2}(C)|=2$, say $\Theta _{\mathbb {Z}/2}(C)=\{s,s'\}$ for some distinct $s,s'\in \operatorname {\mathbb {Q}P}^1$. By Lemma 2.8, $\operatorname {\mathcal {S}}_C=\{s,s'\}$. By Lemma 2.5, $C$ is $s$- and $s'$-consistent. Then, modulo 2, either $\operatorname {g}(c,c')=0$ or $\operatorname {g}(c',c)=0$ for any two lines $c,c'\in C$ with $s(c)=s$ and $s(c')=s'$. Thus, condition (2) of Lemma 2.7 is met, and thus $\Theta _{\mathbb {Z}/2}(C)$ is a (closed) interval, contradicting our initial assumption.

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

Proposition 2.11 If $\Theta _{\mathbb {Z}}(C)$ is an interval, $\Theta _{\mathbb {Z}}(C)=\Theta _{\mathbb {Z}/2}(C)$.

Proof. If $|\operatorname {\mathcal {S}}_C|=1$, this follows from the observation that a line set is $s$-consistent with respect to $G=\mathbb {Z}/2$ if it is $s$-consistent with respect to $G=\mathbb {Z}$. If $|\operatorname {\mathcal {S}}_C|\geq 2$ and $\Theta _{\mathbb {Z}}(C)$ is an interval then by Lemma 2.7, $\Theta _{\mathbb {Z}/2}(C)$ is an interval with the same endpoints. Moreover, whether an endpoint is contained in $\Theta _G(C)$ is independent of $G$.

### 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.

Definition 2.12 We call a line set $C$ *exceptional* if $\operatorname {\mathcal {S}}_C=\{s,s'\}$ for distinct slopes $s,s'\in \operatorname {\mathbb {Q}P}^1$, $C$ is $s$- and $s'$-consistent, but there are lines $c,c'\in C$ with $s(c)=s$ and $s(c')=s'$ such that neither $\operatorname {g}(c,c')$ nor $\operatorname {g}(c',c)$ are equal to 0.

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

Proposition 2.13 If $\Theta _{\mathbb {Z}}(C)=\{s,s'\}$ with $s\neq s'$, then $\operatorname {\mathcal {S}}_C=\{s,s'\}$ and $\Theta _{\mathbb {Z}/2}(C)=[s,s']$ or $[s',s]$.

Proof. If $|\operatorname {\mathcal {S}}_C|=1$, $|\Theta _{\mathbb {Z}}(C)|\neq 2$ by Lemma 2.6. For the case $|\operatorname {\mathcal {S}}_C|\geq 2$, the statement follows from the same arguments as the proof of the second statement of Theorem 2.9.

#### Theorem 2.14 $G$-thin gluing theorem

Let $(C,D)$ be a pair of non-trivial line sets. Suppose not both $C$ and $D$ are exceptional. Then $(C,D)$ is thin if and only if:

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

(2) for all $s\in \partial \Theta (C)\cap \partial \Theta (D)$, at least one of $C$ and $D$ is $s$-rational.

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

Lemma 2.15 Let $(C,D)$ be a pair of non-trivial line sets. Suppose $\Theta (C) \cup \Theta (D) =\operatorname {\mathbb {Q}P}^1$. Then $\operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D=\partial \Theta (C)\cap \partial \Theta (D)$.

Proof. The inclusion $\supseteq$ follows from the first part of Observation 2.10. For the inclusion $\subseteq$, we distinguish four cases, depending on the size of $|\operatorname {\mathcal {S}}_C|$ and $|\operatorname {\mathcal {S}}_D|$. If $|\operatorname {\mathcal {S}}_C|=1=|\operatorname {\mathcal {S}}_D|$, either $\operatorname {\mathcal {S}}_C\cap \operatorname {\mathcal {S}}_D=\varnothing$, so there is nothing to show, or $\operatorname {\mathcal {S}}_C=\{s\}=\operatorname {\mathcal {S}}_D$ for some slope $s$, in which case $\Theta (C) \cup \Theta (D) = \operatorname {\mathbb {Q}P}^1\smallsetminus \{s\} \subsetneq \operatorname {\mathbb {Q}P}^1$ by Lemma 2.6 and the non-triviality of $C$ and $D$. Suppose $|\operatorname {\mathcal {S}}_C|>1$ and $|\operatorname {\mathcal {S}}_D|=1$, say $\operatorname {\mathcal {S}}_D=\{s\}$. If $D$ is not $s$-consistent, the hypothesis is not satisfied by the non-triviality of $D$, Lemma 2.6, and Theorem 2.9. If $D$ is $s$-consistent, $\Theta (D)=\operatorname {\mathbb {Q}P}^1\smallsetminus \{s\}$ by Lemma 2.6, so in particular $s\in \partial \Theta (D)=\operatorname {\mathcal {S}}_D$. Moreover, the hypothesis implies that $s\in \Theta (C)$. If $s\not \in \operatorname {\mathcal {S}}_C$, there is nothing to show, whereas if $s\in \operatorname {\mathcal {S}}_C$, then also $s\in \partial \Theta (C)$ by Lemma 2.7. If $|\operatorname {\mathcal {S}}_C|=1$ and $|\operatorname {\mathcal {S}}_D|>1$, we repeat the argument with the roles of $C$ and $D$ reversed. Thus, it remains to consider the case that $|\operatorname {\mathcal {S}}_C|, |\operatorname {\mathcal {S}}_D|>1$. Combining Lemma 2.7 with the hypothesis shows that $\Theta (C)$ and $\Theta (D)$ are two intervals. The claim now follows from the second part of Observation 2.10.

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

(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

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,

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 $\Theta (C)$.

Corollary 2.16 Let $(C,D)$ be a pair of non-trivial line sets for which $\mathring {\Theta }(C)\cup \mathring {\Theta }(D)=\operatorname {\mathbb {Q}P}^1$. Then $(C,D)$ is thin.

Proof. If $\mathring {\Theta }(C)\cup \mathring {\Theta }(D)=\operatorname {\mathbb {Q}P}^1$, then $\Theta (C)\cup \Theta (D)=\operatorname {\mathbb {Q}P}^1$ and $\partial \Theta (C)\cap \partial \Theta (D)=\varnothing$. Thus, under the assumption that not both $C$ and $D$ are exceptional, the corollary follows from Theorem 2.14. However, we may drop this assumption, because the only case in the proof of Theorem 2.14 in which we use it is Case 2(a), which supposes $|\operatorname {\mathcal {S}}_{C}\cap \operatorname {\mathcal {S}}_{D}|=2$. Here, however, $\operatorname {\mathcal {S}}_{C}\cap \operatorname {\mathcal {S}}_{D}=\varnothing$ by Lemma 2.15.

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,

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,

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.

## 3. The tangle invariant $\operatorname {HFT}$

We review some properties of the immersed curve invariant $\operatorname {HFT}$ of Conway tangles due to the third author [Reference ZibrowiusZib20]; see also [Reference ZibrowiusZib19, Reference ZibrowiusZib23b].

### 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].

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].

Theorem 3.1 For all $\tau \in \operatorname {Mod}(S^2_4)$, $\operatorname {HFT}(\tau (T)) = \tau (\operatorname {HFT}(T))$. In other words, the invariant $\operatorname {HFT}$ commutes with the action of the mapping class group of the four-punctured sphere.

Example 3.2 Figure 5(b) shows the four-punctured sphere $S^2_4$, drawn as the plane plus a point at infinity minus the four punctures labeled ${\mathsf {1}}$, ${\mathsf {2}}$, ${\mathsf {3}}$, and ${\mathsf {4}}$, together with the standard parametrization that identifies $S^2_4$ with $\partial B^3\smallsetminus \partial T$. The dashed curve along with the two immersed curves winding around the punctures form the invariant $\operatorname {HFT}(P_{2,-3})$ for the $(2,-3)$-pretzel tangle [Reference ZibrowiusZib20, Example 2.26].

Definition 3.3 A (parametrized) tangle is called *rational* if it is obtained from the trivial tangle by adding twists to the tangle ends.

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.

#### Theorem 3.4 [Reference ZibrowiusZib20, Theorem 6.2]

A tangle $T$ is rational if and only if $\operatorname {HFT}(T)$ consists of a single embedded component carrying the unique one-dimensional local system.

### 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

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

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.

Definition 3.6 For a curve $\gamma$ in $S^2_4$, its *mirror* $\operatorname {m} (\gamma )$ is the image under the involution of $S^2_4$ that fixes the punctures and arcs and interchanges the gray and white faces from Figure 5(b).

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]).

Lemma 3.7 For any Conway tangle $T$, $\operatorname {HFT}(T^*)=\operatorname {m}(\operatorname {HFT}(T))$.

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 $\mathbf {s}_n({p}/{q};{\mathsf {i}},{\mathsf {j}})$.

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].

Theorem 3.9 For a Conway tangle $T$ the underlying curve of each component of $\operatorname {HFT}(T)$ is either rational or special. Moreover, if it is special, its local system is equal to an identity matrix.

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].

#### Theorem 3.10 Conjugation symmetry

Let $({\mathsf {i}}, {\mathsf {j}} , {\mathsf {k}}, {\mathsf {l}} )$ be some permutation of $({\mathsf {1}},{\mathsf {2}},{\mathsf {3}},{\mathsf {4}} )$ and let ${p}/{q}\in \operatorname {\mathbb {Q}P}^1$. Then, for any Conway tangle $T$, the numbers of components $\mathbf {s}_n({p}/{q};{\mathsf {i}},{\mathsf {j}})$ and $\mathbf {s}_n({p}/{q};{\mathsf {k}},{\mathsf {l}})$ in $\operatorname {HFT}(T)$ in any given $\delta$-grading agree.

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

Lemma 3.11 Each rational component of $\operatorname {HFT}(T)$ separates the four punctures into two pairs, which agrees with how the two open components of $T$ connect the tangle ends.

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. Heegaard Floer thin fillings

We now turn our attention to gradings. Following [Reference ZibrowiusZib20, Definitions 4.28 and 5.1], the $\delta$-grading of an immersed multicurve $\Gamma$ is a function

where $\operatorname {\mathcal {G}}(\Gamma )$ is the set of intersection points between the four parametrizing arcs in $S^2_4$ and $\Gamma$, assuming that this intersection is minimal. The function $\delta$ is subject to the following compatibility condition: suppose $x,x'\in \operatorname {\mathcal {G}}(\Gamma )$ are two intersection points such that there is a path $\psi$ on $\Gamma$ which connects $x$ to $x'$ without meeting any parametrizing arc (except at the endpoints). We distinguish three cases, which are illustrated in Figure 7: a path can turn left (a), can go straight across (b), or can turn right (c). Then

Given a Conway tangle $T$, the generators of the underlying module of the invariant $\operatorname {CFT^\partial }(T)$ are in one-to-one correspondence with elements of $\operatorname {\mathcal {G}}(\operatorname {HFT}(T))$. Moreover, these generators are homogeneous with respect to the $\delta$-grading, so the relative $\delta$-grading on $\operatorname {CFT^\partial }(T)$ determines the relative $\delta$-grading on the corresponding multicurve $\operatorname {HFT}(T)$.

Like link Floer homology, the invariant $\operatorname {HFT}$ comes with a relative bigrading. In this paper we are not concerned with the Alexander grading; our focus is exclusively on the $\delta$-grading. However, we note that the treatment of the grading that follows runs along similar lines to that of [Reference Lidman, Moore and ZibrowiusLMZ22] used to study the Alexander grading.

### 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.

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 $S^2_4$.

Definition 4.2 Suppose $\Gamma =\{\bar {\gamma }_1,\ldots,\bar {\gamma }_n\}$ is a set of curves in $\mathbb {R}^2\smallsetminus \mathbb {Z}^2$ avoiding the integer lattice points such that $P\cup \Gamma =P\cup \bar {\gamma }_1\cup \cdots \cup \bar {\gamma }_n$ is a planar graph whose vertices have all valence four. $P\cup \Gamma$ divides the plane into polygons, which we call *regions*. A *domain* is a formal linear combination of regions. In other words, a domain is an element of $H_2(\mathbb {R}^2,P\cup \Gamma )$.

Let us fix a metric on the plane such that $P\cup \Gamma$ is geodesic and the angles at the vertices of $P$ are ${\pi }/{2}$. We then define the *Euler measure* $e(D)$ of a domain $D$ to be ${1}/{2\pi }$ times the integral over $D$ of the curvature of the metric.

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:

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

Remark 4.4 For readers familiar with Heegaard Floer homology, it can be helpful to think of the curve $\bar {\gamma }$ as playing the role of a $\beta$-curve and $P$ playing the role of an $\alpha$-curve.

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

Proof. For the domains consisting of just the single regions shown in Figure 7, the lemma follows directly from the definition of the $\delta$-grading: the Euler measure of $\varphi$ in these three cases is $\frac {1}{4}$, 0, and $-\frac {1}{4}$, respectively. Now let us consider a general connecting domain $\varphi$ from $\tilde {x}$ to $\tilde {x}'$. By hypothesis, $\partial \varphi \cap \bar {\gamma }$ is a one-chain connecting $\tilde {x}$ to $\tilde {x}'$. Let us first assume that this one-chain corresponds to a path from $\tilde {x}$ to $\tilde {x}'$. That is, either there are no cycles in the one-chain or, in the case $\tilde {x}=\tilde {x}'$, this path is the only cycle. In this case the path can be written as the intersection of $\bar {\gamma }$ with the boundary of a connecting domain $\psi$, which is a sum of finitely many of the basic regions in Figure 7 that we have just considered. The difference $\varphi -\psi$ is a domain whose boundary lies entirely in $P$, so it consists entirely of square regions and, hence, the Euler measure vanishes.

Finally, suppose the one-chain $\partial \varphi \cap \bar {\gamma }$ connecting $\tilde {x}$ to $\tilde {x}'$ also has cycles. Each of them is the boundary of some domain, and we claim that its Euler measure vanishes. To see this, we can apply the previous argument with $\tilde {x}=\tilde {x}'$ being some intersection point of this cycle with $P$ the connecting path being the whole cycle.

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

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

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:

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]).

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,

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,

Combining all three relations, we see that

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

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

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

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,

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

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