Instead of considering the strong involution on $K \# K^{r}$, one might consider the obvious periodic involution $\tau^{p}_{0}$ on $K \# K$, shown in Figure <ref>. This action was already computed by Juhász,  and Zemke in <cit.>. Using a slightly different argument it is possible to show that $\underline{d}_{\tau}(S^{3}_{p}(K \# K), \tau^{p}_{0})$, $\bar{d}_{\tau}(S^{3}_{p}(K \# K), \tau^{p}_{0})$ are also concordance invariants.

§ EXAMPLES

In this Section, we give explicit computation for the action of symmetry on $CFK^{\infty}$ for two classes of symmetric knots. This in turn leads to computations of $\underline{d}_{\tau}$ and $\bar{d}_{\tau}$ invariants.  But before moving ahead, for ease of computation, we would like to show that we can use any chain complex filtered homotopic to that coming from $CFK^{\infty}(\mathcal{H}_{K})$ for a given $\H_K$. This is similar to the observation made in [18].

A tuple $\mathfrak{C}_{p}=(C,\mathcal{F},M, \partial, \tau_{p})$ is called $CFKI_{\tau_{p}}$ data if $C$ is a free, finitely generated $\mathbb{Z}_2[U,U^{-1}]$ module with a $\mathbb{Z}$ grading $M$, and a $\mathbb{Z} \oplus \mathbb{Z}$-filtration $\mathcal{F}(i,j)=(i,j)$. The module $C$ is further equipped with a differential $\partial$ which preserves the filtration $\mathcal{F}$ and decreases the grading $M$ by $1$. Finally, $\tau_{p}$ is a grading preserving filtered quasi-isomorphism on $C$. Similarly one can define a tuple $\mathfrak{C}_{s}=(C,\mathcal{F},M, \partial, \tau_{s})$ with exact same properties except in this case $\tau_{s}$ is a skew-filtered.

With this in mind one can define a notion of quasi-isomorphism between two such  $CFKI_{\tau_{p}} $-data (respectively $CFKI_{\tau_{s}}$) by following [18], leading to the following Proposition:

Given a $CFKI_{\tau_{p}}$ data $\mathfrak{C_{p}}=(C,\mathcal{F},M, \partial, \tau_{p})$-data, let $(C^{\prime},\mathcal{F}^{\prime},M^{\prime},\partial^{\prime})$ be another chain complex but without the quasi-isomorphism $\tau^{\prime}_{p}$, which is filtered homotopy equivalent to $(C,\mathcal{F},M, \partial)$. Then there is a grading preserving filtered quasi-isomorphism $\tau^{\prime}_{p}$ on $C^{\prime}$ such that $\mathfrak{C}_{p}$ and $(C^{\prime},\mathcal{F}^{\prime},M^{\prime},\partial^{\prime},\tau^{\prime}_{p})$ are quasi-isomorphic as $CFKI_{\tau_{p}}$-data.
Analogous conclusions also holds for the $CFKI_{\tau_{s}}$-data.

We also note that chain isomorphism in Theorem <ref> can be made absolutely graded after a grading shift. This in turn results in the following relation as given below, see <cit.> for a discussion.

\begin{equation}\label{gradingshift}
\begin{aligned}
\underline{d}_{\circ}(S^{3}_{p}(K),[0]) &= \frac{p-1}{4} + \underline{d}_{\circ}(A^{-}_{0},[0]), \\
\bar{d}_{\circ}(S^{3}_{p}(K),[0]) &= \frac{p-1}{4} + \bar{d}_{\circ}(A^{-}_{0},[0]).
\end{aligned}
\end{equation}

where $\circ \in \{ \tau , \iota \tau \}$. We now state the examples.

§.§ Strongly invertible L-space knots and their mirrors

As discussed in the Introduction, many L-space knots admit a strong involution. Here we show that we can explicitly compute the involution for those knots and their mirrors.

Ozsváth and Szabó showed that the knot Floer homology of these knots is determined by their Alexander polynomial [38]. Specifically, the knot Floer homology $CFK^{\infty}(K)$ of an L-space knot is chain homotopic to $C \otimes \mathbb{Z}_{2}[U,U^{-1}]$, where $C$ is a staircase complex as shown in Figure <ref>. In a similar fashion, the knot Floer homology of the mirror of an L-space knot can be taken to be copies of staircases shown in Figure <ref>.

Examples of model $CFK^{\infty}$ for L-space knots (on the left), and mirror of L-space knots (on the right).

We define the quantity $n(K)$ as follows. Let $\Delta_{K}(t)$ denote the Alexander polynomial for an L-space knot, written as

\[
\Delta_{K}(t)= (-1)^{m} + \sum_{i=1}^{m} (-1)^{m-i} (t^{n_{i}} + t^{-n_{i}})
\]

where $n_{i}$ are a strictly increasing sequence of positive integers. Then $n(K)$ defined as

\[
n_{K}= n_m - n_{m-1} + \cdots + (-1)^{m-2} n_2 + (-1)^{m-1} n_1.
\]

Recall that in <cit.>, Hendricks and Manolescu computed the $\spinc$-conjugation action $\iota_K$ for L-space knots and its mirrors. Analogously, we have the following computation.

Let $K$ be an L-space knot (or the mirror of an L-space knot) that is strongly invertible with the strong-involution $\tau_{K}$. We have

\[
\iota_{K} \simeq \tau_{K}.
\]

In particular, we get

* $\underline{d}_{\tau}(S^{3}_{p}(K),[0])=\bar{d}_{\tau}(S^{3}_{p}(K),[0])= \frac{p-1}{4} - 2n(K)$,    when $K$ is an L-space knot,

* $\underline{d}^{\tau}(S^{3}_{p}(K),[0])=\frac{p-1}{4}$, $ \bar{d}^{\tau}(S^{3}_{p}(K),[0])=\frac{p-1}{4} + 2n(K)$,   when $K$ is mirror of an L-space knot.

Here $p$ is an integer such that $p \geq g(K)$.

Recall that if $\tau_{K}$ is a strong involution on a knot $K$, the induced map on $CFK^{\infty}$ is a skew filtered map that squares to the Sarkar map (see Proposition <ref>). Furthermore, the Sarkar map is the identity for staircase complexes, [44]. Firstly, by Proposition <ref> we transfer both $\iota_K$ and $\tau_{K}$ to the model complexes depicted in Figure <ref>. As seen in [18], the fact that $\iota_K$ is grading preserving skew-filtered and it squares to identity is enough to uniquely determine it for both L-space knots and the mirrors of L-space knots. Hence the claim follows from <cit.> after observing the relation between $\underline{d}$ and $\bar{d}$ invariants with the $\underline{V}_{0}$ and $\overline{V}_{0}$ invariants.

Using the above we also prove Proposition <ref>.

Montesinos trick [29] imply $(S^{3}_{n}(\overline{T}_{p,q}),\tau)$ is equivariantly diffeomorphic to double branch cover $(\Sigma_{2}(L_{n,p,q}),\tau_{n,p,q})$. The proof then follows from the Theorem <ref>, <cit.> and Proposition <ref> after observing that $n \geq (p-1)(q-1)/2$ is a large surgery for the mirrored torus knots $\overline{T}_{p,q}$.

§.§ Periodic involution on  L-space knots and their mirrors

Several L-space knots admit a periodic involution. For Example, it is easy to visualize that $(2,2n+1)$-torus knots admit a periodic involution. For such knots, we have the following.

Let $K$ be an L-space knot (or mirror of an L-space knot) with a periodic involution $\tau_K$. we have

\[
\tau_K \simeq \mathrm{id}.
\]

In particular, we get,

* $\underline{d}_{\tau}(S^{3}_{p}(K),[0])=\bar{d}_{\tau}(S^{3}_{p}(K),[0])=\frac{p-1}{4} - 2n(K)$,    when $K$ is an L-space knot,

* $\underline{d}_{\tau}(S^{3}_{p}(K),[0])= \bar{d}_{\tau}(S^{3}_{p}(K),[0])=\frac{p-1}{4}$,   when $K$ is mirror of an L-space knot.

Here $p$ is an integer such that $p \geq g(K)$.

Note that when $\tau_K$ is a periodic involution, the induced map on the knot Floer complex is a grading preserving filtered map that squares to the Sarkar map (see Proposition <ref>). For the case in hand, we get $\tau^{2}_{K} \simeq \textrm{id}$. The constraints then imply $\tau_K= \textrm{id}$. Finally note that $d(S^{3}_{p}(K),[0])=\frac{p-1}{4} - 2n(K)$ if $K$ is an L-space knot, and $d(S^{3}_{p}(K),[0])=\frac{p-1}{4}$ of $K$ is the mirror of an L-space knot and that $\tau= \textrm{id}$ implies $\underline{d}^{\tau}=d=\bar{d}^{\tau}$ <cit.>.

Using Corollary <ref>, we can also compute the $\underline{d}_{\iota \tau}$ and $\bar{d}_{\iota \tau}$ invariants for L-space knots. Similarly, as a parallel to <cit.>, we can compute $\underline{d}_{\tau}$, $\bar{d}_{\tau}$  for thin knots using   Theorem <ref>. We omit the computation for brevity.

§.§ Computing $\tau$-local equivalence class using the surgery formula

Associated to $(Y,\tau)$, an integer homology sphere $Y$ equipped with an involution $\tau$, the invariant $h_{\tau}:= [(CF^{-}(Y), \tau)]$ was studied in [8]. Here $ [(CF^{-}(Y), \tau)]$ represents the local equivalence class of the $\iota$-complex $(CF^{-}(Y), \tau)$, see Definition <ref>. In [8], $h_{\tau}$-invariant was bounded using equivariant cobordism and monotonicity results in local equivalence associated with it. Here, we show that in certain situations the equivariant surgery formula is enough to bound $h_{\tau}$ (in the sense of the partial order shown in [9]).

We consider the right-handed trefoil $T_{2,3}$ and $(-1)$-surgery on $T_{2,3}$. Note that $T_{2,3}$ has two different symmetries, one strong $\tau^{s}$ and one periodic $\tau^{p}$, as seen in Figure <ref>.

Symmetries of $T_{2,3}$.

Proposition <ref> and the Proposition <ref> immediately imply that  $\tau^{s}$ and $\tau^{p}$ action on $HF^{+}(S^{3}_{+1}(\overline{T}_{2,3}))$ are as in Figure <ref>.

It follows that, $\tau$-local equivalence class $h_{\tau}(S_{-1}^{3}(T_{2,3}), \tau^{s}) < 0$ and $h_{\tau}(S_{-1}^{3}(T_{2,3}), \tau^{p}) = 0$. This gives an alternative proof of <cit.>. Note that we are implicitly using that $\tau$-action is well-behaved under orientation reversal isomorphism, the proof of which follows from a discussion similar to that in <cit.>.

Readers may compare the Example above with  <cit.>.

We now look at the $(+1)$-surgery on figure-eight knot $4_1$, and the periodic involution $\tau^{p}$ on it, as in Figure <ref>. We identify this involution on the knot Floer complex of $4_1$ using Theorem <ref>. The resulting action on $HF^{-}(S^{3}_{p}(4_1))$ is shown in Figure <ref>. As before we get that the local equivalence class $h_{\tau}((S^{3}_{+1}(4_1),\tau^{p}) < 0$, recovering <cit.>.

Here we consider the periodic involution $\tau^{p}$ on the Stevedore knot $6_1$ as depicted in Figure <ref>. Again as a consequence Theorem <ref>, we obtain the resulting action on the $CFK^{\infty}$ complex.

Left: Periodic symmetry $\tau^{p}$ of $6_1$. Right: action of $\tau^{p}$ on $CFK^{\infty}(6_1)$.

It follows that $\tau^{p}$ acts on $HF^{-}(S^{3}_{+1}(6_1))$ as shown in Figure <ref>.

Left: The action of $\tau^{p}$ on $HF^{-}(S^{3}_{+1}(6_1))$.

The above computation also implies that the local equivalence class $h_{\tau}(S^{3}_{+1}(6_1),\tau^{p})$ is trivial.

In fact, it is known that $(6_1, \tau^{p})$ is equivariantly slice <cit.>. By a standard argument, it can be shown that this implies the induced involution extends over the contractible manifold that $S^{3}_{+1}(6_1)$ bounds. On the contrary, it was shown in <cit.> that the two strong involutions on $S^{3}_{+1}(6_1)$ coming from the two strong involutions on $6_1$ does not extend over any homology ball that $S^{3}_{+1}(6_1)$ may bound.

In Lemma <ref>, we show that for diagonally supported thin knots there is at most one periodic type action. However, sometimes the grading and filtration restrictions allow us to deduce the uniqueness of periodic type actions for thin knots that are not diagonally supported. For example, the action of the periodic involution $\tau^{p}$ on the $\overline{6}_2$ knot, shown in Figure <ref> can be uniquely determined from the constraints. We then compute the action of $\tau^{p}$ on $HF^{+}(S^{3}_{2}(\overline{6}_2),[0])$ in Figure <ref>.

Left: Periodic involution $\tau^{p}$ on $\overline{6}_2$, Right: $CFK^{\infty}(\overline{6}_2)$, with $\tau^{p}$ action.

The action of  $\tau^{p}$ on $HF^{+}(S^{3}_{2}(\overline{6}_2),[0])$.

In <cit.> it was shown two different strong involutions $\tau$ and $\sigma$ acting on figure-eight knot induce different actions on $HF^{-}(S^{3}_{+1}(4_1))$. This was done by showing $h_{\tau}(S^{3}_{+1}(4_1)) < 0$, while $h_{\sigma}(S^{3}_{+1}(4_1))=0$. By the equivariant surgery formula, we at once obtain that $\tau$ and $\sigma$ acts differently on $CFK^{\infty}(4_1)$. This is in contrast to the periodic type and $\iota_K$ action on $4_1$, both of which are unique. Interested readers may consult <cit.> for the explicit computation of the $\tau$, $\sigma$ actions on $CFK^{\infty}(4_1)$.

§.§ Identifying Cork-twists involutions

As discussed in the introduction, the surgery formula enables us to identify the cork-twist involution for corks that are obtained as surgeries on symmetric knots with the action of the symmetry on the knot. Note that we do need that the $3$-genus of the knot to be $1$. Such type of corks include the $(+1)$- surgery on the Stevedore knot [8], $(+1)$-surgery on the $P(-3,3,-3)$ pretzel knot, [8] and the positron cork [13]. For example, it was shown in <cit.> that two strong involutions $\tau$ and $\sigma$ of the Stevedore knot results in strong corks. Hence, the equivariant surgery formula identifies those cork-twists with the action of $\tau$ and $\sigma$ on $A^{-}_{0}(6_1)$. We refer readers to <cit.> for an explicit computation of these actions.

§.§ Strong involution on $T_{2,3} \# T^{r}_{2,3}$

Here, using Theorem <ref>, we compute the strong involution $\tilde{\tau}_{sw}$ on connected sum of two copies of the right-handed trefoil (See Subsection <ref> for the definition of the $\tilde{\tau}_{sw}$ action). We also compute the concordance invariant associated with this action, see Section <ref>. We state the computation below.

Let $K$ represent the strongly invertible knot $ T_{2,3} \# T^{r}_{2,3}$. We have,

\begin{align*}
\underline{d}_{\tau}(A^{-}_{0}(K), \tilde{\tau}_{sw})& = \bar{d}_{\tau}(A^{-}_{0}(K), \tilde{\tau}_{sw}) = - 2. \\
\underline{d}_{\iota \tau}(A^{-}_{0}(K), \tilde{\tau}_{sw}) &= -4, \; \bar{d}_{\iota \tau}(A^{-}_{0}(K), \tilde{\tau}_{sw}) = - 2.
\end{align*}

From Proposition <ref>, we obtain that the action of $\tilde{\tau}_{sw}$ on $CFK^{\infty}(K \# K^{r})$ in  can be identified with the action of $(\mathrm{id} \otimes \mathrm{id} + \Psi \otimes \Phi) \circ \tilde{\tau}_{exch}$ on $CFK^{\infty} \otimes CFK^{\infty}(K^{r})$ (although we used the full version $\mathcal{CFK}$ for the Proposition, it can be checked that the maps involved have $0$ Alexander grading shift, hence they induce maps between the ordinary $CFK^{\infty}$ complexes). The knot Floer complex for $T_{(2,3)}$ and $T^{r}_{(2,3)}$ is shown in Figure <ref>.

$CFK^{\infty}(T_{2,3})$ on the left. For $CFK^{\infty}(T^{r}_{2,3})$, we use the complex that is obtained by reflecting the stair along the $y=x$ diagonal.

The complex $CFK^{\infty}(T_{2,3}) \otimes CFK^{\infty}(T^{r}_{2,3})$ is shown in Figure <ref>.

Left:  $CFK^{\infty}(T_{2,3}) \otimes CFK^{\infty}(T^{r}_{2,3})$, Right: Action of $\tilde{\tau}_{sw}$ and $\Psi \otimes \Phi$ on the basis elements.

Note that for the case in hand, $\tilde{\tau}_{exch}$ map can be computed explicitly. It exchanges the two factors, followed by a skew-filtered map in each factor. For example, we get $\tilde{\tau}_{exch}(x_1 \otimes x^{\prime}_2)=x_1 \otimes x^{\prime}_2$, while $\tilde{\tau}_{exch}(x_1 \otimes x^{\prime}_0)=x_0 \otimes x^{\prime}_2$ and so on. Also note that the map $\Psi \otimes \Psi$ is non-trivial only for the generator $a$. Hence the action of $\tilde{\tau}_{sw}$ is as shown in Figure <ref>. Using Equation <ref> we also compute the action of $\iota_K \tilde{\tau}_{sw}$.

Now, by looking at the $A^{-}_{0}$-complex of the tensor product, it follows that $[i]$ and $[j]$ both are in grading $-2$ and are fixed by $(\mathrm{id} \otimes \mathrm{id} + \Psi \otimes \Phi) \circ \tilde{\tau}_{exch}$. Hence from the Definition of $\underline{d}^{\tau}$, we get,

\[
\underline{d}_{\tau}(A^{-}_{0}(K), \tilde{\tau}_{sw})=-2.
\]

By a similar reasoning as in <cit.>, we can also see that the $gr(i)$ determines the $\bar{d}_{\tau}$ invariant. Hence we get,

\[
\bar{d}_{\tau}(A^{-}_{0}(K), \tilde{\tau}_{sw})=-2.
\]

For $\underline{d}_{\iota \tau}$, we observe that the $i$, and $j$ are the only two generators in grading $-2$, which are $U$-non-torsion. However, neither of them is fixed by $\iota_K \tilde{\tau}_{sw}$. It can be checked that $[Ui]$ is the highest graded $U$-non-torsion element which is fixed by $\iota_K \tilde{\tau}_{sw}$, hence we get

\[
\underline{d}_{\tau}(A^{-}_{0}(K), \tilde{\tau}_{sw})=-4.
\]

Calculation for $\bar{d}_{\iota \tau}$ is similar to the $\bar{d}$. Indeed, although the actions of $\iota_K$ and $\iota_K \tilde{\tau}_{sw}$ different, we see that $gr(i)$ still determines the $\bar{d}_{\iota \tau}$ invariant. We get,

\[
\bar{d}_{\tau}(A^{-}_{0}(K), \tilde{\tau}_{sw})=-2.
\]

Let us now compare the above invariants with similar invariants from the literature. The $d$-invariant defined by Ozsváth and Szabó [32], satisfies $d(A^{-}_0(K)) = -2$. While the involutive correction terms are $\bar{d}(A^{-}_{0}(K))= -2$ and $\underline{d}(A^{-}_{0}(K))=-4$, <cit.>. Hence this motivates the following question:

* Are there knots $K$, such that:

\[
\underline{d}_{\tau}(A^{-}_{0}(K \# K^{r}), \tilde{\tau}_{sw}) \neq {d}(A^{-}_{0}(K \# K^{r})) \; \text{or} \; \bar{d}_{ \tau}(A^{-}_{0}(K \# K^{r}), \tilde{\tau}_{sw}) \neq {d}(A^{-}_{0}(K \# K^{r}))?
\]

* Are there knots $K$, such that:

\[
\underline{d}_{\iota \tau}(A^{-}_{0}(K \# K^{r}), \tilde{\tau}_{sw}) \neq \underline{d}(A^{-}_{0}(K \# K^{r})) \; \text{or} \; \bar{d}_{\iota \tau}(A^{-}_{0}(K \# K^{r}), \tilde{\tau}_{sw}) \neq \bar{d}(A^{-}_{0}(K \# K^{r}))?
\]

Perhaps it is note worthy that

\begin{align*}
\iota_{K \# K^{r}} \circ \tilde{\tau}_{sw} &= (\mathrm{id} \otimes \mathrm{id} + \Phi \otimes \Psi) \circ (\iota_{K} \otimes \iota_{K^{r}}) \circ (\mathrm{id} \otimes \mathrm{id} + \Psi \otimes \Phi) \circ \tilde{\tau}_{exch} \\
&\simeq (\mathrm{id} \otimes \mathrm{id} + \Phi \otimes \Psi) \circ (\mathrm{id} \otimes \mathrm{id} + \Phi \otimes \Psi) \circ (\iota_{K} \otimes \iota_{K^{r}}) \circ \tilde{\tau}_{exch} \\
&\simeq \iota_{K} \otimes \iota_{K^{r}} \circ \tilde{\tau}_{exch}.
\end{align*}

Where in the second line we have used that $\iota_K$ interchanges $\Phi$ and $\Psi$, and in the third line we have used $\Phi^{2}=\Psi^{2} \simeq 0$, see <cit.>. Hence the map $\iota_{K \# K^{r}} \circ \tilde{\tau}_{sw}$  do indeed (non-trivially) differ from $\iota_{K \# K^{r}}$, which makes the case for an affirmative answer to Question <ref>, number $(2)$.

§ AN INVARIANT OF EQUIVARIANT KNOTS
We end with an observation, which leads us to define a Floer theoretic invariant of the equivariant knots. Let us begin by recalling that given a doubly-based knot $K$ inside a $\mathbb{Z}HS^{3}$, $Y$, Hendricks and Manolescu <cit.> considered a mapping complex $CI$ of the map

\[
Q.(\mathrm{id} + \iota_K): CFK^{\infty}(Y, K, w, z) \rightarrow Q.CFK^{\infty}(Y, K, w, z)[-1].
\]

They show that $CI(Y,K,w,z)$ is an invariant of the tuple $(Y,K,w,z)$. However, this mapping cone does not have a $\mathbb{Z} \oplus \mathbb{Z}$-filtration induced from that of $CFK^{\infty}(Y,K,w,z)$. This is because $\iota_K$ is skew-filtered, but the identity map is filtered, so there is no natural filtration in the mapping cone.
On the contrary, if we start with a periodic knot $(Y,K, \tau, w, z)$ and consider a similar mapping cone complex $CI_{\tau}$ of the map;

\[
Q.(\mathrm{id} + \tau_K): CFK^{\infty}(Y, K, w, z) \rightarrow Q.CFK^{\infty}(Y, K, w, z)[-1],
\]

there is a natural $\mathbb{Z} \oplus \mathbb{Z}$-filtration on it. This holds because the action $\tau_K$ is filtered. In particular, we obtain an invariant of periodic knots that has more information than its involutive counterpart. Recall the notion of equivalence between two periodic knots,

We call two periodic knots $(Y_1, K_1, \tau_1, w_1, z_1)$ and $(Y_2, K_2, \tau_2, w_2, z_2)$ equivalent if there exist a diffeomorphism

\[
\phi:(Y_1, K_1, w_1, z_1) \rightarrow (Y_2, K_2, w_2, z_2),
\]

which intertwines with $\tau_1$ and $\tau_2$.

Let $(Y,K, \tau, w,z)$ be a periodic knot inside a $\mathbb{Z}HS^{3}$, $Y$. Then the $\mathbb{Z} \oplus \mathbb{Z}$-filtered quasi isomorphism type of the mapping cone complex $CI_{\tau}(Y, K, \tau, w, z)$ (defined above) over $\mathbb{Z}_{2}[Q, U, U^{-1}]/(Q^{2})$ is an invariant of the equivalence class of $(Y,K,\tau, w,z)$.

Let $(Y_1,K_1,\tau_1)$ and  $(Y_2,K_2,\tau_2)$ be two tuples in the same equivalence class. The diffeomorphism invariance of the link cobordisms proved by Zemke <cit.> implies that the following diagram commutes up to chain homotopy.

\[\begin{tikzcd}[column sep =large, row sep =large]
CFK^{+}(Y_1, K_1) \arrow{r}{\phi} \arrow{d}{\tau_{K_1}} & CFK^{+}(Y_2, K_2) \arrow{d}{\tau_{K_2}} \\
CFK^{+}(Y_1, K_1) \arrow{r}{\phi}  & CFK^{+}(Y_2, K_2)
\end{tikzcd}
\]

Hence this induces a quasi-isomorphism between the corresponding mapping cone complexes which respects the filtration.

In a similar manner, for a strongly invertible knot $(Y,K, \tau, w,z)$, we refer to the mapping cone of

\[
Q.(\mathrm{id} + \iota_K \tau_K): CFK^{\infty}(Y, K, w, z) \rightarrow Q.CFK^{\infty}(Y, K, w, z)[-1]
\]

as $CI_{\iota \tau}(Y, K, \tau, w, z)$. We have:

Let $(Y,K, \tau, w,z)$ be a strongly invetible knot inside a $\mathbb{Z}HS^{3}$, $Y$. Then the $\mathbb{Z} \oplus \mathbb{Z}$-filtered quasi isomorphism type of the mapping cone complex $CI_{\iota \tau}(Y, K, \tau, w, z)$ over $\mathbb{Z}_{2}[Q, U, U^{-1}]/(Q^{2})$ is an invariant of the equivalence class of $(Y,K,\tau, w,z)$.

Hence we have:

Follows from Proposition <ref> and Proposition <ref>.

It follows from the computations in Section <ref> that for any L-space knots both $CI_{\iota \tau}$ (for strongly invertible knots) and $CI_{\tau}$ (for periodic knots), become trivial i.e. determined by the $CFK^{\infty}(K)$. On the other hand for thin knots, both invariants are potentially non-trivial owing to the Proposition <ref>. Moreover, by sending a conjugacy class of $[(Y, K ,\tau, w, z)]$ to the conjugacy class of $[(Y_{p}(K),\tau,[0])]$ (for a large $p$, which is an odd number), we see that equivariant involutive correction terms are invariants of $[(Y, K ,\tau, w, z)]$. Hence they can be used to demonstrate the non-triviality of $CI_{\tau}$ and $CI_{\iota \tau}$.

Let $(4_1, \tau^{p})$ be the figure-eight knot with a periodic symmetry $\tau^{p}$, as in Figure <ref>. Then the quasi-isomorphism type of the mapping cone $CI_{\tau}(4_1, \tau^{p}, w, z)$ is non-trivial. One may see this by computing the mapping cone explicitly, staring from the description of $\tau^{p}$ action on $CFK^{\infty}(K)$. However, instead of going through the full computation, we show that the invariant is non-trivial by appealing to the invariant $\underline{d}_{\tau}$ and showing that it differs from the $d$-invariant. By examining the action in Figure <ref>, we see that $\underline{d}_{\tau}(A^{-}_{0})=-2$, while $d(A^{-}_{0})=0$. The grading shift from the Relation <ref> then implies

\[
\underline{d}_{\tau}(S^{3}_{p}(4_1),[0])=\frac{p-1}{4}-2, \; \; d(S^{3}_{p}(4_1),[0])=\frac{p-1}{4}.
\]

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# Modelling Paralinguistic Properties in Conversational Speech to Detect
Bipolar Disorder and Borderline Personality Disorder

###### Abstract

Bipolar disorder (BD) and borderline personality disorder (BPD) are two
chronic mental health conditions that clinicians find challenging to
distinguish based on clinical interviews, due to their overlapping symptoms.
In this work, we investigate the automatic detection of these two conditions
by modelling both verbal and non-verbal cues in a set of interviews. We
propose a new approach of modelling short-term features with visibility-
signature transform, and compare it with widely used high-level statistical
functions. We demonstrate the superior performance of our proposed signature-
based model. Furthermore, we show the role of different sets of features in
characterising BD and BPD.

Index Terms—  bipolar disorder, borderline personality disorder, speech
analysis, paralinguistic modelling, path signature

##  1 Introduction

Bipolar disorder (BD) is a mood disorder characterised by extreme mood swings
between manic highs and depressive lows that can last from days to weeks each.
Borderline personality disorder (BPD) is a type of personality disorder marked
by a long-term pattern of varying moods, self-image and behaviour. They both
can seriously affect the patients’ ability to function in work and social
activities [1, 2, 3]. They also can co-occur in 10%-20% of the cases and since
the symptomatology of these two disorders is very similar, differentiating
between them poses a diagnostic challenge for the clinicians [4, 5]. However,
accurate diagnosis is crucial, as the most effective treatment for BD being
pharmacological while for BPD psychotherapeutic. Standard diagnostic
assessment for BD and BPD rely primarily on clinical interviews where the
patients have to describe their accounts of symptoms. As a result it can be
potentially influenced by patients’ retrospective recall biases and cognitive
limitations [6].

In recent years a number of machine learning based studies have investigated
the use of acoustic features from speech for automated assessment across
several psychiatric disorders showing diagnostic potentials [6, 7, 8].
Existing work on bipolar disorder showed mood episodes affecting patients’
speech and in turn acoustic and dialogue features extracted from speech can be
used in detecting mood states [9, 10, 11]. In this work, we aim to model both
verbal and non-verbal cues in non-clinical interviews for the distinction
between BD and BPD, which remains understudied.

One key step of dialogue-based automated assessment system is modelling a
series of conversational utterances and aggregate features from each segment
to a fixed-size representation for down-stream classification or regression.
Traditionally, a very popular approach has been to apply a set of high-level
statistical functions (e.g. mean, max, variance, linear regression
coefficients, etc.) for feature aggregation [9, 10, 11, 12], where the role of
these aggregation functions is to describe the global characteristics of given
spoken conversation. However, the conversational dynamics is not effectively
modelled during this process and important sequential information may be
ignored as a result. Recurrent neural networks (RNN) are designed to
explicitly model sequential data, however, they are prone to overfit on small
datasets. Signature transform, initially introduced in stochastic analysis, is
a non-parametric approach of encoding sequential data and capturing the order
information in the data. It has been proven effective in a range of machine
learning tasks involving sequential modelling [13, 14, 15]. A recent study
[16] proposed summarising linguistic and turn-taking behaviour features from
recorded interviews using signature. However, minimal description of its
working was provided, and no comparative evaluation was given.

The contributions of this work are as follows: (1) We study three types of
acoustic features for the classification of BD and BPD patients; (2), We
propose using visibility transform to enhance the signature representation of
conversational speech; (3), We evaluate and compare between high-level
statistical functions (HSF) and signature transform (SIG) based models, and
show the superiority of SIG.

##  2 AMoSS-I Dataset

The Automated Monitoring of Symptoms Severity Interview (AMoSS-I) dataset [16]
contains 50 participants, who were interviewed and transcribed to gather
qualitative feedback of the original AMoSS study. Among these participants, 21
had a BD diagnosis, 17 had been diagnosed with BPD and 12 were healthy
controls. Each participant was interviewed only once.

Among the 50 one-on-one qualitative interviews in AMoSS-I, 32 were recorded in
the meeting room while the remaining 18 were phone interviews. As summarised
in Table 1, the phone interviews are more likely to be shorter than the ones
conducted in the meeting room. We also take the difference of the peak and
trough values for Root-mean-square (RMS) level measured over a 10 ms window,
showing the difference of loudness within each audio recording. Comparing to
the meeting room interviews, on average the phone interviews are shown to have
significant larger difference of loudness within each one, mainly due to the
way of recording. Additionally, we find the noise level in the room interviews
to be higher as seen in their much lower signal-to-noise ratio of 21.16 dB,
computed by Waveform Amplitude Distribution Analysis (WADA-SNR) [17].

Table 1: Differences in sample size and acoustics between the room and phone interviews. $N$ is the number of the interviews. Audio length in minutes summarised in the form of the median +/- the interquartile range. The percentages of clipped samples are averaged over each set of interviews. RMS diff is the difference between the peak and trough values for RMS level measured over a 10 ms window, in dBFS. Signal-to-noise ratio (SNR) measures the amount of non-speech in a speech signal in decibels (dB). Env | $\bf{N}$ | Length | %Clipped | RMS diff | SNR
---|---|---|---|---|---
Room | 32 | $23\pm{12}$ | 10.28% | 64.91 | 21.16
Phone | 18 | $19\pm{10}$ | 3.48% | 83.10 | 95.30
Both | 50 | $22\pm{11}$ | 7.83% | 71.46 | 47.85

###  2.1 Data Preprocessing

The two different recording environments also resulted in different levels of
clipping. As shown in Table 1, clipping occurs much more often in the meeting
room interview recordings than the phone interviews, with an average of 10.28%
of the audio signals being clipped from their maximum range. We have reduced
the level of clipping by extrapolating the clipped parts of the audio using an
open-source digital audio editor Audacity111https://www.audacityteam.org/. In
order to alleviate the effect of loudness difference shown in Table 1, we
scale the audio signal for each speaker turn separately, and make sure each
turn is in the range of -1 and 1. We apply a domain-adversarial neural network
based voice activity detection model [18] for intra-speaker-turn segmentation.
We also exclude speaker turns that are shorter than 2 seconds to extract
robust acoustic features.

##  3 Feature Extraction

Instead of using more complex features such as mel frequency cepstral
coefficients (MFCCs) or high-dimensional embeddings from pretrained speech
encoders, we identify a set of acoustic and (non-verbal) turn-taking behaviour
related features for their interpretability.

Prosodic features: During these dyadic interviews, participants often
exhibited changes in prosodic variables such as the tone and intonation of
their speech when they are recollecting their experience especially from the
intensive week of the study. We compute Legendre polynomial coefficients for
fundamental frequency ($\textup{F}_{0}$) and energy contours. Together with
segment duration they form 13 dynamic features to capture the prosodic
variations. Prosodic features have been proven effective in identifying mood
states in BD [6, 9].

Rhythm features: We also extract 7 rhythm features from each speaker-turn
using algorithm proposed by Tilsen and Arvaniti [19] that is based on
empirical mode decomposition of the vocalic energy amplitude envelope. The
envelope is decomposed into two intrinsic mode functions (IMF). They show the
IMF-based rhythm features can capture information about periodicities that
likely correspond to different linguistic constructs, and thus are useful for
examining rhythmicity in speech. Previous studies also used them for mood
state detection [10, 11].

Phonation features: Voice quality has an important role in signalling
paralinguistic information [20], and previous study showed significant
difference in the speaker’s voice quality when comparing people who suffer
from psychological disorder to healthy controls [21]. We compute 7 phonation-
based dynamic features from sustained vowels and continuous speech utterances,
including jitter, shimmer, amplitude perturbation quotient and pitch
perturbation quotient.

Dialogue features: To model high-level interactive patterns in the dialogue,
we extract the set of 13 turn-taking behaviour related features following the
work in [11, 16], including relative floor control, turn hold offset, number
of consecutive turns, turn switch offset, speech overlaps and number of words
per second, per turn.

All acoustic and dialogue features are Z-normalised either: 1) using the mean
and standard deviation of each interview respectively, denoted as “Person”, or
2) using the mean and standard deviation of all training samples, denoted as
“Global”. Previous studies [10, 22] suggest normalisation by “Person” focuses
on each individual baseline and can reduce the potential extraneous effect,
for example, caused by different recording environment.

##  4 Feature Aggregation

The paralinguistic properties drawn from a patient’s response in an interview,
represented by the features described in Section 3, can be indicative of the
state of the patient at the time of speaking. However, the state of the
patient also evolves while being affected by the highly dynamic nature of a
dyadic conversation. The common approach of applying high-level statistical
functions (HSFs) does not effectively capture this dynamics. In this section
we describe the method of signature transform (SIG), which we use to aggregate
frame-level acoustic or turn-level dialogue features and encode nonlinear
time-dependent interactions in the feature set, $S(X)$.

A sequentially ordered data stream may be thought of as a discretisation of a
path of finite length $X:[a,b]\to\mathbb{R}^{d}$, where $a\leq b$ and
$d\in\mathbb{N}$. For example, a speech sequence represented by its phonation-
based dynamic features can be thought of as a path of $d=7$. Signature
transform, also known as path signature, describes a graded sequence of
statistics characterising the underlying path [23, 24], and thus provides an
effective feature set for capturing its total ordering (i.e., incremental
effect) while ignoring the positional effect. More specifically, consider a
$d$-dimensional path $X$ over the time interval $[a,b]$, the signature $S(X)$
of this path is the infinite collection of statistics222We refer the readers
to [15] Appendix A for a formal definition of signature transform, and [25]
for a primer on its use in machine learning.:

$\displaystyle S(X)_{a,b}=\left(\left\\{S(X)_{a,b}^{i_{1}}\right\\}_{1\leq
i_{1}\leq d},\left\\{S(X)_{a,b}^{i_{1},i_{2}}\right\\}_{1\leq i_{1},i_{2}\leq
d},\dots\right)$

where each term is a $n$-fold iterated integral of x with multi-index
$i_{1},\dots,i_{k}$:

$S(X)_{a,b}^{i_{1},\dots,i_{k}}=\underset{\underset{t_{1},\dots,t_{k}\in[a,b]}{t_{1}<\dots<t_{k}}}{\int\dots\int}dX_{t_{1}}^{i_{1}}\otimes\dots\otimes
dX_{t_{k}}^{i_{k}}$

where $k\in\mathbb{N}$. $S(X)_{a,b}^{i_{1},\dots,i_{k}}$ is termed as the
$k$th level of the signature. In practice we truncate the signature to order
$n$ to have a finite dimensional representation. Note the positional
information in the data stream can be informative in many applications. For
example, it’s useful to know the beginning and end of each speaker turn when
modelling the conversation. The visibility transform, initially introduced in
[26], is able to embed the effect of the absolute positions of the input
sequence into signature transform, which is otherwise not included in the
signature as it is translation-invariant map333The mathematical definition of
the visibility transform can be found in [27]..

Consider a $d$-dimensional data sequence of length $n$, i.e.,
$\textup{x}=\left(x_{1},\dots,x_{n}\right)$. The visibility transform in prior
to computing signature, adds two time steps and a binary coordinate to the
input sequence x that is equal to 1 until the second-to-last time step:

$\phi\left(\textup{x}\right)=\left(\mbox{ap}_{1}(x_{1}),\dots,\mbox{ap}_{1}(x_{n-1}),\mbox{ap}_{1}(x_{n}),\mbox{ap}_{0}(x_{n}),\mathbf{0}\right),$

where $\mbox{ap}_{c}:\mathbb{R}^{d}\to\mathbb{R}^{d+1}$ is an operator
expanding a $d$-dimensional vector to $d+1$ dimensions by appending scalar $c$
at the end, and $\mathbf{0}\in\mathbb{R}^{d+1}$. The resulting sequence
$\phi\left(\textup{x}\right)$ is $d+1$-dimensional and of length $n+2$. We
apply visibility transform to our frame-level acoustic features per speaker-
turn before feature aggregation. This way the start and end positions of each
turn can be embedded in the interview-level signature representation. We name
this approach as VT-SIG.

##  5 Experiments and Analysis

Following previous work we choose nested leave-one-subject-out as the
evaluation scheme, and logistic regression with L2 regularisation for
classification. For each fold, we first apply VT-SIG to each feature type, and
keep only the first three levels of the path signature444We use iisignature
Python library, https://pypi.org/project/iisignature/.. We conduct feature
selection on signature-transformed interview-level features through computing
Pearson Correlation Coefficients (PCC) with the IPDE scores555The
International Personality Disorder Examination (IPDE) [28] is a semi-
structured clinical interview designed to assess major categories of
personality disorders. on the training data and retain the features with
$p$-values less than 0.001. This results in a small number of features. The
selected features are then fed to the classifier for 3 separate binary tasks:
(1), BD vs. healthy controls, (2), BPD vs. healthy controls, and (3), BD vs.
BPD patients.

###  5.1 Analysis of the selected features

Five most significant and commonly selected features from each task are
briefly summarised in Table 7 as examples. Each interview-level feature (a
signature term) is represented as a linear combination of the original frame-
level or turn-level features. We see almost all of the selected features are
volume integrals, i.e., they are triple integrals of three acoustic/dialogue
features. We notice the importance of the binary coordinate (i.e., c in Table
7) from visibility transform, to the task of H vs. BD. For example, (c, apq,
logE) represents the nonlinear effect between apq and logE only at the last
frame of each turn of speech666If c is indexed first in the signature term,
then this term only captures the effect from the coordinates of the following
indices at the last time step. This is proved in Theorem 5 of [27]. while its
feature values before the last frame are shown to be unimportant. (IMF1_m, c,
c) and (c, IMF1_m), which represent the linear incremental effect of IMF1_m
and the linear effect of IMF1_m at the last frame (of each turn) respectively,
are also among the selected features.

While most of the important features for H vs. BD represent either purely
incremental or (last-)positional effects, the selected signature terms for the
other two tasks are of nonlinear mixed effects from both aspects. Overall, we
notice the importance of the phonation features in two tasks involving BD and
in particular, BD vs BPD. Rhythm features, which have previously used for mood
state detection, are selected in distinguishing between the BD/BPD patients
and healthy controls. For each acoustic/dialogue feature, we also analyse its
potential interaction effect with the recording environment, who the
interviewer is and the gender of the participant, which we treat as control
variables, by fitting a linear model with diagnosis as response. We find the
‘recording environment’ variable significantly changed the effect of (apq, c,
logE) ($p<0.05$). As a result we remove it from the final feature set for
classification, this way we only model effects that are not systematically
variable across situational factors.

Table 2: Top-5 most significant and commonly selected features during LOOCV.
Features belong to the dialogue category are colored in red; rhythm features
are in green and phonation features are in blue.777c: binary coordinate added
during visibility transform; apq: amplitude perturbation quotient, logE:
logaritmic energy; DF0/DDF0: first/second derivative of the fundamental
Frequency; IMF1_m: mean within-utterance instantaneous freq. of IMF1; IMF12:
ratio between IMF2 and IMF1; SPBr: ratio between power in envelope spectrum
bands (1/3.5/10 Hz); CNTR: envelope spectrum centroid computed over 1-10 Hz
band; TL: duration of each speaker turn; n_OVL: number of speech overlaps;
TSO: latency between speaker turn transitions; RFC_t: relative floor control
(time); SP_m: average length of short pauses; n_LP: number of long pauses
($>$500 ms).

H vs BD | H vs BPD | BD vs BPD
---|---|---
(c, apq, logE) | (SPBr, IMF12, IMF2_m) | (DF0, Jitter, logE)
(c, logE, apq) | (CNTR, IMF12, IMF2_m) | (logE, DDF0, Jitter)
(apq, c, logE) | (TL, n_OVL, TSO) | (logE, DF0, Jitter)
(c, IMF1_m) | (n_OVL, RFC_t, SP_m) | (logE, Jitter, DDF0)
(IMF1_m, c, c) | (n_OVL, RFC_t, n_LP) | (ppq, logE, DF0)

###  5.2 Results and Discussion

We first summarise the classification results obtained by using visibility
transform enhanced signature (VT-SIG) in Table 3. With the (late) fusion of
acoustic and dialogue features extracted from participants’ speech only, we
obtain a AUROC of 0.717 in H/BD, 0.841 in H/BPD and 0.716 in BD/BPD. The
classification performance drop sharply when we switch to interviewers’ speech
instead, due to the non-clinical nature of the interviews. Modelling the
interviews as a sequence of turns from both speakers (named “Both”) also
result in worse performance than learning from only the participants.

Table 3: Classification results for three binary tasks: H vs. BD, H vs. BPD
and BD vs. BDP, using logistic regression. Results shown are macro-averaged
$\mathrm{F_{1}}$ and AUROC scores across all interviews. $p$-value used for
feature selection: ‘*’$<$ 0.005, or else $<$ 0.001 is used.

| H/BD | H/BPD | BD/BPD
---|---|---|---
Subject | $\mathbf{F_{1}}$ | AUC | $\mathbf{F_{1}}$ | AUC | $\mathbf{F_{1}}$ | AUC
Participant | 0.738 | 0.738 | 0.827 | 0.841 | 0.710 | 0.716
Interviewer | 0.581 | 0.583 | 0.102* | 0.100* | 0.552* | 0.556*
Both | 0.477 | 0.488 | 0.512* | 0.515* | 0.683 | 0.686

We also compare results obtained by using different feature normalisation and
aggregation methods described in Section 3 and 4. First, we notice VT-SIG in
general has the better performance and is more reliable across all three
tasks. As for HSF though we have to increase the p-value feature selection
threshold from 0.001 to 0.005 or even 0.01 to have any feature for H/BD and
H/BPD, it still obtains relatively poor performance. Secondly, it is also
shown normalising features per interview (“Person”) has led to increased
performance for signature-based models with and without the use of visibility
transform (namely, VT-SIG and SIG). Different feature normalisation methods
have not made any significant impact on the overall performance of the HSF-
based models.

Table 4: Performance comparison among different feature aggregation
(Aggr)888For reasonable comparison we adopt a wide range of HSFs previously
used in [10]. and normalisation methods (Norm). Results shown are average
AUROCs across all interviews. $p$-value codes: ‘**’$<$0.001; ‘*’$<$0.005;
‘$\dagger$’$<$0.01.

Aggr | Norm | H/BD | H/BPD | BD/BPD
---|---|---|---|---
HSF | None | 0.381† | 0.544† | 0.681**
HSF | Global | 0.405† | 0.515† | 0.686**
HSF | Person | 0.575* | 0.556* | 0.543**
SIG | None | 0.554* | 0.699** | 0.644**
SIG | Global | 0.435* | 0.699** | 0.662**
SIG | Person | 0.661* | 0.686** | 0.716**
VT-SIG | None | 0.608** | 0.728** | 0.710**
VT-SIG | Global | 0.554** | 0.811** | 0.633**
VT-SIG | Person | 0.738** | 0.841** | 0.716**

##  6 Conclusions and Future Work

In this paper, we demonstrate the potential of using speech from non-clinical
interviews for detecting BD and BPD. Modelling short-term features and
generating final representation is key for any machine learning based mental
health assessment model. We propose the use of visibility-signature transform
that embeds sequential ordering for feature aggregation. We show the better
performance obtained by the proposed approach comparing with widely used high-
level statistical functions. For future work, we plan for new data collection
with more participants and multiple interviews per subject, from two different
locations, which will allow for longitudinal studies and cross-site
validation.

##  7 Acknowledgements

This work was supported by the MRC Mental Health Data Pathfinder award to the
University of Oxford [MC_PC_17215], by the NIHR Oxford Health Biomedical
Research Centre and by the The Alan Turing Institute under the EPSRC grant
EP/N510129/1. We would also like to thank John Gideon and Fasih Haider for
sharing code and thoughts.

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