# A note on the involutive invariants of certain pretzel knots

Matthew Issac<EMAIL_ADDRESS>and  Nicholas McConnell
<EMAIL_ADDRESS>
###### Abstract.

We compute the involutive knot invariants for pretzel knots of the form
$P(-2,m,n)$ for $m\geq n\geq 3$.

###### Contents

  1. 1 Introduction
  2. 2 Background on Heegaard Floer homology and involutive Heegaard Floer homology
    1. 2.1 Heegaard Floer homology for knots
    2. 2.2 Involutive Heegaard Floer homology for knots
  3. 3 Involutive invariants of $(-2,m,n)$ pretzel knots
    1. 3.1 The complex $\mathit{CFK}^{\infty}(P(-2,m,n))$
    2. 3.2 Computations for model complexes

##  1\. Introduction

Heegaard Floer homology is a collection of invariants of three-manifolds and
knots and links introduced by Ozsváth and Szabó [OS04c, OS04b, OS04a] and in
the knot case independently by Rasmussen [Ras03] in the early 2000s. The knot
version associates to a knot $K\subseteq S^{3}$ a
$(\mathbb{Z}\oplus\mathbb{Z})$-filtered, $\mathbb{Z}$-graded chain complex
over $\mathbb{F}[U,U^{-1}]$ called $\mathit{CFK}^{\infty}(K)$. This chain
complex recovers the data of the classical Alexander polynomial [OS04a] and
detects the knot genus [OS06] and whether the knot is fibred [Ghi08, Ni07].
Furthermore, a plethora of interesting invariants of knot concordance have
been extracted from it [OS06, OS08, Ras04, Hom14, OSS17].

In 2015, Hendricks and Manolescu [HM17] introduced a refinement of Heegaard
Floer homology called involutive Heegaard Floer homology, which incorporates
the data of a conjugation symmetry on the Heegaard Floer chain complexes. In
the knot case, this takes the form of a skew-filtered automorphism

$\iota_{K}\colon\mathit{CFK}^{\infty}(K)\rightarrow\mathit{CFK}^{\infty}(K)$

which is order four up to filtered chain homotopy equivalence. From this
additional data, they construct two new concordance invariants
$\underline{V\mkern-3.0mu}\mkern 3.0mu_{0}(K)$ and $\overline{V}_{0}(K)$.
These invariants are particularly interesting in that, unlike many concordance
invariants from Heegaard Floer homology and the related knot homology theory
Khovanov homology, they can take nonzero values on knots of finite concordance
order. For example, $\underline{V\mkern-3.0mu}\mkern 3.0mu_{0}(4_{1})=1$; the
invariants therefore detect the nonsliceness of the figure eight knot. The
authors give combinatorial computations of the involutive concordance
invariants for $L$-space knots (which include the torus knots) and thin knots
(which include alternating and quasi-alternating knots).

Following a similar strategy to Hendricks and Manolescu’s computation for thin
knots [HM17, Section 8], in this note we compute the involutive concordance
invariants of $P(-2,m,n)$ pretzel knots for $m$ and $n$ odd, and their
mirrors. The complexes $\mathit{CFK}^{\infty}$ associated to these knots were
computed by Goda, Matsuda, and Morifuji [GMM05]. The reader should compare the
statement of our result below with [HM17, Proposition 8.2]. We include the
values of the ordinary concordance invariant $V_{0}(K)$ in the statement for
ease of comparison.

######  Theorem 1.1.

Let $m,n$ be odd numbers such that $m\geq n\geq 3$. The involutive knot
concordance invariants of the pretzel knots $K=P(-2,m,n)$ are as follows.

  * •

If $m\not\equiv n\pmod{4}$, then $V_{0}(K)=\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(K)=0$ and $\overline{V}_{0}(K)=-\frac{m+n}{4}$.

  * •

If $m\equiv n\pmod{4}$, then $V_{0}(K)=\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(K)=0$ and $\overline{V}_{0}(K)=-\frac{m+n-2}{4}$.

Moreover, the involutive knot concordance invariants of the mirrors
$\overline{K}=P(2,-m,-n)$ are as follows.

  * •

If $m\not\equiv n\pmod{4}$, then
$V_{0}(\overline{K})=\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(\overline{K})=\overline{V}_{0}(\overline{K})=\frac{m+n}{4}$.

  * •

If $m\equiv n\equiv 3\pmod{4}$, then
$V_{0}(\overline{K})=\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(\overline{K})=\overline{V}_{0}(\overline{K})=\frac{m+n-2}{4}$.

  * •

If $m\equiv n\equiv 1\pmod{4}$, then $\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(\overline{K})=\frac{m+n+2}{4}$ and
$V_{0}(\overline{K})=\overline{V}_{0}(\overline{K})=\frac{m+n-2}{4}$.

This computation comes from analyzing four essentially distinct cases for the
structure of the chain complex $\mathit{CFK}^{\infty}(K)$, corresponding to
the values of $m$ and $n$ modulo four, as we explain further in Section 3.2.

######  Remark 1.2.

In the case that $n=3$, the involutive concordance invariants were already
known. The knots $P(-2,m,3)$ are mirrors of $L$-space knots [OS05]. Hendricks
and Manolescu computed the involutive concordance invariants of L-space knots
and their mirrors [HM17, Section 7]; we review the results of their
computation in Section 2. We however include the case $n=3$ above for
completeness.

######  Remark 1.3.

The computation of knot Floer homology for the pretzel knots discussed in this
note is particularly simple because they are $(1,1)$ knots [GMM05]. In
general, $(1,1)$ knots admit Heegaard diagrams depending only on a set of four
integer parameters, whose knot Floer homology may be computed combinatorially
[Doy05, Rac15]. A possible future research direction is to attempt to give a
general, hopefully combinatorial, formula for the filtered chain homotopy
equivalence class of $\iota_{K}$ for knots admitting such diagrams.

### Organization

This paper is organized as follows. In Section 2 we review some necessary
background; in particular, in Section 2.1 we review some properties of
Heegaard Floer homology for knots, and in Section 2.2 we recall involutive
Heegaard Floer homology for knots and the construction of the involutive
concordance invariants. In Section 3.1 we recall the Heegaard Floer complexes
associated to the pretzel knots $P(-2,m,n)$ and carry out a convenient change
of basis. In Section 3.2 we present the computation of the involutive
invariants associated to these knots and conclude the proof of Theorem 1.1.

### Acknowledgments

This project was carried out during the Rutgers DIMACS REU in Summer 2020; we
thank the organizers of the REU for their support and Kristen Hendricks for
supervising this project. We are also grateful to Jennifer Hom for comments on
a draft of this paper. Both authors were supported by NSF CAREER Grant
DMS-2019396.

##  2\. Background on Heegaard Floer homology and involutive Heegaard Floer
homology

In this section we recall some background on Heegaard Floer homology for knots
and involutive Heegaard Floer homology for knots.

###  2.1. Heegaard Floer homology for knots

We briefly review the algebraic aspects of knot Floer homology; for a more
detailed look at the construction, see [OS04a], or [Hom17] for an expository
view.

Let $\mathbb{F}$ be the field of two elements and $U$ a variable of degree
$-2$. To a knot $K$, knot Floer homology associates a free finitely-generated
$(\mathbb{Z}\oplus\mathbb{Z})$-filtered chain complex
$\mathit{CFK}^{\infty}(K)$ over $\mathbb{F}[U,U^{-1}]$ [OS04a, Ras03] as
follows. As an $\mathbb{F}[U,U^{-}]$ complex, $\mathit{CFK}^{\infty}(K)$ is
generated by finitely many elements $[x;i,j]$, where
$(i,j)\in\mathbb{Z}\oplus\mathbb{Z}$. Each such element has a _homological_ or
_Maslov_ grading denoted $\operatorname{gr}([x;i,j])=M([x;i,j])$ and an
_Alexander_ grading $A([x;i,j])=j$. The action of the $U$-variable is given by

$U[x;i,j]=[x;i-1,j-1]$

and the effect on the gradings of $U$-multiplication is

$M(U[x;i,j])=M([x;i-1,j-1])=M([x;i,j])-2$ $A(U[x;i,j])=A([x;i-1,j-1])=j-1.$

The $(i,j)$ level of the $\mathbb{Z}\oplus\mathbb{Z}$ filtration is
$\mathcal{F}_{(i,j)}=\\{[x;i^{\prime},j^{\prime}]\in\mathit{CFK}^{\infty}(K):(i^{\prime},j^{\prime})\leq(i,j)\\}$,
where $\mathbb{Z}\oplus\mathbb{Z}$ is given the dictionary order. The
differential $\partial$ on $\mathit{CFK}^{\infty}(K)$ respects the filtration
and is $U$-equivariant; moreover if

$\partial([x;i,j])=\bigoplus[y;i^{\prime},j^{\prime}]$

then for each $[y;i^{\prime},j^{\prime}]$ we have

$M([y;i^{\prime},j^{\prime}])=M([x;i,j])-1.$

######  Example 2.1.

The knot Floer complex associated to the right-handed trefoil is shown in
Figure 1. As an $\mathbb{F}[U,U^{-1}]$-module, it has three generators
$a=[a;0,0]$ in homological grading $-1$, $b=[b;-1,0]$ in homological grading
$-2$, and $c=[c;0,-1]$ in homological grading $-2$, with differential given by

$\displaystyle\partial a=b+c\qquad\qquad\partial b=\partial c=0.$

$U^{-1}b$$U^{-1}a$$U^{-1}c$$b$$a$$c$$Ub$$Ua$$Uc$$\iddots$$\iddots$ Figure 1.
The knot Floer complex associated to the right-handed trefoil.

######  Example 2.2.

The knot Floer complex associated to the figure-eight knot is shown in Figure
2. As an $\mathbb{F}[U,U^{-1}]$-module, it has five generators $a=[a;0,0]$ in
homological grading $0$, $b=[b;-1,0]$ in homological grading $-1$,
$c=[c;0,-1]$ in homological grading $-1$, $e=[e;0,0]$ in homological grading
$0$, and $x=[x;0,0]$ with nonzero differentials given by

$\displaystyle\partial a=b+c\qquad\qquad\partial b=\partial
c=Ue\qquad\qquad\partial e=\partial x=0.$

$U^{-1}x$$U^{-1}e$$x$$e$$U^{-1}b$$U^{-1}c$$a$$Ux$$b$$c$$\iddots$$\iddots$
Figure 2. The knot Floer complex associated to the figure-eight knot.

If $K$ is a knot and $\overline{K}$ is its mirror, then
$\mathit{CFK}^{\infty}(\overline{K})$ is the dual complex
$\mathit{CFK}^{\infty}(K)^{*}$ over $\mathbb{F}[U,U^{-1}]$.

######  Example 2.3.

The knot Floer complex associated to the left-handed trefoil is the dual of
the complex associated to the complex associated to the right-handed trefoil,
and shown in Figure 3. As an $\mathbb{F}[U,U^{-1}]$-module, it has three
generators $a=[a;0,0]$ in homological grading $1$, $b=[b;1,0]$ in homological
grading $2$, and $c=[c;0,1]$ in homological grading $2$, with differential
specified by

$\displaystyle\partial b=\partial c=a\qquad\qquad\partial a=0.$

$U^{-1}b$$U^{-1}a$$U^{-1}c$$b$$a$$c$$Ub$$Ua$$Uc$$\iddots$$\iddots$ Figure 3.
The knot Floer complex associated to the left-handed trefoil.

We often consider subquotient complexes of $\mathit{CFK}^{\infty}(K)$. Given a
subset $X\in\mathbb{Z}\oplus\mathbb{Z}$, let $CX$ denote the complex generated
by elements with planar grading $(i,j)\in X$. The complex $CX$ inherits the
structure of a chain complex if it is a subquotient (that is, a quotient of a
subcomplex) of $\mathit{CFK}^{\infty}(K)$. Some complexes $CX$ of particular
importance to us include

  * •

The complex $A_{0}^{-}=C\\{i,j\leq 0\\}$, consisting of the portion of
$\mathit{CFK}^{\infty}(K)$ lying in the third quadrant of the plane.

  * •

$B^{-}_{0}=C\\{i\leq 0\\}$, consisting of the portion of
$\mathit{CFK}^{\infty}(K)$ lying in the second and third quadrants of the
plane.

  * •

$C\\{i=0\\}=C\\{i\leq 0\\}/C\\{i<0\\}$, consisting of the portion of
$\mathit{CFK}^{\infty}(K)$ lying on the $i$-axis.

Let us begin by considering the first two complexes. The homology
$H_{*}(A_{0}^{-})$ is always of the form

$\mathbb{F}[U]\oplus\left(\bigoplus_{i=1}^{k}\mathbb{F}[U]/U^{n_{i}}\right)$

and the homology $H_{*}(B_{0}^{-})$ is a copy of $\mathbb{F}[U]$ with the
property that $\operatorname{gr}(1)=0$. (In particular, $H_{*}(B_{0}^{-})$ is
the three-manifold Heegaard Floer invariant $\mathit{HF}^{-}(S^{3})$.) There
is a chain map $v_{0}\colon A_{0}^{-}\rightarrow B_{0}^{-}$ given by
inclusion. The induced map on homology $H_{*}(A_{0}^{-})\to H_{*}(B^{-}_{0})$
maps the tower $\mathbb{F}[U]\in H_{*}(A_{0}^{-})$ to
$H_{*}(B_{0}^{-})=\mathbb{F}[U]$ via multiplication by $U^{V_{0}(K)}$ for some
nonnegative integer $V_{0}(K)$. Equivalently, we have

$V_{0}(K)=-\frac{1}{2}\max\\{r:\exists\ x\in H_{r}(A_{0}^{-}),\forall\ n,\
U^{n}x\neq 0\\}.$

Peters [Pet10] and Rasmussen [Ras04] showed that $V_{0}(K)$ is an invariant of
knot concordance.

Now consider the complex $C\\{i=0\\}$, which is naturally
$\mathbb{Z}$-filtered by the Alexander filtration, and is equal to the knot
Floer complex $\widehat{\mathit{CFK}}.$ The homology of the associated graded
object of $\widehat{\mathit{CFK}}$ is

(2.1)
$\begin{split}\widehat{\mathit{HFK}}(K)&=\bigoplus_{w\in\mathbb{Z}}\widehat{\mathit{HFK}}(K,w)\\\
&=\bigoplus_{w\in\mathbb{Z}}H_{*}(C\\{i=0,j=w\\})\end{split}$

and is the _knot Floer homology_. If we include the homological grading $k$,
we get a further decomposition

$\widehat{\mathit{HFK}}(K)=\bigoplus_{w\in\mathbb{Z},k\in\mathbb{Z}}\widehat{\mathit{HFK}}_{k}(K,w).$

This theory is symmetric in the sense that
$\widehat{\mathit{HFK}}_{k}(K,w)\simeq\widehat{\mathit{HFK}}_{k-2w}(K,w)$ and
furthermore it detects the knot genus via

$g(K)=\mathrm{max}\\{w:\widehat{\mathit{HFK}}(K,w)\neq 0\\}.$

Finally, the graded Euler characteristic of the knot Floer homology is the
Alexander polynomial of the knot, that is,

$\Delta_{K}(t)=\sum_{w}\chi(\widehat{\mathit{HFK}}(K,w))t^{w}.$

We also consider the vertical and horizontal homologies associated to
$\mathit{CFK}^{\infty}(K)$, as follows. Let $\partial=\sum_{i,j}\partial_{ij}$
where $\partial_{ij}$ is the term in the differential which decreases the two
planar gradings by $i$ and $j$ respectively. Then the vertical differential is
$\partial_{\mathrm{vert}}=\sum_{j}\partial_{j,0}$. The vertical homology is
the $\mathbb{F}[U,U^{-1}]$ module

$H_{*}(\mathit{CFK}^{\infty}(K),\partial_{\mathrm{vert}})=\bigoplus_{i^{\prime}}H_{*}(C\\{i=i^{\prime}\\}).$

Likewise the horizontal differential is
$\partial_{\mathrm{horz}}=\sum_{i}\partial_{i,0}$ and the horizontal homology
is

$H^{*}(\mathit{CFK}^{\infty}(K),\partial_{\mathrm{horz}})=\bigoplus_{j^{\prime}}H_{*}(C\\{j=j^{\prime}\\}).$

For any knot $K$, the ranks of the vertical and horizontal homologies are one.

The action of a Dehn twist around the knot $K$ in $S^{3}$ induces a filtered
chain homotopy equivalence $\sigma$ on $\mathit{CFK}^{\infty}(K)$ with the
property that $\sigma^{2}\sim\operatorname{Id}$, where $\sim$ denotes filtered
chain homotopy equivalence [Sar15]. Zemke [Zem17] shows that $\sigma$ admits a
simple combinatorial computation, as follows. Let

$\Phi=\sum_{\begin{subarray}{c}i,j\geq 0\\\ i\text{
odd}\end{subarray}}\partial_{ij}\qquad\qquad\Psi=\sum_{\begin{subarray}{c}i,j\geq
0\\\ j\text{ odd}\end{subarray}}\partial_{ij}.$

We then have

$\sigma=\operatorname{Id}+U^{-1}\Phi\circ\Psi.$

We briefly mention two special types of knots. A knot $K$ is said to be _$L$
-space_ if it admits a positive surgery to a three-manifold with Heegaard
Floer homology of minimal rank (called a _Heegaard Floer $L$-space_). Ozsváth
and Szabó showed that if $K$ is an $L$-space knot, the complex
$\mathit{CFK}^{\infty}(K)$ has a particularly simple form, which we first
introduce in slightly greater generality.

A _positive staircase complex_ is a $(\mathbb{Z}\oplus\mathbb{Z})$-graded
$\mathbb{F}$-chain complex generated by elements $z_{0},z_{r}^{1},z_{r}^{2}$
where $r$ ranges from $1$ to some integer $v\geq 1$. The element $z_{0}$ has
planar grading $(0,0)$; moreover, the planar gradings of $z^{1}_{v-2w}$ and
$z^{1}_{v-(2w+1)}$ differ only in the $i$ grading and the planar gradings of
$z^{1}_{v-(2w+1)}$ and $z^{1}_{v-(2w+2)}$ differ only in the $j$ grading. The
planar gradings have the symmetry property that if $z_{r}^{1}$ lies at planar
grading $(i,j)$ then $z_{r}^{2}$ lies at planar grading $(j,i)$. If $v$ is
even, the nonzero differentials in this complex are

$\displaystyle\partial(z_{r}^{s})=z_{r-1}^{s}+z_{r+1}^{s}\text{ for
}1<r<v\text{ odd }\qquad\qquad\qquad\partial(z_{1}^{s})=z_{0}+z_{2}^{s}$

whereas if $v$ is odd they are

$\displaystyle\partial(z_{r}^{s})=z_{r-1}^{s}+z_{r+1}^{s}\text{ for }r>0\text{
even}\qquad\qquad\qquad\partial(z_{0})=z_{1}^{1}+z_{1}^{2}.$

Examples of the two possible forms of a positive staircase are shown in Figure
4. The dual of a positive staircase is a _negative staircase complex_. Again
the generators are elements $z_{0},z_{r}^{1},z_{r}^{2}$ where $r$ ranges from
$1$ to some integer $v\geq 1$. The element $z_{0}$ has planar grading $(0,0)$;
moreover, the planar gradings of $z^{1}_{v-2w}$ and $z^{1}_{v-(2w+1)}$ differ
only in the $j$ grading and the planar gradings of $z^{1}_{v-(2w+1)}$ and
$z^{1}_{v-(2w+2)}$ differ only in the $i$ grading. The filtration levels have
the symmetry property that if $z_{r}^{1}$ lies at planar grading $(i,j)$ then
$z_{r}^{2}$ is at $(j,i)$. If $v$ is even, the nonzero differentials of this
complex are

$\displaystyle\partial(z_{0})=z_{1}^{1}+z_{1}^{2}$
$\displaystyle\partial(z_{r}^{s})=z_{r-1}^{s}+z_{r+1}^{s}\text{ for
}0<r<v\text{ even }$ $\displaystyle\partial(z_{v}^{s})=z_{v-1}^{s}$

whereas if $v$ is odd they are

$\displaystyle\partial(z_{1}^{s})=z_{0}+z_{2}^{s}$
$\displaystyle\partial(z_{r}^{s})=z_{r-1}^{s}+z_{r+1}^{s}\text{ for }1\leq
r<v\text{ odd}$ $\displaystyle\partial(z_{v}^{s})=z_{v-1}^{s}.$

Examples of the two possible forms of negative staircase complexes are shown
in Figure 5.

We now turn to the specific case of an $L$-space knot, which by work of
Ozsváth and Szabó [OS05] has Alexander polynomial

$\Delta_{K}(t)=(-1)^{v}+\sum_{i=1}^{v}(t^{w_{i}}-t^{-w_{i}})$

for some sequence of positive integers $0<w_{1}<w_{2}<\dots<w_{v}=g(K)$. Let
$\ell_{i}=w_{i}-w_{i-1}$ and

$n(K)=w_{v}-w_{v-1}+\cdots+(-1)^{v-2}w_{2}+(-1)^{v-1}w_{1}.$

Then $\mathit{CFK}^{\infty}(K)$ is filtered chain homotopy equivalent to
$C\otimes\mathbb{F}[U,U^{-1}]$ where $C$ is a positive staircase complex with
generators $z_{0},z_{1}^{s},\dots,z_{v}^{s}$ such that the planar grading of
$z_{v}^{1}$ is $(n(K),g(K)-n(K))$, and in general the gradings of
$z^{1}_{v-2w}$ and $z^{1}_{v-(2w+1)}$ differ in the $i$ grading by
$\ell_{v-2w}$ and the planar gradings of $z^{1}_{v-(2w+1)}$ and
$z^{1}_{v-(2w-2)}$ differ in the $j$ grading by $\ell_{v-(2w+1)}$. The complex
associated to the mirror of an $L$-space knot is generated by the elements of
the negative staircase produced by dualizing.

The integer $n(K)$ associated to the staircase complex for a positive
$L$-space knot is in general the sum of the lengths of the horizontal arrows
in the top half of the complex. For an arbitrary knot, if
$\mathit{CFK}^{\infty}(K)$ contains a direct summand generated by a positive
staircase complex, we let this fact be a definition of the quantity $n(K)$. If
$\mathit{CFK}^{\infty}(K)$ contains a direct summand generated by a negative
staircase complex, we let the sum of the lengths of the horizontal arrows in
the top half of the dual of the complex be $n(K)$.

$z_{v-2}^{2}$$z_{v-1}^{2}$$z_{v}^{2}$$\ddots$$z_{0}$$z_{1}^{1}$$z_{1}^{2}$$\ddots$$z_{v-2}^{1}$$z_{v-1}^{1}$$z_{v}^{1}$$z_{v-1}^{1}$$z_{v}^{1}$$\ddots$$z_{0}$$z_{1}^{2}$$z_{1}^{1}$$\ddots$$z_{v-1}^{2}$$z_{v}^{2}$

Figure 4. Positive staircase complexes for the case that $v$ is even (on the
left) and odd (on the right).

$z_{v-2}^{1}$$z_{v-1}^{1}$$z_{v}^{1}$$\ddots$$z_{0}$$z_{1}^{2}$$z_{1}^{1}$$\ddots$$z_{v-2}^{2}$$z_{v-1}^{2}$$z_{v}^{2}$$z_{v-1}^{2}$$z_{v}^{2}$$\ddots$$z_{0}$$z_{1}^{1}$$z_{1}^{2}$$\ddots$$z_{v-1}^{1}$$z_{v}^{1}$

Figure 5. Negative staircase complexes for the case that $v$ is even (on the
left) and odd (on the right).

We now turn to our second special type of knot. A knot is said to be _thin_ if
its knot Floer homology is supported in a single diagonal on the plane, or
equivalently if $M([x;i,j])-A([x;i,j])$ is equal to a constant $k$ for all
$[x;i,j]\in\mathit{CFK}^{\infty}(K)$. More generally, a _thin complex_ is one
where there is a constant $k$, such that generators at filtration level
$(i,j)$ always have homological grading $i+j-k$.

Petkova [Pet13] showed that complexes associated to thin knots have an
especially simple form, which we now review. Let the _square complex_ $C_{s}$,
also known as a _one-by-one box complex_ , refer to an $\mathbb{F}$ chain
complex with generators $a,b,c,Ue$, in filtration levels
$(i+1,j+1),(i,j+1),(j+1,i),(i,j)$ respectively, and homological gradings
$k+2,k+1,k+1,k$ respectively, with differential

$\partial(a)=b+c,~{}~{}~{}~{}\partial(b)=Ue,~{}~{}~{}~{}\partial(c)=Ue,~{}~{}~{}~{}\partial(Ue)=0.$

One readily checks that $\ker\partial$ and $\operatorname{im}\partial$ are
both the vector space span of $b+c$ and $Ue$; hence, $C_{s}$ is acyclic. If
$i=j$ the square complex is said to be _on the main diagonal_. See Figure 6
for a picture of $C_{s}$.

$Ue$$c$$b$$a$ Figure 6. A copy of the square complex $C_{s}$.

######  Lemma 2.4.

[Pet13] Suppose $C$ is a thin complex with horizontal and vertical homologies
of rank at most one. Then $C$ is filtered chain homotopy equivalent to the
tensor product of $\mathbb{F}[U,U^{-1}]$ with a direct sum of square complexes
and at most one staircase complex all of whose steps are length one.

The case analysis involved in the proof also determines the length of the
staircase (that is, the integer $v$) and whether the staircase is positive or
negative.

###  2.2. Involutive Heegaard Floer homology for knots

Recall that if $C$ is a $(\mathbb{Z}\oplus\mathbb{Z})$ -filtered complex, a
chain map $\alpha\colon C\rightarrow C$ is said to be skew-filtered if
$\alpha(\mathcal{F}_{j,i})\subset\mathcal{F}_{i,j}$. In [HM17, Section 6],
Hendricks and Manolescu use the conjugation operation on knot Floer homology
to define a new automorphism

$\iota_{K}\colon\mathit{CFK}^{\infty}(K)\rightarrow\mathit{CFK}^{\infty}(K).$

The map $\iota_{K}$ is in principle difficult to compute. However, it has the
following properties.

######  Lemma 2.5.

[HM17] The square of the map $\iota_{K}$ is filtered chain homotopy equivalent
to the Sarkar involution; that is, we have

${\iota_{K}}^{2}\sim\sigma.$

Note that this implies that ${\iota_{K}}^{4}$ is filtered chain homotopy
equivalent to the identity map. In [HM17, Sections 7 and 8], Hendricks and
Manolescu show that this is enough to uniquely determine $\iota_{K}$ up to
filtered chain homotopy equivalence for $L$-space knots and thin knots. There
are two knot concordance invariants analogous to $V_{0}(K)$ arising from this
complex, as follows. Consider the subcomplex $A_{0}^{-}$ of
$\mathit{CFK}^{\infty}$, which is preserved by $\iota_{K}$. Then we form the
complex

$AI_{0}^{-}=\mathrm{Cone}(A_{0}^{-}\xrightarrow{Q(1+\iota_{K})}QA_{0}^{-})[-1]$

where $Q$ is a variable of degree $-1$ and the $[-1]$ denotes an upward
grading shift by $1$. The involutive correction terms are then

$\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(K)=-\frac{1}{2}\left(\max\\{r:\exists\ x\in
H_{r}(AI_{0}^{-}),\forall\ n,\ U^{n}x\neq 0\ \text{and}\
U^{n}x\notin\operatorname{Im}(Q)\\}-1\right)$

and

$\overline{V}_{0}(K)=-\frac{1}{2}\max\\{r:\exists\ x\in
H_{r}(AI_{0}^{-}),\forall\ n,U^{n}x\neq 0;\exists\ m\geq 0\
\operatorname{s.t.}\ U^{m}x\in\operatorname{Im}(Q)\\}.$

We briefly recall two special cases. The _standard staircase map_ on a
staircase complex $C$ is the reflection across the line $i=j$.

######  Proposition 2.6.

[HM17, Section 7] Let $K$ be an $L$-space knot, so that
$\mathit{CFK}^{\infty}(K)\simeq C\otimes\mathbb{F}[U,U^{-1}]$ for $C$ a
positive staircase complex. The involution on $\mathit{CFK}^{\infty}(K)$ is
generated by the standard staircase involution, and the involutive concordance
invariants are

$V_{0}(K)=\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(K)=\overline{V}_{0}(K)=n(K)$

and

$V_{0}(\overline{K})=\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(\overline{K})=0,\ \overline{V}_{0}(\overline{K})=-n(K).$

Computations are also accessible for thin knots. We first define a standard
map on a pair of square complexes. Consider two square complexes $C_{s}$
generated by $a,b,c,Ue$ and $C_{s}^{\prime}$ generated by
$a^{\prime},b^{\prime},c^{\prime},Ue^{\prime}$, with such that $a$ lies in
filtration level $(i+1,j+1)$ and $a^{\prime}$ lies in filtration level
$(j+1,i+1)$, as in Figure 7. The _standard square map_ between
$C_{s}\otimes\mathbb{F}[U,U^{-1}]$ and
$C_{s}^{\prime}\otimes\mathbb{F}[U,U^{-1}]$ is

$\iota_{K}(a)=a^{\prime}+e^{\prime},~{}~{}~{}~{}\iota_{K}(b)=c^{\prime},~{}~{}~{}~{}\iota_{K}(c)=b^{\prime},~{}~{}~{}~{}\iota_{K}(Ue)=Ue^{\prime}$
$\iota_{K}(a^{\prime})=a,~{}~{}~{}~{}\iota_{K}(b^{\prime})=c,~{}~{}~{}~{}\iota_{K}(c^{\prime})=b,~{}~{}~{}~{}\iota_{K}(Ue^{\prime})=Ue.$

Observe that this is a chain map with the property that
${\iota_{K}}^{2}=\sigma$, as $\sigma(a)=a+e$ and
$\sigma(a^{\prime})=a^{\prime}+e^{\prime}$.

$Ue$$c$$b$$a$$Ue^{\prime}$$c^{\prime}$$b^{\prime}$$a^{\prime}$ Figure 7. A
pair of square complexes.

######  Proposition 2.7.

[HM17, Proposition 8.1] Let $K$ be a thin knot. Up to a change of basis, the
complex $(\mathit{CFK}^{\infty}(K),\iota_{K})$ decomposes as a direct sum of
pairs of square complexes $C_{s}\otimes\mathbb{F}[U,U^{-1}]$ and
$C_{s}\otimes\mathbb{F}[U,U^{-1}]$ interchanged by the standard square map,
and a complex $C\otimes\mathbb{F}[U,U^{-1}]$ preserved by $\iota_{K}$ such
that $C$ consists of a staircase complex and at most one square complex on the
main diagonal.

The key fact used in the proof of Proposition 2.7 is that for a thin knot,
grading considerations ensure that $\iota_{K}$ interchanges the planar
gradings $(i,j)$ and $(j,i)$ and that ${\iota_{K}}^{2}=\sigma$ on the nose
(that is, the filtered chain homotopy map $H$ must be zero). Hendricks and
Manolescu use Proposition 2.7 to compute the involutive correction terms of
thin knots [HM17, Proposition 8.2].

We conclude this subsection with a look at the involutive concordance
invariants of the example complexes introduced so far, all of which fall into
one of the special cases above.

######  Example 2.8.

Let $K$ be the right-handed trefoil, with $\mathit{CFK}^{\infty}(K)$ as in
Figure 1. The automorphism $\iota_{K}$ is given by

$\iota_{K}(b)=c\qquad\iota_{K}(c)=b\qquad\iota_{K}(a)=a.$

The right-handed trefoil $K$ is an $L$-space knot with $n(K)=1$; therefore we
have

$V_{0}(K)=\underline{V\mkern-3.0mu}\mkern 3.0mu_{0}(K)=\overline{V}_{0}(K)=1.$

More precisely, in $H_{*}(A_{0}^{-})$ the element $[b]$ is a a maximally
graded element which is not $U$-torsion, such that
$V_{0}(K)=-\frac{1}{2}\operatorname{gr}[b]=-\frac{1}{2}(-2)=1$. In
$H_{*}(AI_{0}^{-})$, the element $[b+Qa]$ is a maximally-graded element which
is not $U$-torsion and such that $U^{n}[b+Qa]$ is never in the image of $Q$,
so that

$\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(K)=-\frac{1}{2}(\operatorname{gr}[b+Qa]-1)=-\frac{1}{2}(-1-1)=1$

and likewise $[Qb]$ is a maximally-graded element which is not $U$-torsion and
lies in the image of $Q$, such that

$\overline{V}_{0}(K)=-\frac{1}{2}(\operatorname{gr}[Qb])=-\frac{1}{2}(-2)=1.$

For the left-hand trefoil $\overline{K}$ with
$\mathit{CFK}^{\infty}(\overline{K})$ as in Figure 3, we still have

$\iota_{K}(b)=c\qquad\iota_{K}(c)=b\qquad\iota_{K}(a)=a.$

Since $\overline{K}$ is the mirror of an $L$-space knot with $n(K)=1$ we have

$V_{0}(\overline{K})=\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(\overline{K})=0,\overline{V}_{0}(\overline{K})=-1.$

More precisely, in $H_{*}(A_{0}^{-})$ the homology class $[Ub+Uc]$ is a
maximally-graded element which is not $U$-torsion, and
$\operatorname{gr}([Ub+Uc])=0$, so $V_{0}(K)=0$. In the homology
$H_{*}(AI_{0}^{-})$, the element $[Ub+Uc]$ is a maximally-graded element which
is not $U$-torsion and for which no $U$-power lies in the image of $Q$, so

$\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(\overline{K})=-\frac{1}{2}(\operatorname{gr}[Ub+Uc])=-\frac{1}{2}(1-1)=0$

whereas $[a]$ is a maximally-graded element which is not $U$-torsion and such
that $U[a]=[Q(Ub+Uc)]$, so

$\overline{V}_{0}(\overline{K})=-\frac{1}{2}(\operatorname{gr}[a])=-\frac{1}{2}(2)=-1.$

######  Example 2.9.

Let $K$ be the figure-eight knot, with $\mathit{CFK}^{\infty}(K)$ as in Figure
2. The automorphism $\iota_{K}$ is given by

$\iota_{K}(c)=b\qquad\iota_{K}(b)=c\qquad\iota_{K}(e)=e\qquad\iota_{K}(a)=a+x\qquad\iota_{K}(x)=x+e.$

If we consider the the element $[x]$ in $H_{*}(A_{0}^{-}),$ we see that it is
a maximally graded element which is not $U$-torsion, such that
$V_{0}(K)=-\frac{1}{2}\operatorname{gr}[x]=0.$ Moreover, in the homology
$H_{*}(AI_{0}^{-})$, the element $[Ux+Qc]$ is a maximally-graded element which
is not $U$-torsion and for which no $U$-power lies in the image of $Q$, so

$\underline{V\mkern-3.0mu}\mkern
3.0mu_{0}(\overline{K})=-\frac{1}{2}(\operatorname{gr}[Ux+Qc])=-\frac{1}{2}(-1-1)=1$

whereas $[Qx]$ is a maximally-graded element which is not $U$-torsion, so

$\overline{V}_{0}(\overline{K})=-\frac{1}{2}(\operatorname{gr}[Qx])=-\frac{1}{2}(0)=0.$

##  3\. Involutive invariants of $(-2,m,n)$ pretzel knots

###  3.1. The complex $\mathit{CFK}^{\infty}(P(-2,m,n))$

In this subsection we review Goda, Matsuda, and Morifuji’s computation of the
knot Floer homology of $P(-2,m,n)$ [GMM05, Section 5] and prove that there is
a change of basis which allows us to simplify the complex. Let $m\geq n\geq
3$. For our proof, we will require a description of the generators of the
complex and their filtration levels, some information about the differentials
of particular generators, and the final computation of knot Floer homology.

Following [GMM05], we fix the following notation. Let $g=\frac{m+n}{2}$ be the
genus of the pretzel knot, and set $m^{\prime}=\frac{m-3}{2},$
$n^{\prime}=\frac{n-3}{2}$,

$\gamma(g)=\begin{cases}1-\frac{g-1}{2}&g\;\;\mathrm{odd}\\\
1-\frac{g}{2}&g\;\;\mathrm{even}\end{cases}\qquad\text{and}\qquad\delta(g)=\begin{cases}\frac{g-1}{2}&g\;\;\mathrm{odd}\\\
\frac{g}{2}-1&g\;\;\mathrm{even}\end{cases}.$

With respect to these choices, the complex $\mathit{CFK}^{\infty}(K)$ has a
set of generators as follows.

$\displaystyle[y_{1},;i+\gamma(g)-1,i+\delta(g)+1]$
$\displaystyle[y_{2};i+\gamma(g)-1,i+\delta(g)]$
$\displaystyle[y_{3};i+\delta(g),i+\gamma(g)-1]$
$\displaystyle[y_{4};i+\delta(g)+1,i+\gamma(g)-1]$
$\displaystyle[x_{2p+1,2q+1};i+\gamma(g)+p+q+1,i+\delta(g)-p-q]\;\;\text{ for
}0\leq p\leq n^{\prime},0\leq q\leq m^{\prime}$
$\displaystyle[x_{2p+1,2q};i+\gamma(g)+p+q,i+\delta(g)-p-q]\;\;\text{ for
}0\leq p\leq n^{\prime},1\leq q\leq m^{\prime}$
$\displaystyle[x_{2p,2q+1};i+\gamma(g)+m^{\prime}+p-q,i+\delta(g)-m^{\prime}-p-q]\;\;\text{
for }1\leq p\leq n^{\prime},0\leq q\leq m^{\prime}$
$\displaystyle[x_{2p,2q};i+\gamma(g)+m^{\prime}+p-q,i+\delta(g)-m^{\prime}-p+q-1]\;\;\text{
for }1\leq p\leq n^{\prime},1\leq q\leq m^{\prime}$

We refer to the generators $y_{i}$ as the _exceptional generators_ and the
generators $x_{k,\ell}$ as the _ordinary generators_. Observe that the
ordinary generators lie between the lines $j-i=\frac{m+n-2}{2}$ and
$j-i=-\frac{m+n-2}{2}$ and that $[y_{2};\gamma(g)-1,\delta(g)]$ and
$[y_{3};\delta(g),\gamma(g)-1]$ lie on these lines respectively. Furthermore,
$[y_{1};i+\gamma(g)-1,i+\delta(g)+1]$ lies on the line $i-j=-\frac{m+n}{2}$
and $[y_{4};i+\delta(g)+1,i+\gamma(g)-1]$ lies on the line
$i-j=\frac{m+n}{2}.$

The differentials most important to us involve the exceptional generators, and
are as follows.

######  Lemma 3.1.

[GMM05, Section 5] With respect to the basis for
$\mathit{CFK}^{\infty}(P(-2,m,n))$ above, the differentials of the exceptional
generators are

  * •

$\partial[y_{1};i,j]=[y_{2};i,j-1]$

  * •

$\partial[y_{2};i,j]=0$

  * •

$\partial[y_{1};i,j]=0$

  * •

$\partial[y_{4};i,j]=[y_{3};i-1,j].$

Furthermore, the elements whose differential, written as a sum of the
generators above, includes an exceptional generator are

  * •

$\partial[x_{1,1};i,j]=[x_{2,m-2};i,j-1]+[x_{1,2};i,j-1]-[y_{2};i-2,j]$

  * •

$\partial[x_{n-2,m-2};i,j]=-[x_{n-3,1};i-1,j]-[x_{n-2,m-3};i-1,j]+[y_{3};i,j-2].$

We omit the rest of the differential, but note the resulting computation of
the knot Floer homology of the pretzel knots.

######  Lemma 3.2.

[GMM05, Section 5] For $m\geq n\geq 3$, the knot Floer homology of $P(-2,m,n)$
in positive Alexander gradings is equal to

$\widehat{\mathit{HFK}}{(S^{3},K,i)}=\begin{cases}0&g>i\\\
\mathbb{F}_{g+i}&i=g,g-1\\\ 0&i=g-2\\\ \mathbb{F}^{g-2-i}_{g-1+i}&g-n\leq
i<g-2\\\ \mathbb{F}^{n-2}_{g-1+i}&0\leq i<g-n.\end{cases}$

We may now state our first splitting lemma:

######  Lemma 3.3.

Let $K$ be a pretzel knot $P(-2,m,n)$ with $m\geq n>3$. Then $CFK^{\infty}(K)$
is isomorphic to the tensor product of $\mathbb{F}[U,U^{-1}]$ with a direct
sum of a single staircase complex together with some number of copies of the
square complex $C_{s}$.

$x_{1,1}$$y_{2}$$y_{1}$$x_{n-1,m-1}$$y_{3}$$y_{4}$$C^{\prime}$$j-i=\frac{m+n-6}{2}$$j-i=\frac{6-m-n}{2}$

Figure 8. A schematic diagram of the situation in Lemma 3.3. The four
extraordinary generators and all of the differentials involving them are
shown. The remaining generators in $C^{\prime}$ lie strictly between the two
dashed lines.

###### Proof.

In this proof we let
$y_{1}=[y_{1};\frac{-m+n}{2},0],y_{2}=[y_{2};\frac{-m+n}{2},-1],y_{3}=[y_{3};-1,\frac{-m+n}{2}],y_{4}=[y_{4};0,\frac{-m+n}{2}],x_{1,1}=[x_{1,1};\frac{4-m-n}{2},-1],$
and $x_{n-2,m-2}=[x_{n-2,m-2};-1,\frac{4-m-n}{2}].$ These gradings are chosen
so that $y_{1},y_{4},x_{1,1}$, and $x_{n-2,m-2}$ lie in homological grading
$0$ and $y_{2}$ and $y_{3}$ lie in homological grading $-1$. Moreover, we have

$\partial(y_{1})=y_{2}\qquad\qquad\partial(y_{4})=y_{3}.$

Furthermore,

$\partial_{\mathrm{horz}}(x_{1,1})=y_{2}\qquad\qquad\partial_{\mathrm{vert}}(x_{n-2,m-2})=y_{3}.$

A schematic of the situtation appears in Figure 8. We first observe that
$[y_{1}]$ is a nontrivial element of the horizontal homology of
$\mathit{CFK}^{\infty}{}(K)$, since the only differential involving $y_{1}$ in
the complex is a vertical arrow to $y_{2}$. Since the horizontal homology is
one-dimensional, it in fact must be generated by $[y_{1}]$. Similarly, the
vertical homology is generated by $[y_{4}].$ Now, let us consider the complex
$C^{\prime}$ of $\mathit{CFK}^{\infty}(K)$ which is a quotient of $C$ by the
sub-complex generated by the elements $y_{1},y_{2},y_{3},$ and $y_{4}.$ In
doing so, we have deleted the horizontal arrow from $x_{1,1}$ to $y_{2}$ and
the vertical arrow from $x_{n-2,m-2}$ to $y_{3}.$ If we now consider the
horizontal homology of $C^{\prime},$ we see that it is generated by
$[x_{1,1}]$ since now $x_{1,1}$ has trivial horizontal differential and the
horizontal differential is otherwise unchanged. Similarly, the vertical
homology of $C^{\prime}$ is generated by $[x_{n-2,m-2}].$ In particular, both
the vertical and horizontal homologies of $C^{\prime}$ have rank one.
Moreover, we see that since $\widehat{\mathit{HFK}}(C^{\prime})$ is determined
purely by the vertical differentials in $C^{\prime}$, we have that
$\widehat{\mathit{HFK}}(C^{\prime})$ differs only from
$\widehat{\mathit{HFK}}(C)$ by deleting the summands in Alexander gradings
$g,g-1,g-1,$ and $-g$ generated by the four exceptional generators. Therefore
$C^{\prime}$ is thin. Therefore, Petkova’s work [Pet13], recapped earlier in
Lemma 2.4 tells us that $C^{\prime}$ may be decomposed as the tensor product
of $\mathbb{F}[U,U^{-1}]$ with a direct sum of a staircase complex and some
number of one-by-one boxes.

However, the case analysis of [Pet13, Section 3] also recovers the sign and
length of the staircase complex in question. We start by considering an
element lying on the bottom-most occupied diagonal line $i+j=k$ in
$C^{\prime}$, which in this case is $x_{n-2,m-2}$ (or any of its
$U$-translates). This element does not lie in the image of a vertical
differential, which corresponds to Case 2 of Lemma 7 of Petkova [Pet13,
Section3]; in particular, $x_{n-2,m-2}$ must form the lower right corner of a
negative staircase summand. Hence we can split off a staircase complex whose
lower-right corner is $x_{n-2,m-2}$. By symmetry of the staircase across the
main diagonal, the upper left corner must lie in filtration level
$\left(\frac{4-m-n}{2},-1\right)$ and have homological grading $0$; the only
element in $C^{\prime}$ satisfying this is $x_{1,1}$.

Now we include $C^{\prime}$ back into $C.$ We see that $x_{1,1}$ has a
horizontal length two arrow to $y_{2}$ and $x_{n-2,m-2}$ has a vertical length
two arrow to $y_{3}.$ Hence the negative staircase in $C^{\prime}$ extends to
a negative staircase in $C$ which includes the four exceptional generators
(and now has two steps of length two). The conclusion of the lemma follows. ∎

See Figure 9 for the example of the pretzel knot $P(-2,9,9)$ after this change
of basis.

$z_{8}^{1}$$z_{7}^{1}$$z_{6}^{1}$$z_{5}^{1}$$z_{4}^{1}$$z_{3}^{1}$$z_{2}^{1}$$z_{1}^{1}$$z_{0}$$z_{1}^{2}$$z_{2}^{2}$$z_{3}^{2}$$z_{4}^{2}$$z_{5}^{2}$$z_{6}^{2}$$z_{7}^{2}$$z_{8}^{2}$232$Uz_{1}^{1}$$Uz_{0}$$Uz_{1}^{2}$232232

Figure 9. The complex $\mathit{CFK}^{\infty}(P(-2,9,9))$ after a change of
basis. Here an integer $k$ in the center of a box indicates the presence of
$k$ boxes on the corresponding diagonal.

###  3.2. Computations for model complexes

In this subsection we carry out computations for several model complexes. Our
computations are modelled on [HM17, Section 8]. We begin by using the
description of the complexes $\mathit{CFK}^{\infty}(P(-2,m,n))$ given in the
previous section to prove a simplifying lemma.

######  Lemma 3.4.

Let $K$ be a pretzel knot $P(-2,m,n),m\geq n>3$. Then

  1. (1)

The skew-filtered chain map $\iota_{K}$ exchanges the planar gradings $(i,j)$
and $(j,i)$.

  2. (2)

If $H$ is a filtered chain homotopy equivalence which raises the homological
grading by one, $H=0$. Hence ${\iota_{K}}^{2}=\sigma$.

###### Proof.

Let $[x;i,j]$ be a generator of $\mathit{CFK}^{\infty}(K)$ with homological
grading $s$. By recalling that the homological grading of
$x_{1,1}=[x_{1,1};\frac{4-m-n}{2},-1]$ is zero, we see that for any non-
exceptional generator $s=i+j+\frac{m+n}{2}-1$; and furthermore, if $s$ is
even,

$\frac{s-m-n+4}{2}\leq i,j\leq\frac{s-2}{2}$

and if $s$ is odd,

$\frac{s-m-n+5}{2}\leq i,j\leq\frac{s-3}{2}.$

If $[y;i,j]$ is an exceptional generator, then if $s$ is even either
$(i,j)=\left(\frac{s}{2},\frac{s-m-n}{2}\right)$ or
$(i,j)=\left(\frac{s-m-n}{2},\frac{s}{2}\right)$, and if $s$ is odd then
$(i,j)=\left(\frac{s+1}{2},\frac{s-m-n+1}{2}\right)$ or
$(i,j)=\left(\frac{s-m-n+1}{2},\frac{s+1}{2}\right)$. In either case,
$s=i+j+\frac{m+n}{2}$.

We now prove part (1). Suppose we have an element $[z;i,j]$ of homological
grading $s$ which is an ordinary generator. Then $\iota_{K}([z;i,j])$ is a sum
of generators $[x;i^{\prime},j^{\prime}]$ such that $i^{\prime}\leq
j,j^{\prime}\leq i$, and $\operatorname{gr}([x,i^{\prime},j^{\prime}])=s$.
These restrictions imply that for any $[x;i^{\prime},j^{\prime}]$ in this sum
which is an ordinary generator, we must then have $i^{\prime}=j$ and
$j^{\prime}=i$. Now suppose that $[y;i^{\prime},j^{\prime}]$ an exceptional
generator which appears in this sum with nonzero coefficient. Then if $s$ is
even, one of $i^{\prime}$ or $j^{\prime}$ is $\frac{s}{2}$. However,
$i,j\leq\frac{s-2}{2}$, so this contradicts $i^{\prime}\leq j$ and
$j^{\prime}\leq i$. Similarly if $s$ is odd, then one of $i^{\prime}$ or
$j^{\prime}$ is $\frac{s-1}{2}$, but $i,j\leq\frac{s-3}{2}$. Ergo
$\iota_{k}([z;i,j])$ cannot contain any exceptional generators, and indeed
$\iota_{k}([z;i,j])$ must be a sum of ordinary generators
$[x;i^{\prime},j^{\prime}]$ with $i^{\prime}=j$ and $j^{\prime}=i$. An
analogous but simpler argument shows that for the four exceptional generators,
$\iota_{K}([y_{\ell};i,j])$ is either $[y_{4-\ell};j,i]$ or zero. We conclude
that $\iota_{k}$ interchanges the planar gradings $(i,j)$ and $(j,i)$. This
proves (1).

The reader is at this point invited to examine Figure 9 as an example and
compare it to the proof above, for clarity. The point is that for any
generator $[x;i,j]$ in the basis shown, all elements in filtration level
$\mathcal{F}_{j,i}$ have the same type (exceptional or ordinary) as $[x;i,j]$,
and therefore grading considerations imply that $\iota_{K}$ must simply
interchange the planar gradings.

Now let $H$ be a filtered map which raises the homological grading by one.
Suppose the map $H$ sends a generator $[z;i,j]$ in homological grading $s$ to
a nonzero sum of generators $[x;i^{\prime},j^{\prime}]$ in homological grading
$t=s+1$ with $i^{\prime}\leq i$ and $j^{\prime}\leq j$. For any
$[x;i^{\prime},j^{\prime}]$ in this sum, if $[x;i^{\prime},j^{\prime}]$ is an
ordinary generator then $t=s+1=i^{\prime}+j^{\prime}+\frac{m+n}{2}-1$ and if
$[x;i^{\prime},j^{\prime}]$ is an exceptional generator then
$t=s+1=i^{\prime}+j^{\prime}+\frac{m+n}{2}$. We deal with each of these two
possibilities separately.

In the first case, if $s+1=i^{\prime}+j^{\prime}+\frac{m+n}{2}-1$, then since
$i^{\prime}\leq i$ and $j^{\prime}\leq j$, we have $s+1\leq
i+j+\frac{m+n}{2}-1$, that is, $s\leq i+j+\frac{m+n}{2}-2$. This is a
contradiction since for any generator $[z;i,j]$, one of
$s=i+j+\frac{m+n}{2}-1$ or $s=i+j+\frac{m+n}{2}$ is true.

In the second case, if $s+1=i^{\prime}+j^{\prime}+\frac{m+n}{2}$, the element
$[x;i^{\prime},j^{\prime}]$ is an exceptional generator. As before it follows
that $s\leq i+j+\frac{m+n}{2}-1$, so we see that the original generator
$[z;i,j]$ must be ordinary. Moreover, the inequality must actually be an
equality, implying that $i^{\prime}=i$ and $j^{\prime}=j$. Therefore, we must
have that $H$ sends the ordinary generator $[z;i,j]$ to a sum of exceptional
generators $[x;i,j]$ in homological grading $s+1$ lying in the same filtration
level $(i,j)$. However, no such generators exist. For suppose $s$ is even.
Then since $[x;i,j]$ is an exceptional generator with grading $s+1$, either
$i$ or $j$ is $\frac{(s+1)+1}{2}=1+\frac{s}{2}$; and since $[x;i,j]$ is an
ordinary generator with grading $s$, we see that $i,j\leq-1+\frac{s}{2}$. This
is a contradiction because whichever of $i,j$ is equal to $1+\frac{s}{2}$ is
greater than $-1+\frac{s}{2}$. A similar contradiction occurs if $s$ is odd:
this time either $i$ or $j$ is $\frac{s+1}{2}$, and yet
$i,j\leq-1+\frac{s+1}{2}$. This implies that $H=0$, proving (2). ∎

Lemma 3.4 is the important step in proving that the pair
$(\mathit{CFK}^{\infty}(P(-2,m,n)),\iota_{K})$ decomposes into a direct sum of
simple complexes preserved by $\iota_{K}$. Since it is no longer necessary to
distinguish the ordinary and exceptional generators, from here on out we will
use the notation for staircase complexes introduced in Section 2.

######  Lemma 3.5.

Let $K$ be a pretzel knot $P(-2,m,n),m\geq n>3$. Then
$\mathit{CFK}^{\infty}(K)$ splits as the tensor product of
$\mathbb{F}[U,U^{-1}]$ with a direct sum of pairs of square complexes
interchanged by the standard square map, and one of four model complexes,
according to the values of $m$ and $n$, as follows:

  * •

When $m\equiv n\equiv 1\pmod{4}$, we obtain the model complex $C_{1}$
consisting of a negative staircase with $n(K)=\frac{m+n-2}{4}$ and a single
square complex on the main diagonal, as in Figure 10.

  * •

When $m\equiv 3\pmod{4}$ and $n\equiv 1\pmod{4}$, we obtain the model complex
$C_{2}$ consisting of a negative staircase with $n(K)=\frac{m+n}{4}$, as in
Figure 11.

  * •

When $m\equiv n\equiv 3\pmod{4}$, we obtain the model complex $C_{3}$
consisting of a negative staircase with $n(K)=\frac{m+n-2}{4}$, as in Figure
12.

  * •

When $m\equiv 1\pmod{4}$ and $n\equiv 3\pmod{4}$, we obtain the model complex
$C_{4}$, consisting of a negative staircase with $n(K)=\frac{m+n}{4}$, as in
Figure 13.

###### Proof.

The argument that we can split off pairs of square complexes until we are left
with a staircase complex and at most one square complex on the main diagonal
proceeds as in [HM17, Proposition 8.1], using Lemma 3.4. It remains to analyze
the staircases involved and determine whether we have a square complex
remaining on the main diagonal.

We first note that in all cases by Lemma 3.3 we have a negative staircase
which begins with a vertical step of length one followed by a horizontal step
of length two, subsequent to which all steps in the top half of the complex
have length one. Let $v$ be the total number of steps in the top half of the
complex and $u=v-2$ be the number of steps in the top half of the complex
after the first two. Then since the total lengths of the steps in the complex
sum to $g=\frac{m+n}{2}$ by projecting the elements of the staircase to their
$U$-translates in the filtration level $\\{i=0\\}$ of
$\mathit{CFK}^{\infty}(K)$, we have that

$u=\frac{m+n}{2}-3=\frac{m+n-6}{2}.$

We see that if $m\equiv n\pmod{4}$ then $m+n\equiv 2\pmod{4}$, hence $u$ is an
even number, and therefore so is $v$, so the number of steps above the main
diagonal is even. However, if it is the case that $m\not\equiv n\pmod{4}$,
then $4\mid(m+n)$ so there is an odd number of steps above the main diagonal.

Before moving on, we also calculate the number $n(K)$ associated to the
negative staircase summand of the complex. If $m\equiv n\pmod{4}$, then $u$ is
even, and the total lengths of the vertical arrows in the top half of the
complex is equal to $\frac{u}{2}+1$, so we have

$n(K)=\frac{m+n-6}{4}+1=\frac{m+n-2}{4}.$

Conversely if $m\not\equiv n\pmod{4}$, then $u$ is odd, and the total lengths
of the vertical arrows in the top half of the complex is $\frac{u+1}{2}+1$, so
we have

$n(K)=\frac{m+n-4}{4}+1=\frac{m+n}{4}.$

We now turn to the matter of square complexes, which we analyze by considering
the Alexander polynomial of the knot, which we recall is equal to the Euler
characteristic of the knot Floer homology $H_{*}(C\\{i=0\\})$ of the knot. In
general, a square complex on the diagonal $j=i+s$ contributes a factor of $\pm
t^{s}(-t+2-t^{-1})$ to the Alexander polynomial of the knot, where in our case
because the knot Floer complex is thin apart from the four exceptional
generators, the sign is always $(-1)^{s}$. The total Alexander polynomial of
$K$ is

$\displaystyle
t^{g}-t^{g-1}+\sum_{k=1}^{n-3}(-1)^{k-1}kt^{g-k-2}+\sum_{k=n-g}^{g-n}(-1)^{g-k-1}(n-2)t^{k}+\sum_{k=1}^{n-3}(-1)^{k-1}kt^{k+2-g}-t^{1-g}+t^{-g},$

The staircase summand contributes
$t^{g}-t^{g-1}+\sum_{k=3-g}^{g-3}(-1)^{k-1}t^{k}-t^{1-g}+t^{-g}$ to the
polynomial. Therefore, the boxes must contribute the remaining terms, which
are:

$\sum_{k=1}^{n-4}(-1)^{k-1}kt^{g-k-3}+\sum_{k=n-g}^{g-n}(-1)^{g-k-1}(n-3)t^{k}+\sum_{k=1}^{n-4}(-1)^{k-1}kt^{k+3-g}.$

This polynomial factors as

$\left(\sum_{k=1}^{\frac{n-5}{2}}kt^{g-2k-3}+\frac{n-3}{2}\sum_{0}^{g-n}t^{g-n-2k}+\sum_{k=1}^{\frac{n-5}{2}}(-1)^{k}kt^{2k+3-g}\right)(-t+2-t^{-1}).$

In particular, note that the center sum of this polynomial
$\frac{n-3}{2}\sum_{0}^{g-n}t^{g-n-2k}$ can be rewritten as
$\frac{n-3}{2}(t^{g-n}+t^{g-n-2}+\dots+t^{n+2-g}+t^{n-g})$. If $m\not\equiv
n\pmod{4}$, then the constant term in this polynomial is zero, and we conclude
that there are no boxes on the main diagonal in $\mathit{CFK}^{\infty}(K)$. If
$m\equiv n\pmod{4}$, then the constant term in this polynomial is
$\frac{n-3}{2}$, and we conclude that there are $\frac{n-3}{2}$ boxes on the
main diagonal. If $n\equiv 3\pmod{4}$, this is an even number of boxes, which
split off in pairs. If $n\equiv 1\pmod{4}$, this is an odd number of boxes,
all but one of which split off in pairs, leaving a single box on the main
diagonal in the model complex. ∎

$z_{v}^{1}$$z_{v-1}^{1}$$z_{v-1}^{2}$$z_{v}^{2}$$z_{2}^{1}$$z_{2}^{2}$$z_{0}$$z_{1}^{1}$$z_{1}^{2}$$b$$a$$Ue$$e$$c$$U^{-1}z_{v}^{1}$$U^{-1}z_{v}^{2}$

Figure 10. The model complex $C_{1}$, corresponding to knots $P(-2,m,n)$ such
that $m\equiv n\equiv 1\pmod{4}$. The case of $m=n=5$ is shown.

$z_{v}^{1}$$z_{v-1}^{1}$$z_{3}^{1}$$z_{2}^{1}$$z_{1}^{1}$$z_{0}$$z_{1}^{2}$$z_{2}^{2}$$z_{3}^{2}$$z_{v-1}^{2}$$z_{v}^{2}$$Uz_{v}^{1}$$Uz_{3}^{1}$$Uz_{1}^{1}$$Uz_{1}^{2}$$Uz_{3}^{2}$$Uz_{v}^{2}$

Figure 11. The model complex $C_{2}$, corresponding to knots $P(-2,m,n)$ such
that $m\equiv 3\pmod{4}$ and $n\equiv 1\pmod{4}$. The case of $m=7$ and $n=5$
is shown.

$U^{-1}z_{v}^{1}$$U^{-1}z_{4}^{1}$$U^{-1}z_{0}$$U^{-1}z_{4}^{2}$$U^{-1}z_{v}^{2}$$z_{v}^{1}$$z_{v-1}^{1}$$z_{4}^{1}$$z_{3}^{1}$$z_{2}^{1}$$z_{1}^{1}$$z_{0}$$z_{1}^{2}$$z_{2}^{2}$$z_{3}^{2}$$z_{4}^{2}$$z_{v-1}^{2}$$z_{v}^{2}$

Figure 12. The model complex $C_{3}$, corresponding to knots $P(-2,m,n)$ such
that $m\equiv n\equiv 3\pmod{4}$. The case of $m=n=7$ is shown.

$z_{v}^{1}$$z_{v-1}^{1}$$z_{5}^{1}$$z_{4}^{1}$$z_{3}^{1}$$z_{2}^{1}$$z_{1}^{1}$$z_{0}$$z_{1}^{2}$$z_{2}^{2}$$z_{3}^{2}$$z_{4}^{2}$$z_{5}^{2}$$z_{v-1}^{2}$$z_{v}^{2}$$Uz_{v}^{1}$$Uz_{5}^{1}$$Uz_{3}^{1}$$Uz_{1}^{1}$$Uz_{1}^{2}$$Uz_{3}^{2}$$Uz_{5}^{2}$$Uz_{v}^{2}$

Figure 13. The model complex $C_{4}$, corresponding to knots $P(-2,m,n)$ such
that $m\equiv 1\pmod{4}$ and $n\equiv 3\pmod{4}$. The case of $m=9$ and $n=7$
is shown.

###### Proof of Theorem 1.1.

We present the argument for the model complex $C_{1}$ and its dual; the other
cases, being straightforward staircase complexes, are substantially easier.
These computations are similar to Cases 2(e) and 2(c) of [HM17, Proposition
8.2] respectively. (Note that, because of the step of length two in our
staircase complex, we in particular get parallel results to the case of thin
knots in the case that the knot invariant $\tau$ is an even number, even
though in our case $\tau=g=\frac{m+n}{2}$ is odd.) The complex is pictured in
Figure 10, and has generators $a,b,c,e,z_{0},z_{r}^{s}$ for $s\in\\{1,2\\}$
and $1\leq j\leq v$, where $v=\frac{m+n}{2}-1$. The complex has nonzero
differentials

$\displaystyle\partial a=b+c$ $\displaystyle\partial b=\partial c=Ue$
$\displaystyle\partial(z_{0})=z_{1}^{1}+z_{1}^{2}$
$\displaystyle\partial(z^{s}_{v})=z^{s}_{v-1}$
$\displaystyle\partial(z_{r}^{s})=z_{r-1}^{s}+z_{r+1}^{s},\ 0<s<v\text{ even
}$

Up to change of basis, the unique skew-filtered map on $C_{1}$ squaring to the
Sarkar involution is

$\displaystyle\iota_{K}(a)=a+z_{0}$ $\displaystyle\iota_{K}(b)=c+z_{1}^{2}$
$\displaystyle\iota_{K}(z_{0})=z_{0}+e$
$\displaystyle\iota_{K}(c)=b+z_{1}^{1}$
$\displaystyle\iota_{K}(z_{r}^{1})=z_{r}^{2}$
$\displaystyle\iota_{K}(z_{r}^{2})=z_{r}^{1}$ $\displaystyle\iota_{K}(e)=e$

We begin by considering the subcomplex $A_{0}^{-}$ of
$C_{1}\otimes\mathbb{F}[U,U^{-1}]$. We see that this subcomplex has a basis
over $\mathbb{F}[U]$ with generators
$\\{e,a,b,c,z_{0},z^{s}_{1},Uz^{s}_{2},Uz^{s}_{3},U^{2}z^{s}_{4},\dots,U^{n(K)}z^{s}_{v}\\}$
and differentials inherited from the differentials on $C_{1}$. The homology
$H_{*}(A_{0}^{-})$ is

$H_{*}(A_{0}^{-})\cong\mathbb{F}_{(0)}[U]\oplus\mathbb{F}_{(2n(K)-1)}[U]/U^{n(K)}\oplus\mathbb{F}_{(2n(K))}$

where the summand $\mathbb{F}_{(0)}[U]$ is generated by
$[U^{n(K)}(z^{1}_{v}+\dots+z_{2}^{1}+z_{0}+z_{2}^{2}+\dots z^{2}_{v})]$, the
summand $\mathbb{F}_{(2n(K)-1)}$ is generated by $[z_{1}^{1}]$ and the summand
$\mathbb{F}_{(2n(K))}$ is generated by $[e]$. Here, the non-$U$-torsion part
of $H_{*}(A_{0}^{-})$ is generated by an element of grading $0$, so
$V_{0}(K)=0$.

$z_{0}$$z_{1}^{1}$$z_{1}^{2}$$b$$a$$Ue$$e$$c$$Uz_{v-1}^{1}$$Uz_{v-1}^{2}$$U^{n(K)}z_{v}^{1}$$U^{n(K)}z_{v}^{2}$$\iddots$$\iddots$$\iddots$

Figure 14. The $\mathbb{F}[U]$-submodule $A_{0}^{-}$ of the model complex
$C_{1}$.

We now compute the involutive correction terms. We start by computing the
homology of the mapping cone
$\mathrm{Cone}(C_{1}\otimes\mathbb{F}[U,U^{-1}]\xrightarrow{1+\iota_{K}}Q(C_{1}\otimes\mathbb{F}[U,U^{-1}]))$.
After a suitable basis change the differential
$\partial^{\iota}=\partial+Q(1+\iota_{K})$ is given by

$\displaystyle\partial^{\iota}(e)=0$ $\displaystyle\partial^{\iota}(Qe)=0$
$\displaystyle\partial^{\iota}(a)=(b+c)+Qz_{0}$
$\displaystyle\partial^{\iota}(Qa)=Q(b+c)$
$\displaystyle\partial^{\iota}(c)=Ue+Q(b+c)+Qz_{1}^{1}+Q(z_{1}^{1}+z_{1}^{2})$
$\displaystyle\partial^{\iota}(Qc)=QUe$
$\displaystyle\partial^{\iota}(b+c)=Q(z_{1}^{1}+z_{1}^{2})$
$\displaystyle\partial^{\iota}(Q(b+c))=0$
$\displaystyle\partial^{\iota}(z_{0})=z_{1}^{1}+z_{1}^{2}+Qe$
$\displaystyle\partial^{\iota}(Qz_{0})=Q(z_{1}^{1}+z_{1}^{2})$
$\displaystyle\mathclap{\begin{aligned}
&\partial^{\iota}(z_{r}^{1})=\begin{cases}(z_{r-1}^{1}+z_{r+1}^{1})+Q(z_{r}^{1}+z_{r}^{2}),\
r\text{ even}\\\ Q(z_{r}^{1}+z_{r}^{2}),\ r\text{ odd}\end{cases}\\\
&\partial^{\iota}(Qz_{r}^{1})=\begin{cases}Qz_{r-1}^{1}+Qz_{r+1}^{1},\ r\text{
even},\\\ 0\ r\text{ odd}\end{cases}\\\
&\partial^{\iota}(z_{r}^{1}+z_{r}^{2})=\begin{cases}(z_{r-1}^{1}+z_{r-1}^{2})+(z_{r+1}^{1}+z_{r+1}^{2}),\
r\text{ even}\\\ 0,\ r\text{ odd}\end{cases}\\\
&\partial^{\iota}(Q(z_{r}^{1}+z_{r}^{2}))=\begin{cases}Q(z_{r-1}^{1}+z_{r-1}^{2})+Q(z_{r+1}^{1}+z_{r+1}^{2}),\
r\text{ even}\\\ 0,\ r\text{ odd}\end{cases}\end{aligned}}$

$QU^{-1}c$$Qe$$z_{0}$$z_{1}^{1}+z_{1}^{2}$$z_{2}^{1}+z_{2}^{2}$$z_{3}^{1}+z_{3}^{2}$$\dots$$z_{v-1}^{1}+z_{v-1}^{2}$$z_{v}^{1}+z_{v}^{2}$

$a$$b+c$$Qz_{0}$$z_{1}^{1}$$Q(z_{1}^{1}+z_{1}^{2})$$z_{2}^{1}$$Q(z_{2}^{1}+z_{2}^{2})$$z_{3}^{1}$$Q(z_{3}^{1}+z_{3}^{2})$$z_{4}^{1}$$Q(z_{4}^{1}+z_{4}^{2})$$\dots$$z_{v-1}^{1}$$\dots$$Q(z_{v-1}^{1}+z_{v-1}^{2})$$z_{v}^{1}$$Q(z_{v}^{1}+z_{v}^{2})$

$Qa$$Q(b+c)$$c$$Ue$$Qz_{1}^{1}$$Qz_{2}^{1}$$\dots$$Qz_{v}^{1}$

Figure 15. Direct summands appearing in the mapping cone
$\operatorname{Cone}(C_{1}\otimes\mathbb{F}[U,U^{-1}]\xrightarrow{Q(1+\iota_{K})}Q(C_{1}\otimes\mathbb{F}[U,U^{-1}]))$
after a change of basis.

The complex splits into the direct summands shown in Figure 15. The top
summand has one-dimensional homology generated by
$[s]=[z_{0}+(z_{2}^{1}+z_{2}^{2})+\cdots+(z_{v}^{1}+z_{v}^{2})+QU^{-1}c]$. The
middle summand has one-dimensional homology generated by
$[Qz_{0}+z_{1}^{1}]=[Q(z_{0}+z_{2}^{1}+z_{2}^{2}+\cdots+z_{v}^{1}+z_{v}^{2})]$.
The bottom summand is acyclic. Hence, the homology of the mapping cone is
generated as a module over $\mathbb{F}[U,U^{-1}]$ by $[s]$ and
$[Qz_{0}+z_{1}^{1}]$.

$QU^{-1}c$$Qe$$z_{0}$$z_{1}^{1}+z_{1}^{2}$$z_{2}^{1}+z_{2}^{2}$$z_{3}^{1}+z_{3}^{2}$$\dots$$z_{v-1}^{1}+z_{v-1}^{2}$

$a$$b+c$$Qz_{0}$$z_{1}^{1}$$Q(z_{1}^{1}+z_{1}^{2})$$z_{2}^{1}$$Q(z_{2}^{1}+z_{2}^{2})$$z_{3}^{1}$$Q(z_{3}^{1}+z_{3}^{2})$$z_{4}^{1}$$Q(z_{4}^{1}+z_{4}^{2})$$\dots$$z_{v-1}^{1}$$\dots$$Q(z_{v-1}^{1}+z_{v-1}^{2})$

$Qa$$Q(b+c)$$c$$Ue$$Qz_{1}^{1}$$Qz_{2}^{1}$$\dots$$Qz_{v-1}^{1}$

Figure 16. Direct summands appearing in the mapping cone of $A_{0}^{-}$. Here,
every vertex is implicitly decorated with an additional power of $U^{n(K)}$.

Now we compute the homology of the mapping cone $AI_{0}^{-}$. Recall that the
complex $A_{0}^{-}$ is generated by the elements
$\\{e,a,b,c,z_{0},z^{s}_{1},Uz^{s}_{2},Uz^{s}_{3},U^{2}z^{s}_{4},\dots,U^{n(K)}z^{s}_{v}\\}$.
Therefore $A_{0}^{-}$ contains all powers of $U^{n(K)}$ or greater of all of
the generators appearing in Figure 15; for lower $U$-powers, we successively
truncate sets of four elements $z_{t}^{k},z_{t}^{k+1}$ for $k$ odd as the
extremal steps of the staircase fall outside of $A_{0}^{-}$. The first such
truncation is shown in Figure 16. We see that after removing $z_{v}^{t}$ and
$z_{v}^{t-1}$ the top summand is acyclic; it remains acyclic after successive
truncations until the final step, when the homology becomes $[Qe]$. The bottom
summand now has one-dimensional homology generated by
$[Ue]=[Qz_{1}^{1}]=[Qz_{3}^{1}]=\cdots=[Qz_{v-1}^{1}]$, which it retains
through successive truncations. As for the middle summand, it still has one-
dimensional homology generated by $[Qz_{0}+z_{1}^{1}]$. We see that as an
$\mathbb{F}[U]$ module, the homology of the mapping cone is

$H_{*}(AI_{0}^{-})\simeq\mathbb{F}_{(1)}[U]\oplus\mathbb{F}_{(2n(K))}[U]\oplus\mathbb{F}_{(2n(K))}[U]/(U^{n(K)+1})\oplus\mathbb{F}_{(2n(K))}.$

Here the summand $\mathbb{F}_{(1)}[U]$ is generated by

$[U^{n(K)}(z^{1}_{v}+z^{1}_{v-2}+\dots+z_{0}+\dots+z^{2}_{v-2}+z^{2}_{v})+QU^{-1}c]$

and the summand $\mathbb{F}_{(2n(K))}[U]$ is generated by
$[z_{1}^{1}+Qz_{0}]$, which notably has the property that
$U^{n(K)}[z_{1}^{1}+Qz_{0}]=[QU^{n(K)}(z^{1}_{v}+z^{1}_{v-2}+\dots+z_{0}+\dots+z^{2}_{v-2}+z^{2}_{v})]$,
so that

$Q[U^{n(K)}(z^{1}_{v}+z^{1}_{v-2}+\dots+z_{0}+\dots+z^{2}_{v-2}+z^{2}_{v})+QU^{-1}c]=U^{n(K)}[z_{1}^{1}+Qz_{0}].$

As for the torsion summands, the summand $\mathbb{F}_{(2n(K))}[U]/(U^{n(K)})$
is generated by $[e]$, which notably has the property that
$U[e]=Q[z_{1}^{1}]$, and finally the summand $\mathbb{F}_{(2n(K))}$ is
generated by $[Qe]$. This implies that

$\displaystyle\underline{V\mkern-3.0mu}\mkern 3.0mu_{0}(K)$
$\displaystyle=-\frac{1}{2}(\operatorname{gr}([U^{n(K)}(z^{1}_{v}+z^{1}_{v-2}+\dots+z_{0}+\dots+z^{2}_{v-2}+z^{2}_{v})+QU^{-1}c])-1)$
$\displaystyle=-\frac{1}{2}(1-1)$ $\displaystyle=0$

and

$\displaystyle\overline{V}_{0}(K)=-\frac{1}{2}(\operatorname{gr}([z_{1}^{1}+Qz_{0}]))=-\frac{1}{2}(2n(K))=-n(K)=-\frac{m+n-2}{4}.$

$z_{v}^{2}$$z_{v-1}^{2}$$z_{v-1}^{1}$$z_{v}^{1}$$z_{2}^{2}$$z_{2}^{1}$$z_{0}$$z_{1}^{2}$$z_{1}^{1}$$e$$a$$Uz_{v}^{2}$$Uz_{v}^{1}$$b$$Ue$$c$

Figure 17. The dual $\overline{C}_{1}$ of the model complex $C_{1}$.

We now consider the dual $\overline{C}_{1}$ of the model complex $C_{1}$. This
complex, which appears in Figure 17, has generators $a,b,c,e,z_{0}$ and
$z_{r}^{s}$ for $1\leq r\leq v$ and $s\in\\{1,2\\}$. The nonzero differentials
are as follows:

$\displaystyle\partial(a)=b+c$ $\displaystyle\partial(b)=\partial(c)=Ue$
$\displaystyle\partial(z_{r}^{s})=z_{r-1}^{s}+z_{r+1}^{s},\ r\text{ is odd}$

The homology of $\overline{C}_{1}\otimes\mathbb{F}[U,U^{-1}]$ is generated by
$[z_{v}^{1}]$. The involution $\iota_{K}$ is given by:

$\displaystyle\iota_{K}(a)=a+z_{0}$ $\displaystyle\iota_{K}(b)=c$
$\displaystyle\iota_{K}(z_{0})=z_{0}+e$ $\displaystyle\iota_{K}(c)=b$
$\displaystyle\iota_{K}(z_{1}^{1})=z_{1}^{2}+U^{-1}c$
$\displaystyle\iota_{K}(z_{1}^{1})=z_{1}^{2}+U^{-1}c$
$\displaystyle\iota_{K}(z_{t}^{1})=z_{t}^{2}$
$\displaystyle\iota_{K}(z_{t}^{2})=z_{t}^{1}$ $\displaystyle\iota_{K}(e)=e$

The homology is generated as an $\mathbb{F}[U,U^{-1}]$ module by $[z_{0}]$.
Now consider the mapping cone
$\operatorname{Cone}((\overline{C}_{1}\otimes\mathbb{F}[U,U^{-1}])\xrightarrow{Q(1+\iota_{K})}Q(\overline{C}_{1}\otimes\mathbb{F}[U,U^{-1}]))$.
After a change of basis, the differentials are given by

$\displaystyle\partial^{\iota}(e)=0$ $\displaystyle\partial^{\iota}(Qe)=0$
$\displaystyle\partial^{\iota}(a)=(b+c)+Qz_{0}$
$\displaystyle\partial^{\iota}(Qa)=Q(b+c)$
$\displaystyle\partial^{\iota}(c)=Ue+Q(b+c)$
$\displaystyle\partial^{\iota}(Qc)=QUe$ $\displaystyle\partial^{\iota}(b+c)=0$
$\displaystyle\partial^{\iota}(Q(b+c))=0$
$\displaystyle\partial^{\iota}(z_{0})=Qe$
$\displaystyle\partial^{\iota}(Qz_{0})=0$
$\displaystyle\partial^{\iota}(z_{1}^{1})=z_{0}+z^{1}_{2}+Q(z_{1}^{1}+z^{2}_{1})+QU^{-1}c$
$\displaystyle\partial^{\iota}(Qz_{1}^{1})=Qz_{0}+Qz_{2}^{1}$
$\displaystyle\partial^{\iota}(z_{1}^{1}+z_{1}^{2})=z_{2}^{1}+z_{2}^{2}+QU^{-1}(b+c)$
$\displaystyle\partial^{\iota}(Q(z_{1}^{1}+z_{1}^{2}))=Q(z_{2}^{1}+z_{2}^{2})$
$\displaystyle\mathclap{\begin{aligned}
&\partial^{\iota}(z_{r}^{1})=\begin{cases}(Q(z_{r}^{1}+z_{r}^{2}),\ r\text{
even}\\\ z^{1}_{r-1}+z^{1}_{r+1}+Q(z_{r}^{1}+z_{r}^{2}),\ r\text{
odd}\end{cases}\\\ &\partial^{\iota}(z_{r}^{1}+z_{r}^{2})=\begin{cases}0,\
r\text{ even}\\\ z^{1}_{r-1}+z^{2}_{r-1}+z^{1}_{r+1}+z^{2}_{r+1},\ r\text{
odd}\end{cases}\\\ &\partial^{\iota}(Qz_{r}^{1})=\begin{cases}0,\ r\text{
even}\\\ Qz_{r-1}^{1}+Qz_{r+1}^{1},\ r\text{ odd}\end{cases}\\\
&\partial^{\iota}(Q(z_{r}^{1}+z_{r}^{2}))=\begin{cases}0,\ r\text{ even}\\\
Q(z_{r-1}^{1}+z_{r-1}^{2})+Q(z_{r+1}^{1}+z_{r+1}^{2}),\ r\text{
odd}\end{cases}\end{aligned}}$

This yields the direct sum decomposition in Figure 18. The complex on the top
row has one-dimensional homology generated by $[b+c]=[Qz_{0}]$, the middle row
has one-dimensional homology generated by $[z_{0}+QU^{-1}c]$, and the bottom
row has trivial homology.

$b+c$$a$$Qz_{0}$$Qz_{1}^{1}$$Qz_{2}^{1}$$Qz_{3}^{1}$$\dots$$Qz_{v-1}^{1}$$Qz_{v}^{1}$

$Qe$$QU^{-1}c$$z_{0}$$Q(z_{1}^{1}+z_{1}^{2})$$z_{1}^{1}$$Q(z_{2}^{1}+z_{2}^{2})$$z_{2}^{1}$$Q(z_{3}^{1}+z_{3}^{2})$$z_{3}^{1}$$Q(z_{4}^{1}+z_{4}^{2})$$z_{4}^{1}$$\dots$$Q(z_{v-1}^{1}+z_{v-1}^{2})$$\dots$$z_{v-1}^{1}$$z_{v}^{1}+z_{v}^{2}$$z_{v}^{1}$

$e$$U^{-1}c$$QU^{-1}(b+c)$$QU^{-1}a$$z_{1}^{1}+z_{1}^{2}$$z_{2}^{1}+z_{2}^{2}$$\dots$$z_{v}^{1}+z_{v}^{2}$

Figure 18. Direct summands appearing in the mapping cone
$\operatorname{Cone}(\overline{C}_{1}\otimes\mathbb{F}[U,U^{-1}]\xrightarrow{Q(1+\iota_{K})}Q(\overline{C}_{1}\otimes\mathbb{F}[U,U^{-1}]))$
after a change of basis.

$z_{0}$$e$$a$$b$$Ue$$c$$Uz_{2}^{2}$$Uz_{2}^{1}$$Uz_{1}^{2}$$Uz_{1}^{1}$$U^{v-1}z_{v-1}^{2}$$U^{v-1}z_{v-1}^{1}$$U^{v-1}z_{v}^{2}$$U^{v-1}z_{v}^{1}$$\iddots$$\iddots$$\iddots$

Figure 19. The $\mathbb{F}[U]$-submodule $A_{0}^{-}$ of $\overline{C}_{1}$.

Now we consider the subcomplex $A_{0}^{-}$, which appear in Figure 19. We see
that $A_{0}^{-}$ is generated by
$\\{a,b,c,e,z_{0},Uz_{1}^{r},Uz_{2}^{r},U^{2}z_{3}^{r},\dots,U^{n(K)+1}z_{v-1}^{r},U^{n(K)+1}z_{v}^{r}\\}$.
The homology of $A_{0}^{-}$ is moreover isomorphic to

$\mathbb{F}_{(-2n(K))}\oplus\mathbb{F}_{(-2n(K))}[U]$

where the $\mathbb{F}_{(-2n(K))}$ is generated by $[e]$ and the
$\mathbb{F}_{(-2n(K))}[U]$ summand is generated by $[z_{0}]$. We therefore
have $V_{0}(\overline{K})=n(K)=\frac{m+n-2}{4}$.

We now consider the homology of the mapping cone $AI_{0}^{-}$. We first
observe that every positive $U$-translate of
$[U(z_{2}^{1}+Q(z_{1}^{1}+z_{1}^{2}))]=U[z_{0}+QU^{-1}c]$ and $[b+c]=[Qz_{0}]$
appears in the homology of the mapping cone. The points $a$, $z_{0}$, $e$,
$Qz_{0}$ and $Qe$ at the origin become the summands $a\to Qz_{0}$, $z_{0}\to
Qe$, and $e$. Therefore the homology is of the form

$H_{*}(AI_{0}^{-})\simeq\mathbb{F}_{(-2n(K)-1)}[U]\oplus\mathbb{F}_{(-2n(K))}[U]\oplus\mathbb{F}_{(-2n(K))}$

where the $\mathbb{F}_{(-2n(K)-1)}[U]$ summand is generated by
$[U(z_{2}^{1}+Q(z_{1}^{1}+z_{1}^{2}))]=U[z_{0}+QU^{-1}c]$, the
$\mathbb{F}_{(-2n(K)-1)}[U]\oplus\mathbb{F}_{(-2n(K))}[U]$ is generated by
$[b+c]=[Qz_{0}]$, and the $\mathbb{F}_{(-2n(K))}$ summand is generated by
$[e]$. We therefore see that

$\displaystyle\underline{V\mkern-3.0mu}\mkern 3.0mu_{0}(\overline{K})$
$\displaystyle=-\frac{1}{2}\left(\operatorname{gr}(U[z_{0}+QU^{-1}c])-1\right)$
$\displaystyle=-\frac{1}{2}(-2n(K)-1-1)$ $\displaystyle=n(K)+1$
$\displaystyle=\frac{m+n+2}{4}$

and

$\overline{V}_{0}(\overline{K})=-\frac{1}{2}\left(\operatorname{gr}([Qz_{0}])\right)=-\frac{1}{2}(-2n(K))=n(K)=\frac{m+n-2}{4}$

as promised.∎

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# Coupling disorder in a population of swarmalators

Hyunsuk Hong  Department of Physics, Jeonbuk National University, Jeonju
54896, Korea  Research Institute of Physics and Chemistry, Jeonbuk National
University, Jeonju 54896, Korea  Kangmo Yeo  Department of Physics, Jeonbuk
National University, Jeonju 54896, Korea  Hyun Keun Lee  Department of
Physics, Sungkyunkwan University, Suwon 16419, Korea

###### Abstract

We consider a population of two-dimensional oscillators with random couplings,
and explore the collective states. The coupling strength between oscillators
is randomly quenched with two values one of which is positive while the other
is negative, and the oscillators can spatially move depending on the state
variables for phase and position. We find that the system shows the phase
transition from the incoherent state to the fully synchronized one at a proper
ratio of the number of positive couplings to the total. The threshold is
numerically measured, and analytically predicted by the linear stability
analysis of the fully synchronized state. It is found that the random
couplings induces the long-term state patterns appearing for constant
strength. The oscillators move to the places where the randomly quenched
couplings work as if annealed. We further observe that the system with mixed
randomnesses for quenched couplings shows the combination of the deformed
patterns understandable with each annealed averages.

Dynamic interaction, phase synchronization, coupled oscillators, synchronized
faster and slower rhythm

###### pacs:

05.45.-a, 89.65.-s

##  I  Introduction

In recent studies [1, 2], oscillators that sync and swarm, called
swarmalators, have been considered, and interesting long-term states have been
found. The swarmalators can move in the space under the correlated dynamics of
the phase and space variables of each oscillator. The mobile feature of the
oscillators is found to induce nonstationary states such as the active-phase
wave state and the splintered phase wave state. The collective property of
finite number of swarmalators is studied [3], the implementation in robots is
tried as an application [4], and various steady states by finite interaction-
distance is reported [5]. Mathematical property [6], model extension [7, 8,
9], and minimal modeling [10] are of basic interest. Application to swarming
and flocking of biological colony [11] is a primary interest. Control of
engineering objects like drones or robots is also an application front [12].

In the study of swarmalators, the interesting incoherent states are usually
the consequence of negative-definite coupling strength in phase dynamics (the
positive-definite case is also in Ref. [2] where noise is added). On the other
hand, the coupling characteristic among the constituents of many systems is
rather complex and not so simple. For example, the interaction among the
neurons in the neural network systems is given by the mixture with positive
and negative ones [13, 14], in general. As another example, Japanese frogs’
calling behavior can be understood by considering the mixture of positive and
negative interaction with each other [15]. Probably, the spin glass [16, 17],
to which the various interesting properties of condensed matter is attributed,
can be the representative example for the non-definite sign of the couplings.
Considering those features in nature, the mixed coupling with positive and
negative strength deserves to be considered for understanding the collective
behavior of the real systems, which motivates the present study.

In this paper, we consider a population of the oscillators that can sync and
swarm, governed by the random interaction in the phase dynamics. We explore
how the coupling-disorder affects the long-term states in the system. In
particular, we pay attention to the possibility of the phase transition [18]
in the system, and focus on whether the patterns observed in the absence of
coupling disorder still appear. The effective annealing of the randomly
quenched coupling strengths to their average is suggested in the mobility of
swarmalators to understand the numerical results. Phase transition to the
fully synchronized phase, reproducibility of the phases known in the original
model of the no coupling disorder, and mixture of deformed long-term states by
coupling disorders are explained from the viewpoint of the annealed couplings.

This paper is organized as follows: Section II introduces the model of coupled
swarmalators with random couplings, and Sec. III shows the collective behavior
and the phase transition in the system. In Sec. IV we derive the threshold of
the transition by the linear stability analysis of the fully synchronized
state. Various long-term states including the nonstationary ones are shown in
Sec. V, and the system with couplings of more than one quenched randomness are
understood as the combination of annealed-coupling systems in Sec. VI. A brief
summary is given in Sec. VII.

##  II  Model

The generalized model of $N$-coupled oscillators that we consider here is
given by

$\displaystyle\frac{d\theta_{i}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{j\neq
i}F(r_{ij})K_{ij}\sin(\theta_{j}-\theta_{i}),$ (1)
$\displaystyle\frac{d{\bf{r}}_{i}}{dt}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{j\neq i}G(r_{ij},\theta_{ij})$ (2)

for $i=1,\cdots,N$, where $\theta_{i}$ and ${\bf{r}}_{i}$ represent the phase
$(0\leq\theta_{i}\leq 2\pi)$ and the position vector of the $i$th oscillator,
respectively. $F(r_{ij})$ is a function for the spatial dynamics of the
oscillators with $r_{ij}=|{\bf{r}}_{ij}|$ for
${\bf{r}}_{ij}\equiv{\bf{r}}_{j}-{\bf{r}}_{i}$, and $G(r_{ij},\theta_{ij})$ is
a function for spatial dynamics with $\theta_{ij}\equiv\theta_{j}-\theta_{i}$.
$K_{ij}$ denotes the random coupling strength between the oscillators $i$ and
$j$, having the symmetric property $K_{ij}=K_{ji}$. It represents a sort of
bond coupling between the oscillators, instead of the site one ($K_{i}$) that
affects the oscillator itself. For simplicity, we randomly chose $K_{ij}$ from
the two-peaks distribution

$h(K_{ij})=p\delta(K_{ij}-K_{p})+(1-p)\delta(K_{ij}-K_{n}),$ (3)

where $p$ is the probability of the positive coupling, and $K_{p}>0$ and
$K_{n}<0$. For convenience, the ratio $Q\equiv-K_{n}/K_{p}(>0)$ has been
chosen for the control parameter. With the functions $F$ and $G$ in the model,
the phase variable $\theta$ and the spatial one $r$ become correlated, which
means the oscillators can move around in the space. Such correlation in the
dynamics of the space and the phase of the oscillators has been also studied
in Ref. [1], where the authors found five long-term states including the two
nonstationary ones. Note that the phase coupling in Ref. [1] does not have any
disorder. We here consider coupling disorder that is given by the random
values from the distribution in Eq. (3).

We notice that the special case with $F(r_{ij})=1$ and
$G(r_{ij},\theta_{ij})=0$ is equivalent to the mean-field $XY$ model [17] with
random coupling strength, governed by the Hamiltonian
${\cal{H}}=-\frac{1}{2N}\sum_{i\neq j}K_{ij}\cos(\theta_{j}-\theta_{i})$.
Equation (1) with $F=1$ and $G=0$ leads to the overdamped version of the
Hamiltonian dynamics at zero temperature, and interestingly it is found that
the first-order phase transition occurs at the same threshold $p_{c}$ as that
we study here [19].

In this paper, we consider $F=1/r_{ij}$. The function $G$ consists of the
attraction and repulsion forces acting on each oscillator, where the force
functions are taken as the algebraic ones with a power like $r_{ij}^{-a}$
following Ref. [1]. The model is then given by

$\displaystyle{\dot{\theta}}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{j\neq
i}\frac{K_{ij}}{r_{ij}}\sin(\theta_{j}-\theta_{i}),$ (4)
$\displaystyle\dot{\bf{r}}_{i}$ $\displaystyle=$
$\displaystyle\frac{1}{N}\sum_{j\neq
i}\left[\frac{{\bf{r}}_{ij}}{r_{ij}}(A+J\cos(\theta_{j}-\theta_{i}))-B\frac{{\bf{r}}_{ij}}{r^{2}_{ij}}\right],$
(5)

where $A$ and $B$ are respectively the parameters for the attractive force and
the repulsive force. Here, we choose $A=B=1$ for convenience. $J$ is the
parameter which measures how the phase similarity enhances the spatial
proximity. For example, a positive value of $J$ means like attracts like,
i.e., the swarmalators tend to be near the other swarmalators with similar
phases. On the other hand, a negative value of $J$ means the opposite case:
The swarmalators prefer to be near the others with opposite phases. To keep
the attractive force always positive, the values of $J$ are constrained like
$-1\leq J\leq 1$, following the Ref. [1]. The difference from the model of
Ref. [1] is the generalization of the constant $K$ to pair-dependent $K_{ij}$.
For $K_{ij}>0$, the swarmalators prefer to have the similar phase, but for
$K_{ij}<0$, the opposite tendency occurs. We are interested in the effect of
random $K_{ij}$ in the formation of long-term states.

##  III  Phase transition

Figure 1: (Color Online) (a) $R$ is shown as a function of $p$ for various
values of $J$; (b) $S$ vs. $p$ for various values of $J$. Parameters: $Q=1.25$
and $N=200$.

To see the collective states in the system and the effects of random coupling
on the states we perform the numerical simulations on Eqs. (4) and (5). We
first investigate the phase synchronization behavior of the oscillators, by
measuring the complex order parameter defined by [20]

$Re^{i\Theta}=\frac{1}{N}\sum_{j=1}^{N}e^{i\theta_{j}},$ (6)

where $R$ measures the phase coherence, and $\Theta$ is the mean phase angle
of the oscillators. For example, $R=0$ means the incoherent state where the
all oscillators have the random phase $\theta_{i}\in[0,2\pi)$, and $R=1$ is
the fully synchronized state where they all have the same one
$\theta_{j}=\theta_{s}$ for all $j$’s.

Also, to see the correlation between the phase angle $\theta_{j}$ and the
azimutal one $\phi_{j}=\tan^{-1}(y_{j}/x_{j})$ for the $j$th oscillator, which
is induced by the mobile feature of the oscillators in the system, we measure
another order parameter defined by [1]

$S_{\pm}e^{i\Psi_{\pm}}=\frac{1}{N}\sum_{j=1}^{N}e^{i(\phi_{j}\pm\theta_{j})},$
(7)

where $S_{\pm}$ is the magnitude, and $\Psi_{\pm}$ is its mean phase. Here,
the order parameter $S$ measures the “correlation” between the spatial
information (by the $\phi_{j}$) and the phase information (by the
$\theta_{j}$). The system exhibits “assortative and/or commutative”
correlation between the two depending on the initial conditions, where the
$S_{+}$ comes from the assortative correlation, and $S_{-}$ comes from the
commutative correlation. We chose $S$ by taking the maximum value of $S_{+}$
and $S_{-}$, i.e., $S={\rm{max}(S_{\pm})}$ [1].

In order to examine $R$ and $S$, we basically performed the numerical
simulations using the Python or C programs. The total $5\times 10^{4}$ time
steps with the discrete time unit $dt=0.1$ have been considered, where the
first $2.5\times 10^{4}$ steps were discarded for the equilibrium, and then
the time average of the quantities has been done for the later steps. And, the
ten samples have been used for the average. Initial phases and positions were,
respectively, randomly sampled from the uniform distributions over $[0,2\pi)$
and $[-1,1)\times[-1,1)$.

Figure 1 shows the behavior of $R$ and $S$ as a function of $p$ for various
values of $J$. The value of $Q=1.25$ was chosen, and $K_{p}=1$ and
$K_{n}=-1.25$ were chosen for convenience. We find that the system shows the
phase transition behavior from the incoherent state ($R=0$) to the fully
synchronized one ($R=1$) at a finite value of $p$, i.e., $p_{c}~{}\sim 0.6$.
Interestingly, we find that the behavior of $R$ and its threshold $p_{c}$ do
not depend on $J$, which can be easily understood by the absence of $J$ in the
phase dynamics given by Eq. (4). On the other hand, the behavior of $S$ is
different from that of $R$ as shown in Fig. 1: In the regime of the fully
synchronized state ($R=1$) for $p>p_{c}$, the order parameter $S$ is zero,
which implies no correlation between the phase angle and the spatial angle.
However, in the incoherent state with $R=0$ for $p<p_{c}$, $S$ shows finite
value, which means there is some correlation between the phase angle and the
spatial angle. Moreover, the value of $S$ for $p<p_{c}$ shows $S=1$ for a
certain region of $p$, and the range of $p$ of this region depends on the
value of $J$ as shown in Fig. 1 (b): The range gets wider as $J$ increases.

To pin down the threshold $p_{c}$, we now estimate it numerically. We measure
the value of $p$ at which the first derivative $dR/dp$ shows the maximum.
Figure 2 shows the behavior of $dR/dp$ as a function of $p$ for various values
of $Q$, whereas the value of $p$ at which the $dR/dp$ reaches the maximum is
denoted as $p^{*}$. The inset of Fig. 2 shows $p^{*}$ as a function of $Q$,
which shows a good consistency with the theoretical prediction $p_{c}=Q/(1+Q)$
(see Sec. IV).

Figure 2: (Color Online) The first derivative of $R$ over $p$, $dR/dp$, is
shown as a function of $p$ for various values of $Q$, where the value of $p$
at which $dR/dp$ reaches the maximum value is denoted as $p^{*}$. Inset:
$p^{*}$ vs. $Q$ is shown, where the blue solid line is the theoretical
prediction given by $p_{c}=Q/(1+Q)$, and the value of $p^{*}$ is consistent
with the prediction. The system size is $N=800$ and $J=0.8$, and the data have
been averaged over 10 samples.

##  IV  Linear stability of the fully synchronized state ($R=1$)

In this section, we investigate the linear stability of the fully synchronized
state with $R=1$. Let a fully synchronized state have a common phase
$\theta_{i}=\theta_{\rm s}$ for all $i$. In order to examine its linear
stability, we consider a slightly perturbed situation such as
$\theta_{i}=\theta_{\rm s}+\phi_{i}$ with small perturbation $\phi_{i}$ for
all $i$. For this setting, Eq. (4) is linearized to

$\dot{\phi}_{i}=\frac{1}{N}\sum_{j\neq
i}\frac{K_{ij}}{r_{ij}}(\phi_{j}-\phi_{i})\,.$ (8)

Separating the summation according to the values of $K_{ij}$, we rewrite Eq.
(8) as

$\displaystyle\dot{\phi}_{i}=\frac{1}{N}$ $\displaystyle\left(K_{p}\sum_{j\in
C_{p}(i)}\frac{1}{r_{ij}}(\phi_{j}-\phi_{i})\right.$ (9)
$\displaystyle\left.~{}~{}~{}+K_{n}\sum_{j\in
C_{n}(i)}\frac{1}{r_{ij}}(\phi_{j}-\phi_{i})\right),$

where $C_{p}(i)$ is the collection of $j$ having $K_{ij}=K_{p}$ and $C_{n}(i)$
is that for $K_{ij}=K_{n}$.

We here note that $K_{ij}$ is not included in the spatial dynamics [see Eq.
(5)]. Thus, for a fixed $i$, the fraction of such $j$’s giving $K_{ij}=K_{p}$
in the total $N$ would be approximately $p$, where the deviation is expected
to decrease as $N$ increases. Similarly, that for $K_{ij}=K_{n}$ is given by
$(1-p)$. This consideration motivates us to replace $\sum_{j\in C_{p}(i)}$ and
$\sum_{j\in C_{n}(i)}$ with $p\sum_{j(\neq i)}$ and $(1-p)\sum_{j(\neq i)}$,
respectively, in Eq. (9) when $N$ is large enough. Therefore, for large $N$,
Eq. (9) can be cast into

$\dot{\phi}_{i}=\langle K_{ij}\rangle\frac{1}{N}\sum_{j\neq
i}\frac{1}{r_{ij}}(\phi_{j}-\phi_{i})\,.$ (10)

where $\langle K_{ij}\rangle\equiv pK_{p}+(1-p)K_{n}$ is the average of
$K_{ij}$ for the two-peak distribution $h(K_{ij})$.

Introducing $m_{ij}=-1/(Nr_{ij})$ for $i\neq j$ and $m_{ii}=-\sum_{j\neq
i}m_{ij}$, one writes Eq. (10) as

$\dot{\phi}_{i}=-\langle K_{ij}\rangle\sum_{j=1}^{N}m_{ij}\phi_{j}\,.$ (11)

Interestingly, the matrix $m=\\{m_{ij}\\}$ is positive-definite for large $N$
as follows. It holds $|m_{ij}|\sim{\cal O}(1/\sqrt{N})$ for off-diagonal
terms. This is based on the numerical observation that all swarmalators reside
in the two-dimensional spatial region of the ${\cal O}(1)$-area and there is
no concentration of them. Thus, $r_{ij}\sim 1/\sqrt{N}$ holds, from which
$|m_{ij}|\sim{\cal O}(1/\sqrt{N})$ follows. It also follows that
$m_{ii}=-\sum_{j\neq i}m_{ij}\sim{\cal O}(\sqrt{N})$. Then, as $N$ increases,
$m=\\{m_{ij}\\}$ approaches $\bar{m}=\\{\bar{m}_{ij}\\}$, which is given by

$\bar{m}_{ij}=\delta_{ij}\bar{m}_{i}\,,$ (12)

where $\delta_{ij}$ is the Kronecker delta that assigns $1$ when it assigns
$i=j$ while $0$ otherwise. The matrix $\bar{m}$ is positive-definite because
$\bar{m}_{i}\equiv\lim_{N\to\infty}m_{ii}>0$. Therefore, $m$ is also positive-
definite for sufficiently large $N$.

When the positive definiteness of $m$ is considered in Eq. (11), one knows
that the sign of $\langle K_{ij}\rangle$ determines whether $\phi_{i}$ will
decay or not. This is because every eigenvalue of the positive-definite matrix
is positive. Thus, $\phi_{i}$ decays when $\langle K_{ij}\rangle>0$, otherwise
it grows when $\langle K_{ij}\rangle<0$. Since $Q\equiv-K_{n}/K_{p}$, we get
$\langle K_{ij}\rangle=K_{p}(p+(p-1)Q)$. Therefore, $\langle K_{ij}\rangle>0$
corresponds to $p>Q/(1+Q)$ and $\langle K_{ij}\rangle<0$ corresponds to
$p<Q/(1+Q)$, which leads to

$p_{c}=Q/(1+Q)\,.$ (13)

For $Q=1.25$, used for the numerical data shown in Figs. 1 and 2, Eq. (13)
gives $p_{c}=5/9=0.55..\,$. The deviation from the numerical value
$p_{c}\approx 0.6$ in the figures is supposed to be a finite $N$ effect. Note
$p_{c}\approx 0.565$ of Fig. 2 for $N=800$ is closer to the prediction
$p_{c}=5/9$ than the prediction in Fig. 1 for $N=200$. It is interesting that
the threshold value shown in Eq. (13) is equivalent to that for the mean-field
$XY$-type oscillators with random coupling disorder [19]. This means that the
spatial dynamics in the system does not change the threshold value of the
phase transition from the incoherent state to the fully synchronized one.

The crucial step of our argument above is the use of $\sum_{j\in
C_{p}(i)}=p\sum_{j(\neq i)}$ and $\sum_{j\in C_{n}(i)}=(1-p)\sum_{j(\neq i)}$
to write Eq. (10). We here remark this is basically an annealed approximation.
In fact, $K_{ij}$ is quenched by definition in the model and has never been
treated in an annealed way in the numerical work. Thus the consistency between
Eq. (13) and its numerical counterpart is interesting. Equation (5) gives a
clue to understanding how the quenched $K_{ij}$ works as if annealed. The
spatial dynamics governed by Eq. (5) is indifferent to $K_{ij}$. Based on
this, one may expect the distribution of $K_{ij}$ for given $i$ has no
position-dependence, i.e., the fractions for $K_{p}$ and $K_{n}$ are,
respectively, $p$ and $(1-p)$ independent of position. This results in the
annealed average of phase dynamics in Eq. (4). The consistency between the
numerical data and the annealed average, even in the presence of quenched
$K_{ij}$, means that the oscillators move to the proper places. That is, the
effective annealing of $K_{ij}$ to $\langle K_{ij}\rangle$ is a characteristic
property of swarmalator that can move.

We also examine that the sign of $\langle K_{ij}\rangle$ always governs the
onset of the sync state in further numerical study using other $h(K_{ij})$
distributions. These observations suggest that the annealed approximation
could be generally available. We remark that the effective annealing requiring
proper locations of swarmalators is considered for long-term states.

##  V  Non-stationary states in the incoherent regime ($R=0$)

Figure 3: (Color Online) Behavior of $S$, $R$, $V$, and $U$ is shown as a
function of $p$ for (a) $J=0.8$ and (b) $J=0.2$. Snapshots for the
corresponding regime are also shown in the insets. Parameters: $N=200$ and
$Q=0.8$.

In this section, we investigate the incoherent state $(R=0)$ further. In
particular, we pay attention to the possibility of the nonstationary states
such as the active phase wave (APW) state and the splintered phase wave (SPW)
state found in Ref. [1]. Therein, the all coupling strength among the
oscillators was chosen as the negative one with no disorder like $K_{ij}=K$,
which corresponds to $p=0$ in the current study. We note that, in Ref. [1],
various long-term states such as the async state, the APW state, the SPW
state, and the static phase wave state were found depending on the strength of
the negative coupling in the phase dynamics. Especially, the APW state and the
SPW state are the nonstationary states where the oscillators are not static in
both phase and space. It is thus interesting whether the dynamics of Eqs. (4)
and (5) with quenched disorder $K_{ij}$ can induce such a nonstationary
feature. Below, we again resort to the numerical data by the same scheme used
for Fig. 1.

To see the possibility of the nonstationary states for $p>0$, we measure the
following two quantities. One is the mean velocity $V$ defined by [1, 2]

$V=\frac{1}{N}\sum_{j=1}^{N}v_{i},$ (14)

where $v_{i}=\sqrt{{{\dot{x}}_{i}}^{2}+{{\dot{y}}_{i}}^{2}}$. The mean
velocity measures the nonstatic property of the state. In other words, the
finite value of $V$ means that the swarmalators move around in the plane in
the long-term state. The other one is [1, 2]

$U=\frac{N_{\rm{rot}}}{N},$ (15)

where $N_{\rm{rot}}$ is the number of swarmalators circulating at least
$2\pi$, and thus $U$ represents the fraction over the total swarmalators [1,
2]. Note that $U$ can play as the role of the indicator for the active phase
wave state [1]. Figure 3(a) and (b) show the quantities $R$, $S$, $V$, and $U$
as a function of $p$ for $J=0.8$ and $0.2$, respectively. As $p$ increases
from zero, we find that the phase transition occurs from the incoherent state
$(R=0)$ to the fully synchronized state ($R=1$) at a finite value of $p_{c}$,
as shown in the behavior of $R$. Interestingly, the value of $p_{c}$ for two
different values of $J$ are found to be same, which means the threshold does
not depend on $J$. On the other hand, the behavior of $S$, $U$, and $V$ is
found to differ depending on the value of $J$ (See Fig. 3).

When $R=0$, note that the system shows a finite value of $S$ through the
splintered and static phase wave. Also, the system shows the active phase wave
state for small $p$ when $J$ is large. This active phase wave does not occur
when $J$ is small. We find that the system shows both the stationary states
and the nonstationary states depending on $p$ for a given value of $Q$ and
$J$. Especially, when $J$ is large $(J=0.8)$, all five long-term states that
have been reported in Ref. [1] exist depending on $p$. It is interesting to
note that the nonstationary states such as active phase wave sate and
splintered phase wave also exist in the presence of the positive coupling
among the swarmalators in the system ($p>0$).

##  VI  Mixture of random couplings

Above, we have seen that the result by random $K_{ij}$ is comparable with that
by constant $K=K_{ij}$. The long-term state patterns, including the fully
synchronized one, from both settings are similar for large $J$. It is
remarkable that such analogy holds in the presence of finite faction for
negative $K_{ij}$ that may bring about the frustration [21, 22] in the phase
dynamics. This strongly suggests the annealed property conjectured for the
onset of the sync phase at $\langle K_{ij}\rangle=0^{+}$ is also valid for
non-zero $\langle K_{ij}\rangle$ cases. This motivates us to test the long-
term states in the system with such quenched random couplings by mixture of
different randomnesses. We are interested in whether the characters of long-
term states attributed to each randomnesses will last or not.

Consider $N_{1}$ number of swarmalators in total $N$, and call them group 1
(G1). The coupling strength $K_{ij}$, therein, is assigned with $1$ for
probability $p_{1}$ or $-1$ for $1-p_{1}$. Next, consider the other group, G2,
of $N_{2}(=N-N_{1})$ swarmalators, where $K_{ij}=\pm 1$ is similarly assigned
using probability $p_{2}$. We also consider coupling between the swarmalators
in both groups as assigning $1$ to $K_{ij}$ for probability $p_{\rm m}$ or
$-1$ for $1-p_{\rm m}$. Below, we use $p_{\rm m}=p_{2}$ for simplicity. When
$(N_{1},p_{1})=(100,1)$ and $(N_{2},p_{2})=(100,0)$, one may expect the sync
disc in G1 and the APW state in G2 if the couplings that work between the
groups do not overwhelm each character. Note the average coupling in G2,
$\langle K_{ij}\rangle_{2}=-1$, is in the region for the APW state in the
phase diagram reported in Ref. [1]. We obtained numerical data with time unit
$dt=0.01$ and consider $2\times 10^{5}$ time steps.

Figure 4(a) shows the numerical result.

Figure 4: (Color Online) (a) Long-term state pattern for $N_{1}=N_{2}=100$,
$p_{1}=1$, $p_{2}=0$, $p_{\rm m}=p_{2}$, and $J=1$. The inset is the
trajectory of an arbitrarily chosen swarmalator in the U-shaped region, which
sweeps the region. (b) Long-term state pattern for the same setting used in
panel (a) except $p_{2}=0.4$. The black solid curve is the trajectory of a
swarmalator localized there.

Two patterns appear; one is a slightly deformed sync disc while the other is
U-shaped one. We have observed the former vibrates a little (not demonstrated
here) and the latter is an active phase state. The inset in Fig.4(a) is the
trajectory of a swarm in the U-shape. Since it sweeps the region while
changing the phase, we name the state the deformed APW (dAPW). It is
interesting to note that this reminds us of the “chimera state” observed in
the oscillators with identical frequency [23, 24, 25, 26]: The partial group
of the swarmalators in the system exhibits the sync cluster, but the other
group of the swarmalators do not join the sync cluster, showing the U-shape
APW.

We vary $N_{1}$ to see a possible change in pattern. One easily see the
distance between the sync disc and the dAPW increases when comparing Figs.
4(a) and 5(a). We also see the sync disc becomes more circular and static when
the distance increases.

Figure 5: (Color Online) Long-term state patterns composed of sync disc and
dAPW for $N_{1}=60$, $N_{2}=140$, $p_{1}=1$, $p_{2}=0$ in (a) and for
$N_{1}=140$, $N_{2}=60$, $p_{1}=0.7$, $p_{2}=0$ in (b).

We next test whether a sync disc by $p_{1}<1$ in G1 only system ($N_{2}=0$)
will still last after G2 is involved ($N_{2}>0$). The interest is a validity
of the annealed approximation of $K_{ij}$ in G1. Figure 5(b) is the result for
$p_{1}=0.7$, which is also composed of the sync disc and the dAPW. The sync
disc is almost circular and static with enough separation from the dAPW.

We finally test whether G2 can show the SPW state while G1 remains a sync
disc. As the splintered state appears for the negative coupling larger than
that for APW, we use $p_{2}=0.4$, which corresponds to $\langle
K_{ij}\rangle_{2}=-0.2$, the average coupling strength in G2. This value is in
the region for the splintered wave in the phase diagram reported in Ref. [1].
Figure 4(b) is the result when the other settings are same as those for Fig.
4(a). There still appear sync disc and U-shaped region. This time, the
swarmalators of different colors in the U-shaped region are not mixed,
different from the dAPW case. The gradual change of the swarmalators’ colors
in space is a characteristic of the splintered phase wave. The black solid
curve near the bottom is the trajectory of a swarm, which remains near there
as time goes on. As shown, it does not sweep the U region but localized in the
small area while splintering. This is always the case for all swarmalators in
the U-shaped region. We thus regard the U-shaped pattern as a deformed
splintered phase wave (dSPW).

In the above numerical tests, the characters by each random couplings are
preserved more or less in the mixed system. There apparently appear sync disc,
APW, and splintered state though deformed. We here add the result is
qualitatively similar for the various values of $p_{\rm m}$ (not shown here),
and this does not depend on the initial condition. These observations suggest
that the long-term states, demonstrated in Secs IV and V, are still robust
more or less in the mixed system. Interestingly, the patterns survive the
frustration taking place between the subsystems G1 and G2. A mixture with
async state or of more than two groups was not tested this time. Further study
on various interesting properties of the mixed deformed patterns will appear
elsewhere.

##  VII  Summary

We considered the population of swarmalators with random coupling strength,
and explored how the coupling disorder affects the long-term states in the
system. In particular, the possibility of the phase transition and the
robustness of the state patterns are focused. To understand the long-term
states observed in the system with quenched disorder of coupling strength, we
considered the effective annealed approximation, which is mediated by the
mobility of swarmalators. In the viewpoint of annealed couplings, the
numerical observation in the quenched system is explained and, furthermore,
such a system of the mixture of different quenched disorders is also
understood.

We found that the system shows the phase transition from the incoherent state
to the fully synchronized state at a certain threshold $p_{c}$, where the
value of $p_{c}$ is argued in the linear stability analysis of the fully
synchronized state. We also found that, in the regime of the incoherent state
below the threshold, various long-term states are found to exist. Especially,
the nonstationary states such as the dAPW and the dSPW besides the normal APW
and SPW are discovered.

All long-term states known for the bare model in Ref. [1] are realized in the
system of random coupling strength. The values of average-coupling where each
state appears are similar to their counterparts in the bare model. This
strongly suggests the randomly quenched coupling strengths work as if
annealed. The pattern by the supposed annealed coupling is so robust that the
mixture of different random couplings leads to the proper combination of each
deformed patterns.

##  VIII Acknowledgements

This research was supported by NRF Grant No. 2021R1A2B5B01001951 and ‘Research
Base Construction Fund Support Program’ funded by Jeonbuk National University
in 2021 (H.H), and by NRF Grant No. 2018R1D1A1B07049254 (H.K.L.).

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