# Non-semisimple TQFT’s and BPS $q$-series

Francesco Costantino  Institut de Mathématiques de Toulouse, 118 route de
Narbonne, F-31062 Toulouse, France  ,  Sergei Gukov  Walter Burke Institute
for Theoretical Physics, California Institute of Technology, Pasadena, CA
91125, USA  and  Pavel Putrov  International Centre for Theoretical Physics,
Strada Costiera 11, Trieste 34151 Italy

###### Abstract.

We propose and in some cases prove a precise relation between 3-manifold
invariants associated with quantum groups at roots of unity and at generic
$q$. Both types of invariants are labeled by extra data which plays an
important role in the proposed relation. Bridging the two sides — which until
recently were developed independently, using very different methods — opens
many new avenues. In one direction, it allows to study (and perhaps even to
formulate) $q$-series invariants labeled by spinc structures in terms of non-
semisimple invariants. In the opposite direction, it offers new insights and
perspectives on various elements of non-semisimple TQFT’s, bringing the latter
into one unifying framework with other invariants of knots and 3-manifolds
that recently found realization in quantum field theory and in string theory.

###### Contents

  1. 1 Introduction and Summary
  2. 2 Preliminaries
    1. 2.1 ADO invariants of links in a three-sphere
    2. 2.2 Combinatorics of spin and spinc-structures
    3. 2.3 WRT invariants
    4. 2.4 $\mathrm{N}_{r}$ invariants
      1. 2.4.1 $\mathrm{N}_{r}$ for links, ADO and Conway-Alexander polynomials
      2. 2.4.2 $\mathrm{N}_{r}$ for manifolds
      3. 2.4.3 A relation between $\mathrm{N}_{r}$ and $\mathrm{WRT}_{r}$
    5. 2.5 $\widehat{Z}$ invariants
      1. 2.5.1 $\widehat{Z}$ for links
      2. 2.5.2 Examples of good links
      3. 2.5.3 Whitehead link
    6. 2.6 $\widehat{Z}$ for 3-manifolds
    7. 2.7 Physics of non-torsion fluxes
  3. 3 Families of examples
    1. 3.1 Plumbed 3-manifolds
    2. 3.2 Level $r=2\mod 4$
    3. 3.3 Level $r=1\mod 4$
    4. 3.4 Level $r=3\mod 4$
    5. 3.5 Generalization to $b_{1}>0$
    6. 3.6 Surgeries on knots
    7. 3.7 0-surgeries on knots
    8. 3.8 Surgery on a link
  4. 4 The relation between $\mathrm{N}_{r}$ and $\widehat{Z}$
    1. 4.1 Gauss sum vs Laplace transform
    2. 4.2 Generalization to $b_{1}\geq 0$
    3. 4.3 Generalization to the spin case
  5. 5 $\widehat{Z}$ and $\mathrm{N}_{r}$ as decorated TQFTs
    1. 5.1 Hilbert space on a torus
      1. 5.1.1 Quantization
      2. 5.1.2 A prototypical example
    2. 5.2 Decorated TQFTs, gradings, and Fourier transform
      1. 5.2.1 Decorated TQFTs and grading of Hilbert spaces
      2. 5.2.2 Fourier transform of TQFTs
      3. 5.2.3 “$r$-wrapping” of $H^{1}(\cdot;\mathbb{R}/\mathbb{Z})$-TQFTs
  6. 6 Relation to WRT invariants
    1. 6.1 Compatibility of the relations between $\mathrm{N}^{0}_{r}$, $\mathrm{WRT}_{r}$, and $\widehat{Z}$
    2. 6.2 0-surgeries on twist knots
    3. 6.3 From CGP to WRT via $\widehat{Z}$ for 0-surgeries on knots
      1. 6.3.1 Odd level $r$.
      2. 6.3.2 Even level $r$.
  7. A Spin and spinc sign-refined Torsion
  8. B $H^{1}(M;\mathbb{Z}/2\mathbb{Z})$-refined WRT invariant and $\widehat{Z}$
    1. B.1 Level $r=2\mod 4$
    2. B.2 Level $r=1\mod 4$
    3. B.3 Level $r=3\mod 4$
  9. C Graded bases of the CGP TQFT
  10. D Commutativity of limits

##  1\. Introduction and Summary

As part of a larger quest for new quantum invariants of 3-manifolds, we wish
to establish a precise relation between the CGP invariants
$\mathrm{N}_{r}(M,\omega)$ and the GPPV invariants $\widehat{Z}_{b}(M,q)$,
introduced in [17] and [33, 32] respectively.

Here we summarize some of the essential features of these invariants, compare
side-by-side what they depend on, and present a motivation for why one should
expect a connection between these two rather different sets of invariants.
First, perhaps the most obvious part of input data that both sets of
invariants need is a choice of 3-manifold $M$. In either case, however, it
needs to be equipped with additional structure; in the case of CGP invariants
$\mathrm{N}_{r}(M,\omega)$ it involves a choice of $\omega\in
H^{1}(M;\mathbb{C}/2\mathbb{Z})\setminus H^{1}(M;\mathbb{Z}/2\mathbb{Z})$,
whereas in the case of GPPV invariants $\widehat{Z}_{\mathfrak{s}}(M,q)$ it
depends on $\mathfrak{s}\in\text{Spin}^{c}(M)/\mathbb{Z}_{2}$. Although these
structures are clearly different, they both are related111Recall that non-
canonically $\text{Spin}^{c}(M)\cong H_{1}(M;\mathbb{Z})$. to $H_{1}(M)$ and
should be regarded as mutual counterparts in identifying the two sets of
invariants, as we will see below.

Similarly, the dependence of $\mathrm{N}_{r}(M,\omega)$ on a positive integer
$r\neq 0\mod 4$ should be compared to the $q$-dependence of
$\widehat{Z}_{\mathfrak{s}}(M,q)$. Indeed, both sets of invariants are quantum
group invariants of 3-manifolds, with $\xi=e^{\frac{\pi i}{r}}$ and $q$
respectively playing the role of the quantum parameters. The definition of
$\mathrm{N}_{r}(M,\omega)$ is based [17] on the representation theory of the
unrolled restricted quantum group
$\overline{\mathcal{U}}_{\xi}^{H}(\mathfrak{sl}_{2})$, whereas
$\widehat{Z}_{\mathfrak{s}}(M,q)$ should be thought of as a quantum group
invariant associated with ${\mathcal{U}}_{q}(\mathfrak{sl}_{2})$ at generic
$|q|<1$. Indeed, $\widehat{Z}_{\mathfrak{s}}(M,q)$ basically gives a non-
perturbative definition of “$SL(2,\mathbb{C})$ Chern-Simons theory” that
behaves well under surgery and has line operators labeled by Verma modules of
arbitrary complex weight [28, 48].

(1.1) $\begin{array}[]{c@{\;}|@{\;}c@{\;}|@{\;}c}&~{}~{}\
r(M,\omega){}{}&~{}~{}\widehat{Z}_{\mathfrak{s}}(M,q){}{}\\\\[2.84544pt]
\hline\cr\text{ quantum parameter }&~{}\xi=e^{\frac{\pi i}{r}}\text{ root of
unity }&\text{ generic }|q|<1\\\\[5.69046pt] \text{ quantum group
}&{\mathcal{U}}^{H}_{\xi}(\mathfrak{sl}_{2})&{\mathcal{U}}_{q}(\mathfrak{sl}_{2})\\\\[5.69046pt]
\text{ additional structure }&~{}\omega\in
H^{1}(M;\mathbb{C}/2\mathbb{Z})\setminus
H^{1}(M;\mathbb{Z}/2\mathbb{Z}){}&~{}\mathfrak{s}\in\text{Spin}^{c}(M)/\mathbb{Z}_{2}{}\end{array}$

Since the relation between ${\mathcal{U}}_{q}(\mathfrak{sl}_{2})$ and
${\mathcal{U}}^{H}_{\xi}(\mathfrak{sl}_{2})$ involves, among other things,
specializing $q$ to be a root of unity, one might expect
$\mathrm{N}_{r}(M,\omega)$ to be related to limiting values of
$\widehat{Z}_{\mathfrak{s}}(M,q)$ at roots of unity. This relation between
quantum groups was a large part of the motivation in [29], where a relation of
this form was established for knot complements. Namely, it was proposed that
$F_{K}(q,x):=\widehat{Z}(S^{3}\setminus K)$ at roots of unity give polynomial
knot invariants $\text{ADO}_{r}(x;K)$ introduced by Akutsu-Deguchi-Ohtsuki
[1]:

######  Conjecture 1.

(1.2)
$\left.F_{K}(x,q)\right|_{q=\xi^{2}}\;=\;\frac{\text{ADO}_{r}(x/\xi^{2};K)}{\Delta_{K}(x^{r})}\cdot\big{(}x^{1/2}-x^{-1/2}\big{)}$

and the problem of establishing a similar relation for more general
3-manifolds was posed. Addressing this problem requires 3-manifold invariants
which specialize to ADO invariants for knot complements, and CGP invariants
$\mathrm{N}_{r}(M,\omega)$ perfectly fit the bill:

(1.3)
$\text{ADO}_{r}(x^{2}/\xi^{2})\;=\;\frac{x^{r}-x^{-r}}{x-x^{-1}}\mathrm{N}_{r}(S^{3},K_{\alpha})\,,\qquad\text{where}~{}x=e^{\frac{\pi
i\alpha}{r}},\xi=e^{\frac{\pi i}{r}}.$

The composition of (1.2) and (1.3) not only gives us the first instance of the
sought-after relation between CGP and GPPV invariants in a certain class of
3-manifolds, but also provides a clue for the relation between parameters $r$
and $q$ for more general $M$:

(1.4)
$\mathrm{N}_{r}(M,\omega)\quad\stackrel{{\scriptstyle?}}{{\longleftrightarrow}}\quad\left.\widehat{Z}(M,q)\right|_{q=\xi^{2}=e^{2\pi
i/r}}.$

Another useful clue that follows from (1.2)–(1.3) is that the relation between
$\mathrm{N}_{r}(M,\omega)$ and $\widehat{Z}(M,q)$ should be linear,

(1.5)
$\boxed{\mathrm{N}_{r}(M,\omega)\;=\;\sum_{\mathfrak{s}}c^{\text{CGP}}_{\omega,\mathfrak{s}}\widehat{Z}_{\mathfrak{s}}(M,q)\Big{|}_{q\to
e^{\frac{2\pi i}{r}}}}$

much like analogous relations between $\widehat{Z}(M,q)$ and other invariants
of $M$, such as the inverse Turaev torsion [54, 52], Witten-Reshetikhin-Turaev
(WRT) invariants [56, 51], and Rokhlin invariants.

Figure 1. One of our main goals is to explore possible relations between
$\mathrm{N}_{r}(M,\omega)$ and the limiting behavior of $\widehat{Z}(M,q)$ at
roots of unity.

Establishing the relation (1.5) — and, in particular, determining the precise
form of the coefficients $c^{\text{CGP}}_{\omega,\mathfrak{s}}$ that relate
additional structures on $M$ that enter the two sets of invariants — is one of
our main goals. We find

######  Conjecture 2.

Let $M$ be a rational homology sphere. Then, the following relation
holds222Note that $\widehat{Z}_{\mathfrak{s}}(M,q)$, due to the overall factor
$q^{\Delta_{\mathfrak{s}}},\;\Delta_{\mathfrak{s}}\in\mathbb{Q}$, are
multivalued functions of $q$ (the values on different branches differ by an
overall phase). Throughout the paper, when writing the limit in the form
“$q\rightarrow e^{i\phi}$”, it is assumed that the branch is such that
$q^{\Delta_{\mathfrak{s}}}\rightarrow e^{i\phi\Delta_{\mathfrak{s}}}$.
Equivalently, one can consider $\widehat{Z}_{\mathfrak{s}}(M,q)$ as single
values functions of $\tau$, with $q=e^{2\pi i\tau}$. Then “$q\rightarrow
e^{i\phi}$” should be understood as “$\tau\rightarrow\phi/(2\pi)$” (taken from
the upper half plane).:

(1.6)
$\boxed{\mathrm{N}_{r}(M,\omega)\;=\;\sum_{\mathfrak{s}\in\mathrm{Spin}^{c}(M)}c^{\text{CGP}}_{\omega,\mathfrak{s}}\widehat{Z}_{\mathfrak{s}}(M,q)\Big{|}_{q\to
e^{\frac{2\pi i}{r}}}}$

with

(1.7)
$c^{\text{CGP}}_{\omega,\sigma(b,s)}\;=\;\frac{\mathcal{T}(M,[\omega])}{{|H^{1}(M)|}}\;\times\\\
\times\;\begin{cases}-e^{-\frac{\pi i}{2}\mu(M,s)}\,\sum_{a,f}e^{2\pi
i\left(-\frac{r-1}{4}{\ell k}(a,a)+{\ell k}(a,f-b)-\frac{1}{2}\omega(a)+{\ell
k}(f,f)\right)}\,,&\text{if}~{}r=1~{}\text{mod}~{}4,\\\
\sqrt{|H^{1}(M)|}\;\sum_{a}e^{-\frac{\pi ir}{2}\,q_{s}(a)-2\pi i\,{\ell
k}(a,b)-\pi i\,\omega(a)}\,,&\text{if}~{}r=2~{}\text{mod}~{}4,\\\
-e^{\frac{\pi i}{2}\mu(M,s)}\sum_{a,f}e^{2\pi i\left(-\frac{r+1}{4}{\ell
k}(a,a)-{\ell k}(a,f+b)-\frac{1}{2}\omega(a)-{\ell
k}(f,f)\right)}\,,&\text{if}~{}r=3~{}\text{mod}~{}4\end{cases}$

where:

  * •

$\mathcal{T}(M,[\omega])$ is a suitable version of the Reidemeister torsion
(see Appendix A);

  * •

${\ell k}(\cdot,\cdot)$ is the linking form on $H_{1}(M;\mathbb{Z})$, and in
the sum $a,f\in H_{1}(M;\mathbb{Z})$;

  * •

$\sigma$ is the canonical map

(1.8) $\sigma:\text{Spin}(M)\times
H_{1}(M;\mathbb{Z})\longrightarrow\text{Spin}^{c}(M)$

producing a spinc structure $\mathfrak{s}=\sigma(b,s)$ on $M$ from a spin
structure333We use the standard typeface to denote spin structures (e.g.
“$s$”) and Fraktur typeface to denote spinc structures (e.g.
“$\mathfrak{s}$”).  $s$ and $b\in H_{1}(M;\mathbb{Z})$.

  * •

$\mu(M,s)$ is the Rokhlin invariant of $M$ for spin structure $s$444Note that
conjecturally $\frac{1}{4}\mu(M,s)+\frac{1}{2}-{\ell
k}(b,b)=\Delta_{\sigma(b,s)}\mod 1$, where $\Delta_{\mathfrak{s}}$ is the
overall rational shift in the powers of $q$ in $\widehat{Z}_{\mathfrak{s}}$
for a given $\mathfrak{s}\in\mathrm{Spin}^{c}(M)$ [31]..

Note, the invariants $\mathrm{N}_{r}(M,\omega)$ are not defined for $r=0$ mod
4, which is why this case is omitted in (1.7). (For more details about this
case, see Subsection 4.3.)

We prove the above conjecture under some technical hypotheses specified in
Theorem 4.17, which can be stated in a simpler form as follows:

######  Theorem 1.1 (Simplified form of Theorem 4.18).

Let $M$ be a rational homology sphere obtained by integral surgery on a framed
link $L\subset S^{3}$ for which Conjecture 1 holds true. Then, under the
technical hypotheses specified in Theorem 4.18, the Conjecture 2 is true for
any pair $(M,\omega)$ with $\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})\setminus
H^{1}(M;\mathbb{Z}/2\mathbb{Z})$.

In the statement of the theorem though, we point out that the status of the
invariant $\widehat{Z}$ is the following: assuming that Conjecture 1 holds for
$L$ (so that in particular a series $F_{L}$ is defined), there is an explicit
formula providing $\widehat{Z}$ for $M$ but the full proof of the convergence
of the formula and its invariance is not yet available.

An infinite family of cases to which Theorem 4.18 applies is provided in
Example 4.20. The following is a list of cases for which the numerical
evidence supports this result:

  1. (1)

$L$ is a plumbing link (for an infinite list of cases of these plumbing links
the hypotheses of Theorem 4.18 can all be verified: see Example 4.20);

  2. (2)

$L$ is a trefoil knot.

We also provide another kind of numerical evidence for the case of surgeries
on the figure eight knot by cross checking the above conjectures with similar
conjectures relating $\widehat{Z}$ and the WRT invariants (see Section 6.2).

######  Remark 1.2.

Conjecture 2 together with the conjectural relation between
$\widehat{Z}_{\mathfrak{s}}$ and mod-2-cohomology-refined WRT invariant (see
Appendix B) can be used to calculate the limiting values of
$\widehat{Z}_{\mathfrak{s}}$ at $q\rightarrow e^{\frac{2\pi i}{r}}$, $r\neq
0\mod 4$. Namely, assuming that $\mathcal{T}(M,[\omega])\neq 0$ for
$\omega\notin H^{1}(M;\mathbb{Z}/2\mathbb{Z})$, one can invert the linear
transform 1.6 formally substituting
$\mathrm{N}_{r}(M,\omega)/\mathcal{T}(M,[\omega])\rightsquigarrow(-1)^{r}\mathrm{WRT}_{r}(M,\omega)/(i\sqrt{8r})$
when $\omega\in H^{1}(M;\mathbb{Z}/2\mathbb{Z})$. For $r=0\mod 4$ one can
similarly use the relation between $\widehat{Z}_{\mathfrak{s}}$ and spin-
version of CGP invariants (see Section 4.3) together with the spin-refined WRT
invariant (see [31]). This can be especially useful since for many 3-manifolds
$\widehat{Z}_{\mathfrak{s}}$ can be computed as $q$-series, but their modular
properties, needed to determine the limiting values at roots of unity, are not
known. Therefore, the relation to CGP and mod-2-cohomology-refined WRT
invariants can serve as a substitute for modularity properties of
$\widehat{Z}_{\mathfrak{s}}$ in problems that involve limiting values at roots
of unity.

When $\mathcal{T}(M,[\omega])=0$ but $\mathrm{N}_{r}(M,\omega)\neq 0$ for some
$\omega\notin H^{1}(M;\mathbb{Z}/2\mathbb{Z})$ this indicates that
$\lim_{q\rightarrow e^{\frac{2\pi i}{r}}}\widehat{Z}_{\mathfrak{s}}=\infty$ at
least for some $\mathfrak{s}$. (The linear combinations of
$\widehat{Z}_{\mathfrak{s}}$ producing refined WRT invariants can still have
finite limits.)

The study of the case of a manifold $M$ with positive $b_{1}$ brought us to
notice that the invariant $\widehat{Z}$ for such manifolds requires an
additional structure on $M$, namely the choice of a splitting
$H_{1}(M;\mathbb{Z})$ into its torsion part and its free part:
$\text{Tor}(H_{1}(M;\mathbb{Z}))\oplus\text{Free}(H_{1}(M;\mathbb{Z}))$, which
we will denote from now on $H_{1}(M;\mathbb{Z})=T\oplus F$; we will also
denote $b^{\prime}$ (resp. $b^{\prime\prime}$) the projection of $b\in
H_{1}(M;\mathbb{Z})$ in $T$ (resp. $F$). For a fixed spinc structure on $M$,
changing such a choice affects $\widehat{Z}_{\mathfrak{s}}$ by multiplying it
by a power of $q$ (see (2.108) for the precise formulation) which we call the
splitting anomaly.

A version of Conjecture 2 for the case of non-rational-homology-spheres is
then the following:

######  Conjecture 3.

Let $M$ be any closed oriented $3$-manifold. Choose a splitting
$H_{1}(M;\mathbb{Z})$ $=T\oplus F$ where $T$ is the torsion part and $F$ the
free part. Then, the following holds:

(1.9)
$\boxed{\mathrm{N}_{r}(M,\omega)\;=\;\sum_{b=b^{\prime}+r^{\prime}m,\,b^{\prime}\in
T,\,m\in
F}c^{\text{CGP}}_{\omega,\sigma(b,s)}\widehat{Z}_{\sigma(b,s)}(M,q)\Big{|}_{q\to
e^{\frac{2\pi i}{r}}}}$

with

(1.10)
$c^{\text{CGP}}_{\omega,\sigma(b^{\prime}+r^{\prime}m,s)}\;=\;\frac{r^{b_{1}}\,\mathcal{T}(M,[\omega])}{{|\mathrm{Tor}\,H_{1}(M;\mathbb{Z})}|}\times\\\
\times\;\begin{cases}\sum_{\begin{subarray}{c}a^{\prime},f^{\prime}\in
T\end{subarray}}e^{2\pi i\left(-\frac{r-1}{4}{\ell
k}(a^{\prime},a^{\prime})+{\ell
k}(a^{\prime},f^{\prime}-b^{\prime})-\frac{1}{2}\omega(a^{\prime})+{\ell
k}(f^{\prime},f^{\prime})+\omega^{\prime\prime}(m)-\frac{1}{4}\mu(M,s)+\frac{1}{2}\right)}\,,&\text{if}~{}r=1~{}\text{mod}~{}4,\\\
\sqrt{|\mathrm{Tor}\,H_{1}(M;\mathbb{Z})|}\sum_{\begin{subarray}{c}a^{\prime}\in
T\end{subarray}}e^{-\frac{\pi ir}{2}\,q_{s}(a^{\prime})-2\pi i\,{\ell
k}(a^{\prime},b^{\prime})-\pi i\,\omega(a^{\prime})+\pi
i\omega^{\prime\prime}(m)}\,,&\text{if}~{}r=2~{}\text{mod}~{}4,\\\
\sum_{\begin{subarray}{c}a^{\prime},f^{\prime}\in T\end{subarray}}e^{2\pi
i\left(-\frac{r+1}{4}{\ell k}(a^{\prime},a^{\prime})-{\ell
k}(a^{\prime},f^{\prime}+b^{\prime})-\frac{1}{2}\omega(a^{\prime})-{\ell
k}(f^{\prime},f^{\prime})+\omega^{\prime\prime}(m)+\frac{1}{4}\mu(M,s)+\frac{1}{2}\right)}\,,&\text{if}~{}r=3~{}\text{mod}~{}4\end{cases}$

where $r^{\prime}=r$ if $r$ is odd and $\frac{r}{2}$ else and
$\omega^{\prime}$ (resp. $\omega^{\prime\prime}$) is the restriction of
$\omega$ on $T$ (resp. on $F$).

######  Remark 1.3.

In the above conjecture the choice of the splitting is auxiliary: as explained
above, a different choice provides different values for $\widehat{Z}$ but
these choices are compensated by the change in the coefficients
$c^{\text{CGP}}$, so that the left-hand side is indeed independent on the
splitting. Equivalently, the behavior of $\widehat{Z}$ under the choice of
such splitting is controlled by the behavior of the coefficients
$c^{\text{CGP}}$. Clearly, if $M$ is a rational homology sphere the splitting
is irrelevant and the above conjecture reduces to Conjecture 2.

Again, we can prove the above conjecture under suitable technical hypotheses
which are omitted in the following statement:

######  Theorem 1.4.

Under the technical hypotheses of Theorem 4.26, Conjecture 3 holds true for
any pair $(M,\omega)$ with $\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})\setminus
H^{1}(M;\mathbb{Z}/2\mathbb{Z})$.

We provide the following tests:

  1. (1)

$M=\Sigma\times S^{1}$ for some closed oriented surface $\Sigma$;

  2. (2)

$M$ obtained by integral surgery on a framed link $L\subset S^{3}$ for which
Conjecture 1 holds true.

Moreover, an infinite family of cases to which Theorem 4.26 applies is
provided in Example 4.27.

In [7] the invariants $\mathrm{N}_{r}$ where extended to a TQFT defined on a
suitable category of cobordisms decorated with (relative) cohomology classes.
Conjecturally, it should be possible to do the same for the invariant
$\widehat{Z}$. In Section 5.2 we discuss this possibility and define (or
sketch) two “operations on TQFTs” which, if applied to the conjectural TQFT
for $\widehat{Z}$ would produce the TQFT build in [7], thus extending
(unfortunately only partially) Conjecture 2 to the case of cobordisms.

The rest of the paper is organized as follows. In general we tried to be as
self-contained as possible and we ascribed the physical motivations to
dedicated sections so that the paper should be accessible for both
mathematicians and physicists. Section 2 gives a self-contained review of all
the relevant invariants that we wish to relate and introduces a number of
technical tools that are used throughout the paper. After presenting a few
families of concrete examples in Section 3, we then proceed to a more general
and systematic discussion of the relation between $\mathrm{N}_{r}$ and
$\widehat{Z}$ invariants in Section 4. (A reader more interested in a general
argument may prefer to read Section 4 first, before going through the examples
in Section 3.) In Section 5, $\widehat{Z}$ and $\mathrm{N}_{r}$ are considered
as decorated TQFT’s and we propose how a relation between this richer
structure can extend the relation between numerical 3-manifold invariants in
the previous sections. Since both $\widehat{Z}$ and $\mathrm{N}_{r}$ are
related to Witten-Reshetikhin-Turaev (WRT) invariants, it is natural to ask
whether our proposed relation between $\widehat{Z}$ and $\mathrm{N}_{r}$ is
compatible with the previously known relations. This question is answered in
the affirmative in Section 6. Finally, many useful facts about various refined
invariants, gradings in CGP TQFT, and details related to the order of limits
are collected in appendices.

### Acknowledgements

We would like to thank Francesco Benini, Christian Copetti, Boris Feigin, Azat
Gainutdinov, Hiraku Nakajima, Sunghyuk Park, Du Pei, and Nicolai Reshetikhin
for helpful discussions. We also would like to thank the organizers of the
2019 conference “New Developments in Quantum Topology” at UC Berkeley, where
the discussion on the relation between the two invariants was initiated. The
work of S.G. is supported by the U.S. Department of Energy, Office of Science,
Office of High Energy Physics, under Award No. DE-SC0011632, and by the
National Science Foundation under Grant No. NSF DMS 1664227. The work of F.C.
was supported by the French Agence Nationale de la Recherche via the ANR
Project QUANTACT and by the Labex CIMI ANR-11-LABX-0040.

##  2\. Preliminaries

###  2.1. ADO invariants of links in a three-sphere

One of the oldest non-semisimple invariants is the collection — labeled by
integer $r\geq 2$ — of polynomial link invariants introduced by Akutsu-
Deguchi-Ohtsuki [1], or ADO invariants for short. Namely, let $L\subset S^{3}$
be an oriented framed link with $V$ components and $\xi=\exp{\frac{2\pi
i}{2r}}$ for some $r\geq 2$. A coloring of $L$ is an assignment to each
component $L_{I}$ of $L$ of a complex number $\alpha_{i}$. Then, the ADO
invariant of $L$ colored by $\\{\alpha_{i}\\}$ is an element of
$\mathbb{Z}[\xi^{\pm\alpha_{1}},\ldots,\xi^{\pm\alpha_{V}}]$ if $V\geq 2$ and
an element of
$\frac{1}{\xi^{r\alpha}-\xi^{-r\alpha}}\mathbb{Z}[\xi^{\pm\alpha}]$ otherwise.

On the one hand, these polynomial link invariants are close cousins of the
Alexander polynomial, and include the Alexander polynomial as a special case
(corresponding to $r=2$), cf. [43]. On the other hand, the ADO polynomials can
be viewed as close cousins of the quantum group invariants that play a role in
the Reshetikhin-Turaev construction. Indeed, the ADO polynomials were later
re-formulated by Murakami [44] in terms of the R-matrix and quantum groups of
at roots of unity. Specifically, the R-matrix (and its inverse) used in [44]
is a map $V_{\lambda}\otimes V_{\mu}\to V_{\mu}\otimes V_{\lambda}$:

(2.1)
$R_{kl}^{ij}=q^{\frac{1}{2}(\lambda-2i-2n)(\mu-2j+2n)+n(n-1)/2}\frac{\\{i+n;n\\}\\{\mu-j+n;n\\}}{\\{n;n\\}},$
(2.2)
$(R^{-1})_{kl}^{ij}=(-1)^{n}q^{-\frac{1}{2}(\lambda-2i)(\mu-2j)-n(n-1)/2}\frac{\\{j+n;n\\}\\{\lambda-i+n;n\\}}{\\{n;n\\}},$

where $n=l-i=j-k$, $\\{a\\}=q^{a}-q^{-a}$,
$\\{x;n\\}=\prod_{i=0}^{n-1}\\{x-i\\}$ and
$Hv_{i}^{\lambda}=(\lambda-2i)v_{i}^{\lambda}$. We will return to this
expression in Section 2.5 and compare it to the R-matrix that one encounters
in the study of $\widehat{Z}$-invariants.

Also, in Section 2.4 below we review how the ADO invariants were used in [17]
to define invariants of $3$-manifolds $M$ endowed with a cohomology class
$\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})$. Namely, if $L$ is a link such
that $M$ is obtained by integral surgery on $L$, then $\omega$ induces a
coloring on $L$ by setting $\overline{\alpha}_{I}=\omega(\mathfrak{m}_{I})$
where $\mathfrak{m}_{I}$ is the oriented meridian of $L_{I}$.

The study of these invariants remained detached from physics for almost 30
years. This is especially surprising given a large number of close ties that
WRT invariants of knots and 3-manifolds have with quantum field theory and
string theory. The situation started to change about a year ago [29] and we
hope that the present paper can be another step toward bridging this gap, see
also [15, 13, 19, 11, 12, 26, 50, 20, 25] for closely related work.

###  2.2. Combinatorics of spin and spinc-structures

Let $M$ be obtained as an integral surgery on a framed oriented link $L$ in
$S^{3}$ and let $\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})\setminus
H^{1}(M;\mathbb{Z}/2\mathbb{Z}).$ Let us index the components of $L$ by a set
$\mathrm{Vert}$ and denote its $V\times V$ linking matrix by
$B_{IJ},\,I,J\in\mathrm{Vert}$, $V:=|\mathrm{Vert}|$. Then we have the
following identifications

(2.3)
$H_{1}(M;\mathbb{Z})\cong\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}},$
(2.4)
$H^{1}(M;\mathbb{Z}_{2})\cong\\{c\in\mathbb{Z}_{2}^{\mathrm{Vert}}\;|\;\sum_{I,J\in\mathrm{Vert}}B_{IJ}c_{J}=0\mod
2\\},$ (2.5)
$H^{2}(M;\mathbb{Z})\cong\\{h\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}\;\\},$
(2.6)
$\mathrm{Spin}(M)\cong\\{s\in\mathbb{Z}_{2}^{\mathrm{Vert}}\;|\;\sum_{I,J\in\mathrm{Vert}}B_{IJ}s_{J}=B_{II}\mod
2\\},$ (2.7) $\displaystyle\mathrm{Spin}^{c}(M)$
$\displaystyle\cong\mathrm{Char}(B)/2B\mathbb{Z}^{\mathrm{Vert}}$
$\displaystyle=$
$\displaystyle\\{K\in\mathbb{Z}^{\mathrm{Vert}}/2B\mathbb{Z}^{\mathrm{Vert}}\;|\;K_{I}=B_{II}\mod
2\\},$

where
$\mathrm{Char}(B):=\\{K\in\mathbb{Z}^{\mathrm{Vert}}\;|\;K^{T}n=n^{T}Bn\mod
2,\;\forall n\in\mathbb{Z}^{\mathrm{Vert}}\\}$ is the space of characteristic
vectors of the lattice dual to the lattice $\mathbb{Z}^{\mathrm{Vert}}$ with
the quadratic form $B$. Here and in what follows we use $(\ldots)^{T}$ to
denote vector transposition. The one-to-one correspondence between the
elements $a$ and $K$ in (2.18) is given by $a=K-B\varepsilon$, where

(2.8) $\varepsilon:=(1,1,1,\ldots,1)\in\mathbb{Z}^{\mathrm{Vert}}.$

The Bockstein homomorphism $\beta:H^{1}(M;\mathbb{Z}_{2})\to
H^{2}(M;\mathbb{Z})$ writes in the above notation as follows; for
$c\in\mathbb{Z}_{2}^{\mathrm{Vert}}$ let
$\tilde{c}\in\mathbb{Z}^{\mathrm{Vert}}$ be such that $\tilde{c}\cong c\mod
2$. Then

(2.9)
$\beta(c)=\frac{1}{2}B\tilde{c}\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}$

where we observe that since $B\tilde{c}$ is even, division by two is possible
and its result in $\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}$ is
independent on the choice of $\tilde{c}$. We observe that $\mathrm{Spin}(M)$
is affine over $H^{1}(M;\mathbb{Z}_{2})$ via component-wise addition and
$\mathrm{Spin}^{c}(M)$ is affine over $H^{2}(M;\mathbb{Z})$ by defining
$(K+[h])_{I}:=K_{I}+2h_{I},\forall I\in\mathrm{Vert}$. Observe also that there
is a canonical map $i:\mathrm{Spin}(M)\to\mathrm{Spin}^{c}(M)$ which in the
above notation writes:

(2.10) $i(s)=\sum_{I,J\in\mathrm{Vert}}B_{IJ}\tilde{s}_{j}$

where $\tilde{s}$ is any lift of $s\in\mathbb{Z}_{2}^{\mathrm{Vert}}$ to
$\tilde{s}\in\mathbb{Z}^{\mathrm{Vert}}$ so that $s=\tilde{s}\mod 2$.
Furthermore it is clear that $i$ is affine over the Bockstein homomorphism:

(2.11) $i(s+c)=i(s)+\beta(c)\ \forall s\in\mathrm{Spin}(M),c\in
H^{1}(M;\mathbb{Z}_{2}).$

We will also need the linking pairing

(2.12) ${\ell
k}:\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\otimes\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\longrightarrow\mathbb{Q}/\mathbb{Z}$

and its quadratic refinement [36]:

(2.13)
$q_{s}:\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\longrightarrow\mathbb{Q}/2\mathbb{Z},\qquad
q_{s}(a+b)-q_{s}(a)-q_{s}(b)=2\,{\ell k}(a,b).$

depending on a spin structure $s\in\mathrm{Spin}(M)$ as follows:

(2.14) $q_{s+c}(a)-q_{s}(a)=c(a),\;c\in H^{1}(M;\mathbb{Z}_{2}).$

In terms of the identifications (2.3)-(2.18) we have

(2.15) ${\ell k}(a,b)=a^{T}B^{-1}b\mod 1,$ (2.16)
$q_{s}(a)=a^{T}B^{-1}a+s^{T}a\mod 2.$

We will also the use the following expression for the mod 4 reduction of
Rokhlin invariant (see e.g. [37]):

(2.17) $\mu(M,s)=\sigma-s^{T}Bs\mod 4$

where $\sigma$ is the signature of the linking matrix $B$.

A special case is when $M$ is a plumbed manifold, i.e. all the components of
$L$ are unknots which are linked to each other as Hopf links according to the
combinatorial structure of a contractible graph $\Gamma$, the vertices of
which are indexed by $\mathrm{Vert}$. In this case, letting $\deg(I)$ be the
degree of $I\in\mathrm{Vert}$ (i.e. the number of edges containing $v$), we
also have the following identification for spinc-structures :

(2.18)
$\mathrm{Spin}^{c}(M)\cong\\{a\in\mathbb{Z}^{\mathrm{Vert}}/2B\mathbb{Z}^{\mathrm{Vert}}\;|\;a_{I}=\deg(I)\mod
2\\}.$

###  2.3. WRT invariants

Let $\xi=\exp{\frac{\pi i}{r}}$, ${\mathcal{U}}^{H}_{\xi}(\mathfrak{sl}_{2})$
be the so-called “unrolled version” of the quantum $\mathfrak{sl}_{2}$ algebra
as defined in [17], given by generators $E,F,H,K,K^{-1}$ and relations:

(2.19) $\displaystyle KK^{-1}$ $\displaystyle=K^{-1}K=1,$ $\displaystyle
KEK^{-1}$ $\displaystyle=\xi^{2}E,$ $\displaystyle KFK^{-1}$
$\displaystyle=\xi^{-2}F,$ $\displaystyle[E,F]$
$\displaystyle=\frac{K-K^{-1}}{\xi-\xi^{-1}},$ (2.20) $\displaystyle HK$
$\displaystyle=KH,$ $\displaystyle[H,E]$ $\displaystyle=2E,$
$\displaystyle[H,F]$ $\displaystyle=-2F.$

The algebra ${\mathcal{U}}^{H}_{\xi}(\mathfrak{sl}_{2})$ is a Hopf algebra
where the coproduct, counit and antipode are defined as follows

(2.21) $\displaystyle\Delta(E)$ $\displaystyle=1\otimes E+E\otimes K,$
$\displaystyle\varepsilon(E)$ $\displaystyle=0,$ $\displaystyle S(E)$
$\displaystyle=-EK^{-1},$ (2.22) $\displaystyle\Delta(F)$
$\displaystyle=K^{-1}\otimes F+F\otimes 1,$ $\displaystyle\varepsilon(F)$
$\displaystyle=0,$ $\displaystyle S(F)$ $\displaystyle=-KF,$ (2.23)
$\displaystyle\Delta(K)$ $\displaystyle=K\otimes K$
$\displaystyle\varepsilon(K)$ $\displaystyle=1,$ $\displaystyle S(K)$
$\displaystyle=K^{-1},$ (2.24) $\displaystyle\Delta(H)$
$\displaystyle=H\otimes 1+1\otimes H,$ $\displaystyle\varepsilon(H)$
$\displaystyle=0,$ $\displaystyle S(H)$ $\displaystyle=-H.$

Let $\mathcal{C}$ be the category of finite dimensional weight modules (i.e.
modules on which $H$ acts diagonally) and on which $K$ acts as $q^{H}$ and
such that $E^{r}$ and $F^{r}$ act as $0$. We will now recall that the category
$\mathcal{C}$ is a ribbon category. Let $V$ and $W$ be objects of
$\mathcal{C}$. Let $\\{v_{i}\\}$ be a basis of $V$ and $\\{v_{i}^{*}\\}$ be a
dual basis of $V^{*}=\mathrm{Hom}_{\mathbb{C}}(V,\mathbb{C})$. Then

(2.25)
$\displaystyle\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{coev}}}_{V}:\;$
$\displaystyle\mathbb{C}\rightarrow V\otimes V^{*},\text{ given by
}1\mapsto\sum v_{i}\otimes v_{i}^{*},$ (2.26)
$\displaystyle\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{ev}}}_{V}:\;$
$\displaystyle V^{*}\otimes V\rightarrow\mathbb{C},\text{ given by }f\otimes
w\mapsto f(w)$

are duality morphisms of $\mathcal{C}$. In [46] Ohtsuki truncates the usual
formula of the $h$-adic quantum $\mathfrak{sl}_{2}$ $R$-matrix to define an
operator on $V\otimes W$ by

(2.27) $R=\xi^{H\otimes
H/2}\sum_{n=0}^{r-1}\frac{\\{1\\}^{2n}}{\\{n\\}!}\xi^{n(n-1)/2}E^{n}\otimes
F^{n}$

where $\xi^{H\otimes H/2}$ is the operator given by

(2.28) $\xi^{H\otimes H/2}(v\otimes
v^{\prime})=\xi^{\lambda\lambda^{\prime}/2}v\otimes v^{\prime}$

for weight vectors $v$ and $v^{\prime}$ of weights of $\lambda$ and
$\lambda^{\prime}$. The $R$-matrix is not an element in
${\mathcal{U}}^{H}_{\xi}(\mathfrak{sl}_{2})\otimes{\mathcal{U}}^{H}_{\xi}(\mathfrak{sl}_{2})$,
however the action of $R$ on the tensor product of two objects of
$\mathcal{C}$ is a well defined linear map on such a tensor product. So, $R$
gives rise to a braiding $c_{V,W}:V\otimes W\rightarrow W\otimes V$ on
$\mathcal{C}$ defined by $v\otimes w\mapsto\tau(R(v\otimes w))$ where $\tau$
is the permutation $x\otimes y\mapsto y\otimes x$. Also, let $\theta$ be the
operator given by

(2.29)
$\theta=K^{r-1}\sum_{n=0}^{r-1}\frac{\\{1\\}^{2n}}{\\{n\\}!}\xi^{n(n-1)/2}S(F^{n})\xi^{-H^{2}/2}E^{n}$

where $\xi^{-H/2}$ is an operator defined on a weight vector $v_{\lambda}$ by
$\xi^{-H^{2}/2}.v_{\lambda}=\xi^{-\lambda^{2}/2}v_{\lambda}.$ Ohtsuki shows
that the family of maps $\theta_{V}:V\rightarrow V$ in $\mathcal{C}$ defined
by $v\mapsto\theta^{-1}v$ is a twist (see [46]).

Now the ribbon structure on $\mathcal{C}$ yields right duality morphisms

(2.30)
$\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{ev}}}_{V}=\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{ev}}}_{V}c_{V,V^{*}}(\theta_{V}\otimes\mathrm{Id}_{V^{*}})\text{
and
}\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{coev}}}_{V}=(\mathrm{Id}_{V^{*}}\otimes\theta_{V})c_{V,V^{*}}\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{coev}}}_{V}$

which are compatible with the left duality morphisms
$\\{\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{coev}}}_{V}\\}_{V}$
and
$\\{\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{ev}}}_{V}\\}_{V}$.
These duality morphisms are given by

(2.31)
$\displaystyle\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{coev}}}{V}:$
$\displaystyle\mathbb{C}\rightarrow V^{*}\otimes V,\text{ where }1\mapsto\sum
K^{r-1}v_{i}\otimes v_{i}^{*},$ (2.32)
$\displaystyle\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{ev}}}_{V}:$
$\displaystyle V\otimes V^{*}\rightarrow\mathbb{C},\text{ where }v\otimes
f\mapsto f(K^{1-r}v).$

The _quantum dimension_ $\mathrm{qdim}(V)$ of an object $V$ in $\mathcal{C}$
is the
$\mathrm{qdim}(V)={\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{ev}}}_{V}\circ\stackrel{{\scriptstyle\longrightarrow}}{{\operatorname{coev}}}_{V}}=\sum
v_{i}^{*}(K^{1-r}v_{i})$.

For each $n\in\\{0,\ldots,r-1\\}$ let $S_{n}$ be the usual $(n+1)$-dimensional
simple highest weight $U_{\xi}(\mathfrak{sl}_{2})$-module with highest weight
$n$. The module $S_{n}$ is a highest weight module with a highest weight
vector $s_{0}$ such that $Es_{0}=0$ and $Hs_{0}=ns_{0}$. Then
$\\{s_{0},s_{1},\ldots,s_{n}\\}$ is a basis of $S_{n}$ where $Fs_{i}=s_{i+1}$,
$H.s_{i}=(n-2i)s_{i}$, $E.s_{0}=0=F^{n+1}.s_{0}$ and
$E.s_{i}=\frac{\\{i\\}\\{n+1-i\\}}{\\{1\\}^{2}}s_{i-1}$. The quantum dimension
of $S_{n}$ is $\mathrm{qdim}(S_{n})=(-1)^{n}\frac{\\{n+1\\}}{\\{1\\}}$. Next
we consider a larger class of finite dimensional highest weight modules: for
each $\alpha\in\mathbb{C}$ we let $V_{\alpha}$ be the $r$-dimensional highest
weight $U^{H}_{\xi}(\mathfrak{sl}_{2})$-module of highest weight $\alpha+r-1$.
The modules $V_{\alpha}$ has a basis $\\{v_{0},\ldots,v_{r-1}\\}$ action on
which is given by

(2.33) $H.v_{i}=(\alpha+r-1-2i)v_{i},\quad
E.v_{i}=\frac{\\{i\\}\\{i-\alpha\\}}{\\{1\\}^{2}}v_{i-1},\quad
F.v_{i}=v_{i+1}.$

For all $\alpha\in\mathbb{C}$, the quantum dimension of $V_{\alpha}$ is zero:

(2.34)
$\mathrm{qdim}(V_{\alpha})=\sum_{i=0}^{r-1}v_{i}^{*}(K^{1-r}v_{i})=\sum_{i=0}^{r-1}\xi^{(r-1)(\alpha+r-1-2i)}=\xi^{(r-1)(\alpha+r-1)}\frac{1-\xi^{2r}}{1-q^{2}}=0.$

As shown in [18], if $L$ is a framed oriented link in $S^{3}$ colored by
modules $S_{c_{I}}$, where $I$ runs over the components of the link, then

(2.35) $F(L)=J_{{c}}(L)$

where $F$ is the Reshetikhin-Turaev functor and $J_{{c}}$ is the
(unnormalised) skein theoretical colored Jones polynomial of $L$, so that in
particular the value for the $S_{i}$-colored unknot is $(-1)^{i}[i+1]$ (as
usual $[n]:=\\{n\\}/\\{1\\}$). Another well known version of the colored Jones
polynomial, which we shall call “representation theoretical”, is obtained by
considering the full subcategory $\mathcal{C}^{\prime}$ of $\mathcal{C}$
generated by the modules $S_{n}$ with the same braiding and ribbon structure
as above, with the only difference in the following morphisms:

$\displaystyle\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{coev}}}{V}:$
$\displaystyle\mathbb{C}\rightarrow V^{*}\otimes V,\text{ where }1\mapsto\sum
K^{-1}v_{i}\otimes v_{i}^{*},$
$\displaystyle\stackrel{{\scriptstyle\longleftarrow}}{{\operatorname{ev}}}_{V}:$
$\displaystyle V\otimes V^{*}\rightarrow\mathbb{C},\text{ where }v\otimes
f\mapsto f(Kv).$

Then the image of $L$ via the Reshetikhin-Turaev functor associated to the
category $\mathcal{C}^{\prime}$ is the version of the colored Jones considered
for instance in [37], let us denote it $V_{c}(L)$.

######  Remark 2.1.

The definition of representation theoretical Jones polynomial can be extended
to generic values of $q$, as opposed to $\xi=\exp(\frac{i\pi}{r})$.

######  Lemma 2.2.

It holds $V_{{c}}(L)=(-1)^{\sum_{I}(B_{II}+1)c_{I}}J_{{c}}(L).$

###### Proof.

It is sufficient to compare when $L$ is the closure of a braid of $B_{m}$ such
that $B_{II}=0$ for all $I$. Then the only difference between the two
Reshetikhin-Turaev functors associated to $\mathcal{C}^{\prime}$ and
$\mathcal{C}$ is coming from the $m$ maxima: a maximum colored by $n_{I}$ will
contribute to $J_{{c}}(L)$ via $K^{1-r}$ and to $V_{c}(L)$ via $K$ and since
$K^{r}$ acts as $\xi^{rn_{I}}=(-1)^{n_{I}}$ the overall difference is
$(-1)^{\sum n_{I}}$ where the sum is taken over all the $m$ strands of the
braid. But now we claim that this sign coincides with
$(-1)^{\sum_{I}(B_{II}+1)c_{I}}$. Indeed observe that $B_{II}$ is not changed
(modulo $2$) if one switches the crossings of the braid so that we can prove
the claim when the link is actually a disjoint union of unknots. Now for each
such unknot colored by odd $n_{I}$ we observe that each Reidemeister $1$ move
changes by one both the number of maxima and $B_{II}$, while the other
Reidemeister moves do not change these values. Finally for the standard
diagram of the unknot the statement is true. ∎

######  Lemma 2.3 (Symmetry principle for skein theoretical Jones
polynomials).

Let $\xi=\exp(\frac{\pi i}{r})$ and $L$ be a framed oriented link colored by
$S_{c_{1}}\cdots,S_{c_{V}}$ with $0\leq c_{I}\leq r-2$. For each
$a_{I}\in\\{0,1\\}$ and $c_{I}\in\\{0,\ldots r-2\\}$ let
$a_{I}*c_{I}:=a_{I}(r-2-c_{I})+(1-a_{I})c_{I}$ and for ${a}\in\\{0,1\\}^{V}$
let $a*c:=(a_{1}*c_{1},\ldots,a_{V}*c_{V})$. Then it holds:

(2.36)
$J_{a*c}(L)=i^{(r-2)\sum_{I,J}B_{IJ}a_{I}a_{J}}(-1)^{\sum_{I,J}B_{IJ}a_{I}c_{J}}(-1)^{\sum_{I}(B_{II}+1)(r-2)a_{I}}J_{c}(L).$

###### Proof.

The symmetry principle for the representation theoretical Jones polynomial
stated in [37], formula (4.20), is:

(2.37)
$V_{a*c}(L)=i^{(r-2)\sum_{I,J}B_{IJ}a_{I}a_{J}}(-1)^{\sum_{I,J}B_{IJ}a_{I}c_{J}}V_{c}(L).$

By Lemma 2.2, for each component where $a_{I}\neq 0$ we acquire an additional
factor $(-1)^{(B_{II}+1)(c_{I}-(r-2-c_{I}))}.$ ∎

We end this section by recalling the definition of the Witten-Reshetikhin-
Turaev invariants in their TQFT normalisation, according to Blanchet-Habegger-
Masbaum-Vogel [8]:

######  Definition 2.4.

Let $D=\sqrt{\frac{r}{2}}(\sin(\frac{\pi}{r}))^{-1}$ and $M$ be the three-
manifold obtained by surgery on a framed oriented link $L$.

(2.38)
$\mathrm{WRT}_{r}(M)=D^{-b_{0}(M)-b_{1}(M)}U_{+}^{-b_{+}}U_{-}^{-b_{-}}\sum_{{c}}d(c)J_{{c}}(L)$

where $U_{\pm}=\sum_{c=0}^{r-2}\mathrm{qdim}(c)J_{c}(u_{\pm})$, $u_{\pm}$ is
the unknot with framing $\pm$, $\vec{c}$ runs over all the colorings of $L$
with colors in $\\{0,1,\ldots r-2\\}$ and
$\mathrm{qdim}({c})=\prod_{I=1}^{V}\mathrm{qdim}(c_{I})$.

We also recall that there exists a cohomological refinement of
$\mathrm{WRT}_{r}(M)$ defined for cohomology classes $\omega\in
H^{1}(M;\mathbb{Z}/2\mathbb{Z})$ (see [18]). Given such a class and an
oriented link $L\subset S^{3}$ such that $M$ is obtained by surgery over $L$,
then $\omega$ induces a parity on the components of $L$ via
$\omega(L_{I}):=\omega(\mathfrak{m}_{I})\in\mathbb{Z}/2\mathbb{Z}$ where
$\mathfrak{m}_{I}$ is homology class of the meridian of $L_{I}$. Let

(2.39)
$\Delta_{-}^{SO(3)}:=\frac{\Delta_{-}}{(-1)^{r-1}(\xi-\xi^{-1})},\qquad\Delta_{+}^{SO(3)}:=-\frac{\Delta_{+}}{(-1)^{r-1}(\xi-\xi^{-1})},$

where $\Delta_{\pm}$ are defined in (2.44) and (2.45). Then one defines (see
also Appendix B) :

######  Definition 2.5.

(2.40)
$\mathrm{WRT}_{r}(M,\omega)=\frac{D^{-b_{0}(M)-b_{1}(M)}}{(\Delta_{+}^{SO(3)})^{b_{+}}(\Delta_{-}^{SO(3)})^{b_{-}}}\sum_{{c}=\omega\mod
2}d(c)J_{{c}}(L)$

where ${c}$ runs over all the colorings of $L$ with values in $\\{0,1,\ldots
r-2\\}$ which are congruent mod $2$ to $\omega(L_{I})$ for each component of
$L$.

###  2.4. $\mathrm{N}_{r}$ invariants

With the above notation, to calculate $\mathrm{N}_{r}(M,\omega)$ in this case
it is enough to introduce the following definitions555In [17] $q$ was used
instead of $\xi:=e^{\frac{\pi i}{r}}$ and, mainly, the definition of
$d(\alpha)$ differs by a factor $(-1)^{r-1}r$ : this only affects
$\mathrm{N}_{r}$ by an overall scalar.:

(2.41)
$d(\alpha):=\frac{\sin\frac{\pi\alpha}{r}}{\sin\pi\alpha}=\frac{\xi^{\alpha}-\xi^{-\alpha}}{\xi^{r\alpha}-\xi^{-r\alpha}},$
(2.42) $S(\alpha,\beta):=\xi^{\alpha\beta},$ (2.43)
$T(\alpha):=\xi^{\frac{\alpha^{2}-(r-1)^{2}}{2}}.$ (2.44)
$\Delta_{-}:=\left\\{\begin{array}[]{cl}0,&r=0\mod 4,\\\
i\xi^{\frac{3}{2}}r^{\frac{1}{2}},&r=1\mod 4,\\\
(1-i)\xi^{\frac{3}{2}}r^{\frac{1}{2}},&r=2\mod 4,\\\
-\xi^{\frac{3}{2}}r^{\frac{1}{2}},&r=3\mod 4,\end{array}\right.$ (2.45)
$\Delta_{+}=\overline{\Delta_{-}}.$

As before, let us denote by $\mathfrak{m}_{I}\in H_{1}(M;\mathbb{Z})$ the
homology class of the oriented meridian of the component of $L$ indexed by
$I\in\mathrm{Vert}$. Then define

(2.46) $\mu_{I}:=\omega(\mathfrak{m}_{I})\in\mathbb{C}/2\mathbb{Z}.$

We will assume from now on that $\mu_{I}\notin\mathbb{Z}/2\mathbb{Z},\ \forall
I$ (it was proved in [17] that one can always find $L$ presenting $M$ such
that this condition is satisfied if $\omega\in
H^{1}(M;\mathbb{C}/2\mathbb{Z})\setminus H^{1}(M;\mathbb{Z}/2\mathbb{Z})$).
Note that

(2.47) $\sum_{J}B_{IJ}\mu_{J}=0\mod 2.$

####  2.4.1. $\mathrm{N}_{r}$ for links, ADO and Conway-Alexander polynomials

As proved in [1] (but in what follows we use the notation used in [17]), there
is an invariant $\mathrm{N}_{r}(S^{3},L_{\mu})$ associated to each framed
oriented link with a $\mathbb{C}$-coloring of its components, which, up to a
factor depending on the linking form, is valued in
$\mathbb{C}[\xi^{\pm\mu_{1}},\cdots,\xi^{\pm\mu_{V}}]$ if $V>1$ and in
$\frac{\xi_{1}^{\mu_{1}}-\xi_{1}^{-\mu_{1}}}{\xi_{1}^{r\mu_{1}}-\xi_{1}^{-r\mu_{1}}}\mathbb{C}[\xi_{1}^{\pm\mu_{1}}]$
in the knot case.

This invariant is computed by first cutting open $L$ on one of its components
(say the $I^{th}$) to get a $(1,1)$-tangle $T$, then computing the
Reshetikhin-Turaev functor $F$ applied to $T$ by considering a color $\mu_{J}$
as the $r$-dimensional simple projective module $V_{\mu_{J}}$ over
${\mathcal{U}}^{H}_{\xi}(\mathfrak{sl}_{2})$ thus getting

(2.48) $F(T)=<T>\mathrm{Id}_{V_{\mu_{I}}}$

for some scalar $<T>$; then defining
$\mathrm{N}_{r}(S^{3},L_{\mu})=f<T>d(\mu_{I})$ where $f$ is a certain factor
related to the linking form (see later). It can be proved that this value does
not depend on the way $L$ was cut to obtain $T$ and is then an invariant of
the colored oriented framed link $L_{\mu}$.

Comparing with the original definition of the ADO polynomial of links we
remark the following : for each framed oriented link $L\subset S^{3}$ with
$V\geq 2$ components we have

(2.49)
$ADO_{r}(L)(x_{1}=\xi^{-2}\xi^{2\mu_{1}},\ldots,x_{V}=\xi^{-2}\xi^{2\mu_{V}})=\mathrm{N}_{r}(S^{3},L)\xi^{-\frac{\mu^{T}B\mu}{2}+\frac{(r-1)^{2}\,\mathrm{Tr}\,B}{2}}.$

For knots, we remark that $\mathrm{N}_{r}$ is not a polynomial and the
relation with ADO polynomial is as follows:

(2.50)
$ADO_{r}(K)(x=\xi^{-2}\xi^{2\mu})=\frac{\xi^{r\mu}-\xi^{-r\mu}}{\xi^{\mu}-\xi^{-\mu}}\mathrm{N}_{r}(S^{3},K)\xi^{-\frac{f\mu^{2}}{2}+\frac{f(r-1)^{2}}{2}}$

where $f$ is the framing of $K$.

A case of special interest is when $r=2$ where, as proved in [7]
$\mathrm{N}_{r}$ is equivalent to the Alexander-Conway function (see Corollary
6.19, taking into account the difference in the definition of the modified
quantum dimension used here). The precise statement is the following, letting
$\xi=i$:

(2.51)
$\mathrm{N}_{2}(L)=i\nabla_{L}(i\xi^{-\mu_{1}},i\xi^{-\mu_{1}},\ldots,i\xi^{-\mu_{V}})\xi^{\frac{\mu^{T}B\mu}{2}-\frac{{\varepsilon}^{T}B{\varepsilon}}{2}}=\\\
(-1)^{n}i\nabla_{L}(-i\xi^{\mu_{1}},-i\xi^{\mu_{2}},\cdots,-i\xi^{\mu_{V}})\xi^{\frac{\mu^{T}B\mu}{2}-\frac{{\varepsilon}^{T}B{\varepsilon}}{2}}.$

######  Example 2.6.

Let $L$ be a plumbing link the components of which are colored by
$\mu_{I},\,I\in\mathrm{Vert}$, and let $B$ be the linking matrix. Then:

(2.52)
$\mathrm{N}_{r}(S^{3},L)=\xi^{-\frac{\mathrm{Tr}\,B\,(r-1)^{2}}{2}}\xi^{\frac{1}{2}\left({\mu}^{T}B{\mu}\right)}\prod_{I\in\mathrm{Vert}}d(\mu_{I})^{1-\deg(I)}.$

In the case $r=2$, so that $\xi=i$ we get:

(2.53)
$\mathrm{N}_{2}(S^{3},L)=\xi^{-\frac{\mathrm{Tr}\,B}{2}}\xi^{\frac{1}{2}\left({\mu}^{T}B{\mu}\right)}\prod_{I\in\mathrm{Vert}}(\xi^{\mu_{I}}+\xi^{-\mu_{I}})^{\deg(I)-1}$

and

(2.54) ${}\nabla_{L}(x_{1},\ldots
x_{V})=\prod_{I}(x_{I}-x_{I}^{-1})^{\deg(I)-1}.$

####  2.4.2. $\mathrm{N}_{r}$ for manifolds

The definition of $\mathrm{N}_{r}$ for a manifold $M$ represented by integral
surgery on $L$ is as follows:

(2.55)
$\mathrm{N}_{r}(M,\omega)=\frac{1}{\Delta_{+}^{b_{+}}\Delta_{-}^{b_{-}}}\sum_{k\in
H_{r}^{\mathrm{Vert}}}\prod_{I\in\mathrm{Vert}}d(\alpha_{k_{I}})\,T(\alpha_{k_{I}})^{B_{II}}\mathrm{N}_{r}(S^{3},L_{\alpha})$

where

(2.56) $\alpha_{k_{I}}:=\mu_{I}+k_{I}\,,\qquad\qquad
H_{r}=\\{-(r-1),-(r-3),\cdots,(r-3),r-1\\}$

and $b_{\pm}$ are the number of positive/negative eigenvalues of $B$.

######  Remark 2.7.

The invariant $\mathrm{N}_{r}$ defined above differs by a constant with
respect to the invariant $N^{\prime}_{r}$ defined in [17] and its
renormalisation $Z_{r}$ introduced in [6] because of the different definition
of $d(\alpha)$ and consequently of $\Delta_{\pm}$ and of the link invariant.
The general relation for a closed connected manifold is then:

(2.57)
$\mathrm{N}_{r}(M,\omega)=\frac{N^{\prime}_{r}(M,\omega)}{((-1)^{r-1}r)^{b_{1}(M)+b_{0}(M)}}=\frac{r\sqrt{r^{\prime}}(\frac{r^{2}}{\sqrt{r^{\prime}}})^{b_{1}(M)}Z_{r}(M,\omega)}{((-1)^{r-1}r)^{b_{1}(M)+b_{0}(M)}}$

where $r^{\prime}=r$ if $r$ is odd and $\frac{r}{2}$ else.

In the special case of a plumbed $M$ then it reads

(2.58)
$\mathrm{N}_{r}(M,\omega)=\frac{1}{\Delta_{+}^{b_{+}}\Delta_{-}^{b_{-}}}\sum_{k\in
H_{r}^{\mathrm{Vert}}}\prod_{I\in\mathrm{Vert}}d(\alpha_{k_{I}})^{2-\deg(I)}\,T(\alpha_{k_{I}})^{B_{II}}\prod_{(I,J)\in\text{Edges}}S(\alpha_{k_{I}},\alpha_{k_{J}}).$

######  Example 2.8 ($\Sigma\times S^{1}$).

Let $M=\Sigma_{g}\times S^{1}$ for a closed oriented surface $\Sigma_{g}$ of
genus $g$ and let $\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})\setminus
H^{1}(M;\mathbb{Z}/2\mathbb{Z})$. If $\beta=\omega(\\{pt\\}\times S^{1})$ then
as shown in [6] (Thm 5.9) it holds:

(2.59) $\mathrm{N}_{r}(M,\omega)=r^{2g}\sum_{k\in
H_{r}}\left(\frac{\\{r\beta\\}}{\\{\beta+k\\}}\right)^{2g-2}.$

####  2.4.3. A relation between $\mathrm{N}_{r}$ and $\mathrm{WRT}_{r}$

The following result outlines a partial direct relation between the invariants
$\mathrm{N}_{r}$ and $\mathrm{WRT}_{r}$ (cf. [3]):

######  Theorem 2.9.

Let $r\geq 2$ be an integer non divisible by $4$ and $K\subset S^{3}$ be an
oriented zero framed knot and $M$ be the surgery on it. Let
$\alpha\in\mathbb{C}$ be a color on $K$ and $\omega_{\alpha}\in
H^{1}(M;\mathbb{C}/2\mathbb{Z})$ be the unique cohomology class the value of
which on the positive meridian of $K$ is $\alpha\ {\rm mod}\ 2\mathbb{Z}$.
Then the following holds:

(2.60) ${\rm if\ }r{\rm\ is\ odd:}\ \lim_{\alpha\to
0}[r\alpha]^{2}\mathrm{N}_{r}(M,\omega_{\alpha})=D^{2}\,\mathrm{WRT}_{r}(M)$
(2.61) $\displaystyle{\rm else:}\;\lim_{\alpha\to
0}[r\alpha]^{2}\mathrm{N}_{r}(M,\omega_{\alpha})$
$\displaystyle=2D^{2}\,\mathrm{WRT}_{r}(M,\omega_{0})$ (2.62)
$\displaystyle{\rm and}\ \lim_{\alpha\to
1}[r\alpha]^{2}\mathrm{N}_{r}(M,\omega_{\alpha})$
$\displaystyle=2D^{2}\,\mathrm{WRT}_{r}(M,\omega_{1})$

where $D=\sqrt{\frac{r}{2}}(\sin(\frac{\pi}{r}))^{-1}$, $\mathrm{WRT}_{r}(M)$
is the standard WRT invariant of $M$, $\mathrm{WRT}_{r}(M,\omega_{i})$ are the
cohomology refined invariants of $M$ and $\omega_{i}\in
H^{1}(M;\mathbb{Z}/2\mathbb{Z})$ is the cohomology class on $M$ the value of
which on the meridian of $K$ is $i=0,1$.

###### Proof.

Present $K$ as the closure of a $(1,1)$-tangle $T$ and for each absolutely
simple module $V$ (i.e. such that $\mathrm{End}(V)=\mathbb{C}$) over
$U^{H}_{q}(\mathfrak{sl}_{2})$ let $T(V)\in\mathbb{C}$ be the scalar such that
$F(T)=T(V)\mathrm{Id}_{V}$ where $F$ is the Reshetikhin-Turaev functor. In
particular we shall use the $n+1$-dimensional highest weight simple module
$S_{n}$ and the absolutely simple $r$-dimensional module $V_{\alpha}$ with
highest weight $\alpha+r-1$.

r odd case. By definition, for $r$ odd we have:

(2.63) $[r\alpha]^{2}\mathrm{N}_{r}(M,\omega)=\sum_{k=-(r-1),\ \text{by}\
2}^{r-1}\frac{\\{\alpha+k\\}^{2}}{\\{1\\}^{2}}T(V_{\alpha+k}),$ (2.64)
$\mathrm{WRT}_{r}(M)=D^{-2}\sum_{j=0}^{r-2}\frac{\\{j+1\\}^{2}}{\\{1\\}^{2}}T(S_{j}).$

By the symmetry principle for the colored Jones polynomials of knots (Lemma
2.3), see [37] formula 4.20) and by the equality $\\{r-1-j\\}=\\{j+1\\}$ we
have also:

(2.65) $\mathrm{WRT}_{r}(M)=2D^{-2}\sum_{j=0\ by\
2}^{r-3}\frac{\\{j+1\\}^{2}}{\\{1\\}^{2}}T(S_{j}).$

As shown in Proposition 4 of [18], it holds

(2.66) $T(S_{r-1-k})=T(V_{k})=T(V_{-k})\ \forall k\in\\{0,\ldots,r-1\\},$

so we have:

(2.67) $\lim_{\alpha\to
0}[r\alpha]^{2}\mathrm{N}_{r}(M,\omega)=\sum_{k=-(r-1),\ \text{by}\
2}^{r-1}\frac{\\{k\\}^{2}}{\\{1\\}^{2}}T(V_{k})=2\sum_{k=2,\ by\
2}^{r-1}\frac{\\{k\\}^{2}}{\\{1\\}^{2}}T(V_{k})=\\\ =2\sum_{k=2,\ \text{by}\
2}^{r-1}\frac{\\{k\\}^{2}}{\\{1\\}^{2}}T(S_{r-1-k})=2\sum_{j=0,\ \text{by}\
2}^{r-3}\frac{\\{r-1-j\\}}{\\{1\\}}T(S_{j})=2\sum_{j=0,\ \text{by}\
2}^{r-3}\frac{\\{j+1\\}^{2}}{\\{1\\}^{2}}T(S_{j}).$

r even case. By definition, for $r$ even we have:

(2.68) $[r\alpha]^{2}\mathrm{N}_{r}(M,\omega_{\alpha})=\sum_{k=-(r-1),\
\text{by}\ 2}^{r-1}\frac{\\{\alpha+k\\}^{2}}{\\{1\\}^{2}}T(V_{\alpha+k}),$
(2.69)
$\mathrm{WRT}_{r}(M,\omega_{i})=D^{-2}\sum_{j=i}^{r-2+i}\frac{\\{j+1\\}^{2}}{\\{1\\}^{2}}T(S_{j}).$

So

(2.70) $\lim_{\alpha\to
0}[r\alpha]^{2}\mathrm{N}_{r}(M,\omega_{\alpha})=\sum_{k=1,\ \text{by}\
2}^{r-1}2\frac{\\{k\\}^{2}}{\\{1\\}^{2}}T(V_{k})=\sum_{k=1\ by\
2}^{r-1}2\frac{\\{k\\}^{2}}{\\{1\\}^{2}}T(S_{r-1-k})=\\\ =\sum_{j=0,\
\text{by}\ 2}^{r-2}2\frac{\\{r-1-j\\}^{2}}{\\{1\\}^{2}}T(S_{j})=\sum_{j=0,\
\text{by}\
2}^{r-2}2\frac{\\{1+j\\}^{2}}{\\{1\\}^{2}}T(S_{j})=2D^{2}\,\mathrm{WRT}_{r}(M,\omega_{0}).$

Similarly:

(2.71) $\lim_{\alpha\to
1}[r\alpha]^{2}\mathrm{N}_{r}(M,\omega_{\alpha})=\sum_{k=-(r-1),\ \text{by}\
2}^{r-1}2\frac{\\{k+1\\}^{2}}{\\{1\\}^{2}}T(V_{k+1})=\sum_{s=2,\ \text{by}\
2}^{r-2}2\frac{\\{s\\}^{2}}{\\{1\\}^{2}}T(V_{s})=\\\ =\sum_{s=2,\ \text{by}\
2}^{r-2}2\frac{\\{s\\}^{2}}{\\{1\\}^{2}}T(S_{r-1-s})=\sum_{j=1,\ \text{by}\
2}^{r-3}2\frac{\\{r-1-j\\}^{2}}{\\{1\\}^{2}}T(S_{j})=2D^{2}\,\mathrm{WRT}_{r}(M,\omega_{1})$

where in the second equality we used that $\\{r\\}=\\{0\\}=0$. ∎

###  2.5. $\widehat{Z}$ invariants

In this subsection we recall some general facts about the invariants
$\widehat{Z}_{b}$ of 3-manifolds and their version
$F_{L}(q,x):=\widehat{Z}(S^{3}\setminus L)$ for links. We formulate some
conjectures about the link invariant and outline how the $3$-manifold
invariant is build out of the invariant for links. We postpone examples to a
later section.

####  2.5.1. $\widehat{Z}$ for links

A physical construction of new $q$-series invariants of 3-manifolds was
proposed in [33, 32]. Much like invariants $\mathrm{N}_{r}(M)$ reviewed above,
these $q$-series invariants are labeled by extra data, which originally was
interpreted as the choice of abelian flat connection or, equivalently,666In
this paper we only consider $SU(2)$ version of these invariants. The higher-
rank version is also available, see e.g. [16, 47]. an element of $H_{1}(M)$.
Soon, it was realized [30, 29] that the extra data $\mathfrak{s}$ which labels
$\widehat{Z}_{\mathfrak{s}}(M)$ should be understood as a spinc structure on
$M$. Recall, that as a set $H_{1}(M)$ is isomorphic to Spin${}^{c}(M)$, but
the isomorphism is non-canonical. The difference between the two becomes
apparent in cutting and gluing operations, which in part explains why it was
not noticed until the invariants were extended to 3-manifolds with toral
boundaries and link complements, where $\mathfrak{s}$ is a relative spinc-
structure. In particular, if $M$ is the complement of a knot $K$ in rational
homology sphere, the invariant is expected to be read off from a single two-
variable series:

(2.72) $F_{K}(x,q)\in 2^{-c}q^{\Delta}\,\mathbb{Z}[q^{-1},q]][[x^{\pm 1}]]$

for some $c\in\mathbb{Z}_{+}$ and $\Delta\in\mathbb{Q}$. Apart from the
original physics formulation, several approaches toward rigorous constructions
of this invariant have been developed. For example, the approach based on
recursion and resurgence, although in principle can be applied to any knot (or
link) is rather laborious and was used in [30] to produce
$F_{K}(q,x):=\widehat{Z}(S^{3}\setminus K)$ only for torus knots and a single
hyperbolic knot, the figure-eight knot $K={\bf 4_{1}}$.

Another, much more efficient diagrammatic approach based on the R-matrix for
Verma modules and quantum groups at generic $q$ was proposed by Park [48, 49].
Using this approach, one can quickly compute $F_{K}(x,q)$ for many hyperbolic
knots up to roughly 10 crossings and also for many infinite families.
Specifically, the R-matrix used by Park [48] is the universal R-matrix [38]
applied to the lowest weight Verma modules with complex weights:

(2.73) $R^{ij}_{kl}=\bigl{[}\\!\begin{smallmatrix}j\\\
n\end{smallmatrix}\\!\bigr{]}\prod_{l=1}^{n}(1-y^{-1}q^{i+l})x^{-\frac{i}{2}-\frac{n}{4}}y^{-\frac{j-n}{2}+\frac{n}{4}}q^{(j-n)i+\frac{(j-n)n}{2}+\frac{j-n+i+1}{2}}.$

Here, and in relating it to the R-matrix (2.2) we are a little cavalier with
the overall powers of $q$. Indeed, the fact that (2.2) and (2.73) are related
by sending $q$ to a root of unity is the first indication for the relation
(1.2) between the corresponding knot invariants. This also clarifies the
quantum group origin of the $\widehat{Z}$-invariants. Since they basically
provide a non-perturbative formulation777The all-order perturbative
formulation was known for quite some time, see e.g. [23] and references
therein. of the $SL(2,\mathbb{C})$ Chern-Simons TQFT — where $q$ is a
continuous complex variable and so are the highest weights of representations
by which Wilson lines are colored — it was expected for a long time that
$SL(2,\mathbb{C})$ Chern-Simons theory should be described by
$U_{q}({\mathfrak{g}})$ with generic $q$.

In order to relate the R-matrices (2.2) and (2.73), we first need to replace
$q\to q^{-2}$ and then take the root of unity limit. As in the generalized
volume conjecture [28], all combinations of the form $q^{\text{weight}}$ must
be treated as independent variables and kept fixed. In particular, we need to
use the identifications $x=q^{\mu}$ and $y=q^{\lambda}$ and treat them as
independent complex parameters. Then, using identities of the form

(2.74) $\bigl{[}\\!\begin{smallmatrix}j\\\
n\end{smallmatrix}\\!\bigr{]}=\frac{[j]!}{[n]![j-n]!}=\frac{\\{1\\}\cdots\\{j\\}}{\\{1\\}\cdots\\{n\\}\;\\{1\\}\cdots\\{j-n\\}}=\frac{\\{j-n+1\\}\cdots\\{j\\}}{\\{1\\}\cdots\\{n\\}}=\frac{\\{j;n\\}}{\\{n;n\\}}$

we arrive at the desired relation between (2.73) and (2.2). Based on this
relation between the R-matrices, it is not surprising to see the relation
between the corresponding knot invariants [29] (see also [12]):

(2.75)
$\lim_{q\to\exp(2i\pi/r)}F_{K}(x_{1},q)\Delta_{K}(x_{1}^{r})=ADO_{r}(K)(\frac{x_{1}}{q})|_{q=\exp{\frac{2\pi
i}{r}}}$

where $\Delta_{K}$ is the Alexander polynomial of $K$.

To properly deal with the case of links, we consider the set

(2.76) $\mathbb{C}[q^{-1},q]][[x_{1}^{\pm 1},\ldots x_{V}^{\pm 1}]]$

of formal power series in $\\{x_{I}\\}$ with coefficients in Laurent series in
$q$. Such a set is not a ring but is a module over the ring:

(2.77) $\mathbb{C}[q^{-1},q]][x_{1}^{\pm 1},\ldots x_{V}^{\pm 1}]$

of Laurent polynomials in $\\{x_{I}\\}$ with coefficients in Laurent series in
$q$.

Taking into account the different normalisation between $ADO_{r}(L)$ and
$\mathrm{N}_{r}(S^{3},L)$ we propose the following conjecture for the link
case:

######  Conjecture 4.

For each framed link $L$ in $S^{3}$ there exists a non-zero formal power
series $F_{L}\in\mathbb{Z}[q^{-1},q]][[x_{1},\ldots,x_{V}]]$ such that the
following holds for every $r\geq 2$ and for every
$\vec{\alpha}\in\mathbb{C}^{n}$:

(2.78) $\boxed{\lim_{t\to 1}\lim_{q\to\exp(\frac{2\pi
i}{r})}\frac{F^{t}_{L}(q^{\alpha_{1}},\dots,q^{\alpha_{V}},q)}{F^{t}_{L}(q^{r\alpha_{1}},\dots,q^{r\alpha_{V}},q^{r})}=\mathrm{N}_{r}(S^{3},L_{\alpha})q^{-\frac{1}{2}\left({\alpha}^{T}B\alpha-(r-1)^{2}\mathrm{Tr}\,B\right)}}$

where $B$ is the linking matrix of $L$ and $F^{t}_{L}$ is the formal power
series in $t$ obtained by replacing each $x_{I}^{n},\,n>0$ by
$tq^{\alpha_{I}}$ and each $x_{I}^{-n}$ by $tq^{-\alpha_{I}}$. In general the
radius of convergence of $F^{t}_{L}$ might be less than one but we conjecture
that $\lim_{q\to\exp(\frac{2\pi i}{r})}F^{t}_{L}$ is actually a rational
function of $t$ so that the limit $t\to 1$ makes sense via analytic
continuation.

The relation of Conjecture 4 with the previous ones is clarified by the
following:

######  Conjecture 5.

If $L\subset S^{3}$ is a framed oriented link multivariable Alexander
polynomial $\nabla_{L}$ of which is non-zero then the following holds:

(2.79) $\lim_{t\to 1}\lim_{q\to
1}F^{t}_{L}(q^{\alpha_{1}},\ldots,q^{\alpha_{V}},q)\nabla_{L}(q^{\alpha_{1}},\ldots,q^{\alpha_{V}})=1.$

Indeed in the case of a knot $K$ endowed with zero framing and colored by
$\alpha$, Conjecture 4 becomes:

(2.80) $\lim_{t\to 1}\lim_{q\to\exp(\frac{2\pi
i}{r})}\frac{F^{t}_{K}(q^{\alpha},q)}{F^{t}_{K}(q^{r\alpha};q^{r})}=\mathrm{N}_{r}(S^{3},K_{\alpha})$

so that using Conjecture 5 we get:

(2.81) $\lim_{t\to 1}\lim_{q\to\exp(\frac{2\pi
i}{r})}F^{t}_{K}(q^{\alpha},q)\nabla(q^{r\alpha})=\mathrm{N}_{r}(S^{3},K_{\alpha}).$

There are, however, links with vanishing Alexander polynomial, e.g.

${\bf 9_{27}}\,,~{}{\bf 10_{32}}\,,~{}{\bf 10_{36}}\,,~{}{\bf
10_{107}}\,,~{}{\bf 11_{244}}\,,~{}{\bf 11_{247}}\,,~{}{\bf
11_{334}}\,,~{}{\bf 11_{381}}\,,~{}{\bf 11_{396}}\,,~{}{\bf
11_{404}}\,,~{}{\bf 11_{406}}\,,~{}\ldots$

We then set the following definition

######  Definition 2.10 (Good links).

A link $L$ is very good if Conjectures 4 and 5 are verified for $L$ and is
good if only Conjecture 4 is satisfied for it.

We remark that the links with zero multivariable Alexander polynomial cannot
be very good. Also, Conjecture 4 defines $F_{L}$ only up to a scalar, which
can be reduced to an indeterminacy of $\pm 1$ by supposing that the greatest
common divisor of the coefficients of $F_{L}$ is $\pm 1$.

When $L$ is a knot, a version of Conjecture 4 was proven by S. Willets ([55])
in which $F_{L}$ belongs to a suitable ring obtained as completion of the
Laurent polynomials in two variables.

####  2.5.2. Examples of good links

Let $L\subset S^{3}$ be a plumbing link, with the notation introduced in
Example 2.6. In [31] the following formula was provided for $F_{L}$

(2.82) $F_{L}(\\{x_{I}^{2}\\}_{I},q)=\prod_{i}(x_{I}-x_{I}^{-1})^{1-\deg(I)}.$

If we use this formula for $F_{L}$ then it is clear that

(2.83)
$\lim_{q\to\exp(2i\pi/r)}\frac{F_{L}(\\{x_{I}^{2}\\}_{I},q)}{F_{L}(\\{x_{I}^{2r}\\}_{I},q^{r})}=\prod_{I}\frac{(x_{I}-x_{I}^{-1})^{1-\deg(I)}}{(x_{I}^{r}-x_{I}^{-r})^{1-\deg(I)}}=\xi^{-\frac{{\alpha}^{T}B{\alpha}}{2}+\frac{(r-1)^{2}}{2}\mathrm{Tr}\,(B)}\mathrm{N}_{r}(S^{3},L)$

where we used Example 2.6 and we set $x_{I}=\xi^{\alpha_{I}}$, so that
Conjecture 4 holds for these links.

In general, $F_{L}$ is to be considered as a formal power series and in the
above case we do it as follows. Let $t$ be a regularisation parameter and let
$F^{t}_{L}$ be the power series development in $t$ of

(2.84)
$F^{t}_{L}(\\{x_{I}^{2}\\}_{I},q)=\prod_{I:\,\deg(I)>1}\frac{1}{2}\left((x_{I}/t-tx_{I}^{-1})^{1-\deg(I)}+(tx_{I}-x_{I}^{-1}/t)^{1-\deg(I)}\right)\times\\\
\prod_{I:\,\deg(I)=0}\left(tx_{I}-x_{I}^{-1}/t\right).$

Since $F^{t}_{L}$ is a rational function in $t$ the limit $F_{l}=\lim_{t\to
1}F^{t}_{L}$ exists even though the power series $F^{t}_{L}$ might have radius
of convergence in $t$ less than $1$. So that Conjecture 4 holds.

Similarly the Alexander-Conway polynomial of $L$ can be computed directly in
these cases as $\nabla_{L}=F_{L}^{-1}$ (see Example 2.6) so that also
Conjecture 5 holds. The plumbing links are then very good links.

####  2.5.3. Whitehead link

Starting with the expression for $F_{L}(x_{1},x_{2},q)$ obtained by the
R-matrix technique [48, 49]:

(2.85) $F_{L}(x_{1},x_{2},q)\;=\;\sum_{n\geq
0}\frac{(-1)^{n}q^{-\frac{n(n+1)}{2}}(q^{n+1})_{n}(x_{1}^{1/2}-x_{1}^{-1/2})(x_{2}^{1/2}-x_{2}^{-1/2})}{\prod_{j=0}^{n}(x_{1}+x_{1}^{-1}-q^{j}-q^{-j})(x_{2}+x_{2}^{-1}-q^{j}-q^{-j})}$

we get

(2.86)
$F_{L}(x_{1},x_{2},q)\big{|}_{q\to-1}\;=\;\frac{1}{(x_{1}^{1/2}-x_{1}^{-1/2})(x_{2}^{1/2}-x_{2}^{-1/2})}.$

According to the identification of the parameters in [29], we should compare
this with $\text{ADO}_{p}$ with $p=2$, evaluated at $xe^{-2\pi i/p}$. In the
case of knots, $\text{ADO}_{p}$ is also multiplied by
$\frac{x^{1/2}-x^{-1/2}}{\Delta(x^{p})}$.

Murakami works in conventions such that $q=e^{\pi i/p}$ is the $2p$-th root of
unity and his $\text{ADO}_{p}$ with $p=2$ gives the Alexander polynomial:

(2.87) $\text{ADO}_{1}=1,$ (2.88)
$\text{ADO}_{2}=(z_{1}-z_{1}^{-1})(z_{2}-z_{2}^{-1}),$

where we used $z_{1}=q^{\lambda}$ and $z_{2}=q^{\mu}$ in Murakami’s notations.
Comparing this with the multivariable Alexander polynomial

(2.89) $\Delta=(x_{1}^{1/2}-x_{1}^{-1/2})(x_{2}^{1/2}-x_{2}^{-1/2})$

we see that Murakami’s $z_{i}=x_{i}^{1/2}$ and the relation between
$F_{L}(\vec{x},q)$ and $\text{ADO}_{p}(\vec{x})$ for links must be more
complicated; it should convert powers of $x_{i}^{1/2}-x_{i}^{1/2}$ in the
denominator to the powers of $x_{i}^{1/2}-x_{i}^{1/2}$ in the numerator.

Note, if as in case of knots we evaluate the ADO invariant at $xe^{-2\pi
i/p}$, we would have $(x_{1}^{1/2}+x_{1}^{-1/2})(x_{2}^{1/2}+x_{2}^{-1/2})$ in
the numerator. Then, dividing it by $\Delta(x_{i}^{2})$ we would get
$(x_{1}^{1/2}-x_{1}^{-1/2})(x_{2}^{1/2}-x_{2}^{-1/2})$ in the denominator. For
$p=3$, we get

(2.90) $F_{L}(x_{1},x_{2},q)\big{|}_{q\to e^{2\pi i/3}}=\\\
\frac{x_{1}^{2}x_{2}^{2}+x_{1}^{2}x_{2}+x_{1}^{2}+x_{1}x_{2}^{2}+x_{1}x_{2}+i\sqrt{3}x_{1}x_{2}+x_{1}+x_{2}^{2}+x_{2}+1}{(x_{1}^{1/2}-x_{1}^{-1/2})(x_{2}^{1/2}-x_{2}^{-1/2})(1+x_{1}+x_{1}^{2})(1+x_{2}+x_{2}^{2})}.$

Apart from the familiar $\Delta(x_{i})$, in the denominator we have a factor
of

(2.91)
$(x_{1}+1+x_{1}^{-1})(x_{2}+1+x_{2}^{-1})=\frac{\Delta(x_{i}^{p})}{\prod_{i}(x_{i}^{1/2}-x_{i}^{-1/2})}$

so that the entire denominator is basically $\Delta(x_{i}^{p})$.

On the other hand, the numerator of (2.90) is precisely Murakami’s ADO3
evaluated at $x_{i}e^{-2\pi i/p}$, just as in the case of knots. Therefore, we
can write this relation as

(2.92) $F_{L}(x_{1},x_{2},q)\big{|}_{q\to e^{2\pi
i/p}}\;=\;\frac{\text{ADO}_{p}(x_{i}e^{-2\pi i/p})}{\Delta(x_{i}^{p})}.$

For reasons mentioned above, this also checks out in the $p=2$ case. For $p=4$
we get

(2.93) $F_{L}(x_{1},x_{2},q)\big{|}_{q\to e^{2\pi i/4}}\;=\\\
\frac{x_{1}^{-3/2}x_{2}^{-3/2}(x_{1}+1)(x_{2}+1)\left(x_{1}^{2}\left(x_{2}^{2}+1\right)+2ix_{1}x_{2}+x_{2}^{2}+1\right)}{\Delta(x_{i}^{4})},$

again, in perfect agreement with (2.92).

As a side remark, we also note that, in general, ADO polynomials have many
coefficients that are algebraic numbers; the coefficients of
$\text{ADO}_{p}(x_{i}e^{-2\pi i/p})$ are also algebraic integers, but
typically much simpler.

###  2.6. $\widehat{Z}$ for 3-manifolds

Let $M=S^{3}(\mathcal{L})$ where $\mathcal{L}$ is a link with linking matrix
$B$ and the set of components $\mathrm{Vert}$. Let

(2.94)
$F(x,q):=F_{L}(x^{2},q)\prod_{I\in\mathrm{Vert}}(x_{I}-x^{-1}_{I})=\sum_{\ell\in\mathbb{Z}^{\mathrm{Vert}}}F_{\ell}\prod_{I\in\mathrm{Vert}}x_{I}^{\ell_{I}}$

be a somewhat differently normalized $\widehat{Z}$-invariant of the link $L$
or, more precisely, the link complement $S^{3}\setminus L$. For example,
$F(x,q)=F_{K}(x^{2},q)(x-x^{-1})$ for a knot (complement), and
$F(x,q)=\prod_{I\in\mathrm{Vert}}(x_{I}-x_{I}^{-1})^{2-\deg(I)}$ for a
plumbing link (complement). Note, the two notations, $F(x,q)$ and
$F_{L}(x,q)$, are very similar and we hope it will not cause a confusion. In
fact, $F_{L}(x,q)$ is used in most of the paper, and in a few places where
$F(x,q)$ is used we try to remind the reader about the relation between the
two normalizations.

By an $SL(V,\mathbb{Z})$ transform we can bring the integral quadratic form
$B$ to $B^{\prime}\oplus\mathbf{0}_{b_{1}}$ where $\mathbf{0}_{b_{1}}$ is the
trivial quadratic form on $\mathbb{Z}^{b_{1}}\subset\mathbb{Z}^{V}$ [40].
Namely, there exists $U\in SL(V,\mathbb{Z})$ such that

(2.95) $UBU^{T}=\left(\begin{array}[]{cc}B^{\prime}&0\\\
0&0\end{array}\right)$

where the right-hand side shows the block decomposition corresponding to the
partition $V=(V-b_{1})+b_{1}$. With $\varepsilon=(1,1,\ldots,1)$ and $s$ being
mod 2 vectors defined as before
($\varepsilon=(1,1,\ldots,1),\;\sum_{J}B_{IJ}s_{J}=s_{I}\mod 2$)

(2.96) $\left(\begin{array}[]{c}s^{\prime}\\\
s^{\prime\prime}\end{array}\right):=(U^{T})^{-1}s,\qquad\left(\begin{array}[]{c}\varepsilon^{\prime}\\\
\varepsilon^{\prime\prime}\end{array}\right):=(U^{T})^{-1}\varepsilon.$

Then

(2.97) $\widehat{Z}_{\sigma(b^{\prime}\oplus
b^{\prime\prime},s)}:=(-1)^{b_{+}}q^{\frac{3\sigma-\mathrm{Tr}\,B}{4}+\sum_{I}|{\varepsilon^{\prime\prime}}_{I}||b^{\prime\prime}_{I}|}\sum_{\begin{subarray}{c}\ell^{\prime}=2b^{\prime}+B^{\prime}(s^{\prime}-\varepsilon^{\prime})\\\
\mod
2B^{\prime}\mathbb{Z}^{V-b_{1}}\end{subarray}}\;F_{U^{-1}\left(\begin{array}[]{c}\ell^{\prime}\\\
2b^{\prime\prime}\end{array}\right)}\,q^{-\frac{\ell^{\prime
T}B^{\prime-1}\ell^{\prime}}{4}}$

where
$b^{\prime}\in\mathrm{Coker}B^{\prime}\cong\mathrm{Tor}\,H_{1}(M;\mathbb{Z})$
and $b^{\prime\prime}\in\mathbb{Z}^{b_{1}}$. Here we assume that the sum over
$\ell^{\prime}$ in the right-hand side is convergent in the space of formal
power series $2^{-c}q^{\Delta}\mathbb{Z}[q^{-1},q]]$. This means that any
given power of $q$ gets contributions only from a finite number of terms.

If $b_{1}=0$, then $B=B^{\prime}$ and the above formula simplifies to:

(2.98)
$\widehat{Z}_{\sigma(b,s)}(M)=q^{\frac{3\sigma-\mathrm{Tr}\,B}{4}}\sum_{\begin{subarray}{c}\ell=2b+B(s-\varepsilon)\\\
\mod
2B\mathbb{Z}^{V}\end{subarray}}F_{\ell}q^{-\frac{\ell^{t}B^{-1}\ell}{4}}.$

If $b_{1}>0$ one can show that the result is independent of the choice of $U$
preserving $b^{\prime}$ and $b^{\prime\prime}$ and the Kirby moves only up to
the following equivalence relation:

(2.99) $1\sim q^{\text{LCM}(2,\text{GCD}(b^{\prime\prime}))}$

where
$\text{GCD}(b^{\prime\prime}):=\text{GCD}(\\{b_{i}^{\prime\prime}\\}_{i=1}^{b_{1}})$
(which is invariant under $SL(b_{1},\mathbb{Z})$ transformations). In other
words, the invariants should be considered as

(2.100) $\widehat{Z}_{\sigma(b^{\prime}\oplus b^{\prime\prime},s)}\in
2^{-c}\,q^{\Delta}\mathbb{Z}[[q]]/(1-q^{\text{LCM}(2,\text{GCD}(b^{\prime\prime}))})\mathbb{Z}[q]$

It is easy to see that $\widehat{Z}_{\sigma(\;\cdot\;,s)}$ transform
covariantly (as functions on
$H_{1}(M;\mathbb{Z})\cong\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\oplus\mathbb{Z}^{b_{1}}$)
under the automorphisms preserving the splitting. They correspond to changes
of the matrix $U$ of the following form:

(2.101) $U\rightsquigarrow\left(\begin{array}[]{cc}\tilde{\gamma}&0\\\
0&\nu\end{array}\right)U$

where $\tilde{\gamma}\in SL(V-b_{1},\mathbb{Z})$ and $\nu\in
SL(b_{1},\mathbb{Z})$. Then from the Definition (2.97) it follo ws that

(2.102) $\widehat{Z}_{\sigma(b^{\prime}\oplus
b^{\prime\prime},s)}\rightsquigarrow\widehat{Z}_{\sigma(\gamma^{-1}b^{\prime}\oplus\nu^{-1}(b^{\prime\prime}),s)}$

where $\gamma$ is the automorphism of $\mathrm{Tor}\,H_{1}(M;\mathbb{Z})$
represented by $\tilde{\gamma}$. Namely

(2.103) $\gamma(a^{\prime}):=\tilde{\gamma}(a^{\prime})\mod
B^{\prime}\mathbb{Z}^{V-b_{1}},\qquad a^{\prime}\in\mathbb{Z}^{V-b_{1}}.$

However there is an anomaly under the automorphisms changing the splitting.
Modulo the automorphisms preserving the splitting, they are of the form

(2.104)
$\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\oplus\mathbb{Z}^{b_{1}}\stackrel{{\scriptstyle\left(\begin{array}[]{cc}\text{id}_{\mathrm{Tor}\,H_{1}}&\mu\\\
0&\text{id}_{\mathbb{Z}^{b_{1}}}\end{array}\right)}}{{\longrightarrow}}\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\oplus\mathbb{Z}^{b_{1}}$

where $\mu$ is a non-trivial homomorphism

(2.105)
$\mu:\;\mathbb{Z}^{b_{1}}\longrightarrow\mathrm{Tor}\,H_{1}(M;\mathbb{Z}).$

This automorphism is realized by the replacement

(2.106)
$U\rightsquigarrow\left(\begin{array}[]{cc}\mathbf{1}_{V-b_{1}}&\tilde{\mu}\\\
0&\mathbf{1}_{b_{1}}\end{array}\right)U$

where $\tilde{\mu}$ is a $(V-b_{1})\times b_{1}$ matrix. Under the
identification $\mathrm{Tor}\,H_{1}(M;\mathbb{Z})=\text{Coker}\,B^{\prime}$,
we have

(2.107) $\mu(a^{\prime\prime}):=\tilde{\mu}(a^{\prime\prime})\mod
B^{\prime}\mathbb{Z}^{V-b_{1}},\qquad a^{\prime\prime}\in\mathbb{Z}^{b_{1}}.$

The corresponding change of $\widehat{Z}$ depends only on $\mu$, and not the
representative $\tilde{\mu}$, if one takes into account the equivalence
relation (2.99).

However it is not true that simply $\widehat{Z}_{\sigma}(b^{\prime}\oplus
b^{\prime\prime},s)\rightsquigarrow\widehat{Z}_{\sigma}((b^{\prime}-\mu(b^{\prime\prime}))\oplus
b^{\prime\prime},s)$. The covariant transformation is corrected by an
anomalous factor:

(2.108) $\widehat{Z}_{\sigma(b^{\prime}\oplus
b^{\prime\prime},s)}\rightsquigarrow
q^{\mathcal{E}(b^{\prime},b^{\prime\prime})}\widehat{Z}_{\sigma((b^{\prime}-\mu(b^{\prime\prime}))\oplus
b^{\prime\prime},s)}$

where

(2.109)
$\mathcal{E}(b^{\prime},b^{\prime\prime})=\text{GCD}(b^{\prime\prime})\cdot\left\\{\begin{array}[]{ll}\,\text{GCD}(b^{\prime\prime})\,q_{s}(\mu\widehat{b}^{\prime\prime})-2{\ell
k}(b^{\prime},\mu\widehat{b}^{\prime\prime}),&\text{GCD}(b^{\prime\prime})\text{
is odd},\\\ \text{GCD}(b^{\prime\prime})\,{\ell
k}(\mu\widehat{b}^{\prime\prime},\mu\widehat{b}^{\prime\prime})-2{\ell
k}(b^{\prime},\mu\widehat{b}^{\prime\prime}),&\text{GCD}(b^{\prime\prime})\text{
is even},\end{array}\right.$

and

(2.110)
$\widehat{b}^{\prime\prime}:=\frac{b^{\prime\prime}}{\text{GCD}(b^{\prime\prime})}\;\in\mathbb{Z}^{b_{1}}.$

The factor $q^{\mathcal{E}(b^{\prime},b^{\prime\prime})}$ is well defined
modulo the equivalence relation (2.99).

###  2.7. Physics of non-torsion fluxes

For a given spinc structure $b$, let ${\mathcal{B}}_{b}$ be the corresponding
boundary condition of the 6d fivebrane theory on $M\times
D^{2}\times_{q}S^{1}$. This defines the boundary condition in 4d gauge theory
on $M\times\mathbb{R}_{+}$ (obtained by projecting
$D^{2}\times_{q}S^{1}\to\mathbb{R}_{+}$) as well as the boundary condition in
3d theory $T[M]$ on $D^{2}\times_{q}S^{1}$ (obtained by reducing on $M$):

(2.111) $\leavevmode\hbox to342.05pt{\vbox
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In each of these descriptions, including the original 6d system viewed from
the enumerative perspective of Calabi-Yau 3-fold and M2-branes, the ambiguity
is naturally associated with the partition function on $M\times T^{2}\times
I$, where both $M\times T^{2}$ boundaries are colored by ${\mathcal{B}}_{b}$.

The effect of ${\mathcal{B}}_{b}$ is two-fold: $i)$ first, it effectively
abelianizes the theory, and $ii)$ it also puts it in a non-trivial background,
so that even abelian fluxes with one “leg” along $M$ and another “leg” along
$I$ now carry a non-trivial $q$-degree. Note, this latter effect is absent
when $b$ is torsion. The sum over such fluxes gives a $q$-series unbounded in
both directions:

(2.112) $\sum_{m\in H^{1}(M)}q^{m\cdot b^{\prime\prime}}$

that we would like to remove or factor out, in order to make the partition
function on $M\times D^{2}\times_{q}S^{1}$ well-defined. It would be
interesting to explore various ways to do this. Relegating a more systematic
study of this question for future work, here we merely sidestep the issue by
imposing the identification $q^{\text{GCD}(b^{\prime\prime})}\sim 1$ that
leads to (2.99)-(2.100).

Let us make a few comments on this interesting phenomenon by examining it from
various perspectives. Reduction to 4d gauge theory (2.111) yields a system of
Kapustin-Witten PDE’s on $M\times\mathbb{R}_{+}$, with a Nahm pole boundary
condition at $y=0$, where $y$ is a coordinate along $\mathbb{R}_{+}$ [57]. The
boundary condition at $y=\infty$ breaks the gauge group $G$ to a Levi subgroup
$\mathbb{L}\subseteq G$, which in applications to $\widehat{Z}$-invariants is
a maximal torus of $G$. In particular, $\mathbb{L}=U(1)$ for $G=SU(2)$. Other
choices of $\mathbb{L}$ are also interesting, and lead to a generalization of
$\widehat{Z}$-invariants labeled by complex coadjoint orbits of
$G_{\mathbb{C}}$ or, equivalently, by $\rho:\mathfrak{sl}(2)\to\mathfrak{g}$
[34].

When $\mathbb{L}=\mathbb{T}$, there are infinitely many different topological
sectors labeled by monopole numbers
$b_{i}^{\prime\prime}\in\Lambda_{\text{cochar}}=\text{Hom}(U(1),\mathbb{T})$
or, more precisely, by spinc structures. In order to keep track of dependence
on $b$, one can introduce a topological term $\exp(2\pi i\eta b)$ in the
action of 4d gauge theory. Then, using Pontryagin duality,

(2.113) $x=e^{2\pi i\eta}\in\text{Hom}\left(H_{1}(M),\mathbb{T}\right)$

can be identified with the variable by the same name in $F_{K}(x,q)$ and in
the integrand of $\widehat{Z}$-invariants. To summarize, on a closed
3-manifold with $b_{1}>0$ the boundary condition ${\mathcal{B}}_{b}$ breaks
the gauge group $G$ to $\mathbb{L}=\mathbb{T}$ and creates a “flux” $b$. When
this flux is non-torsion, the solutions to Kapustin-Witten PDE’s on
$M\times\mathbb{R}_{+}$ can not approach a constant field configuration at
$y=+\infty$. At best one can require solutions to approach a field
configuration periodic in the $y$-direction, which leads us to conclude that
the anomaly in question is controlled by the moduli space of solutions on
$M\times S^{1}$ with gauge group $\mathbb{L}$. Below we give another
interpretation of this claim from the perspective of 3d-3d correspondence.

From the point of view of 3d $\mathcal{N}=2$ theory $T[M]$, the boundary
condition ${\mathcal{B}}_{b}$ labels the background momentum / charge sectors
of the 2d boundary theory, cf. [21]. In the partition function on
$D^{2}\times_{q}S^{1}$, the parameter $q$ keeps track of the spin with respect
to the rotation symmetry of $D^{2}$. It is defined up to spins of BPS states
in 2d boundary theory or, equivalently, 3d theory $T[M,\mathbb{T}]$ on a slab
$T^{2}\times I$ with boundary conditions ${\mathcal{B}}_{b}$ on both sides.
This theory is a close cousin of $T[M\times S^{1},\mathbb{T}]$ in the
background of spinc structure $b$ on $M$. Indeed, both theories exhibit a
qualitative change in behavior depending on whether $b_{1}=0$ or $b_{1}>0$. In
the latter case the BPS states come in towers infinite in both direction, with
spins in each tower being multiples of $b^{\prime\prime}$, which leads again
to the ambiguity (2.99).

Similarly, from the curve counting perspective on $T^{*}M$, when $b_{1}>0$ in
addition to open BPS states one also has a non-trivial “closed sector.”

##  3\. Families of examples

In this section we illustrate the proposed relation with a number of
instructive examples.

###  3.1. Plumbed 3-manifolds

Here we consider the case when $M$ is a rational homology sphere given by a
weakly negative definite plumbing graph $\Gamma$ [32, 30]. We can then assume
that

(3.1) $\omega\in H^{1}(M;\mathbb{Q}/2\mathbb{Z})\setminus
H^{1}(M;\mathbb{Z}/2\mathbb{Z}).$

With the notation of Section 2.2, Formula (2.58) becomes:

(3.2)
$\mathrm{N}_{r}(M,\omega)=\frac{1}{\Delta_{+}^{b_{+}}\Delta_{-}^{b_{-}}}\sum_{k\in
H_{r}^{\mathrm{Vert}}}\prod_{I\in\mathrm{Vert}}d(\alpha_{k_{I}})^{2-\deg(I)}\,T(\alpha_{k_{I}})^{B_{II}}\prod_{(I,J)\in\text{Edges}}S(\alpha_{k_{I}},\alpha_{k_{J}})$

where

(3.3) $\alpha_{k_{I}}:=\mu_{I}+k_{I}$

and $b_{\pm}$ are the number of positive/negative eigenvalues of $B$. In this
section we will be somewhat cavalier with taking the limits and about
convergence of infinite series. Such technical details will be properly
addressed in Section 4.1 and Appendix D.

It is instructive to separate 3 factors:

(3.4) $\mathrm{N}_{r}(M,\omega)=\mathcal{A}\cdot\mathcal{B}\cdot\mathcal{C},$
(3.5)
$\mathcal{A}=r^{-V/2}\,\xi^{\frac{3\sigma-\mathrm{Tr}\,B}{2}}\cdot\left\\{\begin{array}[]{cl}e^{\frac{\pi
i(\sigma+\mathrm{Tr}\,B)}{2}},&r=1\mod 4,\\\ 2^{-V/2}\,e^{-\frac{\pi
i\sigma}{4}},&r=2\mod 4,\\\ e^{-\frac{\pi
i}{2}\mathrm{Tr}\,B}\,(-1)^{\sigma},&r=3\mod 4,\end{array}\right.$

where $\sigma$ is the signature of $B$,

(3.6) $\mathcal{B}=F(\\{e^{\pi i\mu_{I}}\\}_{I\in\mathrm{Vert}})^{-1},$ (3.7)
$\mathcal{C}=\sum_{k\in
H_{r}^{\mathrm{Vert}}}F(\\{\xi^{\mu_{I}+k_{I}}\\}_{I\in\mathrm{Vert}})\cdot\xi^{\frac{1}{2}(\mu+k)^{T}B(\mu+k)},$

and

(3.8)
$F(x):=\prod_{I\in\mathrm{Vert}}(x_{I}-1/x_{I})^{2-\deg(I)}=\sum_{\ell\in\mathbb{Z}^{\mathrm{Vert}}}F_{\ell}\prod_{I}x_{I}^{\ell_{I}}$

is a slightly different normalization of the invariant $F_{L}(x,q)$ for the
case of the plumbing link $L$. Consider a contribution of a monomial
$\prod_{I}x_{I}^{\ell_{I}}$ from $F(x)$ into (3.7):

(3.9) $\mathcal{C}_{\ell}:=\sum_{k\in
H_{r}^{\mathrm{Vert}}}\xi^{\ell^{T}(\mu+k)}\cdot\xi^{\frac{1}{2}(\mu+k)^{T}B(\mu+k)}=\sum_{n\in\mathbb{Z}^{\mathrm{Vert}}/r\mathbb{Z}^{\mathrm{Vert}}}e^{\frac{\pi
i}{2r}(\tilde{\mu}+2n)^{T}B(\tilde{\mu}+2n)+\frac{\pi
i}{r}\ell^{T}(\tilde{\mu}+2n)}$

where

(3.10) $\tilde{\mu}:=\mu+(r-1)\varepsilon.$

We can now use the following version of Gauss reciprocity formula [22, 35]:

(3.11)
$\sum_{n\;\in\;\mathbb{Z}^{\mathrm{Vert}}/r\mathbb{Z}^{\mathrm{Vert}}}\exp\left(\frac{2\pi
i}{r}\,n^{T}Bn+\frac{2\pi i}{r}\,p^{T}n\right)=\\\ \frac{e^{\frac{\pi
i\sigma}{4}}\,(r/2)^{V/2}}{|\det{B}|^{1/2}}\sum_{\tilde{a}\;\in\;\mathbb{Z}^{\mathrm{Vert}}/2B\mathbb{Z}^{\mathrm{Vert}}}\exp\left(-\frac{\pi
ir}{2}(\tilde{a}+\frac{p}{r})^{T}B^{-1}(\tilde{a}+\frac{p}{r})\right).$

Applying it to (3.9) we have

(3.12)
$\mathcal{C}_{\ell}=\xi^{-\frac{\ell^{T}B^{-1}\ell}{2}}\,\frac{e^{\frac{\pi
i\sigma}{4}}(r/2)^{V/2}}{|\det
B|^{1/2}}\,\underbrace{\sum_{\tilde{a}\in\mathbb{Z}^{\mathrm{Vert}}/2B\mathbb{Z}^{\mathrm{Vert}}}e^{-\frac{\pi
ir}{2}\,\tilde{a}^{T}B^{-1}\tilde{a}-\pi
i\,\tilde{a}^{T}B^{-1}(\ell+B\tilde{\mu})}}_{=:\mathcal{C}_{\ell}^{\prime}}.$

Let us make the change of variables $\tilde{a}=BA+a$,
$A\in\mathbb{Z}^{\mathrm{Vert}}/2\mathbb{Z}^{\mathrm{Vert}}$,
$a\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}$:

(3.13) $\mathcal{C}_{\ell}^{\prime}=\\\
\sum_{{a}\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}\sum_{A\in\mathbb{Z}^{\mathrm{Vert}}/2\mathbb{Z}^{\mathrm{Vert}}}e^{-\frac{\pi
ir}{2}\,{a}^{T}B^{-1}{a}-\pi ir\,A^{T}a-\frac{\pi ir}{2}\,A^{T}BA-\pi
i\,a^{T}B^{-1}(\ell+B\tilde{\mu})-\pi i\,A^{T}(\ell+B\tilde{\mu})}=\\\
=\sum_{{a}\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}\sum_{A\in\mathbb{Z}^{\mathrm{Vert}}/2\mathbb{Z}^{\mathrm{Vert}}}\exp\left\\{-\frac{\pi
ir}{2}\,{a}^{T}B^{-1}{a}-\pi ir\,A^{T}a-\frac{\pi ir}{2}\,A^{T}BA\right.\\\
\left.-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}(s+\mu-r\varepsilon)-\pi
i\,A^{T}B(s-r\varepsilon)\right\\}$

where in the last line we used the fact that (3.8) only contains powers
$\prod_{I}x_{I}^{\ell_{I}}$ satisfying $\ell_{I}=\deg(I)\mod 2$, and therefore
one can introduce $b\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}$
and
$s\in\mathbb{Z}^{\mathrm{Vert}}/2\mathbb{Z}^{\mathrm{Vert}},\;\sum_{J}B_{IJ}s_{J}=B_{II}\mod
2$, such that

(3.14) $\ell=2b+B(s-\varepsilon)\mod 2B\mathbb{Z}^{\mathrm{Vert}}.$

We also used the property (2.47). At this point we will need to consider the
cases with different $r\mod 4$ values separately.

###  3.2. Level $r=2\mod 4$

Using the fact that $r$ is even, while $r/2$ is an odd integer, and condition
on $s$ the sum (3.13) simplifies to:

(3.15) $\mathcal{C}_{\ell}^{\prime}=\\\
2^{V}\sum_{{a}\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}\exp\left\\{-\frac{\pi
ir}{2}\,{a}^{T}B^{-1}{a}-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}(s+\mu)\right\\}.$

Combining everything together we then have:

(3.16) $\mathrm{N}_{r}(M,\omega)=\frac{F(\\{e^{\pi
i\mu_{I}}\\}_{I\in\mathrm{Vert}})^{-1}}{|\det
B|^{1/2}}\,\xi^{\frac{3\sigma-\mathrm{Tr}\,B}{2}}\times\\\
\sum_{\ell\in\mathbb{Z}^{\mathrm{Vert}}}\sum_{{a}\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}F_{\ell}\xi^{-\frac{\ell^{T}B^{-1}\ell}{2}}\,e^{-\frac{\pi
ir}{2}\,{a}^{T}B^{-1}{a}-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}(s+\mu)}.$

Using the following expressions for $\widehat{Z}$ and Reidemeister
torsion888In principle the torsion $\mathcal{T}(M,\alpha)$, defined for
$\alpha\in H^{1}(M;\mathbb{Q}/\mathbb{Z})\stackrel{{\scriptstyle{\ell
k}}}{{\cong}}H_{1}(M;\mathbb{Z})$, has sign ambiguity, if no additional
structures on $M$ are introduced. One can fix the sign for example by
introducing a spin structure $s\in\mathrm{Spin}(M)$ on $M$, cf. [42]. The
change of the spin structure $s\rightarrow s+c$, $c\in
H^{2}(M;\mathbb{Z}_{2})$ then changes the sign by $(-1)^{c(\tilde{\alpha})}$
where $\tilde{\alpha}\in H_{1}(M;\mathbb{Z})$ is dual to $\alpha$. However,
since in our case $\alpha=\omega\mod 1$, $\tilde{\alpha}$ is even, and the
dependence on spin structure drops out. $\mathcal{T}$ for plumbed manifolds
(see formula (A.7) in Appendix A)

(3.17)
$\widehat{Z}_{a}(M)=(-1)^{b_{+}}\,q^{\frac{3\sigma-\mathrm{Tr}\,B}{4}}\,\sum_{\ell=a\mod
2B\mathbb{Z}^{\mathrm{Vert}}}F_{\ell}q^{-\frac{\ell^{T}B^{-1}\ell}{4}},\qquad
a_{I}=\deg(I)\mod 2,$ (3.18)
$\mathcal{T}(M,[\omega])=(-1)^{b_{+}}\prod_{I\in\mathrm{Vert}}(e^{\pi
i\mu_{I}}-e^{-\pi i\mu_{I}})^{\deg(I)-2},\qquad[\omega]:=\omega\mod
H^{1}(M;\mathbb{Z}/2\mathbb{Z}),$

and the identifications (2.3)-(2.18), (2.15)-(2.16) we can conjecture the
following general relation for a rational homology sphere $M$ and $r=2\mod 4$:

(3.19)
$\boxed{\mathrm{N}_{r}(M,\omega)=\frac{\mathcal{T}(M,[\omega])}{\sqrt{|H_{1}(M;\mathbb{Z})|}}\left.\sum_{a,b\in
H_{1}(M;\mathbb{Z})}e^{-\frac{\pi ir}{2}\,q_{s}(a)-2\pi i\,{\ell k}(a,b)-\pi
i\,\omega(a)}\,\widehat{Z}_{\sigma(b,s)}\right|_{q\rightarrow e^{\frac{2\pi
i}{r}}}}$

where $\sigma$ is the canonical map

(3.20) $\sigma:\text{Spin}(M)\times
H_{1}(M;\mathbb{Z})\longrightarrow\text{Spin}^{c}(M)$

producing a spinc structure on $M$ from a spin structure $c$ and $\tilde{b}\in
H_{1}(M;\mathbb{Z})$. It is induced by the map $B\text{Spin}\times
BU(1)\rightarrow B\text{Spin}^{c}$ between the corresponding classifying
spaces, combined with the isomorphisms $BU(1)\cong B^{2}\mathbb{Z}$,
$H_{1}(M;\mathbb{Z})\cong H^{2}(M;\mathbb{Z})$. In (3.19) we have introduced
an auxiliary spin structure $s\in\mathrm{Spin}(M)$ (see also 3.14). The result
is independent of it due to (2.14), so that the simultaneous change of $b\in
H_{1}(M;\mathbb{Z})$ and $s\in\mathrm{Spin}(M)$ leaving $\sigma(b,s)$
invariant also leaves invariant the exponent in the sum in (3.19).

###  3.3. Level $r=1\mod 4$

The sum (3.13) reads

(3.21)
$\mathcal{C}_{\ell}^{\prime}=\sum_{a\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}\sum_{A\in\mathbb{Z}^{\mathrm{Vert}}/2\mathbb{Z}^{\mathrm{Vert}}}\exp\left\\{-\frac{\pi
ir}{2}\,a^{T}B^{-1}a-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}\mu\right.\\\ \left.-\pi
i\,a^{T}(s-\varepsilon)-\frac{\pi i}{2}\,A^{T}BA+\pi
i\,A^{T}(a+B(s-\varepsilon))\right\\}.$

Applying a version of the Gauss reciprocity formula to the sum over $A$ we can
rewrite it as follows:

(3.22) $\mathcal{C}_{\ell}^{\prime}=\frac{e^{-\frac{\pi
i\sigma}{4}}2^{V/2}}{|\det
B|^{1/2}}\sum_{a,f\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}\exp\left\\{-\frac{\pi
i(r-1)}{2}\,a^{T}B^{-1}a-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}\mu\right.\\\
\left.+2\pi i\,f^{T}B^{-1}f+2\pi i\,f^{T}B^{-1}a+\frac{\pi
i}{2}\,(s-\varepsilon)^{T}B(s-\varepsilon)\right\\}.$

Taking into account that

(3.23)
$\frac{1}{4}\varepsilon^{T}B\varepsilon=\frac{V-1}{2}+\frac{1}{4}\,\mathrm{Tr}\,B\mod
1,$ (3.24) $\frac{1}{2}\varepsilon^{T}Bs=\frac{1}{2}\,\mathrm{Tr}\,B\mod 1,$

and combining everything together we then have:

(3.25) $\mathrm{N}_{r}(M,\omega)=\frac{F(\\{e^{\pi
i\mu_{I}}\\}_{I\in\mathrm{Vert}})^{-1}}{|\det B|}\,e^{\frac{\pi
i}{2}\left(2-\sigma+s^{T}Bs\right)}\,\xi^{\frac{3\sigma-\mathrm{Tr}\,B}{2}}\times\\\
\sum_{\ell\in\mathbb{Z}^{\mathrm{Vert}}}\sum_{a,f\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}F_{\ell}\,\xi^{-\frac{\ell^{T}B^{-1}\ell}{2}}\,e^{-\frac{\pi
i(r-1)}{2}\,a^{T}B^{-1}a-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}\mu+2\pi
i\,f^{T}B^{-1}f+2\pi i\,f^{T}B^{-1}a}.$

As in the case $r=2\mod 4$, we can then conjecture the following general
relation for a rational homology sphere $M$ and $r=1\mod 4$:

(3.26) $\boxed{\begin{aligned} &\mathrm{N}_{r}(M,\omega)=\frac{-e^{-\frac{\pi
i}{2}\mu(M,s)}\,\mathcal{T}(M,[\omega])}{{|H^{1}(M;\mathbb{Z})|}}\;\times\\\
&\left.\sum_{a,b,f\in H_{1}(M;\mathbb{Z})}e^{2\pi i\left(-\frac{r-1}{4}{\ell
k}(a,a)+{\ell k}(a,f-b)-\frac{1}{2}\omega(a)+{\ell
k}(f,f)\right)}\,\widehat{Z}_{\sigma(b,s)}\right|_{q\rightarrow e^{\frac{2\pi
i}{r}}}\end{aligned}}$

where we have used the surgery formula (2.17) for the mod 4 reduction of the
Rokhlin invariant $\mu(M,s)$.

It is interesting to remark that unlike in the case $r=2\ {\rm mod\ 4}$ here
the relation between $\mathrm{N}_{r}$ and $\widehat{Z}$ is based on a _triple_
summation (instead of double).

###  3.4. Level $r=3\mod 4$

This case is analogous to the case $r=1\mod 4$ considered above. When $r=3\mod
4$ the sum (3.13) reads

(3.27)
$\mathcal{C}_{\ell}^{\prime}=\sum_{a\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}\sum_{A\in\mathbb{Z}^{\mathrm{Vert}}/2\mathbb{Z}^{\mathrm{Vert}}}\exp\left\\{-\frac{\pi
ir}{2}\,a^{T}B^{-1}a-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}\mu\right.\\\ \left.-\pi
i\,a^{T}(s-\varepsilon)+\frac{\pi i}{2}\,A^{T}BA+\pi
i\,A^{T}(a+B(s-\varepsilon))\right\\}.$

Applying again the Gauss reciprocity formula to the sum over $A$ we have:

(3.28) $\mathcal{C}_{\ell}^{\prime}=\frac{e^{\frac{\pi
i\sigma}{4}}2^{V/2}}{|\det
B|^{1/2}}\sum_{a,f\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}\exp\left\\{-\frac{\pi
i(r+1)}{2}\,a^{T}B^{-1}a-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}\mu\right.\\\
\left.-2\pi i\,f^{T}B^{-1}f-2\pi i\,f^{T}B^{-1}a-\frac{\pi
i}{2}\,(s-\varepsilon)^{T}B(s-\varepsilon)\right\\}.$

Combining everything together we then have:

(3.29) $\mathrm{N}_{r}(M,\omega)=\frac{F(\\{e^{\pi
i\mu_{I}}\\}_{I\in\mathrm{Vert}})^{-1}}{|\det B|}\,e^{\frac{\pi
i}{2}\left(2+\sigma-s^{T}B^{-1}s\right)}\,\xi^{\frac{3\sigma-\mathrm{Tr}\,B}{2}}\times\\\
\sum_{\ell\in\mathbb{Z}^{\mathrm{Vert}}}\sum_{a,f\in\mathbb{Z}^{\mathrm{Vert}}/B\mathbb{Z}^{\mathrm{Vert}}}F_{\ell}\,\xi^{-\frac{\ell^{T}B^{-1}\ell}{2}}\,e^{-\frac{\pi
i(r-1)}{2}\,a^{T}B^{-1}a-2\pi i\,a^{T}B^{-1}b-\pi i\,a^{T}\mu+2\pi
i\,f^{T}B^{-1}f+2\pi i\,f^{T}B^{-1}a}.$

We can then conjecture the following general relation for a rational homology
sphere $M$ and $r=3\mod 4$:

(3.30) $\boxed{\begin{aligned} &\mathrm{N}_{r}(M,\omega)=\frac{-e^{\frac{\pi
i}{2}\mu(M,s)}\,\mathcal{T}(M,[\omega])}{{|H^{1}(M;\mathbb{Z})|}}\;\times\\\
&\left.\sum_{a,b,f\in H_{1}(M;\mathbb{Z})}e^{2\pi i\left(-\frac{r+1}{4}{\ell
k}(a,a)-{\ell k}(a,f+b)-\frac{1}{2}\omega(a)-{\ell
k}(f,f)\right)}\,\widehat{Z}_{\sigma(b,s)}\right|_{q\rightarrow e^{\frac{2\pi
i}{r}}}.\end{aligned}}$

###  3.5. Generalization to $b_{1}>0$

Let $M$ be, as before, obtained by a surgery on a link with the linking matrix
$B$. However now we will allow $B$ to be degenerate. By an $SL(V,\mathbb{Z})$
transform we can bring the quadratic form $B$ to
$B^{\prime}\oplus\mathbf{0}_{b_{1}}$ where $\mathbf{0}_{b_{1}}$ is the trivial
quadratic form on $\mathbb{Z}^{b_{1}}\subset\mathbb{Z}^{V}$ [40]. The
expression (3.12) then will be modified to

(3.31)
$\mathcal{C}_{\ell}=\xi^{-\frac{{\ell^{\prime}}^{T}{B^{\prime}}^{-1}{\ell^{\prime}}}{2}}\,\frac{e^{\frac{\pi
i\sigma}{4}}(r/2)^{(V-b_{1})/2}}{|\det
B^{\prime}|^{1/2}}\,\sum_{\tilde{a}\in\mathbb{Z}^{V-b_{1}}/2B^{\prime}\mathbb{Z}^{V-b_{1}}}e^{-\frac{\pi
ir}{2}\,\tilde{a}^{T}(B^{\prime})^{-1}\tilde{a}-\pi
i\,\tilde{a}^{T}(B^{\prime})^{-1}(\ell^{\prime}+B^{\prime}\tilde{\mu}^{\prime})}\times\\\
r^{b_{1}}\,e^{\frac{\pi
i{\ell^{\prime\prime}}^{T}\,\tilde{\mu}^{\prime\prime}}{r}}\,\delta_{\ell^{\prime\prime}=0\mod
r}$

where $\ell=\ell^{\prime}\oplus\ell^{\prime\prime}$ and
$\tilde{\mu}=\tilde{\mu}^{\prime}\oplus\tilde{\mu}^{\prime\prime}$ according
to the splitting of $B$ above.

Consider first the case of $r=2\mod 4$. We have

(3.32)
$\mathcal{A}=\xi^{\frac{3\sigma-\mathrm{Tr}\,B}{2}}\,r^{-V/2+b_{1}/2}\,2^{b_{1}/2-V/2}\,e^{-\frac{\pi
i\sigma}{4}}.$

The relation (3.19) generalizes to

(3.33)
$\mathrm{N}_{r}(M,\omega)=\frac{r^{b_{1}}\,\mathcal{T}(M,[\omega])}{\sqrt{|\mathrm{Tor}\,H_{1}(M;\mathbb{Z})|}}\times\\\
\sum_{\begin{subarray}{c}a^{\prime},b^{\prime}\in\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\\\
m\in\mathbb{Z}^{b_{1}}\end{subarray}}e^{-\frac{\pi
ir}{2}\,q_{s}(a^{\prime})-2\pi i\,{\ell k}(a^{\prime},b^{\prime})-\pi
i\,\omega(a^{\prime})+\pi
i\omega^{\prime\prime}(m)}\,\widehat{Z}_{\sigma(b^{\prime}\oplus
rm/2,s)}|_{q\rightarrow e^{\frac{2\pi i}{r}}}=\\\
\frac{r^{b_{1}}\,\mathcal{T}(M,[\omega])}{\sqrt{|\mathrm{Tor}\,H_{1}(M;\mathbb{Z})|}}\times\\\
\int\limits_{H^{1}(M;\mathbb{R}/\mathbb{Z})}\mu(\alpha)\sum_{b\in
H_{1}(M;\mathbb{Z})}e^{-\frac{\pi ir}{2}\,q_{s}(\alpha^{\prime})-2\pi i\,{\ell
k}(\alpha^{\prime},b^{\prime})-\pi i\,\omega(\alpha^{\prime})+2\pi
i\alpha^{\prime\prime}(b^{\prime\prime})}\times\\\
\delta(r\alpha^{\prime\prime}-\omega^{\prime\prime}/2)\,\widehat{Z}_{\sigma(b,s)}|_{q\rightarrow
e^{\frac{2\pi i}{r}}}$

where we chose explicit splittings
$\omega=\omega^{\prime}\oplus\omega^{\prime\prime}\in
H^{1}(M;\mathbb{C}/2\mathbb{Z})\cong\mathrm{Tor}\,{H_{1}(M;\mathbb{Z})}\oplus(\mathbb{C}/2\mathbb{Z})^{b_{1}}$
and $b=b^{\prime}\oplus b^{\prime\prime}\in
H_{1}(M;\mathbb{Z})\cong\mathrm{Tor}\,{H_{1}(M;\mathbb{Z})}\oplus\mathbb{Z}^{b_{1}}$
according to the splitting of the linking matrix $B$ above. It is
straightforward to see that the right-hand side is independent of the choice
of representative of the equivalence class (2.99). The coefficients in the
relation (3.33) are not invariant under the automorphisms (2.104). However
this compensated by the non-covariance of $\widehat{Z}$. Taking into account
(2.108), one can show that the total sum in (3.33) transforms covariantly
(i.e. as a function on
$H^{1}(M;\mathbb{C}/2\mathbb{Z})\cong\mathrm{Hom}(\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\oplus\mathbb{Z}^{b_{1}},\mathbb{C}/2\mathbb{Z})$).
Namely, considering that

(3.34)
$q^{\mathcal{E}(b^{\prime},b^{\prime\prime})}|_{b^{\prime\prime}=\frac{mr}{2}}=e^{\frac{\pi
ir}{2}q_{s}(\mu m)-2\pi i{\ell k}(\mu m,b^{\prime})}$

and shifting the summation variables $b^{\prime}\rightarrow
b^{\prime}+\frac{r}{2}\,\mu m$, $a^{\prime}\rightarrow a^{\prime}-\mu m$, we
have indeed

(3.35)
$\mathrm{N}_{r}(M,\omega^{\prime}\oplus\omega^{\prime\prime})\rightsquigarrow\mathrm{N}_{r}(M,\omega^{\prime}\oplus(\omega^{\prime\prime}+2\mu^{*}\omega))$

where

(3.36)
$\omega^{\prime}\oplus\omega^{\prime\prime}\in\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\oplus(\mathbb{C}/2\mathbb{Z})^{b_{1}}$

and

(3.37)
$\mu^{*}:\;\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\rightarrow(\mathbb{C}/\mathbb{Z})^{b_{1}}$

is the map dual to $\mu$ in (2.105).

For $r=1\mod 4$ we have

(3.38)
$\mathcal{A}=r^{-V/2+b_{1}/2}\,\xi^{\frac{3\sigma-\mathrm{Tr}\,B}{2}}\,e^{\frac{\pi
i}{2}(\mathrm{Tr}\,B+\sigma)}$

and

(3.39)
$\mathrm{N}_{r}(M,\omega)=\frac{r^{b_{1}}\,\mathcal{T}(M,[\omega])}{{|\mathrm{Tor}\,\,H_{1}(M;\mathbb{Z})|}}\;\times\\\
\sum_{\begin{subarray}{c}a^{\prime},b^{\prime},f^{\prime}\in\mathrm{Tor}\,H_{1}(M;\mathbb{Z})\\\
m\in\mathbb{Z}^{b_{1}}\end{subarray}}e^{2\pi i\left(-\frac{r-1}{4}{\ell
k}(a^{\prime},a^{\prime})+{\ell
k}(a^{\prime},f^{\prime}-b^{\prime})-\frac{1}{2}\omega(a^{\prime})+{\ell
k}(f^{\prime},f^{\prime})-\Delta_{\sigma(b^{\prime},s)}-{\ell
k}(b^{\prime},b^{\prime})+\omega^{\prime\prime}(m)\right)}\times\\\
\,\left.\widehat{Z}_{\sigma(b^{\prime}\oplus rm,s)}\right|_{q\rightarrow
e^{\frac{2\pi i}{r}}}.$

######  Example 3.1.

An interesting example is $M=\Sigma_{g}\times S^{1}$. In this case ${\ell
k}=0$ and $q_{s}=0$. Furthermore we have the identity:

(3.40)
$\mathcal{T}(M,[\omega])=(-2)^{b_{1}+1}(\frac{i}{4})^{b_{1}}\frac{i}{2}\mathrm{N}_{2}(M,2\omega).$

Thus replacing in the above formula we get

(3.41) $\frac{\mathrm{N}_{r}(\Sigma_{g}\times
S^{1},\omega)}{\mathrm{N}_{2}(\Sigma_{g}\times
S^{1},2\omega)}={r^{b_{1}}(-2)^{1+b_{1}}\,(\frac{i}{4})^{b_{1}}\frac{i}{2}}\sum_{m\in\mathbb{Z}^{2g+1}}e^{+\pi
i\omega(m)}\widehat{Z}_{\sigma(\frac{rm}{2},s)}|_{q\rightarrow e^{\frac{2\pi
i}{r}}}.$

If we now recall that

(3.42) $\mathrm{N}_{r}(\Sigma_{g}\times S^{1},\omega)=r^{2g}\sum_{k\in
H_{r}}\left(\frac{\\{r\beta\\}}{\\{\beta+k\\}}\right)^{2g-2}$

(so in particular $\mathrm{N}_{2}(\Sigma_{g}\times
S^{1},2\omega)=2^{2g+1}\frac{1}{i^{2g-2}}(i^{2\beta}-i^{-2\beta})^{2g-2}$ )
where $\beta=\omega(S^{1})$ (note the formula does not depend on the
orientation of $S^{1}$), then we get:

(3.43) $\frac{1}{r}\sum_{k\in
H_{r}}\frac{\\{r\beta\\}^{2g-2}}{\left(\\{\beta+k\\}(i^{2\beta}-i^{-2\beta})\right)^{2g-2}}=\sum_{m\in\mathbb{Z}^{2g+1}}e^{+\pi
i\omega(m)}\widehat{Z}_{\sigma(\frac{rm}{2},s)}|_{q\rightarrow e^{\frac{2\pi
i}{r}}}.$

In order to compute $\widehat{Z}_{\mathfrak{s}}$ we observe that the formula
(3.17) providing the value of $\widehat{Z}$ for surgeries over plumbing links
can be generalised to manifolds which are the boundaries of a plumbing of
surfaces in a tree-like fashion. In particular, for $M=\Sigma_{g}\times
S^{1}$,

(3.44) $(x-x^{-1})^{2-2g}=\sum_{\ell\in\mathbb{Z}}F_{\ell}x^{\ell}$

Since $\mathrm{Tor}\,(H_{1}(M))=0$ the first Chern class provides a bijection
between the set of spinc structures on $M$ and $H^{2}(M)=H_{1}(M)$, but
$\widehat{Z}_{\mathfrak{s}}$ is zero for all spinc structures $\mathfrak{s}$
such that $c_{1}(\mathfrak{s})\neq PD(\ell[\\{pt\\}\times S^{1}])$ for some
$\ell\in\mathbb{Z}$. So letting $\ell$ be the spinc structure the first Chern
class of which is Poincaré dual to $\ell[\\{pt\\}\times S^{1}]$ and using
(2.97) with $U=Id$, $B=0$, $\sigma=0=b_{+}$, we have:
$\widehat{Z}_{\ell}=q^{\ell}F_{\ell}.$

Therefore equation (3.43) becomes:

(3.45) $\sum_{k\in H_{r}}\frac{1}{(e^{\frac{\pi i}{r}(\beta+k)}-e^{-\frac{\pi
i}{r}(\beta+k)})^{2g-2}}=r\sum_{m\in\mathbb{Z}}e^{\pi
im\beta}\widehat{Z}_{\sigma(\frac{rm}{2},s)}|_{q\rightarrow e^{\frac{2\pi
i}{r}}}.$

Then we have that the left hand side of (3.45) equals

(3.46) $\sum_{l\in\mathbb{Z}}F_{l}\sum_{k\in H_{r}}e^{l\frac{\pi
i}{r}(\beta+k)}=r\sum_{l\in\mathbb{Z}}F_{rl}e^{\pi il\beta}(-1)^{l(r-1)}$

and the r.h.s. equals:

(3.47) $r\sum_{m\in\mathbb{Z}}e^{\pi im\beta}F_{rm}e^{\pi im}.$

So if $r$ is even then (3.45) is verified directly and if $r$ is odd, then the
equality is true because $F_{rl}=0$ for odd $l$.

###  3.6. Surgeries on knots

Consider $M=S^{3}_{p}(K)$ and assume that $-p\in\mathbb{Z}_{+}$ for
concreteness. Then

(3.48) $\widehat{Z}_{a}[S^{3}_{p}(K)]=q^{-\frac{3+p}{4}}\sum_{\ell=a\mod
2p}F_{\ell}q^{-\frac{\ell^{2}}{4p}}$

where the coefficients $F_{\ell}$ appear in the expansion of

(3.49) $F(x,q):=F_{K}(x^{2},q)(x-x^{-1})=\sum_{\ell}F_{\ell}x^{\ell}$

with $F_{K}$ introduced in [30]. We will use the facts that
$F_{K}(x^{-2},q)=-F_{K}(x^{2},q)$ and that $F_{K}(x^{2},q)\in
x\mathbb{Z}[[x^{\pm 2},q]]$. In particular it follows that
$\widehat{Z}_{a}\equiv 0$ for $a=1\mod 2$. As argued in [29], the series
$F_{K}(x,q)$ gives ADO polynomials at roots of unity (which, in turn, are
related to the CGP invariants for knot complements). Note, this already
establishes a relation between $F_{K}(x,q):=\widehat{Z}(S^{3}\setminus K)$ and
$\mathrm{N}_{r}(S^{3},K)$ for knots, and tells us the relation between the
parameters: in order to obtain $\mathrm{N}_{r}$ on the CGP side this large
class of examples shows that on the GPPV side we need to take $q=e^{2\pi i/r}$
(not $q=e^{\pi i/r}$). The normalizations are such that

(3.50)
$\left.F_{K}(x,q)\right|_{q=\xi^{2}}\;=\;\frac{\text{ADO}_{r}(x/\xi^{2};K)}{\Delta_{K}(x^{r})}\cdot\big{(}x^{1/2}-x^{-1/2}\big{)}\,,\qquad\xi:=e^{\pi
i/r},$

where $\Delta_{K}(t)$ denotes the Alexander polynomial of $K$. We wish to
compose this with the relation between ADO polynomials and CGP invariants for
knot complements:

(3.51)
$\text{ADO}_{r}(x^{2}/\xi^{2})\;=\;\frac{x^{r}-x^{-r}}{x-x^{-1}}\mathrm{N}_{r}(S^{3},K_{\alpha})\,,\qquad\text{where}~{}x=e^{\frac{\pi
i\alpha}{r}}.$

Eliminating the ADO polynomial from the above two relations we get a more
direct relation between GPPV and CGP invariants for knot complements

(3.52) $\mathrm{N}_{r}(S^{3},K_{\alpha})=\frac{F_{K}(e^{\frac{2\pi
i\alpha}{r}},e^{\frac{2\pi i}{r}})\,\Delta_{K}(e^{2\pi i\alpha})}{(e^{\pi
i\alpha}-e^{-\pi i\alpha})}.$

On the other hand, we have

(3.53) $\mathrm{N}_{r}(S^{3}_{p}(K),\omega)=\frac{1}{\Delta_{-}}\sum_{k\in
H_{r}}d(\alpha_{k})\,\mathrm{N}_{r}(S^{3},K_{\alpha_{k}})\,T(\alpha_{k})^{p}$

where $\mathrm{N}_{r}(S^{3},K_{\alpha})$ denotes CGP invariant of a (with zero
framing) knot in $S^{3}$ colored by $\alpha$ (e.g. for unknot $U$ with zero
framing $\mathrm{N}_{r}(S^{3},U_{\alpha})=d(\alpha)$). The other notations are
the same as in Section 2.4. In particular, $\alpha_{k}:=\alpha+\mu$, where
$\mu=\omega(\mathfrak{m})\in\frac{2}{p}\mathbb{Z}/\mathbb{Z}$ and
$\mathfrak{m}\in H_{1}(S^{3}_{p}(K);\mathbb{Z})\cong\mathbb{Z}/p\mathbb{Z}$ is
the generator represented by the meridian of the knot $K$.

We want to check that the surgery formulas for homological blocks
$\widehat{Z}$ and CGP invariant $\mathrm{N}_{r}$ are consistent with the
conjectural relations between these invariants for knot complements and closed
manifolds. In other words, we want to check commutativity of the following
schematic diagram:

(3.54)
${\widehat{Z}_{b}[S^{3}_{p}(K)](q)}$${\mathrm{N}_{r}(S^{3}_{p}(K),\omega)}$${F_{K}(x,q)}$${\mathrm{N}_{r}(S^{3},K)}$$\scriptstyle{q\rightarrow
e^{\frac{2\pi i}{r}}}$$\scriptstyle{\begin{subarray}{c}\text{Laplace}\\\
\text{transform}\end{subarray}}$$\scriptstyle{q\rightarrow e^{\frac{2\pi
i}{r}}}$$\scriptstyle{\begin{subarray}{c}\text{Kirby}\\\
\text{color}\end{subarray}}$

where, more concretely, the left vertical arrow is given by the equations
(3.48)-(3.49), the right vertical arrow is given by (3.53), the top horizontal
arrow is given by (3.26),(3.19),(3.30), and the bottom horizontal arrow is
given by (3.52).

This check can be done by essentially repeating the analysis done for
plumbings in case of a single vertex (with $F$ in (3.8) replaced by (3.49))
and using the following well known relation between the Reidemeister torsion
of $M=S^{3}_{p}(K)$ and the Alexander polynomial of the knot $K$:

(3.55)
$\mathcal{T}(S^{3}_{p}(K),t)=\left.\frac{t\Delta_{K}(t)}{(1-t)^{2}}\right|_{t\in\mathbb{Z}_{p}\subset
U(1)},$

where $t$ is the holonomy of $U(1)$ flat connection along the the meridian
$\mathfrak{m}$ of the knot $K$. But let us write it explicitly anyway.

As for plumbings, after plugging (2.78) into (3.53), it is instructive to
separate the result into three factors:

(3.56)
$\mathrm{N}_{r}(S^{3}_{p}(K),\omega)=\mathcal{A}\cdot\mathcal{B}\cdot\mathcal{C},$

where

(3.57)
$\mathcal{A}=r^{-1/2}\,\xi^{-\frac{3+p}{2}}\cdot\left\\{\begin{array}[]{cl}e^{\frac{\pi
i\,p}{2}}\,e^{-\frac{\pi i}{2}},&r=1\mod 4,\\\ 2^{-1/2}\,e^{\frac{\pi
i}{4}},&r=2\mod 4,\\\ -e^{-\frac{\pi ip}{2}},&r=3\mod 4,\end{array}\right.$
(3.58) $\mathcal{B}=\frac{\Delta(e^{2\pi i\mu})}{(e^{\pi i\mu}-e^{-\pi
i\mu})^{2}},$ (3.59) $\mathcal{C}=\sum_{k\in
H_{r}}F(\xi^{\mu+k},\xi^{2})\cdot\xi^{\frac{p}{2}(\mu+k)^{2}}$

where, as before, $\xi:=e^{\frac{\pi i}{r}}$. Consider a contribution of a
monomial $x^{\ell}$ from $F_{K}(x)$ into (3.59):

(3.60) $\mathcal{C}_{\ell}:=\sum_{k\in
H_{r}}\xi^{\ell(\mu+k)}\cdot\xi^{\frac{p}{2}(\mu+k)^{2}}=\sum_{n\in\mathbb{Z}/r\mathbb{Z}}e^{\frac{\pi
i\,p}{2r}(\tilde{\mu}+2n)^{2}+\frac{\pi i}{r}\ell(\tilde{\mu}+2n)}$

where

(3.61) $\tilde{\mu}:=\mu+(r-1).$

We can now use the following one-dimensional Gauss reciprocity formula:

(3.62) $\sum_{n\;\in\;\mathbb{Z}/r\mathbb{Z}}\exp\left(\frac{2\pi
i}{r}(p\,n^{2}+\ell\,n)\right)=e^{\frac{\pi
i\,\mathrm{sign}(p)}{4}}\,\sqrt{\frac{r}{2|p|}}\sum_{\tilde{a}\;\in\;\mathbb{Z}/2p\mathbb{Z}}\exp\left(-\frac{\pi
ir}{2p}(\tilde{a}+\frac{\ell}{r})^{2}\right).$

Applying it to (3.60) we have

(3.63) $\mathcal{C}_{\ell}=\xi^{-\frac{\ell^{2}}{2p}}\,e^{-\frac{\pi
i}{4}}\,\sqrt{\frac{r}{2|p|}}\sum_{\tilde{a}\in\mathbb{Z}/2p\mathbb{Z}}e^{-\frac{\pi
ir}{2p}\,\tilde{a}^{2}-\pi i\,\tilde{a}(\ell/p+\tilde{\mu})}.$

Therefore, taking into account (3.48) we can write

(3.64) $\mathcal{C}=\sum_{\ell}\mathcal{C}_{\ell}F_{\ell}=e^{-\frac{\pi
i}{4}}\,\xi^{\frac{3+p}{2}}\sqrt{\frac{r}{2|p|}}\,\left.\sum_{\begin{subarray}{c}b\in\mathbb{Z}/p\mathbb{Z}\\\
\tilde{a}\in\mathbb{Z}/2p\mathbb{Z}\end{subarray}}e^{-\frac{\pi
ir}{2p}\tilde{a}^{2}-\frac{2\pi i\tilde{a}b}{p}-\pi
i\tilde{\mu}\tilde{a}}\,\widehat{Z}_{2b}[S^{3}_{p}(K)]\right|_{q\rightarrow\xi^{2}}.$

Combining this together with (3.57) and (3.58) we get

(3.65)
$\mathrm{N}_{r}(S^{3}_{p}(K),\omega)=\frac{1}{\sqrt{{|p|}}}\,\frac{\Delta(e^{2\pi
i\mu})}{(e^{\pi i\mu}-e^{-\pi
i\mu})^{2}}\,\left\\{\begin{array}[]{rl}2^{-1},&r=2\mod 4\\\
2^{-1/2}\,e^{\mp\frac{\pi ip}{2}}\,e^{\mp\frac{3\pi i}{4}},&r=\pm 1\mod
4\end{array}\right\\}\times\\\
\left.\sum_{\begin{subarray}{c}b\in\mathbb{Z}/p\mathbb{Z}\\\
\tilde{a}\in\mathbb{Z}/2p\mathbb{Z}\end{subarray}}e^{-\frac{\pi
ir}{2p}\tilde{a}^{2}-\pi i(r-1)\tilde{a}-\frac{2\pi i\tilde{a}b}{p}-\pi
i{\mu}\tilde{a}}\,\widehat{Z}_{2b}[S^{3}_{p}(K)]\right|_{q\rightarrow
e^{\frac{2\pi i}{r}}}.$

Using the formula (3.55) with $t=e^{2\pi i\mu}$ we indeed arrive at the
conjectural formula (3.19),(3.26) or (3.30), depending on the value $r\mod 4$,
in the case of $M=S^{3}_{p}(K)$. For example, when $r=2\mod 4$ the sum over
$\tilde{a}$ reduces to the sum over $a=\tilde{a}\mod p$:

(3.66)
$\mathrm{N}_{r}(S^{3}_{p}(K),\omega)=\frac{1}{\sqrt{{|p|}}}\,\,\frac{\Delta(e^{2\pi
i\mu})}{(e^{\pi i\mu}-e^{-\pi i\mu})^{2}}\,\times\\\
\left.\sum_{\begin{subarray}{c}b\in\mathbb{Z}/p\mathbb{Z}\\\
a\in\mathbb{Z}/p\mathbb{Z}\end{subarray}}e^{-\frac{\pi ir}{2p}{a}^{2}-\pi
ia-\frac{2\pi iab}{p}-\pi
i{\mu}a}\,\widehat{Z}_{2b}[S^{3}_{p}(K)]\right|_{q\rightarrow e^{\frac{2\pi
i}{r}}}.$

Similarly, for $r=1$ mod 4:

(3.67) $c^{\text{CGP}}_{a,b}\;=\;\frac{1}{|p|}\frac{\Delta(e^{4\pi
ia/p})}{(e^{2\pi ia/p}-e^{-2\pi ia/p})^{2}}e^{-2\pi
i\frac{3\,\text{sign}(p)-p}{4}}\sum_{c,f=0}^{p-1}e^{-\frac{2\pi
i}{p}\big{(}-f^{2}+ac+(b-f)c+(r-1)\frac{c^{2}}{4}\big{)}}$

and for $r=3$ mod 4:

(3.68) $c^{\text{CGP}}_{a,b}\;=\;\frac{1}{|p|}\frac{\Delta(e^{4\pi
ia/p})}{(e^{2\pi ia/p}-e^{-2\pi ia/p})^{2}}e^{2\pi
i\frac{3\,\text{sign}(p)-p}{4}}\sum_{c,f=0}^{p-1}e^{-\frac{2\pi
i}{p}\big{(}f^{2}+ac+(b+f)c+(r+1)\frac{c^{2}}{4}\big{)}}.$

######  Example 3.2.

As a concrete example, consider $p=-3$ surgery on the right-handed trefoil
$K={\bf 3_{1}^{r}}$. In this case, $H_{1}(S^{3}_{p}(K))=\mathbb{Z}_{3}$ and,
therefore, there are two independent $\widehat{Z}$-invariants, which can be
expressed in terms of the false theta-functions (cf. [13, 30]):

(3.69)
$\widehat{Z}_{0}=q^{\frac{71}{72}}\left(\tilde{\Psi}_{18}^{(1)}+\tilde{\Psi}_{18}^{(17)}\right)=q+q^{5}-q^{6}-q^{18}+q^{20}+\ldots,$
(3.70) $\widehat{Z}_{\pm
1}=-\frac{1}{2}q^{\frac{71}{72}}\left(\tilde{\Psi}_{18}^{(5)}+\tilde{\Psi}_{18}^{(13)}\right)=-\frac{1}{2}q^{4/3}\left(1+q^{2}-q^{7}-q^{13}+q^{23}+\ldots\right),$

where the factor $\frac{1}{2}$ in the latter expression and “$\pm$” in its
label appear precisely because we write these expressions in the unfolded
form. Evaluating the right-hand side of (3.66) for various values of $r$, we
find

(3.71) $\begin{array}[]{cc}r=5:&\qquad 4.85591\,-6.4514i\\\ r=6:&\qquad
5.30731\,-4.45336i\\\ r=7:&\qquad 1.89035\,+3.49675i\\\
r=9:&\qquad-6.77162-0.394402i\\\ r=10:&\qquad-24.2779+7.76375i\\\
r=11:&\qquad-6.01733+3.60533i\\\ r=13:&\qquad-37.2754-1.28057i\\\
r=14:&\qquad-11.4885+28.4093i\\\ r=15:&\qquad-15.3891+13.8158i\\\
r=17:&\qquad-11.2632+37.6555i\\\ r=18:&\qquad 17.4965\,+18.5452i\\\
r=19:&\qquad 59.3259\,+18.3538i\\\ \vdots&\end{array}$

which match the corresponding values of $\mathrm{N}_{r}(S^{3}_{-3}({\bf
3_{1}^{r}}))$.

###  3.7. 0-surgeries on knots

In general, the invariant of a knot complement is usually written as

(3.72) $F_{K}(x,q)\;=\;\frac{1}{2}\sum_{\begin{subarray}{c}m\geq 1\\\
\text{odd}\end{subarray}}f_{m}(q)\cdot(x^{\frac{m}{2}}-x^{-\frac{m}{2}})$

which after multiplying by $(x^{\frac{1}{2}}-x^{-\frac{1}{2}})$ in the surgery
formula gives

(3.73)
$(x^{\frac{1}{2}}-x^{-\frac{1}{2}})F_{K}(x,q)\;=\;\frac{1}{2}\sum_{\begin{subarray}{c}m\geq
1\\\
\text{odd}\end{subarray}}f_{m}(q)\cdot(x^{\frac{m}{2}+\frac{1}{2}}-x^{\frac{m}{2}-\frac{1}{2}}-x^{-\frac{m}{2}+\frac{1}{2}}+x^{-\frac{m}{2}-\frac{1}{2}})$

which means (with $n\in\mathbb{Z}$):

(3.74)
$\text{Coeff}_{x^{n}}\Big{[}(x^{\frac{1}{2}}-x^{-\frac{1}{2}})F_{K}(x,q)\Big{]}\;=\;\begin{cases}f_{2n-1}-f_{2n+1}\,,&\text{if}~{}n\geq
1,\\\ -2f_{1}\,,&\text{if}~{}n=0,\\\
f_{2|n|-1}-f_{2|n|+1}\,,&\text{if}~{}n\leq-1.\end{cases}$

For the unknot we have $f_{m}(q)=\delta_{m,1}$, so that
$\widehat{Z}_{n}=\\{\ldots,0,0,1,-2,1,0,0,\ldots\\}$.

######  Example 3.3.

Trefoil knot:

(3.75) $\mathrm{N}_{r}(S^{3}_{0}({\bf 3_{1}}))\;=\;\sum_{k\in
H_{r}}d(\alpha+k)\frac{r\xi^{9(r-1)^{2}/4}}{\\{2r\alpha+2rk\\}}\sum_{n\in
H_{r}}\xi^{3n\alpha+3nk+\frac{3n^{2}}{4}}\\{2\alpha+2k+n\\}\;=\\\
=\;(-1)^{r-1}\frac{r^{2}\xi^{9(r-1)^{2}/4}}{\\{r\alpha\\}\\{2r\alpha\\}}\sum_{k\in
H_{r}}\sum_{n\in
H_{r}}(-1)^{k}\\{\alpha+k\\}\\{2\alpha+2k+n\\}\xi^{3n\alpha+3nk+\frac{3n^{2}}{4}}.$

For the trefoil knot $f_{m}=\epsilon_{m}q^{\frac{m^{2}+23}{24}}$ and so

(3.76)
$\widehat{Z}_{n}\;=\;\begin{cases}\epsilon_{2n-1}q^{\frac{(2n-1)^{2}+23}{24}}-\epsilon_{2n+1}q^{\frac{(2n+1)^{2}+23}{24}}\,,&\text{if}~{}n\geq
1,\\\ -2\epsilon_{1}q\,,&\text{if}~{}n=0,\\\
\epsilon_{2|n|-1}q^{\frac{(2|n|-1)^{2}+23}{24}}-\epsilon_{2|n|+1}q^{\frac{(2|n|+1)^{2}+23}{24}}\,,&\text{if}~{}n\leq-1.\end{cases}$

Again, we verified numerically and analytically that the proposed relations
hold.

###  3.8. Surgery on a link

Let $L$ be the link $L_{9}27$ in the Thistlethwaite table of links (see also
Knot Atlas). Let $M$ be obtained by $a,b,c\in\mathbb{N}$ integral surgery on
the three components of $L$, where $a$ corresponds to the blue component, $b$
to the purple and $c$ to the green one in Figure 2. The linking matrix of $L$
is diagonal with entries $a,b,c$ so that a cohomology class $\omega\in
H^{1}(M;\mathbb{C}/2\mathbb{Z})$ is described by a three-uple
$(\alpha,\beta,\gamma)$ with $\alpha\in\\{\frac{2k}{a},k=0,\ldots a-1\\}$,
$\beta\in\\{\frac{2k}{b},k=0,\ldots b-1\\}$ and
$\gamma\in\\{\frac{2k}{c},k=0,\ldots c-1\\}$.

Figure 2. The $L_{9}27$ link (Knotscape image).

$L$ is one of the first links with the peculiar property that its multicolored
Alexander polynomial is zero. Thus we have : $\mathrm{N}_{2}(M,\omega)=0$ for
any $\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})$.

On the other hand, a direct computer-based calculation gives:

(3.77) $ADO_{3}(\alpha,\beta,\gamma)=q^{2\gamma}+(1-i\sqrt{3})+q^{-2\gamma}$

so that in particular it does not depend on $\alpha,\beta$ and we will denote
it $P(\gamma)$. As a consequence we have:

(3.78)
$\mathrm{N}_{3}(M,\omega)=\Delta_{+}^{-3}\left(\sum_{k_{1}\in\\{-2,0,2\\}}q^{\frac{a}{2}\left((\alpha+k_{1})^{2}-(r-1)^{2}\right)}d(\alpha+k_{1})\right)\times\\\
\left(\sum_{k_{2}\in\\{-2,0,2\\}}q^{\frac{b}{2}\left((\beta+k_{2})^{2}-(r-1)^{2}\right)}d(\beta+k_{2})\right)\times\\\
\left(\sum_{k_{3}\in\\{-2,0,2\\}}q^{\frac{c-1}{2}\left((\gamma+k_{3})^{2}-(r-1)^{2}\right)}d(\gamma+k_{3})P(\gamma+k_{3})\right)$

(in the last factor of the above expression we used $c-1$ as the self linking
of the green component is $1$ in the given diagram). Choosing for instance
$a=b=c=4$ and $\alpha=\beta=\gamma=\frac{1}{2}$ one can compute directly that
$\mathrm{N}_{3}(M,\omega)\neq 0$. Since the torsion is $0$ as the
multivariable Alexander polynomial is, then in this case the invariants
$\widehat{Z}$ should have infinite limits when $q\to\exp(2i\frac{\pi}{3})$.

##  4\. The relation between $\mathrm{N}_{r}$ and $\widehat{Z}$

In this section we present a more general and systematic discussion of the
proposed relation.

###  4.1. Gauss sum vs Laplace transform

Let $M$ be the rational homology sphere obtained by a surgery on a framed link
$L$ in $S^{3}$ with a linking matrix $B$. We will use the shorthand notations
$x\equiv\\{x_{I}\\}_{I\in\mathrm{Vert}}$, $x^{\pm}\equiv\\{x_{I}^{\pm
1}\\}_{I\in\mathrm{Vert}}$. In particular $K[[x^{\pm}]]$ denotes the space of
formal Laurent series in $x_{I}^{\pm}$ with coefficients in $K$, considered as
a module over the ring of Laurent polynomials $K[x^{\pm}]$.

As before, $\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})$ and
$\mu_{I}=\omega(\mathfrak{m}_{I})$, which we assume to be fixed. We also use
notation

(4.1) $\xi\equiv e^{\frac{\pi i}{r}}$

in what follows.

Let $K$ be a field extension over $\mathbb{C}$ (some of the relevant cases are
$K=\mathbb{C}$, $K=\mathbb{C}((q^{1/p}))$, $K=\mathbb{C}((q^{1/p}))(x)$). By
$K(t)^{\prime}$ we will denote the localization of $K[t]$ by the subset of
polynomials in $t$ non-vanishing at $t=1$. This is a subset of the field
$K(t)$ of rational functions in $t$ closed under multiplication. One can then
consider the following ring homomorphism:

######  Definition 4.1.

(4.2) $\begin{array}[]{rrcl}\lim\limits_{t\rightarrow
1}:&K(t)^{\prime}&\longrightarrow&K,\\\
&\frac{P(t)}{Q(t)}&\longmapsto&\frac{P(1)}{Q(1)}.\end{array}$

Similarly, by $K(x)^{\prime}$ we will denote the localization of $K[x^{\pm}]$
by the subset of Laurent polynomials non-vanishing at $x=\xi^{\mu+k}$ for any
$k\in H_{r}^{\mathrm{Vert}}$. This is a subset of the field $K(x)$ of rational
functions in $x$ closed under multiplication. We can then define the following
$K$-linear “$\omega$-twisted Gauss sum” operation:

######  Definition 4.2 ($\omega$-twisted Gauss sum).

(4.3)
$\begin{array}[]{rrcl}\mathfrak{G}_{\omega}:&K(x)^{\prime}&\longrightarrow&K,\\\
\\\ &\frac{P(x)}{Q(x)}&\longmapsto&\sum\limits_{k\in
H_{r}^{\mathrm{Vert}}}\frac{P(\xi^{\mu+k})}{Q(\xi^{\mu+k})}\,\xi^{\frac{1}{2}\,(\mu+k)^{T}B(\mu+k)}.\end{array}$

Denote $p:=4|\det B|$ and let
$\mathbb{C}((q^{1/p}))\equiv\mathbb{C}[[q^{1/p},q^{-1/p}]$ be the field of
fractions of $\mathbb{C}[[q^{1/p}]]$. We then define the
$\mathbb{C}((q^{1/p}))$-linear “$\omega$-twisted Laplace transform” on a
subspace of $\mathbb{C}((q^{1/p}))[[x^{\pm}]]$ by the following formula:

######  Definition 4.3 ($\omega$-twisted Laplace transform).

(4.4)
$\begin{array}[]{rrcl}\mathfrak{L}_{\omega}:&\mathbb{C}((q^{1/p}))[[x^{\pm}]]&\longrightarrow&\mathbb{C}((q^{1/p})),\\\
\\\ &\sum\limits_{\begin{subarray}{c}\ell\in\mathbb{Z}^{\mathrm{Vert}}\\\
m\in\mathbb{Z}\end{subarray}}A_{\ell,m}\,q^{\frac{m}{p}}\,x^{\ell}&\longmapsto&\sum\limits_{n\in\mathbb{Z}}q^{\frac{n}{p}}\sum\limits_{\begin{subarray}{c}\ell\in\mathbb{Z}^{\mathrm{Vert}}\\\
m\in\mathbb{Z}\\\
4m-p\ell^{T}B^{-1}\ell=4n\end{subarray}}\,A_{\ell,m}\cdot\mathcal{C}_{\ell}^{\omega},\end{array}$

where

(4.5) $\mathcal{C}_{\ell}^{\omega}=\frac{e^{\frac{\pi
i\sigma}{4}}(r/2)^{V/2}}{|\det
B|^{1/2}}\,\sum_{\tilde{a}\in\mathbb{Z}^{\mathrm{Vert}}/2B\mathbb{Z}^{\mathrm{Vert}}}e^{-\frac{\pi
ir}{2}\,\tilde{a}^{T}B^{-1}\tilde{a}-\pi
i\,\tilde{a}^{T}B^{-1}(\ell+B(\mu+(r-1)\varepsilon))}.$

We say that $\mathfrak{L}_{\omega}$ is well defined if the interior sum in the
right hand side of (4.4) has a finite number of non-zero terms.

######  Remark 4.4.

$\mathfrak{L}_{\omega}$ restricted on $\mathbb{C}((q^{1/p}))[x^{\pm}]$ is
well-defined.

######  Remark 4.5.

Apart from $\omega\in H^{1}(M;\mathbb{C}/2\mathbb{Z})$ (equivalently
$\mu\in(\mathbb{C}/2\mathbb{Z})^{\mathrm{Vert}}$ s.t. $B\mu=0\mod
2\mathbb{Z}^{\mathrm{Vert}}$), the operations $\mathfrak{L}_{\omega}$ and
$\mathfrak{G}_{\omega}$ also (implicitly) depend on $B$ and $r$.

Next we introduce the following limit operation (a morphism of
$\mathbb{C}$-algebras):

######  Definition 4.6.

(4.6) $\begin{array}[]{rrcl}\lim\limits_{q\rightarrow e^{\frac{2\pi
i}{r}}}:&\mathbb{C}((q^{1/p}))&\longrightarrow&\mathbb{C},\\\ \\\
&\sum\limits_{m}A_{m}\,q^{\frac{m}{p}}&\longmapsto&\lim\limits_{q\rightarrow
e^{\frac{2\pi i}{r}}}\sum\limits_{m}A_{m}\,q^{\frac{m}{p}}.\end{array}$

We say that the operation is well defined if the power series in $q^{1/p}$ are
convergent for $0<|q^{1/p}|<1$ and the limit, taken to the root of unity along
the radial direction, exists and is finite.

######  Remark 4.7.

This operation can be extended to the polynomials/series with coefficients in
$\mathbb{C}((q^{1/p}))$ by applying it coefficient-wise.

######  Proposition 4.8.

Let $A(q,x)\in\mathbb{C}((q^{1/p}))[x^{\pm}]$ such that $\lim_{q\rightarrow
e^{\frac{2\pi i}{r}}}A(q,x)\in\mathbb{C}[x^{\pm}]$ exists. Then

(4.7) $\mathfrak{G}_{\omega}\,\lim_{q\rightarrow e^{\frac{2\pi
i}{r}}}A(q,x)=\lim_{q\rightarrow e^{\frac{2\pi
i}{r}}}\mathfrak{G}_{\omega}\,A(q,x)=\lim_{q\rightarrow e^{\frac{2\pi
i}{r}}}\mathfrak{L}_{\omega}\,A(q,x).$

###### Proof.

The first equality follows from the definition of $\mathfrak{G}_{\omega}$,
which involves taking a finite linear combination of the evaluations at
certain values of $x$. The second equality is shown by applying the Gauss
reciprocity formula (3.11) to individual monomials in $x$, the number of which
is finite. ∎

######  Proposition 4.9.

Let $A(x)\in K(x)^{\prime},B(x^{r})\in K(x)^{\prime}\cap K(x^{r})$. Then

(4.8) $\mathfrak{G}_{\omega}(B(x^{r})A(x))=B((-1)^{r-1}\,e^{\pi
i\mu})\,\mathfrak{G}_{\omega}(A(x)).$

###### Proof.

Follows from the definition of $\mathfrak{G}_{\omega}$. ∎

######  Remark 4.10.

The statement of the Prop. 4.9 is directly extended to the series in
$K^{\prime}(x)((t))$ by applying it coefficient-wise.

We also define the following limiting operation on a subalgebra of
$\mathbb{C}((q^{1/p}))((t))$:

######  Definition 4.11.

(4.9) $\begin{array}[]{rrcl}\lim\limits_{t\rightarrow
1}:&\mathbb{C}((q^{1/p}))((t))&\longrightarrow&\mathbb{C}((q^{1/p})),\\\ \\\
&\sum\limits_{n}(\sum\limits_{m}A_{n,m}\,q^{\frac{m}{p}})t^{n}&\longmapsto&\sum\limits_{m}(\sum\limits_{n}A_{n,m})q^{\frac{m}{p}}.\end{array}$

We say that the operation is well defined if there is a finite number of non-
vanishing coefficients $A_{n,m}$ for any fixed $m$.

######  Remark 4.12.

When restricted on the subspace $C((q^{1/p}))(t)^{\prime}\subset
C((q^{1/p}))((t))$ and is well-defined, the result of the operation in Def.
4.11 coincides with the result of the operation in Def. 4.1.

With such definitions, consider the following diagram of (partially defined)
algebra homomorphisms, where the dotted arrow means that we make the
hypothesis that in the cases of interest the image of $\lim_{q\to e^{2\pi
i/r}}$ map is contained in the subspace $\mathbb{C}(t)^{\prime}$:

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