Involutive knot Floer homology and bordered modules

We prove that, up to local equivalences, a suitable truncation of the involutive knot Floer homology of a knot in $S^3$ and the involutive bordered Heegaard Floer theory of its complement determine each other. In particular, given two knots $K_1$ and $K_2$, we prove that the $\mathbb{F}_2[U,V]/(UV)$-coefficient involutive knot Floer homology of $K_1 \sharp -K_2$ is $\iota_K$-locally trivial if $\widehat{CFD}(S^3 \backslash K_1)$ and $\widehat{CFD}(S^2 \backslash K_2)$ satisfy a certain condition which can be seen as the bordered counterpart of $\iota_K$-local equivalence. We further establish an explicit algebraic formula that computes the hat-flavored truncation of the involutive knot Floer homology of a knot from the involutive bordered Floer homology of its complement. It follows that there exists an algebraic satellite operator defined on the local equivalence group of knot Floer chain complexes, which can be computed explicitly up to a suitable truncation.


Introduction
Given a closed, connected, and oriented 3-manifold Y , the minus-flavored Heegaard Floer theory, defined by Ozsváth and Szabó [OS04b], associates to Y a chain complex CF − (Y ) over the ring F 2 [U ], whose homotopy type is an invariant of the oriented diffeomorphism class of Y .Furthermore, if we are given a knot K inside Y , then the knot Floer theory [OS08b,Zem19b] associates to K a homotopy class of a chain complex CF K U V (Y, K) over the ring F 2 [U, V ], from which CF − (Y ) can be recovered by taking the specialization (U, V ) = (1, 0), or equivalently, (U, V ) = (0, 1).
Like Seiberg-Witten Floer homology, whose intrinsic Pin(2)-symmetry was used by Manolescu [Man16] to disprove the triangulation conjecture in high dimensions, Heegaard Floer theory has an intrinsic Z 2 -symmetry, which is induced by the involution (Σ, α, β, z) → (−Σ, β, α, z) on the space of pointed Heegaard diagrams representing the given 3-manifold Y .This action, which preserves all relevant counts of holomorphic disks, induces a homotopy-involution ι Y on CF − (Y ), which is well-defined up to homotopy, as observed first in [HM17].Involutive Heegaard Floer theory exploits this involution to give new 3-manifold invariants to define new homology cobordism invariants.Those invariants were then used extensively to solve various problems regarding the structures of homology cobordism groups and knot concordance groups [DHST18, HMZ18, HKPS20, HHSZ20, AKS20, HHSZ21].Moreover, as observed by Hendricks and Manolescu [HM17], a similar construction can also be applied to knot Floer theory.Recall that knot Floer homology starts by representing a pair (Y, K) of a 3-manifold Y and an oriented knot K ⊂ Y as a doubly pointed Heegaard diagram, i.e.Heegaard diagram with two basepoints.Then we have symmetry (Σ, α, β, z, w) → (−Σ, β, α, w, z) on the space of doubly pointed Heegaard diagrams representing (Y, K).However, since the basepoints are swapped to compensate the change of orientation on K occurred by reversing the given orientation on the Heegaard surface Σ, a half-twist along K is needed to define a well-defined homotopy skew -autoequivalence ι K of CF K U V (Y, K).Due to the presence of a half-twist in the definition of ι K , it is no longer a homotopy involution, but satisfies the condition ι 2 K ∼ ξ K , where ξ K denotes the Sarkar map along K.The theory of CF K U V (Y, K) together with ι K is called involutive knot Floer homology, which was used to prove the existence of a linearly independent infinite family of rationally slice knots in [HKPS20].
On the other hand, given a compact oriented 3-manifold M with a suitably parametrized torus boundary, bordered Heegaard Floer theory [LOT16] associates to M a differential module CF D(M ) and an A ∞ -module CF A(M ) over the torus algebra A(T 2 ).When M is the 0-framed exterior S 3 \K of a knot K ⊂ S 3 , we know from [KWZ20] that the homotopy type of those modules is determined by the homotopy type of the truncation CF K R (S 3 , K) of CF K U V (S 3 , K) by taking U V = 0, and vice versa.Furthermore, we know from [HL19] that mimicking the construction of involutive Heegaard Floer theory defines homotopy equivalences Hence, it is natural to ask how the knot involution ι K on CF K R (S 3 , K) is related to the bordered involution ι S 3 \K of its 0-framed knot complement.The following theorem answers this question in the coarse affirmative, by showing that ι K and ι S 3 \K determine each other up to a certain equivalence relation; this equivalence relation is called the ι K -local equivalence, which can be seen as the involutive algebraic counterpart of knot concordance.
Theorem 1.1.Given two knots K 1 and K 2 , consider the involutions ι K1 −K2 of CF K R (S 3 , K 1 − K 2 ), as well as any choice of bordered involutions ι S 3 \K1 ∈ Inv D (S 3 \K 1 ) and ι S 3 \K2 ∈ Inv D (S 3 \K 2 ).Then (CF K R (S 3 , K 1 − K 2 ), ι K1 −K2 ) is ι K -locally equivalent to the trivial complex if and only if there exists a type-D morphism g : CF D(S 3 \K 1 ) → CF D(S 3 \K 2 ) between type-D modules of 0-framed knot complements, such that the diagram CF DA(AZ) CF D(S 3 \K 1 ) is homotopy-commutative and the induced chain map is a homotopy equivalence, and a similar type-D morphism also exists in the opposite direction.Here, T ∞ denotes the ∞-framed solid torus, and S 3 \K 1 and S 3 \K 2 are endowed with the 0-framing on their boundaries.Furthermore, the statement also holds if "any choice of bordered involutions" is replaced with "some choice of bordered involutions".
We now consider involutive knot Floer homology for satellite knots.Given two knots K 1 and K 2 whose knot Floer chain complexes are locally equivalent, it is very unclear whether the satellite knots P (K 1 ) and P (K 2 ) should also have locally equivalent knot Floer chain complexes, where P is any pattern in S 1 × D 2 .Using Theorem 1.1, we prove the existence of a satellite operator in the local equivalence group of knot Floer chain complexes.
A very natural question is then how can one explicitly compute ι K from ι S 3 \K .Using the bordered quasi-stabilization constructions, we prove the following theorem which provides a formula to compute the hat-flavored truncation of ι K from ι S 3 \K up to orientation reversal.
Theorem 1.3.Let ν be the longitudinal knot in the ∞-framed solid torus T ∞ .Then there exists a type-D morphism such that for any knot K and for any choice of ι S 3 \K ∈ Inv D (S 3 \K), the induced map is homotopic to the truncation of either ι K or its homotopy inverse ι −1 K to the hat-flavored complex CF K(S 3 , K) under the natural identification CF A(T ∞ , ν) CF D(S 3 \K) CF K(S 3 , K) induced by the pairing theorem [LOT18, Theorem 11.19], where S 3 \K is endowed with the 0-framing on its boundary.
Theorem 1.3 can also be used to explicitly compute ι S 3 \K for some nontrivial knots K.The case when K is the figure-eight knot is computed in Example 5.8.Note that CF D(S 3 \K) is not rigid, i.e. it has more than one homotopy classes of homotopy autoequivalences; Example 5.8 gives the first example of explicitly computing bordered involutive Floer homology for homotopically non-rigid bordered 3-manifolds.
Furthermore, together with the proof of Theorem 1.2, Theorem 1.3 can also be considered as an involutive satellite formula.In particular, given a pattern P ⊂ S 1 × D 2 , if CF DA((S 1 × D 2 )\P ) is homotopy-rigid and one already knows the action of ι S 3 \K , then one can explicitly compute the hat-flavored involutive knot Floer homology of the satellite knot P (K).
Remark 1.4.When P is the (p, 1)-cabling pattern for some p > 0, the bimodule CF DA((S 1 × D 2 )\P ), with respect to some boundary framings, can be computed from the type DAA trimodule of S 3 × (pair-of-pants), which was explicitly computed in [HW15, Table 1], by taking a box tensor product on its ρ-boundary with the type D module of the 1 p -framed solid torus.It is easy to observe, via manual computation, that the resulting bimodule is homotopy-rigid.Hence Theorem 1.3 gives a hat-flavored involutive (p, 1)-cabling formula, which computes the involutive action of the cable knot K p,1 from ι S 3 \K .
Organization.This article is organized as follows.In Section 2, we recall some results regarding involutive Heegaard Floer homology and bordered Floer homology.In Section 3, we develop a theory of involutive knot Floer homology with a free basepoint and discuss its relationship with involutive bordered Floer homology of 0-framed knot complements.In Section 4, we prove Theorem 1.1 and use it to prove Theorem 1.2.Finally, in Section 5, we prove Theorem 1.3 and discuss its explicit applications.
Acknowledgements.The author would like to thank Kristen Hendricks, Robert Lipshitz, and JungHwan Park for helpful conversations, and Abhishek Mallick, Monica Jinwoo Kang, and Ian Zemke for numerous helpful comments.This work was supported by Institute for Basic Science (IBS-R003-D1).

Involutive Heegaard Floer homology for knots and 3-manifolds
We assume that the reader is familiar with Heegaard Floer theory [OS03, OS04b, OS06, OS04a] of knots and 3-manifolds, as well as bordered Heegaard Floer theory [LOT18].Throughout the paper, we will only work with F 2 coefficients.Furthermore, we will often consider 3-manifolds M endowed with torsion Spin c structures.In such cases, the Heegaard Floer chain complexes CF − (M, s) and CF (M, s) are chain complexes of free modules over F 2 [U ] and F 2 , respectively, and absolutely Q-graded.
2.1.Involutive Heegaard Floer homology and ι-complexes.Recall that the definition of Heegaard Floer homology of any flavor starts with choosing an admissible pointed Heegaard diagram H = (Σ, α, β, z) representing M .The theory of involutive Heegaard Floer homology, as defined first in [HM17], starts by considering the conjugate diagram H = (−Σ, β, α, z).Then we have a canonical identification map Since H also represents M , it is related to H by a sequence of Heegaard moves.Such a sequence induces a homotopy equivalence By the naturality of Heegaard Floer theory [JTZ12], the homotopy class of Φ H,H does not depend on our choice of a sequence of Heegard moves from H to H. Thus the homotopy autoequivalence is well-defined up to homotopy, and the image of its restriction ι s to CF − (M, s) is CF − (M, s).In particular, when s is self-conjugate, i.e. spin, then ι M,s is a homotopy autoequivalence of CF − (M, s).
The involution ι M satisfies the following properties.
Inspired by the above properties, the notion of ι-complex was defined in [HMZ18] as follows.An ι-complex is a pair (C, ι) which satisfies the following properties.
• C is a chain complex of finitely generated free modules over F 2 [U ], such that the localized complex ) and (C 2 , ι 2 ) exist in both directions, we say that the given two ι-complexes are ι-locally equivalent.The set of ι-local equivalence classes of ι-complexes forms a group I under the tensor product operation, which is called the local equivalence group.
The notion of ι-complexes and local equivalences between them can be weakened, as shown in [DHST18], in the following way.An almost ι-complex is a pair (C, ι) which satisfies the following properties.
• C is a chain complex of finitely generated free modules over F 2 [U ], such that the localized complex and ι 2 ∼ id mod U .
Remark 2.1.The definition of ι-local maps, local equivalences, and their "almost" versions also work when we drop the condition that U −1 C is homotopy equivalent to F 2 [U ±1 ].We will sometimes use this generalized notion throughout this paper.
2.2.Involutive knot Floer homology and ι K -complexes.The involutive theory for knot Floer homology is a bit more complicated than the 3-manifold case.For simplicity, we only consider knots K in S 3 .Consider a doubly pointed Heegaard diagram H = (Σ, α, β, z, w) representing (S 3 , K).By counting holomorphic disks while recording their algebraic intersection numbers with z and w by formal variables U and V , respectively, one gets an absolutely Z-bigraded (called Alexander and Maslov grading, respectively) chain complex CF K U V (S 3 , K) of finitely generated free modules over the ring Consider the conjugate diagram H = (−Σ, β, α, w, z) of H; note that, in addition to flipping the orientation of Σ and exchanging α and β curves, we are also exchanging the basepoints z and w.Then, as in the 3manifold case, we have a canonical conjugation map which is a chain skew-isomorphism, i.e. intertwines the actions of U and V on its domain with the actions of V and U on its codomain.Then we consider a self-diffeomorphism of S 3 that acts on a tubular neighborhood of K by a "half-twist", so that it fixes K setwise and maps z and w to w and z, respectively.It induces a chain isomorphism Now, the diagrams φ( H) and H both represent the knot K together with two prescribed basepoints z and w on K, so they are related by a sequence of Heegaard moves.Such a sequence induces a homotopy equivalence whose homotopy class is independent of our choice of a sequence of Heegaard moves from φ( H) to H, due to naturality.Thus we have a homotopy skew-equivalence which is well-defined up to homotopy.Note that such a construction can also be applied for links as well; given a link L, where each component K ⊂ L has one z-basepoint and one w-basepoint (which correspond to formal variables U K and V K ), following the above construction gives a homotopy skew-equivalence ι L which intertwines the actions of U K and V K for each component K.
The homotopy skew-equivalence ι K satisfies the following properties, as shown in [Zem19a].
, where Φ and Ψ are the formal derivatives of the differential ∂ of CF K U V (S 3 , K) with respect to the formal variables U and V , respectively.• The localized map (U, V ) −1 is homotopic to identity.
Using the above properties, the notion of ι K -complexes was defined in [Zem19a] as follows.An ι K -complex is a pair (C, ι K ) which satisfies the following properties.
• C is a chain complex of finitely generated free modules over where Φ and Ψ are the formal derivatives of the differential ∂ of C with respect to the formal variables U and V , respectively.
Given two chain complexes C 1 and C 2 of free modules over F 2 [U, V ], a chain map f : C 1 → C 2 is said to be a local map if the maps induce injective maps in homology.Given two ι K -complexes (C 1 , ι 1 ) and (C 2 , ι 2 ), a local map f : maps between two ι K -complexes exist in both directions, then we say that they are ι K -locally equivalent.The set of ι K -local equivalence classes of ι K -complexes form a group I K when endowed with the addition operation As in the 3-manifold case, the construction of involutive knot Floer homology gives a canonical map C → I K .
We will sometimes work with knot Floer homology with coefficient ring F 2 [U, V ]/(U V ), which is denoted as R, rather than the full two-variable ring F 2 [U, V ].Note that although ι K -local maps and ι K -local equivalences are well-defined, it is unclear whether ι K -local equivalence classes of involutive R-coefficient knot Floer chain complexes form a well-defined group, since the basepoint actions might not be uniquely determined from the R-coefficient differential.

Involutive bordered Floer homology.
Let M be a bordered 3-manifold with one boundary; for simplicity, we will assume that ∂M is a torus.Choose a bordered Heegaard diagram H = (Σ, α, β, z) representing M and consider its conjugate diagram H = (−Σ, β, α, z).Then we have canonical identification maps between the type-D and type-A modules associated to H and H, respectively.Note that we are using the same name for the type-D and type-A identification maps for convenience.
In contrast to the case of closed 3-manifolds, there does not exist a sequence of Heegaard moves from H to H.The reason is that H is α-bordered, whereas H is β-bordered.To remedy this problem, Hendricks and Lipshitz [HL19] uses the Auroux-Zarev piece AZ and its conjugate AZ, which satisfies the property that AZ ∪ AZ represents a trivial cylinder T 2 × I.A Heegaard diagram representing AZ is shown in Figure 2.1.One starts with the [LOT11, Theorem 4.6], which implies that AZ ∪ H and H ∪ AZ are related to H by a sequence of Heegaard moves.Choosing such sequences give homotopy equivalences Recall that we have pairing maps induced by time dilation, as discussed in [LOT18, Chapter 9], which are defined uniquely up to homotopy: Then we can define the bordered involution ι M , in both type-D and type-A modules, as follows: Now suppose that we are given a bordered 3-manifold N whose boundary consists of two torus components.Choose an α-α-bordered Heegaard diagram H representing N .Then it follows again from [LOT11, Theorem 4.6] that AZ ∪ H ∪ AZ is related to H by a sequence of Heegaard moves.Choosing such a sequence gives a homotopy equivalence Thus we can define a bordered involution ι N as follows.
Unlike the cases of knots and closed 3-manifolds, we do not know whether the homotopy classes of ι M and ι N are independent of our choices of sequences of Heegaard moves.This is because a naturality result for bordered Heegaard Floer homology is currently unknown.However, we can instead consider the sets of all possible involutions coming from any possible choices of sequences of Heegaard moves, as shown in the definition below.Definition 2.2.Given a bordered 3-manifold M with one torus boundary, we denote the set of all possible involutions induced by choosing a sequence of Heegaard moves from AZ ∪ H and H ∪ AZ to H as Inv D (M ) and Inv A (M ), respectively.Furthermore, given a bordered 3-manifold N with two torus boundaries, we similarly denote the set of all possible involutions induced by choosing a sequence of Heegaard moves from AZ ∪ H ∪ AZ to H as Inv(N ).
Recall that, given two bordered 3-manifolds M 1 and M 2 , we have a pairing theorem Due to the pairing theorem for triangles [LOT16, Proposition 5.35], it is clear that the homotopy equivalence used in Equation (2.1) is well-defined up to homotopy.[HL19, Theorem 5.1] tells us that for any ι 1 ∈ One also has another pairing formula involving morphism spaces between type-D modules.Given two bordered 3-manifolds M 1 and M 2 with one torus boundary, one can also obtain the hat-flavored Heegaard Floer homology of −M 1 ∪ M 2 as follows[LOT11, Theorem 1]: Unlike the box tensor product version of pairing formula, the well-definedness of homotopy equivalence up to homotopy in the above formula is not entirely obvious.This is because its proof relies on the following isomorphism: In particular, the homotopy equivalence CF A(M 1 ) CF D(M 1 ) CF AA(AZ), which is induced by a sequence of Heegaard moves from M 1 to M 1 ∪ AZ, may not be well-defined due to the lack of naturality.However, if we have two such sequences which induce two identification maps then by the pairing theorem for triangles, the map is the homotopy autoequivalence induced by a loop of Heegaard moves, which should be homotopic to identity due to naturality.Therefore the homotopy equivalence used in Equation (2.2) is well-defined up to homotopy.Now it follows from the proof of [HL19, Theorem 8.5] that the map

Involutive knot Floer homology with a free basepoint
Given a knot K, instead of choosing a doubly-pointed Heegaard diagram representing K, we consider a multipointed Heegaard diagram H = (Σ, α, β, {z, z f ree }, w), where z and w are points on K and z f ree is a free basepoint, which lies outside K. Given such a diagram, we define its 2-variable knot Floer homology where the differential ∂ is defined using the formula Here, M(φ) denotes the moduli space of holomorphic curves representing the given homotopy class φ of Whitney disks from x to y, and n z f ree (φ), n z (φ), and n w (φ) denote the algebraic intersection number of φ with the codimension 2 submanifolds given by z f ree , z, and w, respectively.Note that the naturality result for Heegaard Floer homology [JTZ12] also applies to this case, so that chain homotopy autoequivalences of CF K U V (S 3 , K, z f ree ) induced by any loop of Heegaard moves connecting Heegaard diagrams representing (S 3 , K, z f ree ) are homotopic to the identity map.
As in involutive knot Floer homology, we can define the conjugate diagram H of H as follows: We have a canonically defined chain skew-isomorphism: We then consider the half-twist self-diffeomorphism φ of (S 3 , K, z f ree ) which maps z and w to w and z, respectively.It induces a diffeomorphism map Then, since φ( H) and H both represent (S 3 , K, z f ree ), there exists a sequence of Heegaard moves between them, which induces a homotopy equivalence which is well-defined up to chain homotopy, due to naturality.Composing the above three maps thus gives which is again well-defined up to chain homotopy.Given a doubly-pointed Heegaard diagram H K representing K, we can perform a free-stabilization on H K near the basepoint z, as shown in the left of Figure 3.1, to get a new diagram H st K representing (S 3 , K, z f ree ).Then, by [Zem19b, Lemma 7.1], the differential of CF U V (H st K ) is given by the matrix where we are using an identification We now assume that K is boundary-parallel to the Heegaard surface Σ and the self-diffeomorphism φ acts as identity near the free-stabilization locus.Then φ( Hst K ) is also a free-stabilization on φ( HK ) near the basepoint z, and for any sequence For each i, the Heegaard move H i → H i+1 is either an isotopy, a handleslide, or a stabilization.Since we can always start with sufficiently stabilized diagrams and replace an isotopy by a sequence of handleslides, we may further assume that all Heegaard moves that we use are handleslides.Recall that the chain homotopy equivalences associated to handleslides are defined by counting holomorphic triangles in a Heegaard triple diagram.If the homotopy equivalence is defined by counting triangles in a triple diagram H st T which is obtained by free-stabilizing H T near z, as shown in the right of Figure 3.1.Thus, by [Zem15, Theorem 6.7], we know that ree , so we deduce that S + z f ree is well-defined up to homotopy and is the hat-flavored free-stabilzation map ) induces an injective map in homology.Therefore S + z f ree is local.We now interpret involutive knot Floer theory with a free basepoint in terms of bordered Floer homology.Consider the triply-pointed bordered Heegaard diagram X = (Σ, α, β, {z, z f ree }, w), defined as in Figure 3.2.This diagram represents the longitudinal knot lying inside the ∞-framed solid torus, together with a prescribed free basepoint z f ree on the boundary torus.Note that, for any bordered Heegaard diagram H of M \K, where K is a framed knot inside a closed 3-manifold M and the framing is denoted as ν, the glued diagram H ∪ X is a Heegaard diagram representing the core curve inside the Dehn surgery M ν (K), together with a free basepoint.
We now consider the new diagram φ( X), where X denotes the conjugate diagram of X, defined as X = (−Σ, β, α, {w, z f ree }, z), and φ denotes the "half-twist" self-diffeomorphism of Σ along the longitudinal knot, so that it maps z to w and w to z, respectively.
Proof.Denote the bordered Heegaard diagram representing the 0-framed solid torus as H, and its conjugate as H.It is proven in [LOT11, Figure 8 and 9] that AZ ∪ H and H are related by a sequence of handleslides and a destabilization.Since H is simply X without the basepoints z, w, and the α-and β-curves surrounding them, it is clear that the sequence of handleslides (and a single destabilization) from AZ ∪ H to H induces sequence of handleslides and a single destabilization from AZ ∪ φ( X) to X.A detailed process is drawn in Figure 3.3.
Choose a nice diagram X 0 which is related by X by a sequence of Heegaard moves; such a diagram always exists by Sarkar-Wang algorithm [LOT18, Proposition 8.2], and it is always provincially admissible.Then X 0 has a well-defined bordered Floer homology.In particular, if we write X 0 = (Σ, α, β, {z, z f ree }, w), then we have a well-defined type-D structure CF D U V (X 0 ) and a type-A structure CF A U V (X 0 ) over the module F 2 [U, V ], defined by counting holomorphic disks which do not intersect algebraically with z f ree , while recording their algebraic intersection numbers with z and w by formal variables U and V , respectively.
Recall from [LOT18, Chapter 10] that, given a bordered 3-manifold Y with boundary Z, the associated type-A module CF A(Y ) is graded by a transitive G(Z)-set, and for a doubly-pointed bordered Heegaard diagram H = (Σ, α, β, z, w) with the same boundary, the associated type-D module CF A − (H) admits an enhanced grading by a transitive (G(Z) × Z)-set, where the grading on the Z component is given by n w − n z .We can define a grading on CF A U V (X 0 ) by the group G(T 2 ) × Z is a similar manner, as follows.
Write X 0 = (Σ, α, β, {z, z f ree }, w).Then for any choice of Floer generators x and y and a homology class B ∈ π 2 (x, y) of curves connecting x to y, we define the relative grading g(x, y) ∈ G(T 2 ) × Z as where λ is the central element (1; 0, 0) of G(T 2 ) and g denotes the quantity determined by [LOT18, Formula 10.31].This endows CF A U V (X 0 ) with a grading by a transitive (G(T 2 ) × Z)-set.After taking a box tensor product with CF D(S 3 \K), where K is a knot, the gradings on CF D(S 3 \K) and CF A U V (X 0 ) induce a grading on the tensor product.
Lemma 3.2.Given a knot K ⊂ S 3 , denote the bordered 3-manifold representing its 0-framed complement as S 3 \K.Then we have a pairing formula Furthermore, the induced grading on the left hand side matches the bigrading (i.e.Maslov and Alexander) on the right hand side.
Proof.Choose a nice bordered Heegaard diagram H representing S 3 \K.Since the proof of pairing theorem [LOT18, Theorem 1.3] works trivially for admissible diagrams, we have The Heegaard diagram H ∪ X 0 represents K, together with a free basepoint z f ree lying outside K, we get the desired homotopy equivalence.The statement about gradings follows directly from the arguments used in the proof of [LOT18, Theorem 1.3].
Remark 3.3.In the proof of Lemma 3.2, the term CF U V (H ∪ X 0 ) is the Floer chain complex coming from cylindrical reformulation of Heegaard Floer homology, due to Lipshitz [Lip06].The original setting of cylindrical reformation is only for Heegaard diagrams with one basepoint, so it is natural to ask whether it also works for general diagrams (Σ, α, β, z), where the number of α-curves may exceed the genus of Σ (in which case we have more than one basepoints).Fortunately, the cylindrical reformation also works in those generalized settings; see [OS08a, Section 5.2] for details.
Proof.Write X 0 = (Σ, α, β, {z, z f ree }, w).Since truncating by V = 1 is equivalent to forgetting the basepoint w, we have a following homotopy equivalence of type-D modules: Since we no longer have w as a basepoint, the bordered Heegaard diagram (Σ, α, β, {z, z f ree }) is isotopic to the diagram we obtain by stabilizing a bordered Heegaard diagram representing the 0-framed solid torus near its basepoint.Since we are not counting holomorphic disks intersecting the stabilization region, it is clear, even without a neck-stretching argument, that we have a canonical isomorphism CF A(Σ, α, β, {z, z f ree }) CF A(0-framed solid torus) ⊗ F 2 2 , which proves the lemma.
Let X0 be the conjugate diagram of X 0 , defined in the same way as X.Then, by Lemma 3.1, we know that AZ ∪ φ( X0 ) is related by a sequence of Heegaard moves to X 0 .As in the proof of Lemma 3.2, it is clear that we have a pairing formula so any choice of a sequence of Heegaard moves from AZ ∪ φ( X0 ) to X 0 induces a type-A morphism Note that ι X is a homotopy equivalence of type-A modules over F 2 , but not over F 2 [U, V ]; this is because it intertwines the actions of U and V .Thus ι X is a type-A homotopy skew-equivalence.
The definition of ι X depends on the choices that we have made in its construction.Choosing a different sequence of Heegaard moves may result in another homotopy equivalence which is not homotopic to ι X , due to the lack of naturality for bordered Floer homology.However it will not affect the results of this paper; we only have to choose one sequence of Heegaard moves, once and for all.
Given a knot K ⊂ S 3 and a bordered Heegaard diagram H for the 0-framed complement of K, recall that we can choose a homotopy equivalence which is an element of Inv D (S 3 \K).Furthermore, we have the following conjugation maps: We consider the following composition of homotopy equivalences, which we will denote as F K .
Lemma 3.5.For any choice of ι S 3 \K ∈ Inv D (S 3 \K), the induced homotopy equivalence F K is homotopic to ι K,z f ree .
Proof.One can use the argument used in the proof of [HL19, Theorem 5.1] verbatim.
For later use, we prove the following lemma.Lemma 3.6.Given a knot K, suppose that there exists a local chain map which preserves the Alexander and Maslov gradings, such that f • ι K,z f ree ∼ f .Then there also exists a local (bidegree-preserving) chain map g : CF K R (S 3 , K) → R.
Proof.Consider the free-stabilization map , and the maps f and S + z f ree are local, we deduce that f R is also local.
Recall that the differential on CF K U V (S 3 , unknot, Since U V = 0 in R, we can define a projection map Furthermore, g is a local map due to grading reasons.Therefore g is the desired map.

Involutive knot Floer homology and involutive bordered Floer homology
Recall that, for any two bordered 3-manifold M, N with the same boundary, we have a pairing formula Note that the cycles in the morphism space correspond to type-D morphisms, and boundaries correspond to nullhomotopic morphisms.Consider the case when M is the 0-framed complement of a knot K and N is the 0-framed solid torus.Then we have S 3 0 (−K) −M ∪ N , so the pairing formula induces a homotopy equivalence CF (S 3 0 (−K)) Mor( CF D(S 3 \K), CF D(T 0 )), where T 0 denotes the 0-framed solid torus.Now, by Lemma 3.2, we get a chain map: On the other hand, by pairing with CF A(∞-framed solid torus) instead of CF A U V (X 0 ), we also get a chain map F : CF (S 3 0 (−K)) → Hom( CF (S 3 ), CF (S 3 )) = F 2 .Lemma 4.1.Let X 0 (−K) be the punctured 0-trace of the knot −K, i.e. the 4-manifold obtained by attaching a 0-framed 2-handle to S 3 × I along −K × {1}.Then the map x → F (x)(1) : CF (S 3 0 (−K)) → CF (S 3 ) is the hat-flavored cobordism map induced by the cobordism X 0 (−K), flipped upside-down.
Proof.Discussions in [LOT11, Section 1.5] tells us that the map F : CF (S 3 0 (−K)) ⊗ CF (S 3 ) → CF (S 3 ) is the cobordism map induced by the 4-manifold W 0 given by where denotes a triangle with edges e 1 , e 2 , e 3 , and T denotes a torus.Note that W 0 has three boundary components given by S 3 0 (−K) = −(S 3 \K) ∪ T 0 , S 3 = T 0 ∪ T ∞ , and S 3 = (S 3 \K) ∪ T ∞ .Hence the cobordism map induced by 4-manifold W obtained by gluing a 4-ball to the second boundary, i.e.
is the given map x → F (x)(1).Since W is diffeomorphic to X 0 (K), flipped upside-down, the lemma follows.
The following example explains Lemma 4.1 in the case when K is the unknot.
Example 4.2.Let K be the unknot.Then S 3 \K T 0 , S 3 0 (−K) S 1 × S 2 , and X 0 (−K) D 2 × S 2 .The type-D module of the 0-framed solid torus T 0 is freely generated over the torus algebra A(T 2 ), which is generated (over F 2 ) by the set {ι 0 , ι 1 , ρ 1 , ρ 2 , ρ 3 , ρ 12 , ρ 23 , ρ 123 }, by a single element x, and the differential is given by ∂x = ρ 12 x.The identity morphism induced by D 2 × S 2 which bounds S 1 × S 2 is a map of degree − 1 2 , which maps the 1 2 -graded generator (which corresponds to the identity morphism) to 1 and the − 1 2 -graded generator to 0. Lemma 4.3.Let K be a knot such that (CF K R (S 3 , K), ι K ) is locally equivalent to the trivial complex.Then there exists a cycle x ∈ HF (S 3 0 (−K)) of absolute Q-grading 1 2 , which is mapped to the unique homotopy autoequivalence [id] ∈ H * (Hom( CF (S 3 ), CF (S 3 ))) under the map F .Proof.By Lemma 4.1, we know that the map x → F (x)(1) : CF (S 3 0 (−K)) → CF (S 3 ) is the hat-flavored cobordism map induced by the cobordism W by flipping the 0-framed 2-handle attaching map along −K upside-down.Recall from the involutive mapping cone formula [HHSZ20, Section 22.9] that the Heegaard Floer homology of S 3 0 (−K) is homotopy equivalent to a complex of the form and the involution ι S 3 0 (−K) takes the form ι A + ι B + H, where ι A and ι B are the involutions on Â0 and B0 , respectively, induced by ι −K , and H is a certain homotopy between ι B • D 0 and ) is given by the projection onto Â0 , composed with the inclusion map of Â0 into B0 .
Let g : CF K R (S 3 , unknot) → CF K R (S 3 , K) be a local map such that ι K • g ∼ g • ι unknot .Following the proof of [HHSZ20, Proposition 3.15(3)] shows that choosing a homotopy between ι K •g and g •ι unknot induces a local map Denote by x 0 the unique generator of the 1 2 -graded piece of HF (S 1 × S 2 ).Since projection to A 0 clearly homotopy-commutes with F g , we see from Example 4.2 that F g (x 0 ) is a ι S 3 0 (−K) -invariant element of HF (S 3 0 (−K)) which is mapped to the generator of HF (S 3 ) under the cobordism map induced by W , proving the lemma.Now we can prove Theorem 1.1.
Since we have we have a pairing theorem Denote by F x : CF D(S 3 \K 1 ) → CF D(S 3 \K 2 ) the type-D morphism which corresponds to x.Then we have the following homotopy-commutative diagram for any choice of ι S 3 \K1 ∈ Inv D (S 3 \K 1 ) and ι S 3 \K2 ∈ Inv D (S 3 \K 2 ): Furthermore, since F (x) corresponds to the identity morphism of CF (S 3 ), we see that the induced map is homotopic the identity morphism.
Now suppose that we have a type-D morphism g : CF D(S 3 \K 1 ) → CF D(S 3 \K 2 ) which satisfies the conditions of Theorem 1.1 for some choices of ι S 3 \K1 ∈ Inv D (S 3 \K 1 ) and ι S 3 \K2 ∈ Inv D (S 3 \K 2 ).By taking a box tensor product with an involution ι T∞\P ∈ Inv(T ∞ \P ) of the type DA bimodule CF DA(T ∞ \P ) of the exterior of the connected-sum pattern P induced by −K 1 , we may replace K 1 with K 1 − K 1 and K 2 with K 2 − K 1 without any loss of generality (see the discussion below the proof for details).Then, after pairing with CF DA(X 0 ), we get the following homotopy-commutative diagram.
By Lemma 3.5, the compositions of vertical maps on the two columns of the above diagram are ι K1 −K1,z f ree and ι K2 −K1,z f ree , respectively, which implies that Hence, by Lemma 3.6, we have an ι K -local chain map f : R → CF K R (S 3 , K).Now, since our argument can also be applied to −K instead of K, we should also have an ι K -local chain map Then our choice of ι M and ι N induces a homotopy equivalence ι ι M ,ι N for CF D(M ∪ N ) as follows: Following the proof of [HL19, Theorem 5.1], we immediately see that Using this fact, we can now prove Theorem 1.2.
Proof of Theorem 1.2.Let K 1 and K 2 be two knots satisfying the given assumptions.Then, by Theorem 1.1, there exists a type D morphism which fits into the following homotopy-commutative diagram for any choice of Furthermore, the induced chain map is a homotopy equivalence.Now let N 1 = T ∞ \P be the 0-framed exterior of the given pattern P inside the ∞-framed solid torus.Then the union of N (glued along its 0-framed boundary) with T ∞ is again T ∞ .Hence, if we denote the type D morphism by g 0 , then the induced map is homotopic to identity.Furthermore, we have a following homotopy-commutative diagram.

CF DA(AZ) CF D(S
The compositions of vertical maps on both sides of the above diagram are ι ι N ,ι S 3 \K 1 and ι ι N ,ι S 3 \K 2 , which are contained in Inv D (S 3 \P (K 1 )) and Inv D (S 3 \P (K 2 )), respectively.Also, since our assumption is symmetric on the choices of K 1 and K 2 , we can repeat our argument with K 1 and K 2 swapped.Hence, by Theorem 1.1, we deduce that (CF K R (S 3 , P (K 1 ) − P (K 2 )), ι P (K1) −P (K2) ) is ι K -locally equivalent to the trivial complex.
5. An explicit formula for the hat-flavored truncation of ι K Recall that we had the bordered Heegaard diagram X; write X = (Σ, α, β, {z, z f ree }, w).We can add one more free basepoint w f ree to the component of Σ\ (∪ c∈α∪β c) containing z f ree to get a new diagram Y = (Σ, α, β, {z, z f ree }, {w, w f ree }).As we modified X by Heegaard moves to get a nice diagram X 0 , we can do the same process to Y to get a nice diagram Y.By counting holomorphic disks on Y 0 which does not algebraically intersect z f ree and w f ree , and recording their algebraic intersection numbers with z and w by formal variables U and V , respectively, we can get a well-defined type-A module CF A U V (Y 0 ).Note that, by construction, we have Recall that the proof of the pairing theorem relies on the observation that −M 1 ∪ AZ −M 1 .Denote by Y ∞ the 4-pointed nice bordered diagram obtained by gluing Y 0 with a cylinder whose boundaries have framing 0 and ∞.Since Y ∞ should also satisfy CF D(−Y ∞ ∪AZ) CF D(−Y ∞ ) and the type-D and type-A modules associated to Y 0 and X 0 are homotopy equivalent, we see that where ν denotes the longitudinal knot inside the ∞-framed solid torus T ∞ and L 2 denotes the 2-component link p, q × S 1 in S 2 × S 1 for two points p, q ∈ S 2 .
Here, L 2 is endowed with an orientation so that its total homology class [L 2 ] ∈ H 1 (S 2 × S 1 ; Z) vanishes.Hence L 2 is nullhomologous, which tells us that its link Floer homology (at the unique spin structure of S 2 × S 1 has well-defined Z-valued Maslov and (collapsed) Alexander gradings.These gradings should be compatible with the natural gradings of CF D(Y ∞ ) and CF D(T ∞ , ν); note that the grading on CF D(Y ∞ ) can be defined as in Equation (3.1).
Proof.Write L 2 = A ∪ B and choose z-basepoint z 1 , z 2 and w-basepoints w 1 , w 2 on L 2 so that z 1 , w 1 ∈ A and z 2 , w 2 ∈ B. We will compute the link Floer homology CF L U V (S 2 × S 1 , L 2 ) of the basepointed link (L 2 , {z 1 , z 2 }, {w 1 , w 2 }), where the differential records the algebraic intersections of holomorphic disks with the basepoints z 1 , w 1 , z 2 , w 2 by U, V, 0, 0, respectively.Note that truncating it by U = V = 0 and taking homology gives HF L(S 2 × S 1 , L 2 ).
Consider the Heegaard diagram in Figure 5.1.Since we are counting disks which does not intersect z 2 and w 2 algebraically, the given diagram is nice, so all relevant holomorphic disks are represented by either bigons or squares which do not contain z 2 and w 2 .Thus we see that CF L U V (S 2 × S 1 , L 2 ) is generated by the intersection points xc, xd, yc, yd, and the differential is given by Since U and V act on the bigrading by (−2, −1) and (0, 1), and the differential ∂ lowers the Maslov grading by 1 and leaves the collapsed Alexander grading invariant, we see that xd and xd + yc have bidegree (0, 0), xc has bidegree (1, 1), and yd has bidegree (−1, −1).Therefore, after truncating by U = V = 0, we get four generators xd, xc, yd, xd + yc of HF L(S 2 × S 1 , L 2 ), which lie on bidegrees (0, 0), (0, 0), (1, 1), (−1, −1), respectively, as desired.We define G ∞ as the composition of the above two maps.Then, for any knot K, the induced map is homotopic to the cobordism map F K induced by the trivial saddle cobordism from K ∪ unknot to K, as drawn in Figure 5.3.Furthermore, we can also define type-D endomorphisms are not natural, i.e. depends on the choices of auxiliary data.However, by the pairing theorem for triangles, we know that the map is homotopic to the basepoint action Φ K∪unknot,K corresponding to the basepoint z on the link K ∪ unknot, for any knot K.A similar statement also holds for Ψ D Y as well.
unknot Figure 5.3.A decoration on the trivial saddle cobordism from K ∪ unknot to K. Note that this cobordism can be seen as the composition of a quasi-stabilization followed by a saddle move.
Lemma 5.2.For any knot K, we have Proof.Bypass relation [Zem19a, Lemma 1.4], applied as shown in Figure 5.4, gives the equality where Ψ K∪unknot,unknot denotes the basepoint action associated to w f ree ∈ unknot.Since the basepoint actions for the unknot are trivial, the lemma follows.The same argument also proves the commutation result for Φ actions.
Proof.By Lemma 5.1, we only have to show that the homotopy classes given type-D morphisms are linearly independent, so assume that they are linearly dependent.Then for any knot K, the endomorphisms F K , F K • Φ K∪unknot,K , F K • Ψ K∪unknot,K , and F K • (1 + Φ K∪unknot,K Ψ K∪unknot,K ) should be linearly dependent up to homotopy.By Lemma 5.2 and the fact that F K has a homotopy right inverse (which follows from the fact that the trivial saddle cobordism from K ∪ unknot to K has a right inverse) would imply that the endomorphisms id, Φ K , Ψ K , 1 + Φ K Ψ K of CF K(S 3 , K) should also be linearly dependent up to homotopy.Now consider the case when K is the figure-eight knot.Then CF K(S 3 , K) is generated by five elements, say a, b, c, d, x.The basepoint actions are given by Φ K (a) = b, Φ K (c) = d, Ψ K (a) = c, Ψ K (b) = d, and all other generators are mapped to zero.Thus we see that the endomorphisms id, Φ K , Ψ K , 1 + Φ K Ψ K are linearly independent up to homotopy, a contradiction.
Recall that mimicking the construction of ι X gives a bordered involution which is a homotopy equivalence which satisfies the property that the induced map is homotopic to the involution ι K∪unknot of the link Floer homology of K ∪ unknot.
On the other hand, the type-D module CF D(T ∞ , ν) is generated by a single element, say x, and the differential is trivial.This implies that CF DA(AZ) CF D(T ∞ , ν) is not homotopy equivalent to CF D(T ∞ , ν).In fact, CF DA(AZ) CF D(T ∞ , ν) is homotopy equivalent to a type-D module generated by five elements, say a, b, c, d, e, where the differential is given by (5.1) Since a is a cycle, the map defined by f (x) = a commutes with the differential on both sides, and thus is a well-defined type-D morphism.
Lemma 5.4.The type- CF L(S 3 , K), which we will denote as g K .Then, by construction, we have where f is the map defined as where Ω is the homotopy equivalence CF DA(AZ) CF DA(AZ) CF DA(I), which is unique up to homotopy due to homotopy rigidity [HL19, Lemma 4.4].It is easy to check, using a bypass relation, that We now consider the case when K is the unknot.Then the 0-framed knot complement S 3 \K is the 0-framed solid torus T 0 .Recall that CF DA(AZ) CF D(T ∞ , ν) is homotopic to the type-D module M D generated by a, b, c, d, e, where the differential is given as in Equation (5.1), and the image of the generator x of CF D(T ∞ , ν) is a.This means that there exists a type-D homotopy equivalence On the other hand, the type-A module CF A(S 3 \K), which is homotopy equivalent to CF DA(AZ) CF DA(AZ) via ι S 3 \K , is generated by one element, say y, and the A ∞ operations are given by Hence the chain map maps the generator 1 of CF K(S 3 , K) F 2 to y a.Furthermore, the chain complex CF A(S 3 \K) M D is generated by three elements, namely y a, y c, and y d and the differential is given by Hence there exists a homotopy equivalence such that (h C •m)(1) = 1.However, since ι S 3 \K is a homotopy equivalence and any homotopy autoequivalence of CF K(S 3 , K) F 2 is homotopic to the identity, we should have Therefore f is homotopic to the identity map.Since it is obvious that ι unknot is also homotopic to the identity map, we get g unknot ∼ F unknot , which implies that g itself should not be nullhomotopic.Since box-tensoring with CF DA(AZ) is an equivalence of categories and g clearly has bidegree (0, 0), we can apply Lemma 5.3 to see that g should be chain homotopic to one of the following three morphisms: We have already seen that g unknot is not nullhomotopic, which is a contradiction since Φ unknot and Ψ unknot are both nullhomotopic.Therefore g Now we are ready to prove Theorem 1.3.
Proof of Theorem 1.3.Denote the homotopy autoequivalence of CF K(S 3 , K) defined in the theorem as ιK .By Lemma 5.4, we know that f Since the trivial saddle cobordism from K ∪ unknot to K clearly has a right inverse, its associated cobordism map F K admits a homotopy right inverse.Hence, by precomposing with the homotopy right inverse of F K , we see that ιK should be homotopic to either ι K or ι −1 K , as desired.
Remark 5.5.The proof of the pairing theorem (Equation (2.2)) also works in the following way: The reason is that, although CF D(T ∞ , ν) is not homotopy equivalent to CF DA(AZ) CF D(T ∞ , ν), CF K(S 3 , K) is homotopy equivalent to CF DA(AZ) CF K(S 3 , K).Hence, given an involution ι M ∈ Inv D (S 3 \K), one can consider the following map Here, f is the type-D morphism given in Theorem 1.3.Following the proof of Theorem 1.3, it is straightforward to see that the above map is homotopic to either ι K or ι −1 K .This gives a more applicable interpretation of Theorem 1.3, since type-D modules are easier to work with than type-A modules.
b a It is known [HM17, Section 8] that the action of ι K is given by the reflection along the diagonal, i.e. fixes a and exchanges b and c.
On the bordered side, we know from [LOT18, Theorem 11.26] that the Floer chain complex of K determines CF D(S 3 \K).Thus we see that CF D(S 3 \K) is generated by 7 elements e 0 , f 0 , f 1 , g 0 , g 1 , h 1 , k 1 , where the differential is given as follows.Since one of such homotopy equivalences can be computed explicitly using the proof of [HRW18, Theorem 37], we deduce that it also gives an explicit description of ι S 3 \K .Applying Theorem 1.3 then recovers the hat-flavored action ι K (a) = a, ι K (b) = c, ι K (c) = b in CF K(S 3 , K), which is consistent with the action of ι K on CF K U V (S 3 , K).
Remark 5.7.In general, one can prove that CF D(S 3 \K) is homotopy-rigid whenever K is an L-space knot, which means that one can explicitly compute ι S 3 \K for such knots by computing the box tensor product CF DA(AZ) CF D(S 3 \K) and finding a sequence of homotopy equivalences which connects it to CF D(S 3 \K).One can check using Theorem 1.3 that the hat-flavored action of ι K is given by "reflection with respect to the diagonal".This is consistent with the action of ι K on CF K U V (S 3 , K), which was first determined in [HM17, Section 7].
Theorem 1.3 can also be used in the reverse way to compute ι S 3 \K from ι K , as shown in Example 5.8.
We claim that the type-D morphisms id, K 1 , K 2 , K 2 • K 1 , and K 3 are linearly independent up to homotopy and thus form a basis of V 1 2 .To prove the claim, we take a tensor product with CF A(T ∞ , ν), and consider the maps id K 1 and id K 2 , which are now considered as chain endomorphisms of CF K(S 3 , K).One can easily see that (id K 1 )(a) = x, (id K 1 )(everything else) = 0, (id K 2 )(x) = d, (id K 2 )(everything else) = 0, (id K 3 )(x) = x, (id K 3 )(everything else) = 0.
Hence we see that id g for g = id, K 1 , K 2 , K 2 • K 1 , K 3 induce linearly independent endomorphisms of HF K(S 3 , K), and so the claim is proven.
Given a type-D morphism m : CF DA(AZ) CF D(S 3 \K) → CF D(S 3 \K), we define an endomorphism E m of CF K(S 3 , K) as follows.Since F • ι −1 S 3 \K is an element of V 1 2 , which is generated by id, K 1 , K 2 , K 2 • K 1 , and K 3 , we deduce that ι S 3 \K ∼ (id + K 1 + K 2 ) • F.

Figure 3
Figure 3.1.Left, a free-stabilization of a Heegaard diagram near a basepoint z.Right, a free-stabilization of a Heegaard triple-diagram near the same basepoint z.

Figure 3
Figure 3.3.Top-left, the diagram X. Top-middle, the diagram AZ ∪ X. Top-right, the diagram AZ ∪ φ( X).Bottom-left, A diagram obtained from the one on the top-right by a sequence of handleslides, followed by a destabilization.Bottom-middle, A diagram obtained from the one on the bottom-left by another sequence of handleslides.Bottom-right, the diagram obtained by isotopy from the one on the bottom-middle.Note that this is the same as the original diagram X.
Now suppose that we have two bordered 3-manifolds M and N , where M has one torus boundary ∂M and N has two torus boundaries, ∂ 1 N and ∂ 2 N .Choose any ι M ∈ Inv D (M ) and ι N ∈ Inv(N ), so that we have type-D and type-DA homotopy equivalences ι M : CF DA(AZ) CF D(M ) → CF D(M ), ι N : CF DA(AZ) CF DA(N ) CF DA(AZ) → CF DA(N ), where the boundary components ∂ 1 N and ∂ 2 N are considered as type-A and type-D boundaries, respectively.Recall that we have a pairing theorem for computing CF D(M ∪ N ), where we identify ∂M with ∂ 1 N : CF D(M ∪ N ) CF DA(N ) CF D(M ).

Figure 5
Figure 5.1.4-pointed Heegaard diagram representing the 2-component link L 2 .We define a type-D morphism G ∞ : CF D(Y ∞ ) → CF D(T ∞ , ν) as follows.We start with a Heegaard diagram Y 0 .If we denote by H 0 = (Σ, α, β, z f ree , w) the doubly-pointed Heegaard diagram for the pair (T ∞ , ν) and the diagram we get by quasi-stabilizing it as H qst 0 , then we have a 2-handle map CF D(Y 0 ) → CF D(H qst 0 ).Furthermore, the proof of [Zem17, Proposition 5.3] tells us that we can define the "quasi-destabilization map" CF D(H qst 0 ) → CF D(T ∞ , ν).

Figure 5 . 2 .
Figure 5.2.Upper left, the diagram H 0 .Upper right, the diagram H qst 0 .Lower left, a result of performing a handleslide to H qst 0 .Lower right, the diagram Y.
destabilization maps and (similarly defined) quasi-stabilization maps, as follows.Given a bordered diagram H Y = (Σ, α, β, {z, z f ree }, {w, w f ree }) representing Y ∞ , we (α-)quasi-stabilize it near the point z to get a new diagram H qst Y , which introduces a new pair (z , w ) of basepoints, and then we quasi-destabilize it to eliminate the basepoints z, w and rename z , w as z, w, respectively, to obtain H Y again.We define the resulting map as Ψ Y , i.e.Ψ Y : CF D(Y ∞ ) = CF D(H Y ) quasi-stabilization −−−−−−−−−−−→ CF D(H qst Y ) quasi-destabilization − −−−−−−−−−−−− → CF D(H Y ) = CF A(Y 0 ).We omit the construction of Φ D Y , since it is similar to the construction of Ψ D Y .The definition of Φ D Y and Ψ D Y

Figure 5
Figure 5.4.A Bypass relation applied to the saddle cobordism from K ∪ unknot to K, with a decoration as shown in Figure 5.3.
Then for any knot K, we have an induced map CF L(S 3 , K ∪ unknot) CF A(S 3 \K) CF DA(AZ) CF DA(AZ) CF D(Y ∞ ) id id g − −−−−− → CF A(S 3 \K) CF DA(AZ) CF DA(AZ) CF D(T ∞ ) It can be seen via straightforward computation that there are only two homotopy classes of degree-preserving type-D endomorphisms of CF D(S 3 \K), represented by 0 and id.Hence CF D(S 3 \K) is homotopy-rigid, i.e. it admits a unique homotopy class of homotopy autoequivalences.This means that there exists only one homotopy class of homotopy equivalences CF DA(AZ) CF D(S 3 \K) → CF D(S 3 \K).
Unlike the trefoil case (covered in Example 5.6), the type-D module CF D(S 3 \K) is not homotopy-rigid, so we cannot find a random homotopy equivalence between CF DA(AZ) CF D(S 3 \K) and CF D(S 3 \K) and claim that it is homotopic to ι S 3 \K .Denote by M and N the type-D submodule of CF D(S 3 \K) generated by z and everything else (i.e. e 0 , • • • , h 1 ), respectively, so that we have a splittingCF D(S 3 \K) M ⊕ N.Using the proof of [HRW18, Theorem 37], one can explicitly construct homotopy equivalencesF M : CF DA(AZ) M → M, F N : CF DA(AZ) N → N. Consider F = F M ⊕ F N : CF DA(AZ) CF D(S 3 \K) → CF D(S 3 \K).Then F • ι −1S 3 \K is a homotopy autoequivalence of CF D(S 3 \K).Recall that we have a pairing theorem Mor( CF D(S 3 \K), CF D(S 3 \K)) CF (−(S 3 \K) ∪ (S 3 \K)) CF (S 3 0 (K − K)).
almost local maps exist in both directions, we say that the given two almost ι-complexes are almost locally equivalent.Again, the set of almost local equivalences of almost ι-complexes form a group Î, which is called the almost local equivalence group.The construction of involutive Heegaard Floer homology gives a canonical map