$$ = \beta^{6n} \left( \frac{1}{2} \prod_{j=1}^{n-1} [b_j; b_{j-1}, \dots, b_1] \right)^N \sum_{s \in \mathbb{Z}_2^N} (2[b_n; b_{n-1}, \dots, b_1])^{|s|} e_0 \prod_{j=1}^{N} (\mathbf{f}_j^+ \mathbf{f}_j^-)^{s_j}. \qquad (7.14) $$

*Proof.* We claim first that the left hand side of (7.14) equals

$$ (-1)^{Nn} \beta^{2n} 2^{-N} \left( \prod_{j=1}^{n-1} [b_j; b_{j-1}, \dots, b_1] \right)^N \prod_{j=1}^{N} (e_0 + 2[b_n; b_{n-1}, \dots, b_1] e_0 \mathbf{f}_j^+ \mathbf{f}_j^-), $$

and we will proceed now to prove this via induction. Clearly $e_{0\text{TOP}} \in Z_{\text{ev}}$. For any $m \in \mathbb{N}$, we have

$$ T^m S = \begin{pmatrix} -m & 2 \\ -\frac{1}{2} & 0 \end{pmatrix}. $$

This implies the base case for induction on $n$, i.e. we have

$$ (\mathcal{T}_{Z_\Lambda}^{b_1} \circ \mathcal{S}_{Z_\Lambda})(e_{0\text{TOP}}) = (-1)^N \beta^2 2^{-N} \prod_{j=1}^{N} (e_0 + 2b_1 e_0 \mathbf{f}_j^+ \mathbf{f}_j^-). $$

Assume now that it is true up to $n$. Then

$$ \begin{aligned} & \left( (-1)^{Nn} \beta^{2n} 2^{-N} \left( \prod_{j=1}^{n-1} [b_j; b_{j-1}, \dots, b_1] \right)^N \right)^{-1} \left( \prod_{i=n+1}^{1} \mathcal{T}_{Z_\Lambda}^{b_i} \circ \mathcal{S}_{Z_\Lambda} \right) (e_{0\text{TOP}}) \\ &= (\mathcal{T}_{Z_\Lambda}^{b_{n+1}} \circ \mathcal{S}_{Z_\Lambda}) \prod_{j=1}^{N} (e_0 + 2[b_n; b_{n-1}, \dots, b_1] e_0 \mathbf{f}_j^+ \mathbf{f}_j^-) \\ &= \beta^2 \prod_{j=1}^{N} \left( 2e_0 \mathbf{f}_j^+ \mathbf{f}_j^- + 2[b_n; b_{n-1}, \dots, b_1] \left( -b_{n+1}e_0 \mathbf{f}_j^+ \mathbf{f}_j^- - \frac{1}{2}e_0 \right) \right) \\ &\stackrel{(*)}{=} (-1)^N \beta^2 \left( [b_n; b_{n-1}, \dots, b_1] \right)^N \prod_{j=1}^{N} (e_0 + 2[b_{n+1}; b_n, \dots, b_1] e_0 \mathbf{f}_j^+ \mathbf{f}_j^-), \end{aligned} $$

as claimed. In step (*) we used that by definition of the continued fraction in (3.14) we have

$$ [b_{n+1}; b_n, \dots, b_1] = b_{n+1} - \frac{1}{[b_n; b_{n-1}, \dots, b_1]} . $$

Multiplying out the product and using $\beta^4 = (-1)^N$ yields (7.14). $\square$

*Proof of Proposition 7.2 Part 2.* Because of the assumptions on the continued fraction, Proposition A.1 gives

$$ \prod_{j=1}^{n-1} [a_j; a_{j-1}, \dots, a_1] = \frac{q}{p} \prod_{j=1}^{n} [a_j; a_{j-1}, \dots, a_1] = q. $$

The formula then follows from Lemma 7.4. $\square$
7.3. **Categorical trace: Lyubashenko invariant.** By (3.25), it remains to compute $\varepsilon_{\mathcal{L}} \circ f(a)$ which we know from (7.6) to be equal to $\langle \Lambda | \tilde{f}(a) \rangle$ (recall that $\alpha = 1$ for $\mathbb{Q}(N, \beta)$).

Since by (7.8), $\Lambda$ is proportional to $\text{TOP}^*$, it can only see part of $\tilde{f}(a)$ which lives in $Z_{\Lambda}$. Using Proposition 7.2 we can simply read off

$$ \langle \Lambda | \tilde{f}(a) \rangle = 2\beta^{6n} \sum_{s \in \mathbb{Z}_2^N} p^{|s|} \left(\frac{q}{2}\right)^{N-|s|} \langle \text{TOP}^* | e_0 \prod_{j=1}^N (\tilde{f}_j^+ \tilde{f}_j^-)^{s_j} \rangle = \beta^{6n} p^N $$

Combining this with $\mathcal{D} = 1$ and $\delta = \beta^{-2}$ from Section 5.1.3, altogether (3.25) gives

$$ \text{Lyu}_{\Lambda}(\mathcal{L}(p, q)) = p^N. \qquad (7.15) $$

7.4. **Modified trace: Renormalised Lyubashenko invariant.** Here we compute the renormalised Lyubashenko invariant $L'_{\Lambda,t}(\mathcal{L}_x(\alpha, P))$, $x \in \{*, \circ\}$ with t the modified trace on the projective ideal. To do so, according to Proposition 7.1 we need to compute the central elements $\zeta_x$ and the pairings $(-, -)_U$.

We start with the explicit expressions for $\zeta_x$ for which we will need two preliminary lemmas. For $z \in Z(\mathbb{Q})$ write

$$ F(z) = \langle \Lambda | S^{-1}(\Phi_1 \Psi_1) \alpha z \Phi_2 M_1 \Psi_2 \rangle \cdot \Phi_3 M_2 \Psi_3 . $$

Note that $F(\tilde{f}(a)) = \zeta_0$.

**Lemma 7.5.** $F(z) = \langle \Lambda | z M_1 \rangle \cdot M_2$

*Proof.* From the expression (5.3) for the coassociator of $\mathbb{Q}$, we immediately see that the claim is true if $z$ is in the even sector, and therefore we must only show it for $z = e_1 z$. The first and second tensor factor of $\Phi^{\pm 1}$ in $F(z)$ are then multiplied by $e_1$, and we have

$$ (e_1 \otimes e_1 \otimes 1) \cdot \Phi^{\pm 1} = e_1 \otimes e_1 \otimes (e_0 K^N + e_1 \beta_{\pm}) . $$

Inserting this simplified expression in $F(z)$ gives

$$ \begin{aligned} F(z) &= \langle \Lambda | z M_1 \rangle (e_0 K^N + e_1 \beta_+) M_2 (e_0 K^N + e_1 \beta_-) \\ &= \langle \Lambda | z M_1 \rangle (e_0 K^N M_2 K^N + e_1 \beta_+ M_2 \beta_-), \end{aligned} \qquad (7.16) $$

where in the second step we used that $e_0, e_1$ are central. Since $z$ is central, it in particular commutes with $K$ and hence contains an even number of $\text{fs}$. Furthermore, $\Lambda \propto \text{TOP}^*$ and so the amount of $\text{fs}$ in the monodromy factors that contribute to the above expression is even as well. It follows that the non-zero summands $M_2$ commute with $K$ and therefore also with $K^N$ and $\beta_{\pm}$. Since $K^{2N} e_0 = e_0$ and $\beta_+ \beta_- = 1$, the claim follows. $\square$

Next we compute $F(z)$ for some special values of $z$:
**Lemma 7.6.** We have

$$
\begin{align*}
F(\phi_{X_1^{\pm}}) &= \pm 2^{-N} \phi_{P_0^+} + \frac{1}{2}(\phi_{X_1^+} + \phi_{X_1^-}), \\
F(\phi_{P_0^+}) &= 2^{N-1}(\phi_{X_1^+} - \phi_{X_1^-}), \\
F(e_0 \prod_{j=1}^N (f_j^+ f_j^-)^{s_j}) &= \beta^2 2^{N-2|s|} (-1)^{|s|} e_0 \prod_{k=1}^N (f_k^+ f_k^-)^{1-s_k} \quad \text{for } s \in \mathbb{Z}_2^N.
\end{align*}
$$

Proof. Recall from Section 5.1.2 that the monodromy matrix of $\mathcal{M}$ can be written as $\mathcal{M} = \sum_{I \in X} g_I \otimes f_I$, where the index set $X$ is $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \mathbb{Z}_2^N \times \mathbb{Z}_2^N$, and for $I = (a, b, c, d) \in X$, $f_I$ and $g_I$ are as in (5.7). Substituting (7.8) and (5.18), one computes

$$
\begin{align*}
F(\phi_{X_1^{\pm}}) &= \pm \beta^6 2^2 \sum_{I \in X} f_I \times \langle \mathrm{TOP}^* | e_1^{\pm} g_I \rangle \\
&\stackrel{(a)}{=} \pm \beta^6 2^2 \sum_{b,c} (-\beta^2)^b 2^{2|c|} (-1)^{b|c|} \cdot \mathcal{K} e_b \prod_{k=1}^N (f_k^- f_k^+)^{c_k} \times \langle \mathrm{TOP}^* | e_1^{\pm} \mathcal{K}^b \prod_{k=1}^N (f_k^+ f_k^-)^{c_k} \rangle \\
&\stackrel{(b)}{=} \pm \beta^6 2 \sum_{b,c} (-\beta^2)^b 2^{2|c|} (-1)^{b|c|} \cdot \mathcal{K} e_b \prod_{k=1}^N (f_k^- f_k^+)^{c_k} \\
&\qquad \times \left( \delta_{b,0} \delta_{c,1} \langle \mathrm{TOP}^* | e_1 \rangle \pm i \delta_{b,1} \langle \mathrm{TOP}^* | e_1 \prod_{k=1}^N (\delta_{c_k,0} \mathbf{1} - (2\delta_{c_k,0} + \delta_{c_k,1}) \mathbf{f}_k^+ \mathbf{f}_k^-) \rangle \right) \\
&\stackrel{(c)}{=} \pm \beta^6 \left( 2^{2N} \mathcal{K} e_0 \prod_{k=1}^N \mathbf{f}_k^- \mathbf{f}_k^+ \mp i (-1)^N \beta^2 \cdot \mathcal{K} e_1 \prod_{k=1}^N \sum_{c_k=0}^1 (-4)^{c_k} (2\delta_{c_k,0} + \delta_{c_k,1}) (\mathbf{f}_k^- \mathbf{f}_k^+)^{c_k} \right) \\
&\stackrel{(d)}{=} \pm \beta^6 (-1)^N 2^{2N} \mathcal{K} e_0 \mathrm{TOP} - i 2^N \mathcal{K} e_1 \prod_{k=1}^N (\mathbf{1} - 2\mathbf{f}_k^+ \mathbf{f}_k^-).
\end{align*}
$$

In step (a) we used that $e_1^\pm e_a = \delta_{a,1}e_1^\pm$, that the evaluation against $\mathrm{TOP}^*$ can be non-zero only if each $f_k^+$ is paired with $f_k^-$, resulting in $d=c$, and that $e_1 f_k^+ \omega_- f_k^- \omega_- = -e_1 f_k^+ f_k^-$, giving an additional factor $(-1)^{|c|}$. In step (b) we substituted the explicit form of $e_1^\pm$ in (5.15) and used $e_1(\mathbf{1} - 2f_k^+ f_k^-)(f_k^+ f_k^-)^{c_k} = e_1(\delta_{c_k,0}\mathbf{1} - (2\delta_{c_k,0} + \delta_{c_k,1})f_k^+ f_k^-)$. For step (c) note that only the product with all $f_k^+ f_k^-$ can be non-zero against $\mathrm{TOP}^*$, resulting in a factor $\prod_{k=1}^N (2\delta_{c_k,0} + \delta_{c_k,1})$. In addition, we replaced $\sum_c \prod_{k=1}^N \rightsquigarrow \prod_{k=1}^N \sum_{c_k=0}^1$. The extra factor of $(-1)^N$ in both summands in step (d) arises from reordering the fs.

To write this in our preferred basis of the centre, note from (5.15) and (7.9) that

$$
\begin{align*}
\phi_{P_0^+} &= 2^{3N} \beta^2 \mathcal{K} e_0 \mathrm{TOP}, \\
\phi_{X_1^+} + \phi_{X_1^-} &= 2^{N+1} (e_1^+ - e_1^-) = -2^{N+1} i e_1 \mathcal{K} \prod_{j=1}^N (\mathbf{1} - 2\mathbf{f}_j^+ \mathbf{f}_j^-), \\
\phi_{X_1^+} - \phi_{X_1^-} &= 2^{N+1} (e_1^+ + e_1^-) = 2^{N+1} e_1.
\end{align*}
$$
From this we read off

$$F(\phi_{X_1^{\pm}}) = \pm 2^{-N} \phi_{P_0^+} + \frac{1}{2}(\phi_{X_1^+} + \phi_{X_1^-}),$$

as claimed. For the remaining two equalities, first note that for $z \in Z(Q)$ in the even sector, i.e. $z = e_0z$, we have

$$F(z) = \beta^6 2^{-N+1} \sum_{I \in X} f_I \langle \text{TOP}^* | z g_I \rangle \\ = \beta^6 2^{-N+1} \sum_{b,c} 2^{2|c|} (-1)^{b|c|} \langle \text{TOP}^* | z \mathbf{K}^b \prod_{k=1}^N (\mathbf{f}_k^+ \mathbf{f}_k^-)^{c_k} \rangle e_b \prod_{k=1}^N (\mathbf{f}_k^- \mathbf{f}_k^+)^{c_k}.$$

This immediately yields

$$F(\phi_{P_0^+}) = 2^{2N} e_1 = 2^{N-1}(\phi_{X_1^+} - \phi_{X_1^-})$$

and

$$F(e_0 \prod_{j=1}^{N} (\mathbf{f}_j^+ \mathbf{f}_j^-)^{s_j}) = \beta^6 2^{-N} \sum_{b,c} 2^{2|c|} \langle \text{TOP}^* | \prod_{j=1}^{N} (\mathbf{f}_j^+ \mathbf{f}_j^-)^{s_j+c_j} \rangle e_0 \prod_{k=1}^{N} (\mathbf{f}_k^- \mathbf{f}_k^+)^{c_k} \\ = \beta^6 2^{N-2|s|} (-1)^{N-|s|} e_0 \prod_{k=1}^{N} (\mathbf{f}_k^+ \mathbf{f}_k^-)^{1-s_k}.$$

<figure><img src="image_14.png" /></figure>

With these preparations, we can compute $\zeta_*$, $\zeta_0$ from Proposition 7.1:

**Lemma 7.7.** We have

$$\begin{aligned} \zeta_* &= \langle \Lambda | \widehat{f(a)} \rangle \cdot \mathbf{1} = \beta^{6n} p^N \cdot \mathbf{1} \\ \zeta_0 &= F(\widehat{f(a)}) = 2^{-2N} \left( c_{n+1}^0 \phi_{P_0^+} + c_{n+1}^+ \phi_{X_1^+} + c_{n+1}^- \phi_{X_1^-} \right) \\ &\quad + \beta^{6n} p^N \sum_{s \in \mathbb{Z}_2^N} \left(-\frac{2q}{p}\right)^{|s|} e_0 \prod_{j=1}^N (\mathbf{f}_j^+ \mathbf{f}_j^-)^{s_j}. \end{aligned}$$

In the expression for $\zeta_0$ we have extended the recursive definition of $c_i^{0,\pm}$ in (7.12) to $n+1$ by setting $a_{n+1} = 0$.

*Proof:* The expression for $\zeta_*$ was already computed in Section 7.3. To get the result for $\zeta_0$, recall from Proposition 7.2 the decomposition of $\widehat{f(a)}$ into summands in $Z(Q) = Z_P \oplus Z_\Lambda$. Using the formulas from the preceding lemma, we get

$$\begin{aligned} F(\widehat{f(a)}|_{Z_\Lambda}) &\stackrel{\text{Prop. 7.2}}{=} 2^N \beta^{6n+2} \sum_{s \in \mathbb{Z}_2^N} p^{|s|} \left(\frac{q}{2}\right)^{N-|s|} F(e_0 \prod_{j=1}^N (\mathbf{f}_j^+ \mathbf{f}_j^-)^{s_j}) \\ &\stackrel{\text{Lem. 7.6}}{=} 2^N \beta^{6n+4} e_0 \prod_{j=1}^N \sum_{s_j=0}^1 p^{s_j} \left(\frac{q}{2}\right)^{1-s_j} 2^{1-2s_j} (-1)^{s_j} (\mathbf{f}_j^+ \mathbf{f}_j^-)^{1-s_j} \\ &= 2^N \beta^{6n+4} e_0 \prod_{j=1}^N (q\mathbf{f}_j^+ \mathbf{f}_j^- - p2^{-1}\mathbf{1}) \end{aligned}$$
$$ = \beta^{6n} p^N \sum_{s \in \mathbb{Z}_2^N} \left(-\frac{2q}{p}\right)^{|s|} e_0 \prod_{j=1}^{N} (\mathbf{f}_j^+ \mathbf{f}_j^-)^{s_j} . $$

On the projective centre $Z_P$, we find

$$ \begin{aligned} F(\widehat{f}(a)|_{Z_P}) &\stackrel{\text{Prop. 7.2}}{=} 2^{-2N} (c_n^0 F(\phi_{P_0^+}) + c_n^+ F(\phi_{X_1^+}) + c_n^- F(\phi_{X_1^-})) \\ &\stackrel{\text{Lem. 7.6}}{=} 2^{-2N} \left( 2^{-N} (c_n^+ - c_n^-) \phi_{P_0^+} + (c_n^0 2^{N-1} + \frac{1}{2}(c_n^+ + c_n^-)) \phi_{X_1^+} \right. \\ &\qquad \left. + (-c_n^0 2^{N-1} + \frac{1}{2}(c_n^+ + c_n^-)) \phi_{X_1^-} \right). \end{aligned} $$

The coefficients match the recursive definition of $c_{n+1}^0$, $c_{n+1}^\pm$ from (7.12) under the convention that $a_{n+1} = 0$. Adding the two expressions gives the claimed result for $\zeta_\circ$. $\square$

This completes the computation of $\zeta_\ast$ and $\zeta_\circ$ and we now turn to the pairings $(-, -)_P$ from (7.4). Let $t$ be the modified trace associated with the symmetrised cointegral in (5.10), which was explicitly given by

$$ \hat{\lambda}^r(\mathbb{K}^{m_{\text{TOP}}}) = c_t \delta_{m,\text{odd}}(\beta^2 + i^m) \quad \text{where } c_t = (-1)^N 2^{-N}. $$

Fix an isotopic decomposition $\mathcal{Q} = \bigoplus_U n_U P_U$ with idempotents $e_U$ and where the sum runs over distinct irreducibles $U$. We abbreviate $\{z, z'\}_U := (z, z')_{n_U P_U}$. In terms of the symmetrised cointegral, the pairing is given by

$$ \{z, z'\}_U = \langle \hat{\lambda}^r | e_U z z' \rangle. \qquad (7.17) $$

Explicitly, $n_{X_0^\pm} = 1$ and $n_{X_1^\pm} = 2^N$, so that

$$ \{z, z'\}_{X_0^\pm} = (z, z')_{P_0^\pm}, \quad \{z, z'\}_{X_1^\pm} = 2^N (z, z')_{X_1^\pm}. \qquad (7.18) $$

The next lemma gives the value of the pairing on the basis (7.11) of $Z(\mathcal{Q})$.

**Lemma 7.8.** (1) For $U = X_0^\pm$ we have

$$ \{e_0, \phi_{P_0^+}\}_{X_0^\pm} = 2^{2N-1} $$

and

$$ \left\{ e_0 \prod_{j=1}^{N} (\mathbf{f}_j^+)^{r_j} (\mathbf{f}_j^-)^{s_j}, e_0 \prod_{j=1}^{N} (\mathbf{f}_j^+)^{t_j} (\mathbf{f}_j^-)^{u_j} \right\}_{X_0^\pm} = \pm \varepsilon \delta_{r+t,1} \delta_{s+u,1} 2^{-N-1} \beta^{-2}, $$

where $\varepsilon = \varepsilon(r, s, t, u)$ is a sign, which is +1 if $r = s$ or $t = u$. For all other combinations of basis elements in (7.11), the pairing is zero.

(2) For $U = X_1^\pm$, the only non-zero values of the pairing are

$$ \{\phi_{X_1^\pm}, \phi_{X_1^\pm}\}_{X_1^\pm} = \pm 2^{2N+1} $$
Proof. Note first of all that the map $z \mapsto \{z, -\}_U$ is zero on the odd part of $z$ if $P_U$ is from the even sector. The same is true when swapping even and odd.

(1) For $U = X_0^\pm$ we have $e_U = e_0^\pm$ as in (5.14). From (7.9) we see that $\phi_{P_0^+} = 2^{3N}\beta^2 K e_0^{\text{TOP}}$ already contains TOP as a factor and so the only non-zero pairing can be with the basis element $e_0$. We compute

$$ \{e_0, \phi_{P_0^+}\}_{X_0^\pm} = 2^{3N} \beta^2 \langle \hat{\lambda}^r | e_0^\pm K^{\text{TOP}} \rangle = 2^{2N-1}. $$

Let now $r, s, t, u \in \mathbb{Z}_2^N$ with $|r| + |s|$ and $|t| + |u|$ even. We have

$$ \left( e_0 \prod_{j=1}^{N} (f_j^+)^{r_j} (f_j^-)^{s_j} \right) \cdot \left( e_0 \prod_{j=1}^{N} (f_j^+)^{t_j} (f_j^-)^{u_j} \right) = \varepsilon e_0 \prod_{j=1}^{N} (f_j^+)^{r_j+t_j} (f_j^-)^{s_j+u_j}, $$

where $\varepsilon$ is a sign depending on $r, s, t, u$, which we do not compute explicitly. If $r = s$ or $t = u$, this sign is +1 as a pair $f_j^+ f_j^-$ commutes with everything in the even sector. This shows that

$$ \left\{ e_0 \prod_{j=1}^{N} (f_j^+)^{r_j} (f_j^-)^{s_j}, e_0 \prod_{j=1}^{N} (f_j^+)^{t_j} (f_j^-)^{u_j} \right\}_{X_0^\pm} = \varepsilon \delta_{r+t,1} \delta_{s+u,1} \langle \hat{\lambda}^r | e_0^{\pm \text{TOP}} \rangle \stackrel{(6.1)}{=} \pm \varepsilon \delta_{r+t,1} \delta_{s+u,1} 2^{-N-1} \beta^{-2}. $$

(2) For the odd sector, recall from (5.18) that $\phi_{X_1^\pm} = \pm 2^{N+1} e_1^\pm$, and since $e_1^\pm$ are orthogonal, we only need to compute:

$$ \{\phi_{X_1^\pm}, \phi_{X_1^\pm}\}_{X_1^\pm} = 2^{2N+2} \langle \hat{\lambda}^r | e_1^\pm \rangle \stackrel{(6.2)}{=} \pm 2^{2N+1} \quad (7.19) $$

as claimed.

<figure><img src="image_15.png" /></figure>

Now we are ready to provide the invariants for all natural endomorphisms of the identity functor, acting on the non-contractible loop in $L(p, q)$ as given in (3.19).

**Theorem 7.9.** We have, for the basis elements in (7.11) and $t \in \mathbb{Z}_2^N$,

$$ \begin{align*} L'_{\Lambda,t}(\mathcal{L}_0(e_0 \prod_{j=1}^N (f_j^+ f_j^-)^{t_j}, P_0^{\pm})) &= \begin{cases} \frac{1}{2} c_{n+1}^0 \beta^{2n} \pm \frac{1}{2} \beta^2 q^N, & t=0, \\ \pm(-1)^{|t|} \beta^2 2^{-|t|-1} q^{N-|t|} |p|^{|t|}, & |t| > 0 \end{cases} \\ L'_{\Lambda,t}(\mathcal{L}_0(\phi_{P_0^+}, P_0^{\pm})) &= 2^{2N-1} p^N \\ L'_{\Lambda,t}(\mathcal{L}_0(\phi_{X_1^\pm}, X_1^{\pm})) &= \pm 2^{1-N} \beta^{2n} c_{n+1}^{\pm}. \end{align*} $$

Note that combining the first formula in the theorem above for $t = 0$ with Remark 7.3 results in the invariant $L'_{\Lambda,t}(\mathcal{L}_0(\text{id}, P_0^\pm))$ as stated in Theorem 3.19.

Proof. Recall from Proposition 7.1 that $L'_{\Lambda,t}(\mathcal{L}_0(\alpha, P)) = \delta^{-\sigma} \mathcal{D}^{-1-n}(\hat{\alpha}, \zeta_0)_P$, with $\sigma = n$ by (3.17) and $\zeta_0$ is given in Lemma 7.7. From Section 5.1.3 we have $\mathcal{D} = 1$
and $\delta = \beta^{-2}$ so the surgery coefficient of $\mathcal{L}_{\circ}$ is $\beta^{2\sigma}$. Let $t \in \mathbb{Z}_2^N$ with $|t| \ge 1$. Using Lemma 7.8, we compute

$$ \begin{aligned} \{e_0 \prod_{j=1}^{N} (f_j^+ f_j^-)^{t_j}, \zeta_\circ\}_{X_0^\pm} &= \beta^{6n} p^N \sum_{s \in \mathbb{Z}_2^N} \left(-\frac{2q}{p}\right)^{|s|} \{e_0 \prod_{j=1}^{N} (f_j^+ f_j^-)^{t_j}, e_0 \prod_{j=1}^{N} (f_j^+ f_j^-)^{s_j}\}_{X_0^\pm} \\ &= \pm \beta^{6n+6} p^N 2^{-N-1} \sum_{s \in \mathbb{Z}_2^N} \left(-\frac{2q}{p}\right)^{|s|} \delta_{s,1-t} \\ &= \pm \beta^{6n+6} p^N 2^{-N-1} \left(-\frac{2q}{p}\right)^{N-|t|} = \pm (-1)^{|t|} \beta^{6n+2} 2^{-|t|-1} q^{N-|t|} p^{|t|}. \end{aligned} $$

If $|t| = 0$, then

$$ \begin{aligned} \{e_0, \zeta_\circ\}_{X_0^\pm} &= 2^{-2N} c_{n+1}^0 \{e_0, \phi_{P_0^+}\}_{X_0^\pm} + \beta^{6n} (-2q)^N \{e_0, e_{0\text{TOP}}\}_{X_0^\pm} \\ &= 2^{-2N} c_{n+1}^0 2^{2N-1} \pm \beta^{6n} (-2q)^N 2^{-N-1} \beta^2 (-1)^N \\ &= \frac{1}{2} c_{n+1}^0 \pm \frac{1}{2} \beta^{6n+2} q^N. \end{aligned} $$

Finally, we have

$$ \begin{aligned} \{\phi_{P_0^\pm}, \zeta_\circ\}_{X_0^\pm} &= \beta^{6n} p^N \sum_{s \in \mathbb{Z}_2^N} \left(-\frac{2q}{p}\right)^{|s|} \{\phi_{P_0^\pm}, e_0 \prod_{j=1}^N (f_j^+ f_j^-)^{s_j}\}_{X_0^\pm} \\ &= \beta^{6n} p^N \{\phi_{P_0^\pm}, e_0\}_{X_0^\pm} = 2^{2N-1} \beta^{6n} p^N, \text{ and} \\ \{\phi_{X_1^\pm}, \zeta_\circ\}_{X_1^\pm} &= \pm 2c_{n+1}^\pm. \end{aligned} $$

Combining with (7.18) to translate $\{-,-\}_U$ into $(-,-)_{P_U}$ results in the statement of the theorem. $\square$

**Remark 7.10.** Let us discuss a quick sanity check of the formulas above. In the category of modules over a factorisable (or, more generally, unimodular) ribbon quasi-Hopf algebra $H$, there is a natural transformation $\alpha: \text{id}_C \Rightarrow \text{id}_C$ such that $\alpha_{P_1}$ is non-zero and factors through 1, cf. Remark 3.17. Indeed, if $c$ is an integral in $H$, then it is central by unimodularity, and thus acting by it yields a natural transformation. In the notation of Section 3.4, we must then necessarily have $L'_{\Lambda,t}(\mathcal{L}_*(c, P_1)) = L'_{\Lambda,t}(\mathcal{L}_*(c, P_1))$. Indeed, this can be checked directly. Let $c$ be the integral of $\mathbb{Q}$ corresponding to $\Lambda^{\text{co}}$ with normalisation such that $\Lambda^{\text{co}} \circ \Lambda = \text{id}_1$. By [FGR2, Sec. 3.5], it is explicitly given by

$$ c = 2^{N+1} \beta^2 e_0^{\text{TOP}} \stackrel{(5.18)}{=} \phi_1 \stackrel{(7.13)}{=} 2^{-2N} \phi_{P_0^+} + 2^N \beta^2 e_0^{\text{TOP}}. $$

We saw in the proof of Lemma 4.5 that $t_{P_0^+}(c) = 1$. On the one hand, combining (3.24) and (7.15) gives

$$ L'_{\Lambda,t}(\mathcal{L}_*(c, P_0^+)) = p^N. $$
On the other hand, using Theorem 7.9 and the expansion of $c$ in the basis of the
centre we get

$$
\begin{align*}
L'_{\Lambda,t}(\mathcal{L}_{\circ}(c, P_0^+)) &= 2^{-2N} L'_{\Lambda,t}(\mathcal{L}_{\circ}(\phi_{P_0^+}^+, P_0^+)) + 2^N \beta^2 L'_{\Lambda,t}(\mathcal{L}_{\circ}(e_0\text{TOP}, P_0^+)) \\
&= \frac{1}{2} p^N + \frac{1}{2} \beta^2 (-1)^N \beta^2 p^N = p^N,
\end{align*}
$$

as expected.

△

7.5. **Pullback trace.** Finally, we consider $H = \mathbb{Q}(2, \beta)$ and $A$ as in Section 5.3, and treat the invariants for lens spaces with one loop coloured by the object $P_\mu$ from the pullback ideal (cf. Proposition 5.4).

We start by computing the values of Res*(t^A)_{P_μ}(z) for z from the 11-dimensional
centre of H (recall (7.10)), and where μ = (a^- b^- a^+ b^+).

**Lemma 7.11.** *The only non-zero values of Res*(t^A)_{P_μ}(z) for z an element from the canonical basis (7.11) of Z(H) are*

$$
\begin{array}{c|ccc}
z \in Z(H) & \mathbf{e}_0 \mathbf{f}_1^+ \mathbf{f}_1^- & \mathbf{e}_0 \mathbf{f}_2^+ \mathbf{f}_2^- & \mathbf{e}_0 \mathbf{f}_1^\gamma \mathbf{f}_2^\epsilon \\
\hline
\mathrm{Res}^*(\mathbf{t}^A)_{P_\mu}(z) & 1 & \det \mu & \delta_{\gamma,+} a^\epsilon - \delta_{\gamma,-} b^\epsilon
\end{array}
$$

where $\gamma, \epsilon \in \{+, -\}$.

*Proof.* Using Proposition 5.7, we compute

$$
\hat{\lambda}^{A,r}(\mathbf{e}_0^+ \mathbf{f}_1^- \mathbf{f}_1^+) = \frac{1}{4} \sum_{m=0}^{3} \hat{\lambda}^{A,r}(\mathbf{K}^m \mathbf{f}_1^- \mathbf{f}_1^+) = \frac{1}{4}(1 - i\beta^2),
$$

so that in our chosen normalisation (5.23), the pullback trace at $P_\mu$ equals 1
on $\mathbf{e}_0^+ \mathbf{f}_1^+ \mathbf{f}_1^-$. From Corollary 5.5, we know that $\mathbf{f}_2^+ \mathbf{f}_2^-$ acts on $P_\mu$ as $\det \mu \cdot \mathbf{f}_1^+ \mathbf{f}_1^-$.
Similarly, from (5.20) one finds for $\gamma, \epsilon \in \{+, -\}$ and $p \in P_\mu$ that $\mathbf{f}_1^\gamma \mathbf{f}_2^\epsilon p =$
$(\delta_{\gamma,+} a^\epsilon - \delta_{\gamma,-} b^\epsilon) \cdot \mathbf{f}_1^+ \mathbf{f}_1^- p$.

Likewise, $\mathbf{f}_1^+\mathbf{f}_1^-\mathbf{f}_2^+\mathbf{f}_2^-$ is seen to act as 0, whence the trace vanishes for $\phi_{P_0^+}$.
Finally, since $P_\mu$ is purely even, $\phi_{X_1^\pm}$ act as zero as well. $\square$

As above, we have a symmetric bilinear pairing $(z, z')_{P_\mu} := \mathrm{Res}^*(\mathbf{t}^A)_{P_\mu}(zz')$ for $z, z' \in Z(H)$. Then Lemma 7.11 shows that the only non-zero values are

$$
(e_0, e_0 \mathbf{f}_1^+ \mathbf{f}_1^-)_{P_\mu} = 1, \quad (e_0, e_0 \mathbf{f}_2^+ \mathbf{f}_2^-)_{P_\mu} = \det \mu,
$$

and

$$
(e_0, e_0 \mathbf{f}_1^\gamma \mathbf{f}_2^\epsilon)_{P_\mu} = \delta_{\gamma,+} a^\epsilon - \delta_{\gamma,-} b^\epsilon .
$$
**Theorem 7.12.** The invariants $L'_{\Lambda, \text{Res}^*(t^A)}(\mathcal{L}_\circ(z, P_\mu))$ for basis elements $z$ of $Z(H)$ as in (7.11) are:

<table><thead><tr><th>z</th><th>$L'_{\Lambda, \text{Res}^*(t^A)}(\mathcal{L}_\circ(z, P_\mu))$</th></tr></thead><tbody><tr><td>$\phi_{P_0^+}, \phi_{X_1^\pm}$</td><td>0</td></tr><tr><td>$e_0$</td><td>$-2pq(1 + \det \mu)$</td></tr><tr><td>$e_0 f_j^+ f_j^-$</td><td>$p^2(\delta_{j,1} + \delta_{j,2} \det \mu)$</td></tr><tr><td>$e_0 f_1^\gamma f_2^\epsilon$</td><td>$p^2(\delta_{\gamma,+} a^\epsilon - \delta_{\gamma,-} b^\epsilon)$</td></tr><tr><td>$e_0 \text{TOP}$</td><td>0</td></tr></tbody></table>

*Proof.* The surgery coefficient is $\beta^{2\sigma}$ as in the proof of Theorem 7.9. By Proposition 7.1 we need to compute $(z, \zeta_\circ)_{P_\mu}$. The zero entries in the table are immediate from Lemma 7.11. Using Lemma 7.7, we first compute

$$ \begin{aligned} (e_0, \zeta_\circ)_{P_\mu} &= \beta^{6n} p^2 \sum_{s \in \mathbb{Z}_2^2} \left(-\frac{2q}{p}\right)^{|s|} (e_0, e_0 \prod_{j=1}^2 (f_j^+ f_j^-)^{s_j})_{P_\mu} \\ &= -\beta^{6n} 2pq (e_0, e_0 f_1^+ f_1^- + e_0 f_2^+ f_2^-)_{P_\mu} \\ &= -2\beta^{6n} pq(1 + \det \mu) . \end{aligned} $$

Since $\sigma = n$, $\beta^{2\sigma+6n} = 1$. Similarly, we have

$$ \begin{aligned} (e_0 f_j^+ f_j^-, \zeta_\circ)_{P_\mu} &= \beta^{6n} p^2 \sum_{s \in \mathbb{Z}_2^2} \left(-\frac{2q}{p}\right)^{|s|} (e_0 f_j^+ f_j^-, e_0 \prod_{j=1}^2 (f_j^+ f_j^-)^{s_j})_{P_\mu} \\ &= \beta^{6n} p^2 (e_0 f_j^+ f_j^-, e_0)_{P_\mu} \\ &= \beta^{6n} p^2 (\delta_{j,1} + \delta_{j,2} \det \mu) . \end{aligned} $$

A completely analogous computation shows

$$ (e_0 f_1^\gamma f_2^\epsilon, \zeta_\circ)_{P_\mu} = \beta^{6n} p^2 (e_0 f_1^\gamma f_2^\epsilon, e_0)_{P_\mu} = \beta^{6n} p^2 (\delta_{\gamma,+} a^\epsilon - \delta_{\gamma,-} b^\epsilon) . $$

□

Theorem 7.12 shows the result stated in Theorem 3.21. In particular, the invariants in the above theorem manage to distinguish all $P_\mu$, by evaluating on all $e_0 f_1^\gamma f_2^\epsilon$. Explicitly, the choices for $\alpha_{j,l}$ in Theorem 3.21 are

$$ \alpha_{1,1} = e_0 f_1^+ f_2^-, \quad \alpha_{1,2} = e_0 f_1^+ f_2^+, \quad \alpha_{2,1} = -e_0 f_1^- f_2^-, \quad \alpha_{2,2} = -e_0 f_1^- f_2^+ . $$

## APPENDIX A. MORE ON CONTINUED FRACTIONS

Consider the $(n \times n)$-matrix $M_a$ with entries

$$ (M_a)_{ij} = \begin{cases} a_i & i = j, \\ 1 & (i < n \land j = i + 1) \lor (i > 1 \land j = i - 1). \end{cases} $$
This is precisely the linking matrix of the surgery link of $\mathcal{L}(p, q)$ as given in (3.15), see e.g. [Lyu, Sec. 5.3]. By definition, the signature $\sigma(\mathcal{L}(p, q))$ of the surgery link is the signature of the linking matrix, i.e. the number of positive minus negative eigenvalues.

**Proposition A.1.** Let $a_1, \dots, a_n \in \mathbb{Z}$ be non-zero such that there are coprime positive integers $p_i, q_i$ with $\frac{p_i}{q_i} = [a_i; a_{i+1}, \dots, a_n]$ for $i = 1, \dots, n$. Then we have:

$$ (1) \det M_a = \prod_{j=1}^{n} [a_j; a_{j+1}, \dots, a_n]. $$

$$ (2) M_a \text{ is positive definite, and so its signature is } n. $$

$$ (3) p_1 = \prod_{j=1}^{n} [a_j; a_{j+1}, \dots, a_n] = \prod_{j=1}^{n} [a_{n-j+1}; a_{n-j}, \dots, a_1]. $$

*Proof. Part 1:* We proceed by induction on $n$. For $n = 1$ this is obvious, and to see what is happening note that for $n = 2$ we have

$$ [a_1; a_2] \cdot a_2 = (a_1 - \frac{1}{a_2})a_2 = a_1 a_2 - 1 = \det \begin{pmatrix} a_1 & 1 \\ 1 & a_2 \end{pmatrix}. $$

Assume now that the statement holds up to $n-1$ for a given $n \ge 2$. Setting $d(a_1, \dots, a_n) = \prod_{j=1}^n [a_j; a_{j+1}, \dots, a_n]$, we note that $d(a_i, \dots, a_n) = [a_i; a_{i+1}, \dots, a_n] \cdot d(a_{i+1}, \dots, a_n)$. Using the Laplace expansion for the first row, we obtain

$$ \begin{aligned} \det M_a &= a_1 \cdot \det M_{a \setminus (a_1)} - 1 \cdot \det \begin{pmatrix} 1 & 1 & 0 & \cdots 0 \\ 0 & 0 & M_{a \setminus (a_1, a_2)} & \cdots 0 \end{pmatrix} \\ &= a_1 \cdot \det M_{a \setminus (a_1)} - \det M_{a \setminus (a_1, a_2)}, \end{aligned} $$

where by $M_{a \setminus b}$ we mean the matrix associated to the list obtained by removing entries of $b$ from $a$. By hypothesis, we know the values of the two determinants on the right hand side, and so we obtain

$$ \begin{aligned} \det M_a &= a_1 \cdot d(a_2, \dots, a_n) - d(a_3, \dots, a_n) \\ &= d(a_2, \dots, a_n) \cdot \left( a_1 - \frac{1}{[a_2; \dots, a_n]} \right) = d(a_2, \dots, a_n) \cdot [a_1; a_2, \dots, a_n], \end{aligned} $$

which equals $d(a_1, \dots, a_n)$, as claimed.

**Part 2:** Positive definiteness now follows from the Sylvester criterion. Namely, consider the lower right submatrices $N_d$, $d = 1, \dots, n$ of $M$. That is, $(N_d)_{st} = (M_a)_{s+n-d, t+n-d}$ and $s, t = 1, \dots, d$.

Using the above notation, we see that $N_d = M_{(a_{n-d+1}, a_{n-d+2}, \dots, a_n)}$. Thus by part 1 we get $\det N_d = \prod_{j=n-d+1}^{n} [a_j; a_{j+1}, \dots, a_n] = \prod_{j=n-d+1}^{n} \frac{p_j}{q_j}$, and all factors are positive by assumption.

**Part 3:** By part 1, the second equality is simply the statement that the similar matrices $M_{(a_1, \dots, a_n)}$ and $M_{(a_n, \dots, a_1)}$ have the same determinant.

To see the first equality, note first that

$$ \prod_{j=1}^{n} [a_j; a_{j+1}, \dots, a_n] = \prod_{j=1}^{n} \frac{p_j}{q_j} = \frac{p_1}{q_n} \prod_{j=1}^{n-1} \frac{p_{j+1}}{q_j}. $$
Since clearly $q_n = 1$, this equals $p_1$ if $q_j = p_{j+1}$ for all $1 \le j < n$. Let us show that this holds. For $j < n$, we have

$$ \frac{p_j}{q_j} = a_j - \frac{q_{j+1}}{p_{j+1}} = \frac{a_j p_{j+1} - q_{j+1}}{p_{j+1}}. $$

Now, for any $c, d, n \in \mathbb{Z}$ with $d|n$, we have $\gcd(n \pm c, d) \le \gcd(c, d)$.<fnref n="15" /> Applying this to our situation, we find $\gcd(a_j p_{j+1} - q_{j+1}, p_{j+1}) \le \gcd(q_{j+1}, p_{j+1}) = 1$, and so the above is in fact an equality of reduced fractions. Thus we even obtain equality of numerators and denominators up to sign. But the sign vanishes in light of our positivity assumption; in particular, we obtain $q_j = p_{j+1}$. □

The proposition above in particular implies (3.17) for the signature of the surgery link. The same expression (with the stronger assumption that all $a_i$ are at least 2) was obtained e.g. in [BCGP, Prop. 6.24].

## REFERENCES

[Ab] T. Abe, A $\mathbb{Z}_2$-orbifold model of the symplectic fermionic vertex operator superalgebra, *Mathematische Zeitschrift* 255 (2007) 755–792 [math/0503472 [math.QA]].

[BBG] A. Beliakova, C. Blanchet, A.M. Gainutdinov, *Modified trace is a symmetrised integral*, *Select. Math. New Ser.* 27 (2021), [1801.00321 [math.QA]].

[BC1] D. Bulacu, S. Caenepeel, *Integrals for (dual) quasi-Hopf algebras*. *Applications*, *J. Algebra* 266 (2003), 552–583, [math/0110063 [math.QA]].

[BC2] D. Bulacu, S. Caenepeel, *On integrals and cointegrals for quasi-Hopf algebras*, *J. Algebra* 351 (2012), 390–425, [1103.2263 [math.QA]].

[BCGP] C. Blanchet, F. Costantino, N. Geer, B. Patureau-Mirand, *Non semi-simple TQFTs*, *Reidemeister torsion and Kashaev's invariants*, *Adv. Math.* 301 (2016) 1–78, [1404.7289 [math.GT]].

[BCPO] D. Bulacu, S. Caenepeel, F. Panaite, F. v.Ostaeyen, *Quasi-Hopf algebras: a categorical approach*, *Encyclopedia Math. Appl.* (2019).

[BGR1] J. Berger, A.M. Gainutdinov, I. Runkel, *Modified traces for quasi-Hopf algebras*, *J. Algebra* 548 (2020) 96–119, [1812.10445 [math.QA]].

[BGR2] __________, *Monadic cointegrals and applications to quasi-Hopf algebras*, *J. Pure Appl. Algebra* 225 (2021), [2003.13307 [math.QA]].

[BK] B. Bakalov, A. Kirilov Jr., *Lectures on tensor categories and modular functors*, *Univ. Lect. Ser.* 21 (2001).

[Br] A. Bruguières, *Tresses et structure entière sur la catégorie des représentations de sl$_N$ quantique*, *Commun. in Algebra* 28 (2000) 1989–2028.

[BT] D. Bulacu, B. Torrecillas, *Factorizable quasi-Hopf algebras – applications*, *J. Pure Appl. Alg.* 194 (2004) 39–84 [math/0312076 [[math.QA/0312076]].

[BV1] A. Bruguières, A. Virelizier, *Hopf monads*, *Adv. in Math.* 215 (2007), 679–733, [math/0604180 [math.QA]].

[BV2] __________, *On the center of fusion categories*, *Pacific J. Math.* 264 (2013), [1203.4180 [math.QA]].

[CDGS] W. Cheong, A. Doser, M. Gray, S.F. Sawin, *Relationship of the Hennings and Chern-Simons Invariants For Higher Rank Quantum Groups*, [1701.01423 [math.GT]].

[CG] T. Creutzig, T. Gannon, *Logarithmic conformal field theory, log-modular tensor categories and modular forms*, *J. Phys. A: Math. Theor.* 50 (2017) 404004, [1605.04630 [math.QA]].

<footnote n="15">¹⁵If $x$ divides $\gcd(d, n \pm c)$, then since $d$ divides $n$ also $x$ divides $n$, and hence $x$ divides $c$.</footnote>
[CGP1] F. Costantino, N. Geer, B. Patureau-Mirand, *Quantum invariants of 3-manifolds via link surgery presentations and non-semi-simple categories*, Journal of Topology 7 (2014) 1005–1053, [1202.3553 [math.GT]].

[CGP2] F. Costantino, N. Geer, B. Patureau-Mirand, *Relations between Witten-reshetikhin-turaev and non semi-simple sl(2) 3-manifold invariants*, Algebr. Geom. Topol. 15 (2015) 1363–1386, [1310.2735 [math.GT]].

[CGR] T. Creutzig, A. M. Gainutdinov and I. Runkel, *A quasi-Hopf algebra for the triplet vertex operator algebra*, Commun. Contemp. Math. 22 (2020) 1950024, [1712.07260 [math.QA]].

[CKS] Q. Chen, S. Kuppum, P. Srinivasan, *On the relation between the WRT invariant and the Hennings invariant*, Math. Proc. of the Cambridge Phil. Soc. 146 (2009) 151–163, [0709.2318 [math.QA]].

[CLR] T. Creutzig, S. Lentner and M. Rupert, *Characterizing braided tensor categories associated to logarithmic vertex operator algebras*, [2104.13262 [math.QA]].

[CYZ] Q. Chen, C-C. Yu, Y. Zhang, *Three-manifold invariants associated with restricted quantum groups*, Math. Z. 272 (2012) 987–999.

[DGGPR1] M. De Renzi, A.M. Gainutdinov, N. Geer, B. Patureau-Mirand, I. Runkel, 3-dimensional TQFTs from non-semisimple modular categories, Selecta Math. 28 (2022) 42, [1912.02063 [math.GT]].

[DGGPR2] __________, *Mapping Class Group Representations From Non-Semisimple TQFTs*, Commun. Contemp. Math. 25 (2023) 2150091, [2010.14852 [math.GT]].

[DGP] M. De Renzi, N. Geer, B. Patureau-Mirand, *Renormalized Hennings invariants and 2+1-TQFTs*, Commun. Math. Phys. 362 (2018), 855–907, [1707.08044 [math.GT]].

[Dr] V. Drinfeld, *Quasi-Hopf algebras*, Leningrad Math. J. 1 (1990), 1419–1457.

[DR] A. Davydov, I. Runkel, $Z/2Z$-extensions of Hopf algebra module categories by their base categories, Adv. Math. 247 (2013) 192–265, [1207.3611 [math.QA]].

[EGNO] P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik, *Tensor categories*, Math. Surveys Monogr. 205 (2015).

[EO] P. Etingof, V. Ostrik, *On Semisimplification of Tensor Categories*, in: V. Baranovsky et al. (eds) *Representation Theory and Algebraic Geometry*, Trends in Mathematics (2022), Birkhäuser (2022) 3–35, [1801.04409 [math.RT]].

[Et] P. Etingof, *On Vafa's theorem for tensor categories*, Math. Res. Let. 9 (2002) 651–657, [math/0207007 [math.QA]].

[FGR1] V. Farsad, A.M. Gainutdinov, I. Runkel, SL(2, $\mathbb{Z}$)-action for ribbon quasi-Hopf algebras, J. Algebra 522 (2019), 243–308, [1702.01086 [math.QA]].

[FGR2] __________, *The symplectic fermion ribbon quasi-Hopf algebra and the SL(2, $\mathbb{Z}$)-action on its centre*, Adv. Math. 400 (2022) 108247, [1706.08164 [math.QA]].

[FGST] B.L. Feigin, A.M. Gainutdinov, A.M. Semikhatov, I.Y. Tipunin, *Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center*, Commun. Math. Phys. 265 (2006) 47–93, [hep-th/0504093 [hep-th]].

[FL] I. Flandoli, S. Lentner, *Logarithmic conformal field theories of type $B_n$, $\ell = 4$ and symplectic fermions*, J. Math. Phys. 59 (2018) 071701, [1706.07994 [math.RT]].

[FOG] A. Fontalvo Orozco, A.M. Gainutdinov, *Module traces and Hopf group-coalgebras*, [1809.01122 [math.QA]].

[FS] J. Fuchs, C. Schweigert, *Hopf algebras and finite tensor categories in conformal field theory*, Rev. Union Mat. Argentina 51 (2010) 43–90 [1004.3405 [hep-th]].

[GK] M.R. Gaberdiel, H.G. Kausch, *A rational logarithmic conformal field theory*, Phys. Lett. B 386 (1996) 131–137 [hep-th/9606050 [hep-th]].

[GKP1] N. Geer, J. Kujawa, B. Patureau-Mirand, *Generalized trace and modified dimension functions on ribbon categories*, B. Sel. Math. New Ser. 17 (2011), 453–404, [1001.0985 [math.RT]].
[GKP2] __________, *Ambidextrous objects and trace functions for nonsemisimple categories*, Proc. Amer. Math. Soc. **141** (2013), 2963–2978, [1106.4477 [math.RT]].

[GKP3] __________, *M-traces in (non-unimodular) pivotal categories*, Algebr. Represent. Theory (2021), [1809.00499 [math.RT]].

[GN] T. Gannon, C. Negron *Quantum SL(2) and logarithmic vertex operator algebras at (p, 1)-central charge*, [2104.12821 [math.QA]].

[GPT] N. Geer, B. Patureau-Mirand, V.G. Turaev, *Modified quantum dimensions and re-normalized link invariants*, Compositio Math. **145** (2009), 196–212, [0711.4229 [math.QA]].

[GPV] N. Geer, B. Patureau-Mirand, A. Virelizier *Traces on ideals in pivotal categories*, Quantum Topol. **4** (2013), 91–124, [1103.1660 [math.QA]].

[GR1] A.M. Gainutdinov, I. Runkel, *Symplectic fermions and a quasi-Hopf algebra structure on $\overline{U_i} s l(2)$*, J. Algebra **476** (2017) 415–458, [1503.07695 [math.QA]].

[GR2] __________, *The non-semisimple Verlinde formula and pseudo-trace functions*, J. Pure Appl. Algebra **223** (2019), 660–690, [1605.04448 [math.QA]].

[GR3] __________, *Projective objects and the modified trace in factorisable finite tensor categories*, Compositio Math. **156** (2020) 770–821, [1703.00150 [math.QA]].

[He] M. Hennings, *Invariants of Links and 3-Manifolds Obtained from Hopf Algebras*, J. London Math. Soc. (2) **54** (1996), No. 3, 594–624.

[HN] F. Hausser, F. Nill, *Integral theory for quasi-Hopf algebras*, [math/9904164 [math.QA]].

[LW] R. Laugwitz, C. Walton, *Constructing non-semisimple modular categories with relative monoidal centers*, Int. Math. Res. Not. IMRN (2021), [2010.11872 [math.QA]].

[Lyu] V.V. Lyubashenko, *Invariants of 3-manifolds and projective representations of mapping class groups via quantum groups at roots of unity*, Commun. Math. Phys. **172** (1995), 467–516, [hep-th/9405167 [hep-th]].

[Ka] H.G. Kausch, *Curiosities at $c = -2$*, [hep-th/9510149 [hep-th/9510149]].

[Ke] T. Kerler, *On the connectivity of cobordisms and half-projective TQFTs*, Commun. Math. Phys. **198** (1998) 535–590, [q-alg/9603017 [math.QA]].

[KL] T. Kerler, V. V. Lyubashenko, *Non-semisimple topological quantum field theories for 3-manifolds with corners*, Lecture Notes in Math. **1765** (2001).

[ML] S. Mac Lane, *Categories for the working mathematician*, Grad. Texts in Math. **5** (1998).

[MR] R. McRae, *Deligne tensor products of categories of modules for vertex operator algebras*, [2304.14023 [math.QA]].

[PS] V.V. Prasolov, A.B. Sossinsky, *Knots, links, braids and 3-manifolds*, Transl. Math. Monogr. **154** (1997).

[RT] N. Reshetikhin, V. Turaev, *Ribbon Graphs and Their Invariants Derived From Quantum Groups*, Commun. Math. Phys. **127** (1990) 1–26.

[Ru] I. Runkel, *A braided monoidal category for free super-bosons*, J. Math. Phys. **55** 041702 (2014) [1209.5554 [math.QA]].

[Sch] P. Schauenburg, *A quasi-Hopf algebra freeness theorem*, Proc. Amer. Math. Soc. **132** (2004) 965–972, [math/0204141 [math.QA]].

[Sh1] K. Shimizu, *On unimodular finite tensor categories*, Int. Math. Res. Notices **2017** (2017) 277–322, [1402.3482 [math.QA]].

[Sh2] K. Shimizu, *The monoidal center and the character algebra*, J. Pure and Appl. Algebra **221** (2017) 2338–2371, [1504.01178 [math.QA]].

[Sh3] K. Shimizu, *Non-degeneracy conditions for braided finite tensor categories*, Adv. Math. **355** (2019) 106778 [1602.06534 [math.QA]].

[Sh4] K. Shimizu, *Integrals for finite tensor categories*, Algebr. Represent. Theory **22** (2019) 459–493, [1702.02425 [math.CT]].
[So] Y. Sommerhäuser, *On the notion of a ribbon quasi-Hopf algebra*, Rev. Unión Mat. Argent. **51** (2010), 177-192, [0910.1638 [math.RA]].

[Tu] V.G. Turaev, *Quantum Invariants of Knots and 3-Manifolds*, de Gruyter Stud. Math. **18** (2016).

[Vi] A. Virelizier, *Kirby elements and quantum invariants*, Proc. Lond. Math. Soc. **93** (2006) 474-514, [math/0312337v2 [math.GT]]