# On Hempel pairs and Turaev–Viro invariants

Yi Liu  Beijing International Center for Mathematical Research, Peking
University
Beijing 100871, China P.R<EMAIL_ADDRESS>
###### Abstract.

Surface bundles arising from periodic mapping classes may sometimes have non-
isomorphic, but profinitely isomorphic fundamental groups. Pairs of this kind
have been discovered by Hempel. This paper exhibits examples of nontrivial
Hempel pairs where the mapping tori can be distinguished by some Turaev–Viro
invariants, and also examples where they cannot be distinguished by any
Turaev–Viro invariants.

###### Key words and phrases:

profinite completion, mapping class group, Turaev–Viro invariant

###### 2020 Mathematics Subject Classification:

Primary 57K31; Secondary 57K16

Partially supported by NSFC Grant 11925101, and National Key R&D Program of
China 2020YFA0712800

##  1\. Introduction

Let $S$ be a connected closed orientable surface. Denote by $\mathrm{Mod}(S)$
the mapping class group of $S$, whose elements are the isotopy classes of
orientation-preserving self-homeomorphisms of $S$.

A _Hempel pair_ as we call refers to a pair of periodic mapping classes
$[f_{A}],[f_{B}]\in\mathrm{Mod}(S)$ of identical order $d\geq 1$, such that
$[f_{B}]=[f_{A}^{k}]$ holds for some integer $k$ coprime to $d$. Hence
$[f_{A}]=[f_{B}^{k^{*}}]$ holds for any congruence inverse $k^{*}$ of $k$
modulo $d$, (that is, $k^{*}k\equiv 1\bmod d$). Hempel studied such pairs
[Hem14], finding out that the fundamental groups of their mapping tori $M_{A}$
and $M_{B}$ always have identical collections of (isomorphism types of) finite
quotient groups. This is equivalent to saying that the profinite completions
of $\pi_{1}(M_{A})$ and $\pi_{1}(M_{B})$ are isomorphic groups. Hempel
discovered examples of such pairs with non-isomorphic fundamental groups. This
is equivalent to saying that $M_{A}$ and $M_{B}$ are not homeomorphic
$3$–manifolds.

We call $[f_{A}]$ and $[f_{B}]$ a _nontrivial_ Hempel pair, if $M_{A}$ and
$M_{B}$ are not homeomorphic. There are no nontrivial Hempel pairs when $S$ is
a sphere or a torus, as the condition forces $d=1$ or $d\in\\{1,2,3,4,6\\}$,
and hence $k=\pm 1$. One may obtain a nontrivial Hempel pair of order $5$ when
$S$ has genus $2$. Nontrivial Hempel pairs are a source of distinct
$3$–manifold pairs that cannot be distinguished by their profinite fundamental
groups. Among $3$–manifolds, the question as to which topological invariants
are determined by the profinite fundamental group has stimulated a lot of
fruitful study in recent years. See [Rei18] and [LiuS22, Section 9] for past
surveys on that fast-growing topic.

Turaev–Viro invariants are topological invariants of closed $3$–manifolds,
originally constructed using quantum 6j-symbols [TurV92]. That they are
generally _not_ profinite invariants is evident from their explicit values on
all lens spaces [Sok97]. Meanwhile, there are many homeomorphically distinct
torus bundles over a circle, whose monodromies are Anosov and mutually
conjugate up to inverse in every congruence quotient of
$\mathrm{Mod}(T^{2})\cong\mathrm{SL}(2,{\mathbb{Z}})$. These torus bundles
have isomorphic profinite fundamental groups. Funar shows that no Turaev–Viro
invariants (associated to any spherical fusion categories) distinguish these
torus bundles [Fun13, Proposition 1.1].

In this paper, we take up the question as to whether Turaev–Viro invariants
distinguish nontrivial Hempel pairs. We shall content ourselves with the
$\mathrm{SU}(2)$ and the $\mathrm{SO}(3)$ Turaev–Viro invariants, as they are
mentioned the most often. The $\mathrm{SU}(2)$ series can be fully listed as
$\mathrm{TV}_{r,s}$, for any integer $r\geq 3$ and any integer $s$ coprime to
$r$; the $\mathrm{SO}(3)$ series can be fully listed as
$\mathrm{TV}^{\prime}_{r,s}$, for any odd integer $r\geq 3$ and any even
integer $s$ coprime to $r$. More economically, one could focus on
$\mathrm{TV}_{r,1}$ ($r$ even), and $\mathrm{TV}^{\prime}_{r,r-1}$ ($r$ odd),
together with $\mathrm{TV}_{3,1}$ and $\mathrm{TV}_{3,2}$, which depend only
on the ${\mathbb{Z}}/2{\mathbb{Z}}$ cohomology ring. These determine all the
other ones. See Section 3 for the notations and more review.

Our conclusion can be summarized as follows.

######  Theorem 1.1.

  1. (1)

For any integer $d\geq 5$ other than $6$, there exists some nontrivial Hempel
pair of order $d$, such that the mapping tori can be distinguished by some
$\mathrm{SU}(2)$ Turaev–Viro invariant, and if $d$ is odd, also by some
$\mathrm{SO}(3)$ Turaev–Viro invariant.

  2. (2)

For any prime integer $p\geq 5$, there exists some nontrivial Hempel pair of
order $p$, such that the mapping tori cannot be distinguished by any
$\mathrm{SU}(2)$ or $\mathrm{SO}(3)$ Turaev–Viro invariants.

Theorem 1.1 is proved in Section 5 by exhibiting concrete families of
examples. Our simplest distinguishable Hempel pair exists with order $d$ on
genus $d-2\geq 3$. Our simplest indistinguishable nontrivial Hempel pair
exists with prime order $p$ on genus $(p-1)/2\geq 2$.

The technical heart of this paper is the following calculation.

######  Theorem 1.2.

Let $a\geq 3$ be an integer and $b_{1},\cdots,b_{n}$ be integers coprime to
$a$. Let $M$ be a Seifert fiber space with orientable orbifold base and
orientable Seifert fibration, and with symbol
$(g;(a,b_{1}),\cdots,(a,b_{n}))$. Suppose $g\geq 0$, and $a>n\geq 0$, and
$b_{1}+\cdots+b_{n}=0$.

  1. (1)

If there exist some integer $b^{*}$ coprime to $a$ and
$\nu_{1},\cdots,\nu_{n}\in\\{\pm 1\\}$, such that
$b^{*}b_{j}\equiv\nu_{j}\bmod a$ holds for all $j\in\\{1,\cdots,n\\}$, then
for any $s$ coprime to $a$,

$\mathrm{TV}_{a,s}(M)=\frac{a^{n+2g-2}}{2^{2n+2g-4}}\cdot\frac{1}{{\sin}^{2n+4g-4}(\pi
b^{*}s/a)},$

and moreover, if $a$ is odd and $s$ is even,

$\mathrm{TV}^{\prime}_{a,s}(M)=\frac{a^{n+2g-2}}{2^{2n+4g-4}}\cdot\frac{1}{{\sin}^{2n+4g-4}(\pi
b^{*}s/a)}.$

  2. (2)

Otherwise, for any integer $r\geq 3$ divisible by $a$ and any integer $s$
coprime to $r$,

$\mathrm{TV}_{r,s}(M)=0,$

and moreover, if $r$ is odd and $s$ is even,

$\mathrm{TV}^{\prime}_{r,s}(M)=0.$

See Section 2 for review on Seifert fiber spaces and the standard notation for
their symbols.

Theorem 1.2 is proved in Section 4, by applying a general formula for
calculating the Witten–Reshetikhin–Turaev invariant $\tau_{r}$ of oriented
closed Seifert fiber spaces. The exact formula we invoke is due to Hansen
[Han01], while similar calculation in special cases or with likewise strategy
also appear in many other places, for instance, see [And13, LawR99, Roz96,
Tan07].

Theorem 1.2 is formulated by first testing samples of Seifert fiber spaces (on
computer), and then observing interesting phenomena. Luckily, we find the
assumptions as in Theorem 1.2, which greatly simplify the situation and the
answer.

In the appendix, we prove a splitting formula

$\mathrm{TV}_{r,s}(M)=\mathrm{TV}_{3,1}(M)\cdot\mathrm{TV}^{\prime}_{r,r-s}(M)$

for $r$ odd and $s$ odd. This complements a former formula

$\mathrm{TV}_{r,s}(M)=\mathrm{TV}_{3,2}(M)\cdot\mathrm{TV}^{\prime}_{r,s}(M)$

for $r$ odd and $s$ even, proved by Detcherry, Kalfagianni, and Yang [DetKY18,
Theorem 2.9]. Our proof in the appendix automatically includes the non-
orientable case, although the orientable case suffices for our application.

This paper is organized as follows. In Section 2, we recall the prelimiary
description of periodic mapping tori as Seifert fiber spaces with vanishing
rational Euler number. In Section 3, we review Turaev–Viro invariants. In
Section 4, we prove Theorem 1.2. In Section 5, we prove Theorem 1.1. In
Section A, we give an elementary proof of the aforementioned splitting formula
regarding $\mathrm{SU}(2)$ Turaev–Viro invariants at odd $r$ and odd $s$.

### Acknowledgment

The author would like to thank Yuxuan Yang for valuable comments on a
preliminary version of this paper.

##  2\. Periodic mapping tori

Let $S$ be a connected orientable closed surface. The mapping class group
$\mathrm{Mod}(S)$ consists of all the isotopy classes of orientation-
preserving self-homeomorphism of $S$. For any mapping class
$[f]\in\mathrm{Mod}(S)$, we denote by $M_{f}$ the mapping torus

$M_{f}=\frac{S\times{\mathbb{R}}}{(f(x),r)\sim(x,r+1)},$

which naturally fibers over the oriented circle ${\mathbb{R}}/{\mathbb{Z}}$
with fiber type $S$ and (backward) monodromy type $[f]$. The mapping torus
$M_{f}$ is a connected orientable closed $3$–manifold, whose homeomorphism
type depends only on $[f]$.

A _periodic mapping class_ refers to a mapping class of finite order. In this
case, the mapping torus is a Seifert fiber space. The Seifert fibration is
orientable over an orientable base orbifold, and has vanishing rational Euler
number. Moreover, the Seifert fibration on any periodic mapping torus is
unique up to isotopy. Conversely, any closed Seifert fiber space with
orientable orbifold base and orientable Seifert fibration of vanishing
rational Euler number arises as the mapping torus of some periodic mapping
class. Moreover, the genus of surface and the conjugacy class of the periodic
mapping class up to inverse are both unique. This way, periodic mapping
classes can be described equivalently by indicating the symbol of its mapping
torus, as a Seifert fiber space. (See [AscFW15, Chapter 1]).

The most general symbol describes any (connected, compact) Seifert fiber space
with all features, allowing possibly non-orientable orbifold base, non-
orientable Seifert fibration, and non-empty boundary. For dicussing periodic
mapping classes, we only need to consider closed Seifert fiber spaces with
orientable orbifold base and orientable Seifert fibration, whose (possibly
non-normalized) symbol is denoted as

(2.1) $(g;(a_{1},b_{1}),\cdots,(a_{n},b_{n}))$

where $g\geq 0$ is an integer, and where $a_{j}\geq 1$ is an integer and
$b_{j}$ is an integer coprime to $a_{j}$, for all $j\in\\{1,\cdots,n\\}$. This
symbol presents a Seifert fiber space (with standard orientation) constructed
as follows.

Take a product $3$–manifold $\Sigma\times S^{1}$ of a connected closed
oriented surface $\Sigma$ of genus $g$ and an oriented circle $S^{1}$; take
$n$ disjoint embedded disks $D_{1},\cdots,D_{n}\subset\Sigma$; remove the
solid tori $D_{j}\times S^{1}$ from $\Sigma\times S^{1}$, and refill with
solid tori in other ways, such that the slopes $a_{j}[\partial
D_{j}]+b_{j}[S^{1}]$ on $\partial D_{j}\times S^{1}$ bound disks in the new
solid tori.

The base of this Seifert fiber space is a connected, closed, oriented
$2$–orbifold of genus $g$ with $n$ cone points of order $a_{1},\cdots,a_{n}$,
(ordinary and negligible if $a_{j}=1$). Its orbifold Euler characteristic
equals $2-2g-\sum_{j}(1-1/a_{j})$. The rational Euler number of the Seifert
fibration equals $-\sum_{j}b_{j}/a_{j}$, where the minus sign comes from our
convention on the orientation of $\partial D_{j}$, (as induced by the
orientation of $D_{j}$, rather than $S\setminus(D_{1}\cup\cdots\cup D_{n})$).

The following operations of the symbol do not change the homeomorphism type of
the resulting Seifert fiber space: re-ordering all $(a_{j},b_{j})$; inserting
or deleting a term $(1,0)$; replacing one $(a_{j},b_{j})$ with
$(a_{j},b_{j}+a_{j})$ and another $(a_{j^{\prime}},b_{j^{\prime}})$ with
$(a_{j^{\prime}},b_{j^{\prime}}-a_{j^{\prime}})$ simultaneously; or replacing
all $(a_{j},b_{j})$ with $(a_{j},-b_{j})$ simultaneously. Note that only the
last operation changes the orientation-preserving homeomorphism type of the
resulting Seifert fiber space.

For homeomorphic periodic mapping tori, their symbols are all related by
finitely many steps of the above operations. This is a special case of the
classification of Seifert fiber spaces (see [Orl72, Theorem 1.10] or [Hem76,
Chapter 12]).

The following Lemmas 2.1 and 2.2 actually appear in [Hem14] in equivalent
forms. We include quick (and slightly different) proofs here for the reader’s
reference.

######  Lemma 2.1.

Let $M$ be a closed Seifert fiber space with orientable orbifold base and
orientable Seifert fibration, and with symbol
$(g;(a_{1},b_{1}),\cdots,(a_{n},b_{n}))$. If the Seifert fibration has
vanishing rational Euler characteristic, then $M$ is homeomorphic to the
mapping torus $M_{f}$ of a periodic mapping class $[f]\in\mathrm{Mod}(S)$.
Moreover, $[f]$ has order $d=\mathrm{lcm}(a_{1},\cdots,a_{n})$, and $S$ has
genus $1+(g-1)d+\sum_{j}(1-1/a_{j})d/2$.

###### Proof.

The orbifold fundamental group of the base $\mathcal{O}$ has a presentation
with generators $x_{1},y_{1},\cdots,x_{g},y_{g},s_{1},\cdots,s_{n}$ and
relatorions $[x_{1},y_{1}]\cdots[x_{g},y_{g}]=s_{1}\cdots s_{n}$, and
$s_{1}^{a_{1}}=\cdots=s_{n}^{a_{n}}=1$. Under the assumption of vanishing
rational Euler number, The assignments $x_{i}\mapsto 0\bmod d$, $y_{i}\mapsto
0\bmod d$, and $s_{j}\mapsto-b_{j}d/a_{j}\bmod d$ yield a well-defined,
surjective homomorphism $\pi_{1}(\mathcal{O})\to{\mathbb{Z}}/d{\mathbb{Z}}$,
where $d=\mathrm{lcm}(a_{1},\cdots,a_{n})$. The kernel of the homorphism
corresponds to a cyclic orbifold covering $S\to\mathcal{O}$ of degree $d$,
where $S$ has no singular cone points. The generator
$1\in{\mathbb{Z}}/d{\mathbb{Z}}$ corresponds to a deck transformation,
representing a periodic mapping class $[f]\in\mathrm{Mod}(S)$ of order $d$. It
is elementary to check that $M_{f}$ is homeomorphic to $M$. The Euler
characteristic of the surface $S$ is equal to $d$ times the rational Euler
number of $\mathcal{O}$, which implies the asserted genus of $S$. ∎

######  Lemma 2.2.

Let $S$ be a connected, close, orientable surface, and $[f]\in\mathrm{Mod}(S)$
be a periodic mapping class of order $d$. If the mapping torus $M_{f}$ has
symbol $(g;(a_{1},b_{1}),\cdots,(a_{n},b_{n}))$, then for any integer $k$
coprime to $d$, the mapping torus $M_{f^{k}}$ of the iterate
$[f^{k}]\in\mathrm{Mod}(S)$ has symbol
$(g;(a_{1},b_{1}k^{*}),\cdots,(a_{n},b_{n}k^{*}))$, where $k^{*}$ is any
integer satisfying $kk^{*}\equiv 1\bmod d$.

###### Proof.

The mapping torus $M_{f^{k}}$ naturally cyclically cover $M_{f}$ of degree
$k$. Since $f$ has order $k$, we also see that $M_{f}$ cyclically covers
$M_{f^{k}}$ of degree $k^{*}$. Consider the covering $M_{f}\to M_{f^{k}}$. The
pull-back of the Seifert fibration of $M_{f^{k}}$ is a Seifert fibration on
$M_{f}$. If $M_{f}$ has symbol $(g;(a_{1},b_{1}),\cdots,(a_{n},b_{n}))$, by
definition, the preimage of $(\Sigma\setminus(D_{1}\cup\cdots\cup
D_{n}))\times S^{1}\subset M_{f^{k}}$ is also a product
$(\Sigma\setminus(D_{1}\cup\cdots\cup D_{n}))\times S^{1}\subset M_{f}$, while
ordinary fibers (that is, $*\times S^{1}$) in $M_{f}$) cyclically covers the
ordinary fibers in $M_{f^{k}}$ with degree $k^{*}$. Since the slope
$a_{j}[D_{j}]+b_{j}[S^{1}]$ on $\partial D_{j}\times S^{1}$ is homotopically
trivial in $M_{f}$, the slope $a_{j}[D_{j}]+b_{j}k^{*}[S^{1}]$ on $\partial
D_{j}\times S^{1}$ is homotopically trivial in $M_{k}$. Therefore,
$(g;(a_{1},b_{1}k^{*}),\cdots,(a_{n},b_{n}k^{*}))$ is a symbol of $M_{f^{k}}$.
∎

##  3\. Turaev–Viro invariants

Turaev–Viro invariants are topological invariants of closed $3$–manifolds,
arising from representation theory of quantum groups at roots of unity.
Throughout this paper, we only discuss Turaev–Viro invariants pertaining to
the most basic quantum group $\mathcal{U}_{q}(\mathfrak{sl}_{2})$. In this
setting, there are essentially two series, namely, the $\mathrm{SU}(2)$
Turaev–Viro invariants $\mathrm{TV}_{r}$ for all integers $r\geq 3$, and the
$\mathrm{SO}(3)$ Turaev–Viro invariants $\mathrm{TV}^{\prime}_{r}$ for all odd
integers $r\geq 3$.

Throughout this paper, a _closed_ manifold only means a compact manifold with
empty boundary, possibly disconnected, and possibly non-orientable.

###  3.1. Abstract versions

Let $q^{1/2}$ be a square root of a root of unity $q\neq\pm 1$. Denote by
$r\geq 3$ the order of $q$. Note that $q^{1/2}$ must have order $2r$ when $r$
is even, but may have order $2r$ or $r$ when $r$ is odd. We denote by
${\mathbb{Q}}(q^{1/2})$ the abstract cyclotomic field generated by $q^{1/2}$.

For any closed $3$-manifold $M$, the Turaev–Viro invariant
$\mathrm{TV}(M;q^{1/2})$ can be defined at $q^{1/2}$; if $r$ is odd, the
refined Turaev–Viro invariant $\mathrm{TV}^{\prime}(M;q^{1/2})$ can be defined
at $q^{1/2}$ of order $r$. Both $\mathrm{TV}(M;q^{1/2})$ and
$\mathrm{TV}^{\prime}(M;q^{1/2})$ are topological invariants of $M$, with
totally real values in ${\mathbb{Q}}(q^{1/2})$. We call $\mathrm{TV}$ and
$\mathrm{TV}^{\prime}$ the _abstract_ $\mathrm{SU}(2)$ and the _abstract_
$\mathrm{SO}(3)$ Turaev–Viro invariants at $q^{1/2}$ (or at level $r-2$),
respectively.

Most details about the actual construction of these invariants are not be
necessary in the sequel, except the appendix. We record them below for the
sake of clarity.

Let $\mathscr{T}=(V,E,F,T)$ be any finite simplicial $3$–complex, where the
items denote the sets of vertices, edges, faces, and tetrahedra, respectively.
Denote $I_{r}=\\{0,1,\cdots,r-2\\}$. A triple $(i,j,k)\in I_{r}\times
I_{r}\times I_{r}$ is said to be _admissible_ if the numbers $(i+j+k)/2$,
$(i+j-k)/2$, $(j+k-i)/2$, $(k+i-j)/2$ all stay in $I_{r}$. An _admissible
coloring_ of $(M,\mathscr{T})$ (of _level_ $r-2$) refers to a map $c\colon
E\to I_{r}$, such that on any face, the edge colors (that is, values of $c)$
form an admissible triple. Denote by
$\mathcal{A}_{r}=\mathcal{A}_{r}(M,\mathscr{T})$ the set of all admissible
colorings of level $r-2$. If $r$ is odd, denote by
$I^{\prime}_{r}=\\{0,2,\cdots,r-3\\}$ the subset of even elements of $I_{r}$.
Denote by $\mathcal{A}^{\prime}_{r}=\mathcal{A}^{\prime}_{r}(M,\mathscr{T})$
the subset of $\mathcal{A}_{r}$ consisting of admissible colorings with values
in $I^{\prime}_{r}$.

Let $M$ be a (possibly disconnected, possibly non-orientable) closed
$3$–manifold. Take a triangulation $\mathscr{T}=(V,E,F,T)$ of $M$. Then
invariants $\mathrm{TV}$ and $\mathrm{TV}^{\prime}$ of $M$ at $q^{1/2}$ can be
expressed in terms of $\mathscr{T}$ as follows:

(3.1)
$\mathrm{TV}\left(M;q^{1/2}\right)=\left(\frac{(q^{1/2}-q^{-1/2})^{2}}{-2r}\right)^{|V|}\cdot\sum_{c\in\mathcal{A}_{r}}\left|\mathscr{T}\right|_{c};$

if $r$ is odd and $q^{1/2}$ has order $r$,

(3.2)
$\mathrm{TV}^{\prime}\left(M;q^{1/2}\right)=\left(\frac{(q^{1/2}-q^{-1/2})^{2}}{-r}\right)^{|V|}\cdot\sum_{c\in\mathcal{A}^{\prime}_{r}}\left|\mathscr{T}\right|_{c},$

where $|V|$ denotes the number of vertices in $\mathscr{T}$, and where
$|\mathscr{T}|_{c}$ is explained in Notation 3.1 below. Note that our
notations follow the reformulation as in [DetKY18, Appendix A], where the
factors in $|e|_{c}$, $|f|_{c}$, and $|t|_{c}$ are grouped slightly
differently than those in [TurV92], so as to avoid unnecessary square roots.

As it turns out, the values of the above expressions in
${\mathbb{Q}}(q^{1/2})$ are independent of the auxiliary choice of the
triangulation $\mathscr{T}$ [TurV92]. Therefore, $\mathrm{TV}(M;q^{1/2})$ and
$\mathrm{TV}^{\prime}(M;q^{1/2})$ are indeed topological invariants of $M$.

######  Notation 3.1.

  1. (1)

Denote

$[n]!=\begin{cases}[1]\cdot[2]\cdot\cdots\cdot[n]&n=1,2,\cdots,r-1\\\
1&n=0\end{cases}$

where

$[n]=\frac{q^{n/2}-q^{-n/2}}{q^{1/2}-q^{-1/2}}.$

Note that the quantum integers $[1],[2],\cdots,[r-1]$ take totally real
nonzero values in ${\mathbb{Q}}(q^{1/2})$, as $q$ is a primitive $r$-th root
of unity ($r\geq 3$).

  2. (2)

For any $e\in E$ and $c\in\mathcal{A}_{r}$,

$|e|_{c}=(-1)^{i}\cdot[i+1]$

where $i$ is the color of $e$ under $c$.

  3. (3)

For any $f\in F$ and $c\in\mathcal{A}_{r}$,

$|f|_{c}=(-1)^{-S}\cdot\frac{[S-i]!\cdot[S-j]!\cdot[S-k]!}{[S+1]!}$

where $i,j,k$ are the edge colors of $f$ under $c$, and where $S=(i+j+k)/2$.

  4. (4)

For any $t\in T$ and $c\in\mathcal{A}_{r}$, denote

$|t|_{c}=\sum_{z}\frac{(-1)^{z}\,[z+1]!}{\prod_{a}[z-T_{a}]!\cdot\prod_{b}[Q_{b}-z]!}$

where $(i,j,k),(i,m,n),(j,l,n),(k,l,m)$ are the edge colors of the faces of
$t$ under $c$, and where $T_{1}=(i+j+k)/2$, $T_{2}=(i+m+n)/2$,
$T_{3}=(j+l+n)/2$, $T_{4}=(k+l+m)/2$, $Q_{1}=(i+j+l+m)/2$,
$Q_{2}=(i+k+l+n)/2$, $Q_{3}=(j+k+m+n)/2$. The index $a$ ranges in
$\\{1,2,3,4\\}$, and $b$ in $\\{1,2,3\\}$, and $z$ from $\max_{a}T_{a}$ to
$\min_{b}Q_{b}$.

  5. (5)

For any $c\in\mathcal{A}_{r}$, denote

$|\mathscr{T}|_{c}=\prod_{e\in E}|e|_{c}\cdot\prod_{f\in
F}|f|_{c}\cdot\prod_{t\in T}|t|_{c}.$

###  3.2. Specialized versions

In the literature, _the_ $\mathrm{SU}(2)$ and _the_ $\mathrm{SO}(3)$
Turaev–Viro invariants often refer to specialization of the abstract versions
at customary complex roots of unity. To be precise, these refer to the
numerical quantities $\mathrm{TV}_{r}$ and $\mathrm{TV}^{\prime}_{r}$ below.

######  Notation 3.2.

Let $r\geq 3$ be an integer and $s$ be an integer coprime to $r$. The
following expressions are all evaluated by specializing
${\mathbb{Q}}(q^{1/2})\to{\mathbb{C}}$.

  1. (1)

Denote

$\mathrm{TV}_{r,s}(M)=\mathrm{TV}\left(M;q^{1/2}=e^{\sqrt{-1}\cdot\pi
s/r}\right),$

  2. (2)

If $r$ is odd and $s$ is even, denote

$\mathrm{TV}^{\prime}_{r,s}(M)=\mathrm{TV}^{\prime}\left(M;q^{1/2}=e^{\sqrt{-1}\cdot\pi
s/r}\right).$

  3. (3)

In informal discussion, we often write $\mathrm{TV}_{r}$ and
$\mathrm{TV}^{\prime}_{r}$, assuming $s$ implicitly fixed.

######  Lemma 3.3.

Let $M,N$ be any closed $3$–manifolds.

  1. (1)

$\mathrm{TV}_{r,s}(M)\in{\mathbb{R}}$.

  2. (2)

$\mathrm{TV}_{r,s}(M\sqcup N)=\mathrm{TV}_{r,s}(M)\cdot\mathrm{TV}_{r,s}(N)$.

  3. (3)

$\mathrm{TV}_{r,s}(S^{2}\times S^{1})=1.$

If $r$ is odd and $s$ is even, the same statements hold with
$\mathrm{TV}^{\prime}_{r,s}$ in place of $\mathrm{TV}_{r,s}$.

The first statement follow immediately from the observation that $\mathrm{TV}$
and $\mathrm{TV}^{\prime}$ can be written as rational functions over
${\mathbb{Q}}$ in $[1],\cdots,[r-1]$ and $(q^{1/2}-q^{-1/2})^{2}$; see (3.1),
(3.2), and Notation 3.1. Note $[n]=\sin(\pi sn/r)/\sin(\pi s/r)$ and
$(q^{1/2}-q^{-1/2})^{2}=-4\sin^{2}(\pi s/r)$ evaluated at
$q^{1/2}=e^{\sqrt{-1}\cdot\pi s/r}$. The second statement is also obvious by
definition. The last statement appears in Turaev and Viro’s original paper
[TurV92, Section 8.1.B].

######  Lemma 3.4.

Let $M$ be any closed $3$–manifold.

  1. (1)

$\mathrm{TV}_{r,s}(M)=\mathrm{TV}_{r,-s}(M)=\mathrm{TV}_{r,s+2r}(M)$. If $r$
is odd and $s$ is even, the same identities hold with $\mathrm{TV}^{\prime}$
in place of $\mathrm{TV}$.

  2. (2)

If $s$ is odd, $\mathrm{TV}_{r,s}(M)$ is Galois conjugate to
$\mathrm{TV}_{r,1}(M)$; if $s$ is even (hence $r$ odd), $\mathrm{TV}_{r,s}(M)$
is Galois conjugate to $\mathrm{TV}_{r,r-1}(M)$. If $r$ is odd and $s$ is
even, the same statements hold with $\mathrm{TV}^{\prime}_{r,s}$ in place of
$\mathrm{TV}_{r,s}$.

  3. (3)

If $r$ is odd,

$\mathrm{TV}_{r,s}(M)=\begin{cases}\mathrm{TV}_{3,2}(M)\cdot\mathrm{TV}^{\prime}_{r,s}(M)&s\mbox{
even}\\\ \mathrm{TV}_{3,1}(M)\cdot\mathrm{TV}^{\prime}_{r,r-s}(M)&s\mbox{
odd}\end{cases}$

  4. (4)

Denote by $\beta_{i}$ the dimension of $H_{i}(M;{\mathbb{Z}}/2{\mathbb{Z}})$
over ${\mathbb{Z}}/2{\mathbb{Z}}$, and by $w_{1}\in
H^{1}(M;{\mathbb{Z}}/2{\mathbb{Z}})$ the first Stifel–Whitney class of $M$.

$\mathrm{TV}_{3,2}(M)=2^{-\beta_{0}(M)+\beta_{2}(M)},$

and

$\mathrm{TV}_{3,1}(M)=2^{-\beta_{0}(M)}\cdot\sum_{t}(-1)^{\langle
t^{3}+w_{1}^{2}t,[M]\rangle}$

where the index $t$ ranges over $H^{1}(M;{\mathbb{Z}}/2{\mathbb{Z}})$.

The first and the second statements are again obvious properties of the
defining expressions.

The third statement is proved when $s$ is even by Detcherry, Kalfagianni, and
Yang [DetKY18, Theorem 2.9], and can be easily derived from well-known facts
when $s$ is odd, assuming $M$ orientable, (see Lemmas 3.6 and 3.7). We supply
a proof without assuming orientability in the appendix Section A.

The formulas for $\mathrm{TV}_{3,s}$ in the fourth statement are due to Turaev
and Viro [TurV92, Section 9.3.A].

###  3.3. Relation to Witten–Reshetikhin–Turaev invariants

For connected oriented closed $3$–manifolds, the Witten–Reshetikhin–Turaev
invariants are invariant under orientation-preserving homeomorphisms. These
invariants were suggested by Witten [Wit89], and the first mathematically
rigorous construction was due to Reshetikhin and Turaev [ResT91].

Following Kirby and Melvin [KirM91], we denote by $\tau_{r}$ the
$\mathrm{SU}(2)$ Witten–Reshetikhin–Turaev invariants, defined for any integer
$r\geq 3$, and by $\tau^{\prime}_{r}$ the $\mathrm{SO}(3)$
Witten–Reshetikhin–Turaev invariants, defined for any odd integer $r\geq 3$.
Note that $\tau_{r}$ is slight modification of the original construction by
Reshetikhin and Turaev [ResT91], differing by a factor of absolute value $1$
(depending on the first Betti number), and $\tau^{\prime}_{r}$ is introduced
by Kirby and Melvin [KirM91, Section 8]. The invariant $\tau_{r}$ corresponds
to the $4r$–th primitive complex root of unity $q^{1/4}=e^{\sqrt{-1}\cdot
2\pi/4r}$ by construction, whereas $\tau^{\prime}_{r}$ is obtained by
modifying the defining expression of $\tau_{r}$ when $r$ is odd. Upon suitable
interpretation, $\tau^{\prime}_{r}$ appears to correspond to the $2r$–th
primitive complex root of unity $q^{1/4}=e^{\sqrt{-1}\cdot\pi(r-1)/4r}$,
(compare [BlaHMV92, Theorem B]). Both $\tau_{r}$ and $\tau^{\prime}_{r}$ take
values in ${\mathbb{C}}$.

The following properties characterize the normalization of $\tau_{r}$ and
$\tau^{\prime}_{r}$.

######  Lemma 3.5.

Let $M,N$ be any connected oriented closed $3$–manifolds.

  1. (1)

$\tau_{r}(M)=\overline{\tau_{r}(-M)}$.

  2. (2)

$\tau_{r}(M\\#N)=\tau_{r}(M)\cdot\tau_{r}(N)$.

  3. (3)

$\tau_{r}(S^{2}\times S^{1})=\sqrt{r/2}\,/\sin(\pi/r)$. Hence,
$\tau_{r}(S^{3})=1$.

If $r$ is odd, the same statements hold with $\tau^{\prime}_{r}$ in place of
$\tau_{r}$, and $\sqrt{r/4}$ in place with $\sqrt{r/2}$.

See [KirM91, Section 1 and (5.11)] and [BlaHMV92, Proposition 6.10].

######  Lemma 3.6.

Let $M$ be any connected oriented closed $3$–manifold. If $r$ is odd,

$\tau_{r}(M)=\begin{cases}\tau_{3}(M)\cdot\tau^{\prime}_{r}(M)&r\equiv 3\bmod
4\\\ \overline{\tau_{3}(M)}\cdot\tau^{\prime}_{r}(M)&r\equiv 1\bmod
4\end{cases}$

See [KirM91, Corollary 8.9 and Theorem 8.10]. See also [KirM91, Theorem 6.11]
for characterization of $\tau_{3}(M)$ in terms of classical topological
invariants.

The Turaev–Viro invariants $\mathrm{TV}_{r}$ and $\mathrm{TV}^{\prime}_{r}$
are essentially the absolute value squares of Witten–Reshetikhin–Turaev
invariants, or precisely as follows.

######  Lemma 3.7.

Let $M$ be a connected closed $3$–manifold. Let $r\geq 3$ be any integer.

  1. (1)

If $M$ is oriented,

$\frac{\mathrm{TV}_{r,1}(M)}{\mathrm{TV}_{r,1}(S^{3})}=|\tau_{r}(M)|^{2},$

and if $r$ is odd,

$\frac{\mathrm{TV}^{\prime}_{r,r-1}(M)}{\mathrm{TV}^{\prime}_{r,r-1}(S^{3})}=|\tau^{\prime}_{r}(M)|^{2}.$

  2. (2)

If $M$ is non-orientable,

$\frac{\mathrm{TV}_{r,1}(M)}{\mathrm{TV}_{r,1}(S^{3})}=\tau_{r}(W),$

and if $r$ is odd,

$\frac{\mathrm{TV}^{\prime}_{r,r-1}(M)}{\mathrm{TV}^{\prime}_{r,r-1}(S^{3})}=\tau^{\prime}_{r}(W),$

where $W$ denotes the orientable connected double cover of $M$. Note that $W$
can be canonically constructed, with a canonical orientation and an
orientation-reserving deck transformation.

  3. (3)

$\mathrm{TV}_{r,1}(S^{3})=\frac{2}{r}\cdot\sin^{2}(\pi/r),$

and if $r$ is odd,

$\mathrm{TV}^{\prime}_{r,r-1}(S^{3})=\frac{4}{r}\cdot\sin^{2}(\pi/r).$

See [Tur94, Theorems 4.1.1 and 4.4.1] for systematic proofs of the formulas
regarding $\mathrm{TV}_{r}$; see also [Rob95] for an elegant proof for the
oriented case regarding $\mathrm{TV}_{r}$. The formulas regarding
$\mathrm{TV}^{\prime}_{r}$ can be easily derived from the $\mathrm{TV}_{r}$
formulas using Lemmas 3.6 and 3.4. The values for $\mathrm{TV}_{r,1}(S^{3})$
and $\mathrm{TV}^{\prime}_{r,r-1}(S^{3})$ can be obtained immediately by
taking $M=S^{2}\times S^{1}$ and applying Lemmas 3.3 and 3.6.

###  3.4. Perspective of TQFT

The Turaev–Viro invariants fit into the general framework of
$(2+1)$–dimensional topological quantum field theories (TQFT) [Ati89]. We
describe below focusing on the $\mathrm{SU}(2)$ Turaev–Viro invariants. The
discussion regarding $\mathrm{SO}(3)$ Turaev–Viro invariants is completely
similar.

Turaev and Viro construct a functor $\mathcal{Z}^{\mathrm{TV}}$ from the
$(2+1)$–dimensional cobordism category to the category of Hermitian vector
spaces (and linear homomorphisms) over the abstract cyclotomic field
$\mathbb{K}={\mathbb{Q}}(q^{1/2})$, with respect to the involution $*$ on
$\mathbb{K}$ as provided by the Galois transformation $q^{1/2}\mapsto
q^{-1/2}$. To any oriented closed surface $S$, there is a finite-dimensional
vector space $\mathcal{Z}^{\mathrm{TV}}(S)$, equipped with a nondegenerate
Hermitian pairing

$\langle\\_,\\_\rangle\colon\mathcal{Z}^{\mathrm{TV}}(S)\times\mathcal{Z}^{\mathrm{TV}}(S)\to\mathbb{K}.$

(Being _Hermitian_ means $\mathbb{K}$–sesquilinear and $*$–symmetric.) To any
cobordism $M$ from $S_{0}$ to $S_{1}$ (that is, an oriented compact
$3$-manifold $M$ with bipartite boundary $\partial
M=\partial_{-}M\sqcup\partial_{+}M=(-S_{0})\sqcup S_{1}$, up to boundary-
fixing homomeorphisms), there is a linear homomorphism

$\mathcal{Z}^{\mathrm{TV}}(M)\colon\mathcal{Z}^{\mathrm{TV}}(S_{0})\to\mathcal{Z}^{\mathrm{TV}}(S_{1}).$

The assignment $\mathcal{Z}^{\mathrm{TV}}$ is functorial, and satisfies
Atiyah’s Hermitian TQFT axioms [Ati89, Section 2]:
$\mathcal{Z}^{\mathrm{TV}}(-S)=\mathcal{Z}^{\mathrm{TV}}(S)^{*}$;
$\mathcal{Z}^{\mathrm{TV}}(-M)=\mathcal{Z}^{\mathrm{TV}}(M)^{*}$;
$\mathcal{Z}^{\mathrm{TV}}(S^{\prime}\sqcup
S^{\prime\prime})=\mathcal{Z}^{\mathrm{TV}}(S^{\prime})\otimes_{\mathbb{K}}\mathcal{Z}^{\mathrm{TV}}(S^{\prime\prime})$;
and $\mathcal{Z}^{\mathrm{TV}}(\emptyset)=\mathbb{K}$, ($\emptyset$ denoting
the empty surface, having a unique orientation by convention).

Turaev and Viro show that $\mathrm{TV}$ comes from a functor
$\mathcal{Z}^{\mathrm{TV}}$ as above, in the sense that the identity

$\mathrm{TV}\left(M;q^{1/2}\right)=\mathcal{Z}^{\mathrm{TV}}(M)$

holds for any closed $3$-manifold $M$. Here, $M$ is treated as a cobordism
between empty surfaces, and
$\mathcal{Z}^{\mathrm{TV}}(\emptyset,\emptyset)=\mathrm{End}_{\mathbb{K}}(\mathbb{K})$
is identified as $\mathbb{K}$. See [TurV92, Section 2.3].

Being a TQFT functor, $\mathcal{Z}^{\mathrm{TV}}$ naturally induces a
$\mathbb{K}$–linear representation

$\mathrm{Mod}(S)\to\mathrm{GL}\left(\mathcal{Z}^{\mathrm{TV}}(S)\right)$

of the mapping class group $\mathrm{Mod}(S)$ for any oriented closed surface
$S$ [TurV92, Section 2.4]. Specializing $q^{1/2}$ to complex roots of unity as
in Notation 3.2, we obtain a complex linear representation, denoted as

(3.3)
$\rho^{\mathrm{TV}}_{r,s}\colon\mathrm{Mod}(S)\to\mathrm{GL}\left(\mathcal{Z}^{\mathrm{TV}}_{r,s}(S)\right)$

for each integer $r\geq 3$ and any integer $s$ coprime to $r$. These
representations preserve the specialized Hermitian pairings
$\langle\\_,\\_\rangle_{r,s}$, whose signatures depend on both $r$ and $s$,
but are not necessarily (Hilbert) unitary. We refer to these as the
$\mathrm{SU}(2)$ _Turaev–Viro TQFT representations_ (at level $r-2$) of
$\mathrm{Mod}(S)$.

The following formula is useful implication of TQFT axioms [Ati89, Section 2].

######  Lemma 3.8.

Let $r\geq 3$ be any integer and $s$ be any integer coprime to $r$. For any
oriented closed surface $\Sigma$, and any mapping class
$[f]\in\mathrm{Mod}(S)$,

$\mathrm{TV}_{r,s}(M_{f})=\mathrm{tr}_{\mathbb{C}}\left(\rho^{\mathrm{TV}}_{r,s}([f])\right),$

where $M_{f}$ denotes the mapping torus of $f$.

If $S$ is connected, the complex dimension of the representation
$\rho^{\mathrm{TV}}_{r,s}$ depends only on $r$ and genus $g$ of $S$. This fact
can be derived from Lemmas 3.8 and 3.4 by considering the mapping torus of
$f=\mathrm{id}_{S}$. Verlinde type formulas for these dimensions can be
derived from Lemma 3.7 and known formulas about Witten–Reshetikhin–Turaev
invariants [BlaHMV95, Corollary 1.16]. However, Witten–Reshetikhin–Turaev
invariants only come from generalized TQFTs, which require extra structures
for resolving framing anomaly [BlaHMV95]. That is why they only naturally lead
to projective linear representations of surface mapping class groups.

##  4\. Calculation

This section is devoted to the proof of Theorem 1.2.

To restate our task, we consider a closed Seifert fiber space $M$ with
orientable orbifold base and orientable fibration, and with symbol
$(g;(a_{1},b_{1}),\cdots,(a_{n},b_{n}))$, such that

$a_{1}=a_{2}=\cdots=a_{n}=a,$

and

$b_{1}+b_{2}+\cdots+b_{n}=0.$

Moreover, we suppose $a\geq 3$ and $a>n\geq 0$. We compute $\mathrm{TV}_{r}$
and $\mathrm{TV}^{\prime}_{r}$ of $M$ for $r=a$, and once they vanish, we show
that they must also vanish for any $r$ divisible by $a$.

We invoke the following explicit formula for computing the
Witten–Reshetikhin–Turaev invariant $\tau_{r}$ of Seifert fiber spaces. Recall
$\tau_{r}(S^{1}\times S^{2})=\sqrt{r/2}/\sin(\pi/r)$ (Lemma 3.5).

######  Lemma 4.1.

Let $M$ be a closed Seifert fiber space with orientable orbifold base and
orientable fibration, and with symbol
$(g;(a_{1},b_{1}),\cdots,(a_{n},b_{n}))$, where $g,n\geq 0$ are integers, and
where $a_{j}\geq 0$ and $b_{j}$ are coprime pairs of integers for
$j=1,\cdots,n$. Orient $M$ by orienting the base and the fibers, such that the
rational Euler number of the Seifert fibration is

$E=-\sum_{j}b_{j}/a_{j}.$

Then,

$\frac{\tau_{r}(M)}{\tau_{r}(S^{2}\times S^{1})}=\frac{r^{g-1}\cdot U_{r}\cdot
Z_{r}}{2^{n+g-1}\sqrt{\prod_{j}a_{j}}}$

where

$Z_{r}=\sum_{(\gamma,\boldsymbol{\mu},\boldsymbol{m})}\left\\{\frac{e^{\sqrt{-1}\cdot\frac{\pi\gamma^{2}E}{2r}}}{{\sin}^{n+2g-2}(\pi\gamma/r)}\cdot\prod_{j}\mu_{j}e^{\sqrt{-1}\cdot\left(\frac{-\pi(2rm_{j}+\mu_{j})\gamma}{a_{j}r}+\frac{-2\pi(rm_{j}^{2}+\mu_{j}m_{j})b^{*}_{j}}{a_{j}}\right)}\right\\}$

and

$U_{r}=(-1)^{n}\cdot
e^{\sqrt{-1}\cdot\left(\frac{3\pi}{2r}-\frac{3\pi}{4}\right)\cdot\mathrm{sgn}(E)}\cdot
e^{\sqrt{-1}\cdot\frac{\pi\left(E+12\cdot\sum_{j}\mathrm{s}(b_{j},a_{j})\right)}{2r}}$

Here, any $j$ ranges over $\\{1,\cdots,n\\}$, and
$(\gamma,\boldsymbol{\mu},\boldsymbol{m})=(\gamma,(m_{1},\cdots,m_{n}),(\mu_{1},\cdots,\mu_{n}))$
ranges over $\\{1,2,\cdots,r-1\\}\times\\{\pm
1\\}^{n}\times{\mathbb{Z}}/a_{1}{\mathbb{Z}}\times\cdots\times{\mathbb{Z}}/a_{n}{\mathbb{Z}}$.
The notation $b_{j}^{*}$ denotes any congruence inverse of $b_{j}$ modulo
$a_{j}$, namely, $b_{j}b_{j}^{*}\equiv 1\bmod a_{j}$; $\mathrm{sgn}(E)$
denotes the sign of $E$, with value $\pm 1$ or $0$; $\mathrm{s}(b_{j},a_{j})$
denotes the Dedekind sum
$(4a_{j})^{-1}\cdot\sum_{l\,\in\\{1,2,\cdots,a_{j}-1\\}}\cot(\pi
l/a_{j})\cot(\pi lb_{j}/a_{j})$.

See [Han01, Theorem 8.4] for this formula; see also Remark 4.2 below for
clarification about different notations and normalizations. Hansen actually
obtains the most general formula that applies to any orientable closed Seifert
fiber space, including those with non-orientable orbifold base. We have only
stated here the case with orientable orbifold base. An equivalent formula for
this case is formerly obtained by Rozansky [Roz96].

######  Remark 4.2.

Our notation $\tau_{r}$ agrees with Kirby–Melvin [KirM91], differing from
Hansen’s notation by our factor $1/\tau_{r}(S^{2}\times S^{1})$. In [Han01,
Section 8], Hansen’s $\tau_{r}(M)$ is defined as
$\tau_{(\mathcal{V}_{t},\mathcal{D})}(M)$ therein; as pointed out in [Han01,
Appendix A], Kirby–Melvin’s $\tau_{r}(M)$ is equal to
$\tau_{A}(M)=\mathcal{D}\cdot\tau_{(\mathcal{V}_{t},\mathcal{D})}(M)$, where
$\mathcal{D}$ is specified as $\sqrt{r/2}/\sin(\pi/r)$ in the equation (38)
therein. More directly, one may check by evaluating the formula in [Han01,
Theorem 8.4] for $S^{3}$ (setting $g=0$, $b=1$, and $n=0$) and $S^{2}\times
S^{1}$ (setting $g=0$, $b=0$ and $n=0$) in the simplest case $r=3$.

######  Lemma 4.3.

Under the assumptions of Theorem 1.2, and assuming $r$ divisible by $a$, the
term $Z_{r}=Z_{r}(M)$ as in Lemma 4.1 becomes

$Z_{r}=\sum_{(\gamma,\boldsymbol{\mu})}\left\\{\frac{e^{\sqrt{-1}\cdot\frac{-\pi\gamma\sum_{j}\mu_{j}}{ar}}\cdot\prod_{j}\mu_{j}}{{\sin}^{n+2g-2}(\pi\gamma/r)}\cdot\prod_{j}\sum_{m_{j}}e^{\sqrt{-1}\cdot\frac{-2\pi(\gamma+b_{j}^{*}\mu_{j})m_{j}}{a}}\right\\}$

where $(\gamma,\boldsymbol{\mu},\boldsymbol{m})$ ranges over
$\\{1,\cdots,r-1\\}\times\\{\pm
1\\}^{n}\times({\mathbb{Z}}/a{\mathbb{Z}})^{n}$.

###### Proof.

In the expression of $Z_{r}$ in Lemma 4.1, if any $a_{j}$ divides $r$, we can
ignore the term $rm_{j}^{2}$ in the exponent of the $j$-th factor in the
product; if $E=0$, we can ignore the factor that involves $\gamma^{2}$ on the
exponent. Therefore, under these conditions, the expression of $Z_{r}$ can be
rearranged into

$\displaystyle Z_{r}(M)$ $\displaystyle=$
$\displaystyle\sum_{(\gamma,\boldsymbol{\mu})}\left\\{\frac{\prod_{j}\mu_{j}e^{\sqrt{-1}\cdot\frac{-\pi\gamma\mu_{j}}{a_{j}r}}}{{\sin}^{n+2g-2}(\pi\gamma/r)}\cdot\sum_{\boldsymbol{m}}e^{\sqrt{-1}\cdot\sum_{j}\left(\frac{-2\pi\gamma
m_{j}}{a_{j}}+\frac{-2\pi b^{*}_{j}\mu_{j}m_{j}}{a_{j}}\right)}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{(\gamma,\boldsymbol{\mu})}\left\\{\frac{e^{\sqrt{-1}\cdot\sum_{j}\frac{-\pi\gamma\mu_{j}}{a_{j}r}}\cdot\prod_{j}\mu_{j}}{{\sin}^{n+2g-2}(\pi\gamma/r)}\cdot\sum_{\boldsymbol{m}}e^{\sqrt{-1}\cdot\sum_{j}\frac{-2\pi(\gamma+b_{j}^{*}\mu_{j})m_{j}}{a_{j}}}\right\\}$
$\displaystyle=$
$\displaystyle\sum_{(\gamma,\boldsymbol{\mu})}\left\\{\frac{e^{\sqrt{-1}\cdot\sum_{j}\frac{-\pi\gamma\mu_{j}}{a_{j}r}}\cdot\prod_{j}\mu_{j}}{{\sin}^{n+2g-2}(\pi\gamma/r)}\cdot\prod_{j}\sum_{m_{j}}e^{\sqrt{-1}\cdot\frac{-2\pi(\gamma+b_{j}^{*}\mu_{j})m_{j}}{a_{j}}}\right\\}$

where $(\gamma,\boldsymbol{\mu})$ ranges in $\\{1,2,\cdots,r-1\\}\times\\{\pm
1\\}^{n}$, and $j$ in $\\{1,\cdots,n\\}$, and $m_{j}$ in
${\mathbb{Z}}/a_{j}{\mathbb{Z}}$. In particular, the simplification applies as
we assume $a_{1}=a_{2}=\cdots=a_{n}=a$, and $b_{1}+b_{2}+\cdots+b_{n}=0$, and
$r$ divisible by $a$. ∎

######  Lemma 4.4.

Under the assumptions of Theorem 1.2, and assuming $r$ divisible by $a$, if
there does not exist any $\boldsymbol{\mu}\in\\{\pm 1\\}^{n}$ that satisfies
the congruence equations

$b_{1}^{*}\mu_{1}\equiv b_{2}^{*}\mu_{2}\equiv\cdots\equiv
b_{n}^{*}\mu_{n}\mod a,$

then

$Z_{r}=0$

where $Z_{r}$ is the term as in Lemma 4.1.

###### Proof.

For any fixed $(\gamma,\boldsymbol{\mu})$, we observe

$\sum_{m_{j}\,\in{\mathbb{Z}}/a{\mathbb{Z}}}e^{\sqrt{-1}\cdot\frac{-2\pi(\gamma+b_{j}^{*}\mu_{j})m_{j}}{a}}=\begin{cases}a&\mbox{if
}\gamma+b_{j}^{*}\mu_{j}\equiv 0\bmod a\\\ 0&\mbox{otherwise}\end{cases}$

Therefore, in the simplified expression of $Z_{r}$ as in Lemma 4.3, the
summand corresponding to $(\gamma,\boldsymbol{\mu})$ is nonzero if and only if
$\gamma+b^{*}_{j}\mu_{j}\equiv 0\bmod a$ holds for all $j\in\\{1,\cdots,n\\}$.
For the sum $Z_{r}$ to be nonzero, there has to be some
$\gamma\in\\{1,2,\cdots,r-1\\}$ that satisfies the above condition for some
$\boldsymbol{\mu}\in\\{\pm 1\\}^{n}$, then there has to be some
$\boldsymbol{\mu}$ that satisfies $b_{1}^{*}\mu_{1}\equiv
b_{2}^{*}\mu_{2}\equiv\cdots\equiv b_{n}^{*}\mu_{n}\mod a$. ∎

######  Lemma 4.5.

Under the assumptions of Theorem 1.2, if there exists some integer $b^{*}$
coprime to $a$, and some $\boldsymbol{\nu}\in\\{\pm 1\\}^{n}$, such that
$b^{*}\equiv b^{*}_{j}\nu_{j}\bmod a$ holds for all $j\in\\{1,\cdots,n\\}$,
then

$Z_{a}=\frac{2\cdot a^{n}\cdot\prod_{j}\nu_{j}}{{\sin}^{n+2g-2}(\pi b^{*}/a)}$

where $Z_{a}$ is the term as in Lemma 4.1 with $r=a$.

###### Proof.

Possibly after permuting $\\{1,\cdots,n\\}$, we may assume $\nu_{j}=1$ for
$j=1,\cdots,m$ and $\nu_{j}=-1$ for $j=m+1,\cdots,n$. We may also assume
$b^{*}\in\\{1,2,\cdots,a-1\\}$ without changing its residue class modulo $a$.

For $r=a\geq 3$, there are only two nonzero summands in $Z_{r}$, and their
corresponding to $(\gamma,\boldsymbol{\mu})$ are

$(\gamma,\boldsymbol{\mu})=(a-b^{*},(\underbrace{1,\cdots,1}_{m},\underbrace{-1,\cdots,-1}_{n-m}))$

and

$(\gamma,\boldsymbol{\mu})=(b^{*},(\underbrace{-1,\cdots,-1}_{m},\underbrace{1,\cdots,1}_{n-m})).$

By our assumption $\sum_{j}b_{j}=0$ in Theorem 1.2, $(2m-n)b\equiv
mb+(n-m)(-b)\equiv\sum_{j}b_{j}=0\bmod a$, so $n-2m$ is divisible by $a$. By
our assumption $a>n\geq 0$ in Theorem 1.2, we must have $|n-2m|<a$, and hence
$n-2m=0$. So, we observe

$(-1)^{n-m}=(-1)^{m},$

which is useful below.

We compute

$\displaystyle Z_{a}(M)$ $\displaystyle=$ $\displaystyle\frac{(-1)^{n-m}\cdot
e^{\sqrt{-1}\cdot\frac{-\pi(2m-n)(a-b^{*})}{a^{2}}}}{{\sin}^{n+2g-2}(\pi(a-b^{*})/a)}\cdot
a^{n}+\frac{(-1)^{m}\cdot
e^{\sqrt{-1}\cdot\frac{-\pi(n-2m)b^{*}}{a^{2}}}}{{\sin}^{n+2g-2}(\pi
b^{*}/a)}\cdot a^{n}$ $\displaystyle=$ $\displaystyle\frac{2\cdot
a^{n}\cdot(-1)^{n-m}}{{\sin}^{n+2g-2}(\pi b^{*}/a)}$ $\displaystyle=$
$\displaystyle\frac{2\cdot a^{n}\cdot\prod_{j}\nu_{j}}{{\sin}^{n+2g-2}(\pi
b^{*}/a)},$

as desired. ∎

######  Lemma 4.6.

Let $M$ be a closed Seifert fiber space with orientable orbifold base and
orientable fibration, and with symbol
$(g;(a_{1},b_{1}),\cdots,(a_{n},b_{n}))$, where $g,n\geq 0$ are integers, and
where $a_{j}\geq 1$ and $b_{j}$ are coprime pairs of integers, for
$j=1,\cdots,n$, and satisfy $b_{1}/a_{1}+\cdots+b_{n}/a_{n}=0$. If
$a_{1},\cdots,a_{n}$ are all odd,

$\mathrm{TV}_{3,1}(M)=\mathrm{TV}_{3,2}(M)=2^{2g}.$

###### Proof.

Denote

$\mathrm{lcm}(a_{1},\cdots,a_{n})=d.$

If $a_{1},\cdots,a_{n}$ are all odd, $d$ is also odd.

The fundamental group of $M$ has a presentation with generators

$x_{1},y_{1},\cdots,x_{g},y_{g},s_{1},\cdots,s_{n},f$

and relations

$\begin{cases}s_{1}\cdots s_{n}=[x_{1},y_{1}]\cdots[x_{g},y_{g}]\\\
x_{i}f=fx_{i}&i=1,\cdots,g\\\ y_{i}f=fy_{i}&i=1,\cdots,g\\\
s_{j}f=fs_{j}&j=1,\cdots,n\\\ s_{j}^{a_{j}}f^{b_{j}}=1&j=1,\cdots,n\\\
\end{cases}$

With ${\mathbb{Z}}/2{\mathbb{Z}}$ coefficients, we can eliminate any $[s_{j}]$
using the relation $a_{j}[s_{j}]+b_{j}[f]=0$, since $a_{j}$ is odd. Then the
first relation is equivalent to $-(b_{1}d/a_{1}+\cdots+b_{n}d/a_{n})[f]=0$
over ${\mathbb{Z}}/2{\mathbb{Z}}$, having no effect by our assumption
$b_{1}/a_{1}+\cdots+b_{n}/a_{n}=0$. It follows that
$H_{1}(M;{\mathbb{Z}}/2{\mathbb{Z}})$ is freely generated by
$[x_{1}],[y_{1}],\cdots,[x_{g}],[y_{g}],[f]$ over
${\mathbb{Z}}/2{\mathbb{Z}}$. Then the ${\mathbb{Z}}/2{\mathbb{Z}}$ Betti
numbers of $M$ are $\beta_{0}=\beta_{3}=1$ and $\beta_{2}=\beta_{1}=2g+1$, by
the Poincaré duality with ${\mathbb{Z}}/2{\mathbb{Z}}$ coefficients.
Therefore, we obtain

$\mathrm{TV}_{3,2}(M)=2^{2g},$

by Lemma 3.4.

Since $M$ is homeomorphic to the mapping torus of a periodic surface
automorphism of order $d$, there is a cyclic cover $\tilde{M}\to M$ of degree
$d$, and $\tilde{M}$ is a product of a closed orientable surface with a
circle. Since $d$ is odd, the induced homomorphism
$H^{*}(M;{\mathbb{Z}}/2{\mathbb{Z}})\to
H^{*}(\tilde{M};{\mathbb{Z}}/2{\mathbb{Z}})$ is injective, (because of the
Poincaré duality pairing with ${\mathbb{Z}}/2{\mathbb{Z}}$ coefficients and
the isomorphism on the top dimension). However,
$H^{1}(\tilde{M};{\mathbb{Z}}/2{\mathbb{Z}})$ contains no element whose cube
is nontrivial, (by the Künneth theorem which determines the cohomology ring of
$\tilde{M}$ over ${\mathbb{Z}}/2{\mathbb{Z}}$). It follows that $t^{3}=0\in
H^{3}(M;{\mathbb{Z}}/2{\mathbb{Z}})$ holds for any $t\in
H^{1}(M;{\mathbb{Z}}/2{\mathbb{Z}})$. Moreover, the first Stiefel–Whitney
class $w_{1}$ of $M$ vanishes, as $M$ is orientable. Therefore, we obtain

$\mathrm{TV}_{3,1}(M)=2^{2g},$

by Lemma 3.4. ∎

Under the assumptions of Theorem 1.2, we compute the Turaev–Viro invariants in
Theorem 1.2 as follows.

Suppose that $b^{*}b_{j}\equiv\nu_{j}\bmod a$ holds for some integer $b^{*}$
coprime to $a$ and some $\nu_{j}\in\\{\pm 1\\}$, and for all
$j\in\\{1,\cdots,n\\}$. Then,

$\displaystyle\mathrm{TV}_{a,1}(M)$ $\displaystyle=$
$\displaystyle\mathrm{TV}_{a,1}(S^{3})\cdot|\tau_{a}(M)|^{2}$ $\displaystyle=$
$\displaystyle|\tau_{a}(M)/\tau_{a}(S^{2}\times S^{1})|^{2}$ $\displaystyle=$
$\displaystyle\left|\frac{a^{g-1}}{2^{n+g-1}a^{n/2}}\cdot\frac{2\cdot
a^{n}}{{\sin}^{n+2g-2}(\pi b^{*}/a)}\right|^{2}$ $\displaystyle=$
$\displaystyle\frac{a^{n+2g-2}}{2^{2n+2g-4}}\cdot\frac{1}{\sin^{2n+4g-4}(\pi
b^{*}/a)}$

by Lemmas 4.1, 4.5, and 3.7.

When $a$ is even, we apply Galois conjugacy (transforming
$e^{\sqrt{-1}\cdot\pi/a}\mapsto e^{\sqrt{-1}\cdot\pi s/a}$) for any $s$
coprime to $a$, obtaining

$\mathrm{TV}_{a,s}(M)=\frac{a^{n+2g-2}}{2^{2n+2g-4}}\cdot\frac{1}{\sin^{2n+4g-4}(\pi
b^{*}s/a)},$

by Lemma 3.4.

When $a$ is odd, we apply Lemmas 3.4 and 4.6, obtaining

$\mathrm{TV}_{a,a-1}(M)=\mathrm{TV}_{3,2}(M)\cdot\mathrm{TV}^{\prime}_{a,a-1}(M)=\frac{\mathrm{TV}_{3,2}(M)}{\mathrm{TV}_{3,1}(M)}\cdot\mathrm{TV}_{a,1}(M)=\mathrm{TV}_{a,1}(M)$

and

$\mathrm{TV}^{\prime}_{a,a-1}(M)=\frac{\mathrm{TV}_{a,1}(M)}{\mathrm{TV}_{3,1}(M)}=\frac{1}{2^{2g}}\cdot\mathrm{TV}_{a,1}(M).$

Again, we apply Galois conjugacy (transforming $e^{\sqrt{-1}\cdot\pi/a}\mapsto
e^{\sqrt{-1}\cdot\pi s/a}$ or $e^{\sqrt{-1}\cdot\pi(a-1)/a}\mapsto
e^{\sqrt{-1}\cdot\pi s/a}$ depending on $s$ odd or even), obtaining for any
$s$ coprime to $a$

$\mathrm{TV}_{a,s}(M)=\frac{a^{n+2g-2}}{2^{2n+2g-4}}\cdot\frac{1}{\sin^{2n+4g-4}(\pi
b^{*}s/a)},$

and if $s$ is even,

$\mathrm{TV}^{\prime}_{a,s}(M)=\frac{a^{n+2g-2}}{2^{2n+4g-4}}\cdot\frac{1}{\sin^{2n+4g-4}(\pi
b^{*}s/a)},$

by Lemma 3.4. This completes the computation of the nonvanishing values in
Theorem 1.2.

Suppose the otherwise case. Then, for all $r\geq 3$ divisible by $a$, we
obtain

$\mathrm{TV}_{r,1}(M)=0,$

by Lemmas 4.1, 4.4, and 3.7. Similarly, we derive
$\mathrm{TV}_{r,r-1}(M)=\mathrm{TV}^{\prime}_{r,r-1}(M)=0$ in this case, using
Lemmas 3.4 and 4.6. Finally, by Galois conjugacy (Lemma 3.4), we see that
$\mathrm{TV}_{r,s}(M)=0$ and $\mathrm{TV}^{\prime}_{r,s}(M)=0$ for any
applicable $s$.

This completes the proof of Theorem 1.2.

##  5\. Examples

In this section, we prove Theorem 1.1 by exhibiting nontrivial Hempel pairs
that can or cannot be distinguished by Turaev–Viro invariants. See Example 5.2
for our distinguishable ones, and Example 5.3 for our indistinguishable ones.

We need the following lemma for verifying our examples. A rationality
statement would be good enough for our application, but the integrality here
is also well-known to experts.

######  Lemma 5.1.

Let $S$ be a connected orientable closed surface of genus $g\geq 0$, and
$[f]\in\mathrm{Mod}(S)$ be a periodic mapping class of order $d\geq 1$. Then,
for any integer $r\geq 3$ coprime to $d$, and integer $s$ coprime to $r$,

$\mathrm{TV}_{r,s}(M_{f})\in{\mathbb{Z}},$

and if $r$ is odd and $s$ is even,

$\mathrm{TV}^{\prime}_{r,s}(M_{f})\in{\mathbb{Z}},$

where $M_{f}$ denotes the mapping torus of $f$.

###### Proof.

By Lemma 3.8, $\mathrm{TV}_{r,s}(M_{f})\in{\mathbb{C}}$ is equal to the trace
of the Turaev–Viro TQFT representation
$\rho^{\mathrm{TV}}_{r,s}([f])\in\mathrm{GL}(\mathcal{Z}^{\mathrm{TV}}_{r,s}(S))$.
If $[f]\in\mathrm{Mod}(S)$ is periodic of order $d$, the eigenvalues of
$\rho^{\mathrm{TV}}_{r,s}([f])$ are all complex roots of unity of order
divisible by $d$. In particular, $\mathrm{TV}_{r,s}(M_{f})$ is an algebraic
integer.

On the other hand, entries of $\rho^{\mathrm{TV}}_{r,s}([f])$ lie in the
cyclotomic subfield ${\mathbb{Q}}(e^{\sqrt{-1}\cdot\pi s/r})$ of
${\mathbb{C}}$. For any roots of unity $\zeta_{m},\zeta_{n}\in{\mathbb{C}}$ of
coprime orders $m,n\geq 1$, respectively, it is an elementary fact that
${\mathbb{Q}}(\zeta_{m})\cap{\mathbb{Q}}(\zeta_{n})$ equals ${\mathbb{Q}}$
[Was97, Chapter 2, Proposition 2.4]. Applying with $m|d$ and $n=r$ (taking
$\zeta_{n}=e^{\sqrt{-1}\cdot\pi s/r}$, or if $r$ is odd and $s$ is even,
$\zeta_{n}=e^{\sqrt{-1}\cdot\pi(r-s)/r}$), we obtain
$\mathrm{TV}_{r,s}(M_{f})\in{\mathbb{Q}}$. Together with the algebraic
integrality, we obtain $\mathrm{TV}_{r,s}(M_{f})\in{\mathbb{Z}}$.

With Lemmas 3.4 and 4.6, one may derive
$\mathrm{TV}^{\prime}_{r,s}(M_{f})\in{\mathbb{Q}}$ from
$\mathrm{TV}_{r,s}(M_{f})\in{\mathbb{Z}}$. To obtain
$\mathrm{TV}^{\prime}_{r,s}(M_{f})\in{\mathbb{Z}}$, it is possible to appeal
to a similar lemma as Lemma 3.8 with a TQFT functor associated to
$\mathrm{TV}^{\prime}$. In fact, this case has been established by Detcherry
and Kalfagianni. We refer to [DetK19, Corollary 6.1] for their proof of this
case. ∎

######  Example 5.2 (Distinguishable pairs).

Let $g\geq 0$ and $d=5$ or $d\geq 7$ be integers, and $k$ be an integer
coprime to $d$. Let $M_{A}$ and $M_{B}$ be closed Seifert fiber spaces with
orientable orbifold base and orientable Seifert fibration. We assign their
symbols as

$\displaystyle M_{A}$ $\displaystyle\colon$
$\displaystyle(g;(d,1),(d,1),(d,-1),(d,-1))$ $\displaystyle M_{B}$
$\displaystyle\colon$
$\displaystyle(g;(d,k^{*}),(d,k^{*}),(d,-k^{*}),(d,-k^{*}))$

where $k^{*}$ is an integer satisfing $k^{*}k\equiv 1\bmod p$. The
$3$–manifold $M_{A}$ is homeomorphic to the mapping torus of some periodic
mapping class $[f_{A}]\in\mathrm{Mod}(S)$ of order $d$, where $S$ is a
connected orientable closed surface of genus $dg+d-2$ (Lemma 2.1). The
$3$–manifold $M_{B}$ is homeomorphic to the mapping torus of the iterate
mapping class $[f_{B}]=[f_{A}^{k}]$ (Lemma 2.2).

By Theorem 1.2, we compute

$\displaystyle\mathrm{TV}_{d,1}(M_{A})$ $\displaystyle=$
$\displaystyle\frac{d^{2g+2}}{2^{2g+4}}\cdot\frac{1}{{\sin}^{4g+4}(\pi/d)}$
$\displaystyle\mathrm{TV}_{d,1}(M_{B})$ $\displaystyle=$
$\displaystyle\frac{d^{2g+2}}{2^{2g+4}}\cdot\frac{1}{{\sin}^{4g+4}(\pi k/d)}$

The values are equal if and only if $k\equiv\pm 1\bmod d$, namely,
$[f_{B}]=[f_{A}]$ or $[f_{B}]=[f_{A}^{-1}]$. For $k$ other than these values,
one may also check that $\mathrm{TV}_{d,s}(M_{A})\neq\mathrm{TV}_{d,s}(M_{B})$
for any integer $s$ coprime to $d$, and if $d$ is odd,
$\mathrm{TV}^{\prime}_{d,s}(M_{A})\neq\mathrm{TV}^{\prime}_{d,s}(M_{B})$ for
any even integer $s$ coprime to $d$. Under our assumption on $d$, such $k$
does exist, so $M_{A}$ and $M_{B}$ form a nontrivial Hempel pair.

######  Example 5.3 (Indistinguishable pairs).

Let $g\geq 0$ be any integers, and $p\geq 5$ be a prime integer, and $k$ be an
integer coprime to $p$. Let $M_{A}$ and $M_{B}$ be closed Seifert fiber spaces
with orientable orbifold base and orientable Seifert fibration. We assign
their symbols as

$\displaystyle M_{A}$ $\displaystyle\colon$
$\displaystyle(g;(p,1),(p,1),(p,-2))$ $\displaystyle M_{B}$
$\displaystyle\colon$ $\displaystyle(g;(p,k^{*}),(p,k^{*}),(p,-2k^{*}))$

where $k^{*}$ is an integer satisfing $k^{*}k\equiv 1\bmod p$. Obtain the
connected orientable closed surface $S$ of genus $pg+(p-1)/2$, and the
periodic mapping classes $[f_{B}]=[f_{A}^{k}]$ of order $p$, similarly as in
the previous example (Lemmas 2.1 and 2.2).

Again, $[f_{A}]$ and $[f_{B}]$ form a nontrivial Hempel pair exactly when
$k\not\equiv\pm 1\bmod p$, existing under the assumption on $p$.

If $r\geq 3$ is divisible by $p$, applying Theorem 1.2 and Lemma 5.1, we see
that $\mathrm{TV}_{r,s}(M_{A})=\mathrm{TV}_{r,s}(M_{B})=0$ holds for any $s$
coprime to $r$, and moreover, if $r$ is odd and $s$ is even,
$\mathrm{TV}^{\prime}_{r,s}(M_{A})=\mathrm{TV}^{\prime}_{r,s}(M_{B})=0$ also
holds.

If $r\geq 3$ is not divisible by $p$, then it is coprime to $p$. By Lemma 5.1,
$\mathrm{TV}_{r,s}(M_{A})$ and $\mathrm{TV}_{r,s}(M_{B})$ are rational. Since
$\mathrm{TV}_{r,s}(M_{A})$ and $\mathrm{TV}_{r,s}(M_{B})$ are the traces of
the Turaev–Viro TQFT representations $\rho^{\mathrm{TV}}_{r,s}$ of the
periodic mapping class $[f_{A}]$ and $[f_{B}]=[f_{A}^{k}]$, respectively, the
eigenvalues of $[f_{A}]$ are roots of unity of order dividing $d$, and the
eigenvalues of $[f_{B}]$ are their Galois conjugates under the transformation
$e^{\sqrt{-1}\cdot 2\pi/d}\mapsto e^{\sqrt{-1}\cdot 2\pi k/d}$. Then by the
rationality, we obtain $\mathrm{TV}_{r,s}(M_{A})=\mathrm{TV}_{r,s}(M_{B})$ for
any $s$ coprime to $r$. Moreover, if $r$ is odd and $s$ is even, we apply
Lemma 3.4 and 4.6 to deduce
$\mathrm{TV}^{\prime}_{r,s}(M_{A})=\mathrm{TV}^{\prime}_{r,s}(M_{B})$.

##  Appendix A Splitting of Turaev–Viro invariants at odd levels

In this appendix section, we prove the formula in Lemma 3.4 about
$\mathrm{TV}_{r,s}$ when $r$ and $s$ are both odd. We restate this part as a
separate theorem , and make a couple of remarks regarding former results.

######  Theorem A.1.

Let $M$ be any closed $3$–manifold. Let $r\geq 3$ be an odd integer and $s$ be
an integer coprime to $r$. Adopt Notation 3.2.

If $s$ is odd, then the following formula holds.

$\mathrm{TV}_{r,s}(M)=\mathrm{TV}_{3,1}(M)\cdot\mathrm{TV}^{\prime}_{r,r-s}(M).$

######  Remark A.2.

  1. (1)

Sokolov obtains a canonical splitting of $\mathrm{TV}_{r,s}$ into the sum of
three refined invariants [Sok96]. When $r$ is odd and $s$ is odd, one may
identify the three refined invariants (the zeroth, the first, and the second
in Sokolov’s definition) as $\mathrm{TV}^{\prime}_{r,r-s}$, and
$(\mathrm{TV}_{r,s}-\mathrm{TV}_{r,r-s})/2$, and
$(\mathrm{TV}_{r,s}+\mathrm{TV}_{r,r-s})/2-\mathrm{TV}^{\prime}_{r,r-s}$. In
this case, the splitting of $\mathrm{TV}_{r,s}$ is proportional to the
splitting of $\mathrm{TV}_{3,1}$. Similarly, when $r$ is odd and $s$ is even,
the splitting of $\mathrm{TV}_{r,s}$ is proportional to the splitting of
$\mathrm{TV}_{3,2}$. Compare [BlaHMV95, Theorem 1.5].

  2. (2)

In the same paper, Sokolov quickly points out Lemma A.6 below, with assumption
of orientability. See the formula (1) in [Sok96, Proof of Lemma 2.2].

The rest of this section is devoted to the proof of Theorem A.1.

Our strategy is to derive needed ingredients from the proof of Detcherry,
Kalfagianni, and Yang for the case with $r$ odd and $s$ even [DetKY18,
Appendix A]. We count sign change from their case for individual terms in the
defining state-sum expression, and verify that overall, the factors
$\mathrm{TV}_{3,1}$ and $\mathrm{TV}_{r,s}$ are result of proportional change
from factors in their case.

We denote by $\mathrm{ev}_{r,s}\colon{\mathbb{Q}}(q^{1/2})\to{\mathbb{C}}$ the
evaluation which assigns the abstract root of unity $q^{1/2}$ to be
$e^{\sqrt{-1}\cdot\pi s/r}$.

For any odd integer $r\geq 3$, recall that $I_{r}=\\{0,1,\cdots,r-2\\}$
denotes the set of colors on this level. It contains the subset of even colors
$I^{\prime}_{r}=\\{0,2,\cdots,r-3\\}$ and also $I_{3}=\\{0,1\\}$. For any
finite simplicial $3$–complex $\mathscr{T}=(V,E,F,T)$. From any coloring
$c\colon E\to I_{r}$, we obtain a pair of colorings $c_{3}\colon E\to I_{3}$
and $c^{\prime}\colon E\to I^{\prime}_{r}$ as follows: For any $e\in E$, we
assign $c_{3}(e)=0$ and $c^{\prime}(e)=c(e)$ if $c(e)$ is even, or
$c_{3}(e)=1$ and $c^{\prime}(e)=r-2-c(e)$ if $c(e)$ is odd. By observation,
this operation preserves admissible colorings, and yields a bijective
correspondence between $\mathcal{A}_{r}$ and
$\mathcal{A}_{3}\times\mathcal{A}^{\prime}_{r}$.

######  Lemma A.3.

Let $\mathscr{T}=(V,E,F,T)$ be any finite simplicial $3$–complex. Let $r\geq
3$ be an odd integer and $s$ be an integer coprime to $r$. Adopt Notation 3.1.
Identify $\mathcal{A}_{r}=\mathcal{A}_{3}\times\mathcal{A}^{\prime}_{r}$. If
$s$ is even, then, for any $x\in E\sqcup F\sqcup T$ and any
$c=(c_{3},c^{\prime})\in\mathcal{A}_{r}$,

$\mathrm{ev}_{r,s}\left(|x|_{c}\right)=\mathrm{ev}_{3,2}\left(|x|_{c_{3}}\right)\cdot\mathrm{ev}_{r,s}\left(|x|_{c^{\prime}}\right),$

and hence,

$\mathrm{ev}_{r,s}\left(|\mathscr{T}|_{c}\right)=\mathrm{ev}_{3,2}\left(|\mathscr{T}|_{c_{3}}\right)\cdot\mathrm{ev}_{r,s}\left(|\mathscr{T}|_{c^{\prime}}\right).$

The identities in Lemma A.3 are key to the proof of Detcherry, Kalfagianni,
and Yang for the $s$ even case [DetKY18, Theorem 2.9]. We refer to [DetKY18,
Lemma A.4] for the proof. Note that the identity (A.1) therein essentially
relies on the parity assumption of $s$.

For proving the $s$ odd case, our next few lemmas examine the sign difference
between $\mathrm{ev}_{r,s}(|x|_{c})$ and $\mathrm{ev}_{r,r-s}(|x|_{c})$, and
between $\mathrm{ev}_{r,s}(|\mathscr{T}|_{c})$ and
$\mathrm{ev}_{r,r-s}(|\mathscr{T}|_{c})$.

######  Lemma A.4.

Let $\mathscr{T}=(V,E,F,T)$ be any finite simplicial $3$–complex. Let $r\geq
3$ be an integer and $s$ be an integer coprime to $r$. Adopt Notation 3.1.
Then, for any $x\in E\sqcup F\sqcup T$ and any $c\in\mathcal{A}_{r}$,

$\mathrm{ev}_{r,s}\left(|x|_{c}\right)=(-1)^{\delta(x,c)}\cdot\mathrm{ev}_{r,r-s}\left(|x|_{c}\right),$

where $\delta(x,c)\in{\mathbb{Z}}$ is assigned as follows.

  1. (1)

For $x\in E$, having color $i$ under $c$,

$\delta(x,c)=i.$

  2. (2)

For $x\in F$, having edge colors $(i,j,k)$ under $c$,

$\delta(x,c)=\frac{i^{2}+j^{2}+k^{2}}{2}.$

  3. (3)

For $x\in T$, having edge colors $(i,j,k),(i,m,n),(j,m,n),(k,l,n)$ on each
face under $c$,

$\delta(x,c)=i+j+k+l+m+n+\frac{il+jm+kn}{2}.$

###### Proof.

We prove for $x\in T$. The formulas with $x\in E$ and $x\in F$ can be proved
by similar means, and are simpler. Suppose the tetrahedron $x$ has edge colors
$i,j,k,l,m,n$ given by $c$. Note that the value of the quantum integer only
change by a sign determined by its parity

$\mathrm{ev}_{r,s}([w])=(-1)^{w-1}\cdot\mathrm{ev}_{r,r-s}([w]).$

For any fixed $z$, the total sign change of the summand is $-1$ to the power

$\displaystyle\frac{(z+1)z}{2}+\sum_{a}\frac{(z-T_{a})(z-T_{a}-1)}{2}+\sum_{b}\frac{(Q_{b}-z)(Q_{b}-z-1)}{2}$
$\displaystyle=$ $\displaystyle
4z^{2}-2zW+\frac{1}{2}\cdot\left(\sum_{a}T_{a}^{2}+\sum_{b}Q_{b}^{2}\right)$

where $W=i+j+k+l+m+n=\sum_{a}T_{a}=\sum_{b}Q_{b}$ is an integer. The first two
terms are even integers, having no effect to the total change of sign, and the
last term is independent of $z$. Therefore, we obtain

$\mathrm{ev}_{r,s}(|x|_{c})=(-1)^{\delta(x,c)}\cdot\mathrm{ev}_{r,r-s}(|x|_{c}).$

where

$\delta(x,c)\equiv\frac{1}{2}\cdot\left(\sum_{a}T_{a}^{2}+\sum_{b}Q_{b}^{2}\right)\mod
2.$

The expression $\sum_{a}T_{a}^{2}+\sum_{b}Q_{b}^{2}$ is equal to the sum of
all the quadratic monomials in $i,j,k,l,m,n$, namely,
$i^{2}+ij+\cdots+in+j^{2}+jk+\cdots+mn$. We rearrange

$\sum_{a}T_{a}^{2}+\sum_{b}Q_{b}^{2}=X^{2}+Y^{2}+Z^{2}+XY+XZ+YZ-il-jm-kn$

where $X=i+l$, $Y=j+m$, and $Z=k+n$. Note that the parity pattern of $(X,Y,Z)$
may only be $(0,0,0)$ or $(1,1,1)$, up to permutation of the components.
Indeed, the only possible patterns of $(i_{3}+l_{3},j_{3}+m_{3},k_{3}+n_{3})$
are $(0,0,0)$, $(1,1,1)$, and $(0,2,2)$, up to permutation of the components,
because of admissible coloring. So, the part
$X^{2}+Y^{2}+Z^{2}+XY+XZ+YZ=(X+Y)(X+Z)+Y^{2}+Z^{2}$ is congruent to $0$ modulo
$4$ if $(X,Y,Z)\equiv(0,0,0)\bmod 2$, or congruent to $2$ modulo $4$ if
$(X,Y,Z)\equiv(1,1,1)\bmod 2$. In both cases, we see that
$(X^{2}+Y^{2}+Z^{2}+XY+XZ+YZ)/2\equiv X+Y+Z\bmod 2$. This yields
$\delta(x,c)\equiv X+Y+Z-(il+jm+kn)/2\equiv i+j+k+l+m+n+(il+jm+kn)/2\bmod 2$,
as desired. ∎

######  Lemma A.5.

Let $\mathscr{T}=(V,E,F,T)$ be any finite simplicial $3$–complex. Let $r\geq
3$ be an odd integer and $s$ be an integer coprime to $r$. Adopt Notation 3.1.
Identify $\mathcal{A}_{r}=\mathcal{A}_{3}\times\mathcal{A}^{\prime}_{r}$. Let
$\delta\colon(E\sqcup F\sqcup T)\times\mathcal{A}_{r}\to{\mathbb{Z}}$ be
expressed as in Lemma A.4.

  1. (1)

For any $x\in E\sqcup F$ and any $c=(c_{3},c^{\prime})\in\mathcal{A}_{r}$,

$\delta(x,c)\equiv\delta(x,c_{3})\mod 2,$

treating $\mathcal{A}_{3}$ as a subset of $\mathcal{A}_{r}$.

  2. (2)

For $x\in T$, having edge colors $(i,j,k),(i,m,n),(j,m,n),(k,l,n)$ on each
face under $c$,

$\delta(x,c)\equiv\delta(x,c_{3})+\lambda(x,c)\mod 2,$

where

$\lambda(x,c)=\frac{i_{3}l^{\prime}+l_{3}i^{\prime}+j_{3}m^{\prime}+m_{3}j^{\prime}+k_{3}n^{\prime}+n_{3}k^{\prime}}{2}$

.

###### Proof.

We make use of the relation
$c(e)=c_{3}(e)\cdot(r-2-c^{\prime}(e))+(1-c_{3}(e))\cdot
c^{\prime}(e)=(r-2)\cdot c_{3}(e)-c^{\prime}(e)-2\cdot c_{3}(e)\cdot
c^{\prime}(e)$, for any $e\in E$. Since $r$ is odd and $i^{\prime},l^{\prime}$
are even, we obtain

$\displaystyle il$ $\displaystyle=$
$\displaystyle((r-2)i_{3}-i^{\prime}-2i_{3}i^{\prime})\cdot((r-2)l_{3}-l^{\prime}-2l_{3}l^{\prime})$
$\displaystyle\equiv$
$\displaystyle(r-2)^{2}i_{3}l_{3}-(r-2)\cdot(i_{3}l^{\prime}+l_{3}i^{\prime})$
$\displaystyle\equiv$ $\displaystyle
i_{3}l_{3}+(i_{3}l^{\prime}+l_{3}i^{\prime})\mod 4$

and similarly we manipulate $jk$ and $kn$. Taking the sum, we obtain

$il+jm+kn\equiv i_{3}l_{3}+j_{3}m_{3}+k_{3}n_{3}+2\cdot\lambda(x,c)\mod 4.$

Moreover, we observe

$i+j+k+l+m+n\equiv i_{3}+j_{3}+k_{3}+l_{3}+m_{3}+n_{3}\mod 2.$

By Lemma A.4, the above congruence equalities imply
$\delta(x,c)\equiv\delta(x,c_{3})+\lambda(x,c)\bmod 2$, as desired. ∎

For any $c_{3}\in\mathcal{A}_{3}$, there is a canonical subsurface
$\mathcal{S}(c_{3})\subset M$, in normal position with respect to
$\mathscr{T}$, such that $c_{3}(e)$ indicates the number of intersection
points of any edge $e\in E$ with $\mathcal{S}(c_{3})$. The subsurface
$\mathcal{S}(c_{3})$ is formed by taking one normal disk in each tetrahedron
that has nonzero color, and then taking their union matching the sides. The
types of the normal disks (among four triangular types and three quadrilateral
types for each tetrahedron) are forced by the admissible coloring $c_{3}$. The
subsurface $\mathcal{S}(c_{3})$ is closed, as $M$ does not have boundary.

######  Lemma A.6.

Let $(M,\mathscr{T})$ be any triangulated closed $3$–manifold. Let $r\geq 3$
be an odd integer and $s$ be an integer coprime to $r$. Adopt Notation 3.1.
Identify $\mathcal{A}_{r}=\mathcal{A}_{3}\times\mathcal{A}^{\prime}_{r}$.
Then, for any $c=(c_{3},c^{\prime})\in\mathcal{A}_{r}$,

$\mathrm{ev}_{r,s}\left(|\mathscr{T}|_{c}\right)=(-1)^{\chi(\mathcal{S}(c_{3}))}\cdot\mathrm{ev}_{r,r-s}\left(|\mathscr{T}|_{c}\right),$

where $\chi(\mathcal{S}(c_{3}))$ denotes the Euler characteristic of the
normal subsurface $\mathcal{S}(c_{3})\subset M$ determined by $c_{3}$.

###### Proof.

For all $x\in E\sqcup F\sqcup T$, the nonempty intersections
$x\cap\mathcal{S}(c_{3})$ give rise to a polygonal cell decomposition of
$\mathcal{S}(c_{3})$. Denote by
$\nu_{0},\nu_{1},\nu_{2,\triangle},\nu_{2,\square}$ the numbers of vertices,
edges, triangular normal disks, and quadrilateral normal disks in
$\mathcal{S}(c_{3})$, respectively.

Let $\delta\colon(E\sqcup F\sqcup T)\times\mathcal{A}_{r}\to{\mathbb{Z}}$ be
expressed as in Lemma A.4. For any $x\in E\sqcup F\sqcup T$, it is direct to
check that $\delta(x,c_{3})=1$ if and only if $x\cap\mathcal{S}(c_{3})$ is a
vertex or an edge or a quadrilateral normal disk of $\mathcal{S}(c_{3})$,
otherwise $\delta(x,c_{3})=0$, treating $\mathcal{A}_{3}$ as a subset of
$\mathcal{A}_{r}$. This means $\nu_{0}=\sum_{x\in E}\delta(x,c_{3})$,
$\nu_{1}=\sum_{x\in F}\delta(x,c_{3})$, and $\nu_{2,\square}=\sum_{x\in
T}\delta(x,c_{3})$. Moreover, the relation
$3\cdot\nu_{2,\triangle}+4\cdot\nu_{2,\square}=2\cdot\nu_{1}$ implies that
$\nu_{2,\triangle}$ is even, as $\mathcal{S}(c_{3})$ is closed. By Lemma A.5,
we obtain

$\displaystyle\chi(\mathcal{S}(c_{3}))$ $\displaystyle=$
$\displaystyle\nu_{0}-\nu_{1}+\nu_{2,\triangle}+\nu_{2,\square}$
$\displaystyle\equiv$ $\displaystyle\sum_{x\in E\sqcup F\sqcup
T}\delta(x,c_{3})$ $\displaystyle\equiv$ $\displaystyle\sum_{x\in E\sqcup
F\sqcup T}\delta(x,c)+\sum_{x\in T}\lambda(x,c)\mod 2.$

Therefore, to derive the asserted formula
$\mathrm{ev}_{r,s}(|\mathscr{T}|_{c})=(-1)^{\chi(\mathcal{S}(c_{3}))}\cdot\mathrm{ev}_{r,r-s}(|\mathscr{T}|_{c})$
from Lemma A.4, it remains to prove

$\sum_{x\in T}\lambda(x,c)\equiv 0\mod 2.$

To this end, we observe that $\sum_{x\in T}\lambda(x,c)$ is the sum of
$c^{\prime}(e)/2$, where $e$ ranges over the edges of $x$ whose opposite edge
in $x$ meets $\mathcal{S}(c_{3})$. For any edge $e^{*}\in E$, the link of
$e^{*}$ refers to the union of all the opposite edges $e_{1},\cdots,e_{h}$ in
all the tetrahedra $t_{1},\cdots,t_{h}\in T$ that contain $e^{*}$, denoted as
$\mathrm{lk}(e^{*})\subset M$. The link $\mathrm{lk}(e^{*})$ is a contractible
loop in $M$ (bounding a disk transverse to $e^{*}$). On the other hand, any
edge either misses $\mathcal{S}(c_{3})$, or meets $\mathcal{S}(c_{3})$ at
exactly one point. Then the number of edges in $\mathrm{lk}(e^{*})$ that meet
$\mathcal{S}(c_{3})$ must be even, and the integer $c^{\prime}(e^{*})/2$
contributes exactly this even number of times to $\sum_{x\in T}\lambda(x,c)$.
Because $\sum_{x\in T}\lambda(x,c)$ is the total of the contribution from each
edge $e^{*}\in E$, we conclude $\sum_{x\in T}\lambda(x,c)\equiv 0\bmod 2$ as
desired, completing the proof. ∎

######  Lemma A.7.

Let $(M,\mathscr{T})$ be any triangulated closed $3$–manifold. Let $r\geq 3$
be an odd integer and $s$ be an integer coprime to $r$. Adopt Notation 3.1.
Identify $\mathcal{A}_{r}=\mathcal{A}_{3}\times\mathcal{A}^{\prime}_{r}$. If
$s$ is odd, then,

$\mathrm{ev}_{r,s}\left(|\mathscr{T}|_{c}\right)=\mathrm{ev}_{3,1}\left(|\mathscr{T}|_{c_{3}}\right)\cdot\mathrm{ev}_{r,r-s}\left(|\mathscr{T}|_{c^{\prime}}\right).$

###### Proof.

By Lemma A.6, we obtain
$\mathrm{ev}_{r,s}(|\mathscr{T}|_{c})=(-1)^{\chi(\mathcal{S}(c_{3}))}\cdot\mathrm{ev}_{r,r-s}(|\mathscr{T}|_{c})$
and
$\mathrm{ev}_{3,1}(|\mathscr{T}|_{c_{3}})=(-1)^{\chi(\mathcal{S}(c_{3}))}\cdot\mathrm{ev}_{3,2}(|\mathscr{T}|_{c})$.
Then the asserted identity follows from the $s$ even case (Lemma A.3). ∎

To complete the proof of Theorem A.1, we observe

(A.1)
$\mathrm{ev}_{r,s}\left(Y_{r}\right)=\begin{cases}\mathrm{ev}_{3,2}\left(Y_{3}\right)\cdot\mathrm{ev}_{r,s}\left(Y^{\prime}_{r}\right)&s\mbox{
even}\\\
\mathrm{ev}_{3,1}\left(Y_{3}\right)\cdot\mathrm{ev}_{r,r-s}\left(Y^{\prime}_{r}\right)&s\mbox{
odd}\end{cases}$

where $Y_{r}=-(q^{1/2}-q^{-1/2})^{2}/2r$ and
$Y^{\prime}_{r}=-(q^{1/2}-q^{-1/2})^{2}/r$, (indeed,
$\mathrm{ev}_{r,s}(q^{1/2}-q^{-1/2})=-\sqrt{-1}\cdot 2\sin(\pi s/r)$).

Let $M$ be any closed $3$–manifold. If $r$ is odd and $s$ is even, we obtain

$\displaystyle\mathrm{TV}_{r,s}(M)$ $\displaystyle=$
$\displaystyle\mathrm{ev}_{r,s}\left(Y_{r}\cdot\sum_{c\in\mathcal{A}_{r}}|\mathscr{T}|_{c}\right)$
$\displaystyle=$
$\displaystyle\mathrm{ev}_{r,s}(Y_{r})\cdot\sum_{c\in\mathcal{A}_{r}}\mathrm{ev}_{r,s}(|\mathscr{T}|_{c})$
$\displaystyle=$
$\displaystyle\mathrm{ev}_{3,1}(Y_{3})\cdot\mathrm{ev}_{r,r-s}(Y^{\prime}_{r})\cdot\sum_{c_{3}\in\mathcal{A}_{3}}\sum_{c^{\prime}\in\mathcal{A}^{\prime}_{r}}\mathrm{ev}_{3,1}(|\mathscr{T}|_{c})\cdot\mathrm{ev}_{r,r-s}(|\mathscr{T}|_{c^{\prime}})$
$\displaystyle=$
$\displaystyle\mathrm{ev}_{3,1}\left(Y_{3}\cdot\sum_{c_{3}\in\mathcal{A}_{3}}|\mathscr{T}|_{c}\right)\cdot\mathrm{ev}_{r,r-s}\left(Y^{\prime}_{r}\cdot\sum_{c^{\prime}\in\mathcal{A}^{\prime}_{r}}|\mathscr{T}|_{c^{\prime}}\right)$
$\displaystyle=$
$\displaystyle\mathrm{TV}_{3,1}(M)\cdot\mathrm{TV}^{\prime}_{r,r-s}(M)$

by (3.1), (3.2), (A.1), and Lemma A.7. This completes the proof of Theorem
A.1.

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