# On a volume invariant of 3–manifolds

Marc Kegel  Arunima Ray  Jonathan Spreer  Em Thompson and Stephan Tillmann
Marc Kegel
Humboldt Universitaet zu Berlin, Rudower Chaussee 25, 12589 Berlin, Germany
Universitaet Heidelberg, Im Neuenheimer Feld 205, 69120 Heidelberg, Germany
<EMAIL_ADDRESS><EMAIL_ADDRESS>
—–  Arunima Ray
Max-Planck-Institut fuer Mathematik, Vivatsgasse 7, 53111 Bonn
<EMAIL_ADDRESS>
—–  Jonathan Spreer
School of Mathematics and Statistics F07, The University of Sydney, NSW 2006
Australia
<EMAIL_ADDRESS>
—–  Em Thompson
School of Mathematics, Monash University, Vic 3800 Australia
<EMAIL_ADDRESS>
—–  Stephan Tillmann
School of Mathematics and Statistics F07, The University of Sydney, NSW 2006
Australia
<EMAIL_ADDRESS>
###### Abstract

This paper investigates a real-valued topological invariant of $3$–manifolds
called _topological volume_. For a given $3$–manifold $M$ it is defined as the
smallest volume of the complement of a (possibly empty) hyperbolic link in
$M$. Various refinements of this invariant are given, asymptotically tight
upper and lower bounds are determined, and all non-hyperbolic closed
$3$–manifolds with topological volume of at most $3.07$ are classified.
Moreover, it is shown that for all but finitely many lens spaces, the volume
minimiser is obtained by Dehn filling one of the cusps of the complement of
the Whitehead link or its sister manifold.

###### keywords:

hyperbolic volume, hyperbolic knots and links

tiny[table]font=tiny

57K10, 57K31, 57K32, 57R65.

##  1 Introduction

Let $M$ be a closed 3–manifold. A topological invariant
$\operatorname{volt}(M)$111The reader may interpret this as a quantification
of how much potential the manifold has. We read this as _topological volume_
or _volume topologique_. The name was coined with permission of Michel Boileau
and Joan Porti. is defined in [BBBMP] in an account of a complete proof of
Thurston’s Geometrisation Conjecture. Roughly speaking, this is the minimal
volume of the complement of a (possibly empty) hyperbolic link in $M.$ Here we
begin the study of this invariant in its own right by establishing some basic
properties and computing it for certain families of non-hyperbolic
3–manifolds.

This introduces an interesting new organising principle for closed
$3$–manifolds: Hyperbolic volume is one of the most widely used topological
invariants for hyperbolic $3$–manifolds. By passing to hyperbolic links in
(non-hyperbolic) manifolds we obtain a unified notion of volume for all
$3$–manifolds. We show in Section 2 that this volume invariant has interesting
properties when studied over the whole class of closed 3–manifolds.
Furthermore, we can partition the set of $3$–manifolds by asking for the
number of components of the smallest volume hyperbolic link in $M$. Naturally,
this number is zero if and only if $M$ is hyperbolic. It is an open question
whether there exist manifolds for which this number is greater than one (cf.
2.5).

As we demonstrate in this paper, topological volume is not just a theoretical
concept. Instead, surprisingly powerful results about $\operatorname{volt}$
can be established. A key ingredient of our upcoming proofs is how hyperbolic
volume behaves under Dehn filling, due to work of Jørgensen and Thurston
[thurston-notes, Gro81, Mai10]. This theory provides not just valuable tools,
but also an insight into how hyperbolic volume is connected to the
construction of general $3$–manifolds using Dehn surgery.

#### Related work

A different but related _link volume_ was introduced by Rieck and Yamashita in
[Rieck-link-2013]. A key difference between their invariant and
$\operatorname{volt}$ is that there are at most finitely many 3–manifolds of
the same topological volume (Proposition 2.6) whilst there are examples of
infinitely many 3–manifolds of the same link volume [Remigio-link-2012,
Corollary 1.2]. In this respect, our definition agrees with hyperbolic volume,
since a given real number is realised as the volume of only finitely many
hyperbolic 3–manifolds by Jørgensen–Thurston theory [thurston-notes, Gro81,
Mai10].

#### Our contributions

We begin, in Section 2.2, by presenting some natural refinements of
$\operatorname{volt}$. In particular, we consider $\operatorname{volt}_{n}(M)$
denoting the minimum over the volumes of all hyperbolic $n$-component links in
$M$ and we define refinements of $\operatorname{volt}$ considering links
representing a specified homology class. In the rest of Section 2, we
establish some basic properties of $\operatorname{volt}$. In Section 2.4 we
investigate how topological volume behaves under taking (branched) covers, and
in Section 2.5 identify sequences of closed 3–manifolds with unbounded
topological volume, based on an observation on how topological volume
interacts with classical topological invariants.

In Section 3, we strengthen these results by presenting upper and lower bounds
for $\operatorname{volt}(M)$ for arbitrary closed $3$–manifolds $M$. In
particular, we prove the following theorems.

######  Theorem 1.1.

Let $M$ be a closed, orientable 3–manifold given by surgery along a nontrivial
$n$-component link $L\subset S^{3}$ with crossing number $c=c(L)$. Then

$\operatorname{volt}(M)\leq\big{(}9c+15n-20\big{)}v_{oct}+4v_{tet}$

where $v_{oct}\approx 3.6638$ and $v_{tet}\approx 1.01494$ are the volumes of
a regular hyperbolic ideal octahedron and a regular hyperbolic ideal
tetrahedron respectively.

######  Theorem 1.2.

Let $M$ be a closed, orientable 3–manifold and $p$ be a prime. Then

$\frac{\operatorname{rk}(H_{1}(M;\mathbb{Z}_{p}))}{168.602}<\operatorname{volt}(M)$

Moreover, we have $\operatorname{volt}(M)>3.08,$ $3.69$, $3.77$ for
$\operatorname{rk}(H_{1}(M;\mathbb{Z}_{2}))>5$, $7$, $10$, respectively.

These bounds are asymptotically tight. In particular, we show in Example 3.6
that for $M$ a Seifert fibered space over a genus $g$ orbifold with $n$
exceptional fibres of index $p$ a prime we have

$\frac{2g+n}{168.602}<\operatorname{volt}(M)<37g+22n-28$

Dunfield’s classification of exceptional Dehn fillings of cusped $3$–manifolds
[Du20] allows us to determine the exact topological volume of all non-
hyperbolic closed $3$–manifolds of topological volume at most $3.07$. See
LABEL:tab:complete and Section 4 for summaries and discussion of observations
arising from this computation.

The last result of this paper observes that for all but finitely many lens
spaces, their volume minimiser arises as a Dehn filling of either the
complement of the Whitehead link $W$ or its sister manifold $P$. Its proof
exhibits the interesting interplay between the computation of topological
volume of non-hyperbolic 3–manifolds and properties of the volume spectrum of
cusped hyperbolic 3–manifolds.

######  Theorem 1.3.

For all but finitely many homeomorphism types of lens spaces $L(p,q)$
($p>q\geq 1$ coprime), the volume minimising link in $L(p,q)$ has complement
one of $W(\frac{p^{\prime}}{q^{\prime}},\infty)$ or
$P(\frac{p^{\prime\prime}}{q^{\prime\prime}},\infty)$, where the filling
slopes $\frac{p^{\prime}}{q^{\prime}}$ and
$\frac{p^{\prime\prime}}{q^{\prime\prime}}$ can be determined from an
efficiently computable finite set. For $P$ this means that we fill along the
unknotted component.

We hope that this survey over the properties of $\operatorname{volt}$, its
refinements, and first results will convince the reader of the usefulness of
this topological invariant as both an organising principle as well as a
meaningful topological property.

Acknowledgements. The authors thank the MATRIX Institute in Creswick and the
Mathematisches Forschungsinstitut Oberwolfach for a collaborative environment
between different continents and time-zones during a tandem workshop in 2021.
This is where this work was begun. They thank the participants of the workshop
for stimulating conversations. Special mention goes to Jonathan Bowden,
Jessica Purcell and Saul Schleimer for comments and contributions during the
initial phase of this project.

Figures 2–5 were created using KLO [KLO].

MK and ST thank the Max-Planck-Institut für Mathematik in Bonn, where parts of
this work were carried out, for its hospitality. MK is partially supported by
the SFB/TRR 191 Symplectic Structures in Geometry, Algebra and Dynamics,
funded by the DFG (Projektnummer 281071066 - TRR 191). Research of JS and ST
is supported in part under the Australian Research Council’s Discovery funding
scheme (project number DP190102259). JS thanks Michael Joswig and Technische
Universität Berlin, where parts of this work were carried out, for their
hospitality. ET is supported by an Australian Government Research Training
Program (RTP) Scholarship.

##  2 Properties, examples and questions

We first define the notion of topological volume and refinements thereof. We
then establish general properties of the topological volume, give examples and
state questions that arise from our census in Section 4.

###  2.1 Definition of topological volume and its spectrum

Let $M$ be a closed, orientable 3–manifold. A link in $M$ is a (possibly
empty, possibly disconnected) closed 1–submanifold $L$ of $M.$ We consider
links up to (not necessarily orientation preserving) diffeomorphisms of their
complement $M\setminus L$. This is motivated by the fact that none of the
invariants we consider in this article differ on non-isotopic links with
diffeomorphic complement.

A link $L$ in $M$ is hyperbolic if its complement admits a complete hyperbolic
structure of finite volume. In this case, let
$\operatorname{vol_{\mathbb{H}}}(M\setminus L)$ denote the hyperbolic volume
of $M\setminus L.$ We often write
$\operatorname{vol_{\mathbb{H}}}(L)=\operatorname{vol_{\mathbb{H}}}(M\setminus
L).$ Following [BBBMP, §1.3.2], let

$\operatorname{volt}(M)=\inf\big{\\{}\;\operatorname{vol_{\mathbb{H}}}(M\setminus
L)\;\mid\;L\subset M\text{ is a hyperbolic link }\;\big{\\}}$ (2.1)

It follows from Mostow–Prasad rigidity that $\operatorname{volt}(M)$ is a
topological invariant of $M$, and we term it the topological volume of $M.$
The invariant $\operatorname{volt}(M)$ is indeed finite since every closed,
orientable 3–manifold contains a hyperbolic knot [My82, My93].222[My93] in
fact established that all compact $3$–manifolds, including non-orientable
$3$–manifolds, contain hyperbolic knots. However we restrict ourselves to the
closed and orientable case. If $M$ admits a complete hyperbolic structure,
then $\operatorname{volt}(M)=\operatorname{vol_{\mathbb{H}}}(M)$, since volume
decreases under Dehn filling.

######  Example 2.1.

As our first example, we observe that
$\operatorname{volt}(S^{3})=2{v}_{0}\approx 2.02988$, where ${v}_{0}$ is the
volume of the ideal regular tetrahedron in hyperbolic $3$-space, and with the
figure eight knot as unique minimiser. To see this, recall that [cao-
meyerhoff] showed that the complement of the figure eight knot $K4a1$, called
$m004$ in the SnapPy [snappy] census,333The naming convention throughout this
paper follows that of the SnapPy census manifolds. Hence the symbol $m$ means
that the corresponding manifolds can be triangulated with at most $5$ ideal
tetrahedra. is a cusped hyperbolic 3–manifold of minimum volume. Indeed, the
cusped hyperbolic 3–manifolds of minimum volume are the once-punctured torus
bundles $m003$ and $m004$ and they are related by a change in monodromy.
However, $m003$ admits no filling to $S^{3}$ and thus does not represent a
knot complement in $S^{3}$. All exceptional fillings of these two manifolds
have the same topological volume.

According to Jørgensen and Thurston (see [thurston-notes, Gro81, Mai10]), the
set of volumes of complete orientable hyperbolic 3–manifolds of finite volume
is a closed, non-discrete, well-ordered subset of the positive reals of order
type $\omega^{\omega}.$ Throughout this article, we refer to this fact, as
well as related results, as Jørgensen–Thurston theory. In particular, there is
always at least one hyperbolic link $L\subset M$ with
$\operatorname{volt}(M)=\operatorname{vol_{\mathbb{H}}}(M\setminus L)$. We
term such a link a volume minimiser. We say that two minimising links are
equivalent if they have diffeomorphic complements. Occasionally we refer to
the least volume hyperbolic link complement in $M$ as a volume minimiser of
$M$ as well.

######  Question 2.2.

Let $M$ be a closed, orientable $3$–manifold $M$. Suppose $L_{1}$ and $L_{2}$
are links realising $\operatorname{volt}(M)$. Is $M\setminus L_{1}$
diffeomorphic with $M\setminus L_{2}$?

The manifold $m003$ appears as complement of a minimiser in $L(5,1)$ in two
different ways – through the fillings $m003(-1,1)$ and $m003(0,1).$ However,
there is a diffeomorphism of $m003$ that sends one surgery slope to the other.
In particular, the minimising hyperbolic knot in $L(5,1)$ is the same. Hence
this is not a counterexample to the following question, which is equivalent to
the previous one if the cosmetic surgery conjecture is true.

######  Question 2.3.

Let $M$ be a closed, orientable $3$–manifold $M$. Suppose $L_{1}$ and $L_{2}$
are links realising $\operatorname{volt}(M)$. Is there a diffeomorphism $M\to
M$ taking $L_{1}$ to $L_{2}$?

It is shown in [Adams_unbounded_volume] that the set of volumes of hyperbolic
knots in any 3–manifold is unbounded. Hence not only determining the minimum
but also the study of the spectrum of $\operatorname{volt}$ is interesting.
Let $\mathcal{V}$ be the set of volumes of all cusped hyperbolic 3–manifolds,
and $\mathcal{N}$ be the set of all closed, non-hyperbolic, orientable
3–manifolds.

######  Question 2.4.

Is $\operatorname{volt}\co\mathcal{N}\to\mathcal{V}$ surjective?

###  2.2 Refinements of topological volume

Let $M$ be a closed, orientable $3$–manifold. For each integer $n\geq 0,$ we
define the invariant

$\operatorname{volt}_{n}(M)=\inf\big{\\{}\;\operatorname{vol_{\mathbb{H}}}(M\setminus
K)\;\mid\;K\subset M\text{ is a hyperbolic $n$--component link }\;\big{\\}}$
(2.2)

For all $n\geq 1$ and all closed, orientable 3–manifolds $M$, we have
$\operatorname{volt}(M)\leq\operatorname{volt}_{n}(M)$. In particular,
$\operatorname{volt}(M)=\operatorname{volt}_{0}(M)<\operatorname{volt}_{n}(M)$
if and only if $M$ is hyperbolic. This also allows us to define an integer
valued invariant

$h(M)=\min\\{\;n\;\mid\operatorname{volt}(M)=\operatorname{volt}_{n}(M)\;\\}$

Hence $M$ is hyperbolic if and only if $h(M)=0.$ The examples presented in
Section 4 and based on the census in [GHMTY21] satisfy $h(M)=1.$

######  Question 2.5.

Is there an explicit family of examples $(M_{n})$ with $h(M_{n})=n$?

Myers [My93] showed that every homology class (indeed every free homotopy
class) in a closed $3$–manifold is represented by some hyperbolic knot. More
generally, he showed that every given finite set of free homotopy classes can
be represented by a hyperbolic link. Hence for each class $c\in H_{1}(M),$
there is an invariant

$\operatorname{volt}(M,c)=\inf\big{\\{}\;\operatorname{vol_{\mathbb{H}}}(M\setminus
L)\;\mid\;L\subset M\text{ is an oriented hyperbolic link with
}[L]=c\;\big{\\}}$ (2.3)

A natural question is to ask which homology classes satisfy
$\operatorname{volt}(M,c)=\operatorname{volt}(M).$ Some examples and a further
discussion of this refinement can be found in Section 4.3.

###  2.3 A finiteness result

According to Jørgensen–Thurston theory [thurston-notes, Gro81, Mai10], for
each positive real number $v$, there are at most finitely many hyperbolic
3–manifolds of volume $v.$ We show that the analogous result holds for
topological volume.

######  Proposition 2.6.

For each positive real number $v$, there are at most finitely many closed,
orientable 3–manifolds of topological volume $v.$

###### Proof.

Suppose there are infinitely many closed 3–manifolds of topological volume
$v$. Since only finitely many of them can be hyperbolic [thurston-notes], this
implies that there are infinitely many closed non-hyperbolic 3–manifolds
$M_{i}$ of topological volume $v.$ In each $M_{i}$, choose a minimiser
$L_{i}$. Then $\operatorname{vol_{\mathbb{H}}}(M_{i}\setminus L_{i})=v$, for
each $i$, and hence there are only finitely many possibilities for
$M_{i}\setminus L_{i}.$ For the rest of the proof suppose the cusped
hyperbolic 3–manifold $M$ is a minimiser for infinitely many closed non-
hyperbolic 3–manifolds $M_{i}$. In particular, for these manifolds, we have
$M_{i}=M\big{(}\frac{p^{i}_{1}}{q^{i}_{1}},\ldots,\frac{p^{i}_{k}}{q^{i}_{k}}\big{)}$.
Since $M$ is a minimiser, leaving $k-1$ cusps unfilled and filling the $j$–th
cusp with surgery coefficient $\frac{p^{i}_{j}}{q^{i}_{j}}$ gives a non-
hyperbolic 3–manifold, since otherwise we would have a hyperbolic link
complement of smaller volume than $v$ in $M_{i}.$ But only finitely many
slopes $\frac{p^{i}_{j}}{q^{i}_{j}}$ on the $j$–th cusp may lead to a non-
hyperbolic filling [thurston-notes]. Observing this for every cusp
independently, we conclude that there are only finitely many possibilities for
$M_{i}.$ ∎

###  2.4 Covers and branched covers

It is straightforward to see that hyperbolic volume is multiplicative under
finite-sheeted covers. The following example shows that the analogous fact is
not true for topological volume in general.

######  Example 2.7.

We show in Section 4 that $m009$ is the unique minimiser of $\mathbb{R}P^{3},$
and hence
$\operatorname{volt}(\mathbb{R}P^{3})=\operatorname{vol_{\mathbb{H}}}(m009)\approx
2.66674$. One sees this by noting that $\mathbb{R}P^{3}$ is obtained as a Dehn
filling of $m009$ in a unique way, but it is not obtained by filling a cusped
hyperbolic 3–manifold of smaller volume. As we saw in Example 2.1, we have
$\operatorname{volt}(S^{3})\approx 2.02988$, so

$\operatorname{volt}(S^{3})<2\operatorname{volt}(\mathbb{R}P^{3})$

This shows that the topological volume is not multiplicative under finite
covers in general. A surgery description of $m009$ is given by performing a
$2$-surgery on one component of the Whitehead link $L5a1$. The complement of
the other link component in this surgery yields $m009$. From this surgery
description one can see that the lift to $S^{3}$ of the minimiser in
$\mathbb{R}P^{3}$ is the complement of $L6a1$ with volume
$2\cdot\operatorname{volt}(\mathbb{R}P^{3})\approx 5.33$.

Nonetheless we have the following behaviour of topological volume under
finite-sheeted covers.

######  Lemma 2.8.

Let $X$ and $Y$ be closed, orientable 3–manifolds. If there is a $d$–fold
cover $X\to Y$, then

$\operatorname{volt}(X)\leq d\cdot\operatorname{volt}(Y)$ (2.4)

###### Proof.

Suppose $X$ and $Y$ are closed, orientable 3–manifolds with $p\co X\to Y$ a
$d$–fold cover. Suppose $L$ in $Y$ is a hyperbolic link. Then $X\setminus
p^{-1}(L)\to Y\setminus L$ is a $d$–fold cover and in particular
$p^{-1}(L)\subset X$ is a hyperbolic link, with
$\operatorname{vol_{\mathbb{H}}}(X\setminus
p^{-1}(L))=d\cdot\operatorname{vol_{\mathbb{H}}}(Y\setminus L)$. A minimiser
$L\subset Y$ hence gives
$\operatorname{volt}(X)\leq\operatorname{vol_{\mathbb{H}}}(X\setminus
p^{-1}(L))=d\cdot\operatorname{vol_{\mathbb{H}}}(Y\setminus
L)=d\cdot\operatorname{volt}(Y).$ ∎

######  Example 2.9.

For each $d>0,$ choose $p$ and $q$ such that $d\cdot p$ and $q$ are coprime.
Then we have a $d$–fold covering of lens spaces $L(p,q)\to L(d\cdot p,q).$ We
show in Theorem 1.3 that for $p$ sufficiently large,
$\operatorname{volt}(L(d\cdot p,q))$ and $\operatorname{volt}(L(p,q))$ are
arbitrarily close to the volume of the Whitehead link complement. Hence the
gap in Equation 2.4 can be arbitrarily large, as $d$ varies.

######  Question 2.10.

If $X\to Y$ is a $d$–fold cover of non-hyperbolic closed, orientable
3–manifolds, is $\operatorname{volt}(X)<d\cdot\operatorname{volt}(Y)$, that
is, is Inequality (2.4) always strict?

As an application of Lemma 2.8 we observe that knowledge about the topological
volume can be leveraged to give bounds on degrees of covers. Recall that a
link $L\subset S^{3}$ is called _universal_ if every closed, orientable
3–manifold arises as a cover of $S^{3}$ branched along $L$. The existence of
universal links was first shown by Thurston in unpublished work. By work of
Hilden, Lozano, and Montesinos [HLM2, HLM4, HLM3], we know that there are
numerous universal links, including the figure eight knot, the Whitehead link,
and the Borromean rings.

######  Corollary 2.11.

Let $L$ be a hyperbolic universal link in $S^{3}$ and let $d_{L}(M)$ denote
the minimal degree of a cover $M\rightarrow S^{3}$ branched along $L$. Then

$\displaystyle\operatorname{volt}(M)\leq
d_{L}(M)\operatorname{vol_{\mathbb{H}}}(L)$

For example, we have

$\displaystyle\operatorname{volt}(M)\leq
d_{K4a1}(M)\operatorname{vol_{\mathbb{H}}}(K4a1)=d_{K4a1}(M)\operatorname{volt}(S^{3})$

In particular, we obtain general lower bounds on $d_{L}(M)$ from the the
topological volumes of $S^{3}$ and $M$. On the other hand, we remark that
$d_{L}(M)$ is algorithmically computable, since the decision problem for
3–manifolds is solved [Kuperberg].

###  2.5 Constructions of unbounded topological volume

The main observation used in this section is that, following
Jørgensen–Thurston theory [thurston-notes, Gro81, Mai10], all hyperbolic
3–manifolds of volume bounded from above by a constant are obtained by Dehn
surgery on a finite set of cusped hyperbolic 3–manifolds. In particular, every
infinite sequence of hyperbolic manifolds of bounded volume must have a
subsequence with converging volumes, and which is obtained from a sequence of
Dehn fillings on a fixed cusped hyperbolic $3$–manifold. This implies bounds
on a number of topological invariants of those manifolds that have bounded
topological volume, such as rank of the fundamental group, rank of homology,
and hence number of JSJ pieces or prime summands in a decomposition of the
manifold. In more precise terms we have the following result.

######  Proposition 2.12.

Let $\mathbf{I}$ be a real valued topological invariant of $3$–manifolds with
the following property: If $M$ denotes a $3$–manifold with at least one torus
boundary component, then $\mathbf{I}(M)\geq\mathbf{I}(M(r))$ for all but
finitely many slopes $r$, where $M(r)$ denotes the Dehn-filling of $M$ with
slope $r$. Let $(M_{n})_{n\in\mathbb{N}}$ be a family of closed, orientable
$3$–manifolds with $\mathbf{I}(M_{n})\rightarrow\infty$ for
$n\rightarrow\infty$. Then $\operatorname{volt}(M_{n})\rightarrow\infty$ for
$n\rightarrow\infty.$

###### Proof.

Suppose that $(\operatorname{volt}(N_{m}))_{m\in\mathbb{N}}$ is a bounded
sequence. Then, by Jørgensen–Thurston theory, as explained at the beginning of
Section 2.5, the elements of some subsequence
$(N_{m_{n}})_{n\in\mathbb{N}}=(M_{n})_{n\in\mathbb{N}}$ are obtained by Dehn
filling a fixed cusped hyperbolic $3$–manifold $M$. This implies that, by
assumption, the values $\mathbf{I}(M_{n})$ must also be bounded. This is a
contradiction. ∎

Proposition 2.12 has the following immediate consequences.

######  Corollary 2.13.

  1. 1.

Let $(M_{n})$ be a sequence of closed, orientable $3$–manifolds with unbounded
Heegaard genus or surgery number. Then $(\operatorname{volt}(M_{n}))$ is
unbounded.

  2. 2.

Let $(M_{n})$ be a sequence of closed, orientable $3$–manifolds with unbounded
rank of first homology or fundamental group. Then
$(\operatorname{volt}(M_{n}))$ is unbounded.

  3. 3.

For each $i$, let $M_{i}$ be a closed, orientable 3–manifold which is not
homeomorphic to $S^{3}$. Then
$\operatorname{volt}(M_{1}\\#\cdots\\#M_{n})\to\infty$ as $n\to\infty$.

  4. 4.

Let $(M_{n})$ be a sequence of closed, orientable Seifert fibered spaces such
that the number of exceptional fibres is unbounded. Then
$(\operatorname{volt}(M_{n}))$ is unbounded.

###### Proof.

$(1)$ and $(2)$ follow directly from Proposition 2.12 since these complexities
are well-known not to decrease under Dehn filling. $(3)$ follows from $(2)$
and the fact that the rank of the fundamental group is additive under
connected sum. Similarly, the rank of the first homology is unbounded if the
number of exceptional fibres in a Seifert fibered space is unbounded and thus
$(4)$ follows. ∎

##  3 Bounds on topological volume

Bounds on the topological volume of manifolds in terms of covers and branched
covers were determined in Section 2.4. We now give upper and lower bounds
arising from topological or algebraic properties of a non-hyperbolic
3–manifold.

###  3.1 Upper bounds from surgery descriptions or triangulations

Suppose the $3$–manifold $M$ is given by surgery on a hyperbolic link
$L\subset S^{3}.$ By definition of the surgery diagram, there is a link
$L^{*}\subset M$ such that $S^{3}\setminus L=M\setminus L^{*}.$ We thus have
$\operatorname{vol_{\mathbb{H}}}(S^{3}\setminus
L)=\operatorname{vol_{\mathbb{H}}}(M\setminus
L^{*})\geq\operatorname{volt}(M)$.

######  Example 3.1.

The 3–torus $T^{3}$ is given by $0$-surgery on all components of the Borromean
rings $B$ (a hyperbolic link). Thus there is a hyperbolic link $B^{*}\subset
T^{3}$ with the same complement as $B$ in $S^{3}.$ In particular, we have
$\operatorname{volt}(T^{3})\leq\operatorname{vol_{\mathbb{H}}}(B)<7.33$.

The following theorem expands on this idea to give a general upper bound.

######  Theorem 1.1.

Let $M$ be a closed, orientable 3–manifold given by surgery along a nontrivial
$n$-component link $L\subset S^{3}$ with crossing number $c=c(L)$. Then

$\operatorname{volt}(M)\leq\big{(}9c+15n-20\big{)}v_{oct}+4v_{tet}$

where $v_{oct}\approx 3.6638$ and $v_{tet}\approx 1.01494$ are the volumes of
a regular hyperbolic ideal octahedron and a regular hyperbolic ideal
tetrahedron respectively.

###### Proof.

The link $L$ may not be hyperbolic. We first show how to construct a
hyperbolic link $\bar{L}$ in $S^{3}$, so that the complement of $L$ is a Dehn
filling of the complement of $\bar{L}$. In other words, the complement of
$\bar{L}$ is the complement of a hyperbolic link in $M$, so by computing an
upper bound for the volume of $\bar{L}$ we obtain an upper bound on the
topological volume of $M$. Note that in general $L$ will not be a sublink of
$\bar{L}$.

To construct $\bar{L}$ we follow an algorithm due to Adams [Adams_Augmented].
In that process we keep track of the new crossing number. Following this
algorithm more carefully in specific examples will usually improve the stated
bound in the theorem. We now describe the algorithm.

  1. 1.

We start with a diagram $D$ of $L$ that has the minimal number of crossings
$c=c(L)$. First, we introduce a circle (an augmentation) around each twist
region of the link diagram. This introduces $4$ new crossings for every twist
region. Since the number of twist regions is bounded from above by the number
of crossings, we obtain a link diagram $D_{1}$, with at most $5c$ crossings,
consisting of the old diagram $D$ and augmentations $C_{1}$ around each twist
region of $D$.

  2. 2.

We remove or change crossings in the twist regions of $D$ until we get a
diagram $D_{2}$, consisting of a diagram of an alternating link $A_{2}$
together with circles $C_{2}=C_{1}$ around each twist region of $A_{2}$. This
does not increase the number of crossings and thus $D_{2}$ has at most $5c$
crossings. Note that here $A_{2}$ is no longer isotopic to our original link
$L$. However by cutting along disks bounded by the components of $C_{1}$ and
twisting, we see that the complement of $L$ arises as a Dehn filling of the
complement of $D_{2}$, which is all we need.

  3. 3.

We add unknotted components to $A_{2}$ (that introduce no new crossings with
the augmentations) to transform it into a non-split alternating diagram. This
will add at most $4(n-1)$ additional crossings to the link diagram $D_{2}$ and
at most $2(n-1)$ twist regions to $A_{2}$, and produce a link diagram $D_{3}$
with at most $5c+4n-4$ crossings that consists of a non-split alternating
diagram $A_{3}$ together with some augmentations $C_{3}=C_{2}$.

  4. 4.

For each reducing sphere of the diagram $A_{3}$, we add one unknotted
component to the diagram to get a non-split, prime, alternating diagram
$A_{4}$. This can be done in such a way that the new unknotted components have
no crossings with the augmentation circles. The diagram $A_{3}$ has at most
$c+4n-4$ crossings and thus it contains at most $(c+4n-4)/3$ reducing spheres.
The unknotted components can be added in such a way that we introduce for each
unknotted component $4$ crossings and two additional twist regions. Thus we
get a diagram $D_{4}$ consisting of a non-split, prime, alternating diagram
$A_{4}$ together with augmentations $C_{4}$ around all but at most
$(2c+14n-14)/3$ twist regions. The diagram $D_{4}$ has at most
$(19c+28n-28)/3$ crossings.

  5. 5.

We add augmentations around the at most $(2c+14n-14)/3$ twists regions of
$A_{4}$ that are not already augmented. This yields a diagram $D_{5}$ of a
link $\bar{L}$ that is augmented alternating and has at most $9c+15n-15$
crossings.

The construction ensures that the link $\bar{L}$ is augmented alternating. It
is thus hyperbolic by a result of Adams [Adams_Augmented]. Hence

$\operatorname{volt}(M)\leq\operatorname{vol_{\mathbb{H}}}(\bar{L})$

The claimed inequality in the theorem statement follows from another result of
Adams [Adams_Bound], based on a construction by Weeks [handbookknottheory,
Section 3], stating that for every hyperbolic link $J$ in $S^{3}$ with more
than $4$ crossings, the volume is bounded from above by
$(c(J)-5)v_{oct}+4v_{tet}$. ∎

######  Remark 3.2.

For a specific given surgery presentation, the bound above can be greatly
improved. For example, in the case that a closed, orientable 3–manifold $M$ is
described as surgery on a knot $K$, the main result of [BAR-altvolume] implies
that $\operatorname{volt}(M)\leq 10(5c(K)-1)v_{tet}$. The key ingredient of
[BAR-altvolume] is a result of [Blair-altaug], showing that the complement of
every knot $K\subset S^{3}$ contains an unknot $U$ so that the link $K\sqcup
U$ is hyperbolic. In [BAR-altvolume] it is shown that the hyperbolic volume of
the link $K\sqcup U$ is bounded above by $10(5c(K)-1)v_{tet}$. Since
$S^{3}\smallsetminus(K\sqcup U)$ is naturally a submanifold of $M$, our claim
follows.

The above bound can be applied to the standard surgery diagram of a Seifert
fibered space. However, in this specific situation the above strategy can be
improved as follows.

######  Corollary 3.3.

Let $SFS(g;r_{1},\ldots,r_{N})$ be the Seifert fibered space over a surface of
genus $g$ with Seifert invariants $(r_{1},\ldots,r_{N})$. Then

$\operatorname{volt}(SFS(g;r_{1},\ldots,r_{N}))\leq\begin{cases}v_{oct}<3.67\,\textrm{
if $(g,n)=(0,0)$ or $(0,1)$}\\\
(10g+6N-9)v_{oct}+4v_{tet}<37g+22N-28\,\textrm{ otherwise}\end{cases}$

###### Proof.

Consider the standard surgery diagram of the Seifert fibered space
$SFS(g;r_{1},\ldots,r_{N})$ shown in Figure 1. The surgery link is a connected
sum of $g$ copies of the Borromean rings and $N$ copies of the Hopf link. Add
$N+g-1$ many further unknots to that surgery diagram as indicated in Figure 1
to obtain a connected prime alternating diagram of a link $L$ that contains
the surgery link of $SFS(g;r_{1},\ldots,r_{N})$ as a sublink. Since $L$ also
contains the Borromean rings as a sublink, it is not a torus link. Now a
result of Menasco [Menasco84] implies that $L$ is hyperbolic.

Since $SFS(g;r_{1},\ldots,r_{N})$ is obtained by surgery on $L$, the volume of
$L$ is an upper bound on the topological volume of
$SFS(g;r_{1},\ldots,r_{N})$. From Figure 1 we see that $L$ has crossing number
$10g+6N-4$ and thus the result of Adams [Adams_Bound] mentioned above implies
the stated bound. ∎

$0$
---
$0$
---
$0$
---
$0$
---
$0$
---
$0$
---
$r_{1}$
---
$r_{2}$
---
$g$
---
$g$
---
$N$
---
$r_{N}$
---
$0$
---
$N$
---
Figure 1: Left: The standard surgery diagram of a Seifert fibered space with
genus $g$ and $N$ exceptional fibres. The surgery link is a connected sum of
$g$ copies of the Borromean rings and $N$ copies of the Hopf link. Right: A
hyperbolic link that contains the standard surgery diagram of a Seifert
fibered space as a sublink.

Although the bound from Corollary 3.3 improves the general bound from Theorem
1.1, it is far from optimal. For example, using SnapPy [snappy] to compute the
volume of $L$ for $g=2$ and $N=3$ gives a bound slightly smaller than $70$,
while the upper bound from Corollary 3.3 is $112$.

######  Proposition 3.4.

Let $M$ be a closed, orientable 3–manifold. Denote the Heegaard genus of $M$
by $g(M).$ There does not exist a function $f:~{}\mathbb{N}\to\mathbb{N}$ such
that all 3–manifolds $M$ satisfy

$\operatorname{volt}(M)\leq f(g(M))$

###### Proof.

It suffices to demonstrate the existence of an infinite family of hyperbolic
manifolds with unbounded volumes but bounded Heegaard genus. Indeed, there
exist Berge knots with arbitrarily large volume [Baker_Berge_knots, Theorem
4.1]. On the other hand, Berge knots have tunnel number one and thus the knot
exterior of a Berge knot has Heegaard genus $2$. Since Dehn surgery cannot
increase the Heegaard genus, we obtain an infinite family of closed hyperbolic
manifolds of Heegaard genus $2$ but arbitrarily large volume. ∎

######  Proposition 3.5.

For a closed, orientable 3–manifold $M$, let $t(M)$ be the triangulation
complexity of $M$, i.e. the minimal number of tetrahedra needed to triangulate
$M$. There exists an increasing function $f:\mathbb{N}\to\mathbb{N}$ such that
for each $M$, we have

$\operatorname{volt}(M)\leq f(t(M))$

###### Proof.

This essentially follows from [My82]. For every triangulation of $M$, an
explicit hyperbolic knot $K\subset M$ is constructed. From this construction
we obtain an upper bound on the number of ideal tetrahedra needed to ideally
triangulate the complement of $K$ in $M$. This bound only depends on the
number of tetrahedra used to triangulate $M$, and not on the topology of $M$
or $M\setminus K$. The upper bound on the number of ideal tetrahedra gives an
upper bound on the topological volume of $M$. ∎

###  3.2 Lower bounds

In Corollary 2.13 (2) we have shown that a sequence of manifolds with
unbounded rank of first homology has unbounded topological volume. In the next
theorem, we develop a lower bound which quantifies this.

######  Theorem 1.2.

Let $M$ be a closed, orientable 3–manifold and $p$ be a prime. Then

$\frac{\operatorname{rk}(H_{1}(M;\mathbb{Z}_{p}))}{168.602}<\operatorname{volt}(M)$

Moreover, we have $\operatorname{volt}(M)>3.08,$ $3.69$, $3.77$ for
$\operatorname{rk}(H_{1}(M;\mathbb{Z}_{2}))>5$, $7$, $10$, respectively.

###### Proof.

A result by Guzman–Shalen [Guzman_Shalen] states that for every finite-volume,
orientable, hyperbolic 3-manifold $N$, and every prime $p$, we have
$\operatorname{rk}(H_{1}(N;\mathbb{Z}_{p}))<168.602\cdot\operatorname{vol_{\mathbb{H}}}(N)$.
Now let $L\subset M$ be a volume minimiser of $M$, and set $N=M\setminus L$.
Since the rank of the first homology cannot increase under filling, the first
inequality follows, i.e.

$\frac{\operatorname{rk}(H_{1}(M;\mathbb{Z}_{p}))}{168.602}\leq\frac{\operatorname{rk}(H_{1}(M\setminus
L;\mathbb{Z}_{p}))}{168.602}<\operatorname{vol_{\mathbb{H}}}(M\setminus
L)=\operatorname{volt}(M)$

For the other inequalities we use the main results of [Culler_Shalen2011,
Culler_Shalen2012, Guzman_Shalen2020] stating that for every closed, finite-
volume, orientable, hyperbolic $3$–manifold $N$ with
$\operatorname{rk}(H_{1}(N;\mathbb{Z}_{2}))>5$, $7$ or $10$ we have
$\operatorname{vol_{\mathbb{H}}}(N)>3.08$, $3.69$ or $3.77$ respectively. Let
$L$ be a volume minimiser for $M$. For every cusp $c_{i}$ of $M\setminus L$ we
choose a sequence of slopes $p^{c_{i}}_{n}/q^{c_{i}}_{n}$ whose lengths
converge to $\infty$ and such that filling $M\setminus L$ with these slopes
yield manifolds $N_{n}$ with
$\operatorname{rk}(H_{1}(N_{n};\mathbb{Z}_{2}))=\operatorname{rk}(H_{1}(M;\mathbb{Z}_{2}))$.
Note that every sequence of even slopes with diverging coefficients will
satisfy these requirements. Since the lengths of the slopes converge to
infinity it follows that (a) the manifolds $N_{n}$ are hyperbolic for $n$
sufficiently large, and (b) the volumes of $N_{n}$ converge to the volume of
$M\setminus L$. Thus, applying the results of [Culler_Shalen2011,
Culler_Shalen2012, Guzman_Shalen2020] to the $N_{n}$ yields the claimed
inequalities for $\operatorname{volt}(M)$. ∎

######  Example 3.6.

Although our general upper and lower bounds are far from sharp, we claim that
their asymptotic behaviour is the same. Let $M$ be the Seifert fibered space
of genus $g$ and $n$ singular fibres $SFS(g;p,\ldots,p)$ for a prime $p$. From
the standard surgery description (see Figure 1) of $M$ we compute
$\operatorname{rk}(H_{1}(M;\mathbb{Z}_{p}))=2g+n$ and thus Theorem 1.2 and
Corollary 3.3 yield

$\frac{2g+n}{168.602}<\operatorname{volt}(M)<37g+22n-28$

We observe that the quotient of the upper bound divided by the lower bound is
bounded.

##  4 The minimal topological volume manifolds

This section classifies all non-hyperbolic $3$–manifolds with topological
volume less than $3.07$, giving their volumes and their volume minimising
hyperbolic links. A number of observations from this census are subsequently
made.

###  4.1 A census

######  Proposition 4.1.

The complete list of non-hyperbolic closed 3–manifolds of topological volumes
at most $3.07$, their explicit volumes and their minimisers are given in
LABEL:tab:complete. In particular, each of these manifolds contains a unique
volume minimiser. In all cases the volume minimiser is a knot.

###### Proof.

From Theorem 1.5 of [GHMTY21], cf. [Mi09], we know the complete list of cusped
hyperbolic 3–manifolds of volume at most $3.07$. These are:

$\mathcal{V}=\big{[}m003,m004,m006,m007,m009,m010,m011,m015,m016,m017,m019,m022,m023,m026\big{]}$

All of these have exactly one cusp. Hence the non-hyperbolic 3–manifolds of
topological volumes at most $3.07$ are precisely the exceptional fillings on
these manifolds, and thus there are finitely many.

We claim that the volumes of the manifolds in the above list satisfy:

$\displaystyle\operatorname{vol_{\mathbb{H}}}(m003)=\operatorname{vol_{\mathbb{H}}}(m004)$
$\displaystyle<\operatorname{vol_{\mathbb{H}}}(m006)=\operatorname{vol_{\mathbb{H}}}(m007)$
$\displaystyle<\operatorname{vol_{\mathbb{H}}}(m009)=\operatorname{vol_{\mathbb{H}}}(m010)$
$\displaystyle<\operatorname{vol_{\mathbb{H}}}(m011)$
$\displaystyle<\operatorname{vol_{\mathbb{H}}}(m015)=\operatorname{vol_{\mathbb{H}}}(m016)=\operatorname{vol_{\mathbb{H}}}(m017)$
$\displaystyle<\operatorname{vol_{\mathbb{H}}}(m019)$
$\displaystyle<\operatorname{vol_{\mathbb{H}}}(m022)=\operatorname{vol_{\mathbb{H}}}(m023)$
$\displaystyle<\operatorname{vol_{\mathbb{H}}}(m026)$

We first note that the above equalities and inequalities are correct if one
computes the volumes with precision _up to two decimal places_. Hence it
remains to show that the stated equalities hold.

We first note that $m003$ and $m004$ are once-punctured torus bundles with
monodromies $-RL$ and $RL$ respectively. Hence they are related by volume
preserving mutation along the fibre and so
$\operatorname{vol_{\mathbb{H}}}(m003)=\operatorname{vol_{\mathbb{H}}}(m004).$
Similarly, $m009$ and $m010$ are once-punctured torus bundles with monodromies
$RRL$ and $-RRL$ respectively; and $m022$ and $m023$ are once-punctured torus
bundles with monodromies $-RRRL$ and $RRRL$ respectively.

For
$\operatorname{vol_{\mathbb{H}}}(m006)=\operatorname{vol_{\mathbb{H}}}(m007)$
and
$\operatorname{vol_{\mathbb{H}}}(m015)=\operatorname{vol_{\mathbb{H}}}(m016)=\operatorname{vol_{\mathbb{H}}}(m017)$
we refer to Appendix A, where we use the gluing and completeness equations to
show that the manifolds claimed to have the same volume are built from
tetrahedra with equal shapes.

In [Du20], Dunfield provides a complete list of all exceptional fillings of
SnapPy [snappy] manifolds with one cusp. This algorithmic search is made
feasible by the $6$-theorem of Agol and Lackenby [agol-bounds, lackenby-6],
and the fact that the low complexity non-hyperbolic closed 3–manifolds that
arise can easily be identified.

Thus we can determine the topological volume of any exceptional filling along
a manifold in $\mathcal{V}$ as follows. First enumerate all exceptional
fillings of the two manifolds of smallest volume. Then enumerate all
exceptional fillings of the manifolds with second smallest volume in the list
and only keep those non-hyperbolic 3–manifolds that have not yet been found.
Then continue this process iteratively. If an exceptional filling is found as
fillings of multiple distinct cusped manifolds of the same volume, or by
filling one manifold with different surgery coefficients, we report all
instances in which it arises. We present the results in LABEL:tab:complete,
where we sort our results by volume. ∎

Table 1: The complete list of non-hyperbolic manifolds of volume less than $3.07$ ordered by their volumes.  Manifold | Volume | Realisations |
---|---|---|---
$S^{3}$ | $2.02988321281931$ | $m004(1,0)$ |
$L(5,1)$ | $2.02988321281931$ | $m003(-1,1)$ | $m003(0,1)$
$L(10,3)$ | $2.02988321281931$ | $m003(1,0)$ |
$SFS[S2:(2,1)(3,2)(3,-1)]$ | 2.02988321281931 | $m003(-2,1)$ | $m003(1,1)$
$SFS[S2:(2,1)(3,1)(7,-6)]$ | $2.02988321281931$ | $m004(-1,1)$ | $m004(1,1)$
$SFS[S2:(2,1)(4,1)(5,-4)]$ | $2.02988321281931$ | $m004(-2,1)$ | $m004(2,1)$
$SFS[S2:(3,1)(3,1)(4,-3)]$ | $2.02988321281931$ | $m004(-3,1)$ | $m004(3,1)$
$T\times I/[-2,-1|-1,-1]$ | $2.02988321281931$ | $m003(-1,2)$ |
$T\times I/[2,1|1,1]$ | $2.02988321281931$ | $m004(0,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,2)]$ | $2.02988321281931$ | $m003(1,2)$ | $m003(-3,2)$
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,0)}SFS[D:(2,1)(3,1)]$ | $2.02988321281931$ | $m004(-4,1)$ | $m004(4,1)$
$L(15,4)$ | $2.56897060093671$ | $m006(0,1)$ |
$L(5,2)$ | $2.56897060093671$ | $m006(1,0)$ |
$L(3,1)$ | $2.56897060093671$ | $m007(1,0)$ |
$SFS[S2:(2,1)(3,2)(4,-3)]$ | $2.56897060093671$ | $m006(-1,1)$ |
$SFS[S2:(2,1)(2,1)(3,2)]$ | $2.56897060093671$ | $m006(1,1)$ |
$SFS[S2:(2,1)(2,1)(5,-2)]$ | $2.56897060093671$ | $m007(1,1)$ |
$SFS[S2:(2,1)(3,1)(7,-5)]$ | $2.56897060093671$ | $m006(-2,1)$ |
$SFS[S2:(3,1)(3,1)(3,-1)]$ | $2.56897060093671$ | $m007(0,1)$ |
$SFS[S2:(2,1)(4,1)(5,-3)]$ | $2.56897060093671$ | $m007(-1,1)$ |
$SFS[S2:(2,1)(3,1)(9,-7)]$ | $2.56897060093671$ | $m007(-2,1)$ |
$SFS[A:(2,1)]/[0,-1|-1,0]$ | $2.56897060093671$ | $m006(-3,1)$ |
$SFS[D:(2,1)(3,1)]\cup_{(-1,1|0,1)}SFS[D:(2,1)(3,1)]$ | $2.56897060093671$ | $m006(1,2)$ |
$SFS[A:(2,1)]/[0,1|1,0]$ | $2.56897060093671$ | $m007(-3,1)$ |
$L(2,1)$ | $2.66674478344906$ | $m009(1,0)$ |
$L(6,1)$ | $2.66674478344906$ | $m010(1,0)$ |
$SFS[S2:(2,1)(2,1)(4,-1)]$ | $2.66674478344906$ | $m010(0,1)$ |
$SFS[S2:(2,1)(3,1)(8,-7)]$ | $2.66674478344906$ | $m009(0,1)$ |
$SFS[S2:(2,1)(4,1)(6,-5)]$ | $2.66674478344906$ | $m009(-1,1)$ |
$SFS[S2:(3,1)(3,1)(5,-4)]$ | $2.66674478344906$ | $m009(-2,1)$ |
$SFS[S2:(3,1)(3,2)(3,-1)]$ | $2.66674478344906$ | $m010(1,1)$ |
$T\times I/[3,2|1,1]$ | $2.66674478344906$ | $m009(1,1)$ |
$T\times I/[-3,-2|-1,-1]$ | $2.66674478344906$ | $m010(-2,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,0)}SFS[D:(2,1)(4,1)]$ | $2.66674478344906$ | $m009(-3,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(-1,2|0,1)}SFS[D:(2,1)(3,1)]$ | $2.66674478344906$ | $m009(3,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(4,3)]$ | $2.66674478344906$ | $m010(2,1)$ |
$L(2,1)\\#L(3,1)$ | $2.66674478344906$ | $m010(-1,1)$ |
$L(9,2)$ | $2.78183391239608$ | $m011(0,1)$ |
$L(13,3)$ | $2.78183391239608$ | $m011(1,0)$ |
$SFS[S2:(2,1)(3,2)(4,-1)]$ | $2.78183391239608$ | $m011(-1,1)$ |
$SFS[S2:(2,1)(2,1)(3,-2)]$ | $2.78183391239608$ | $m011(1,1)$ |
$SFS[S2:(2,1)(3,2)(5,-3)]$ | $2.78183391239608$ | $m011(2,1)$ |
$SFS[D:(2,1)(3,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,2)]$ | $2.78183391239608$ | $m011(-1,2)$ |
$L(19,7)$ | $2.82812208833078$ | $m016(-1,1)$ |
$L(18,5)$ | $2.82812208833078$ | $m016(0,1)$ |
$L(21,8)$ | $2.82812208833078$ | $m017(-1,1)$ |
$L(14,3)$ | $2.82812208833078$ | $m017(0,1)$ |
$L(7,2)$ | $2.82812208833078$ | $m017(1,0)$ |
$SFS[S2:(2,1)(4,1)(7,-5)]$ | $2.82812208833078$ | $m015(0,1)$ |
$SFS[S2:(3,1)(3,1)(5,-3)]$ | $2.82812208833078$ | $m015(1,1)$ |
$SFS[S2:(2,1)(3,1)(11,-9)]$ | $2.82812208833078$ | $m015(-1,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,0)}SFS[D:(2,1)(3,2)]$ | $2.82812208833078$ | $m015(2,1)$ |
$SFS[A:(2,1)]/[0,1|1,-1]$ | $2.82812208833078$ | $m015(-2,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(1,1|0,1)}SFS[D:(2,1)(3,1)]$ | $2.82812208833078$ | $m016(2,1)$ |
$SFS[D:(2,1)(3,1)]\cup_{(-1,1|0,1)}SFS[D:(2,1)(3,2)]$ | $2.82812208833078$ | $m016(-1,2)$ |
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,1)]$ | $2.82812208833078$ | $m017(-2,1)$ |
$SFS[A:(2,1)]/(0,-1|-1,1)$ | $2.82812208833078$ | $m017(2,1)$ |
$L(17,5)$ | $2.944106486676$ | $m019(0,1)$ |
$L(11,3)$ | $2.944106486676$ | $m019(1,1)$ |
$SFS[S2:(2,1)(3,2)(5,-2)]$ | $2.944106486676$ | $m019(-1,1)$ |
$SFS[D:(2,1)(3,2)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,2)]]$ | $2.944106486676$ | $m019(-2,1)$ |
$L(7,1)$ | $2.98912028293$ | $m022(1,0)$ |
$SFS[S2:(2,1)(3,1)(4,-1)]$ | $2.98912028293$ | $m022(0,1)$ |
$SFS[S2:(3,2)(3,2)(4,-3)]$ | $2.98912028293$ | $m022(-1,1)$ |
$SFS[S2:(2,1)(3,1)(9,-8)]$ | $2.98912028293$ | $m023(0,1)$ |
$SFS[S2:(2,1)(4,1)(7,-6)]$ | $2.98912028293$ | $m023(1,1)$ |
$SFS[S2:(3,1)(3,1)(6,-5)]$ | $2.98912028293$ | $m023(2,1)$ |
$TxI/[-4,-3|-1,-1]$ | $2.98912028293$ | $m022(2,1)$ |
$TxI/[4,3|1,1]$ | $2.98912028293$ | $m023(-1,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(5,4)]]$ | $2.98912028293$ | $m022(-2,1)$ |
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,0)}SFS[D:(2,1)(5,1)]$ | $2.98912028293$ | $m023(3,1)$ |
$L(19,4)$ | $3.059338057779$ | $m026(0,1)$ |
$L(8,3)$ | $3.059338057779$ | $m026(1,0)$ |
$SFS[S2:(2,1)(3,2)(5,-4)]$ | $3.059338057779$ | $m026(-1,1)$ |
$SFS[S2:(2,1)(3,1)(3,2)]$ | $3.059338057779$ | $m026(1,1)$ |
$SFS[D:(2,1)(3,1)]\cup_{(-1,1|0,1)}SFS[D:(2,1)(4,1)]$ | $3.059338057779$ | $m026(1,2)$ |

######  Remark 4.2.

Upper bounds on the volume of non-hyperbolic manifolds not appearing in our
census can be computed from Dunfield’s data [Du20]. The sharpness on these
bounds depends on determining further terms in the length spectrum of one-
cusped hyperbolic 3–manifolds. The data ordered for our setting can be
accessed at [data].

###  4.2 Parenthood

Next, we describe all ways in which the fourteen $1$-cusped manifolds in
$\mathcal{V}$ of volumes at most $3.07$ can arise as fillings of the two
$2$-cusped manifolds of minimal volume [agol-2cusped]. These are the Whitehead
link complement $m129$ and the Whitehead link sister $m125$; compare Section
5.

######  Proposition 4.3.

Every manifold in $\mathcal{V}$ can be obtained by filling the first cusp of
$m125$ or $m129$. The complete list of surgeries on $m125$ and $m129$ that
yield a manifold in $\mathcal{V}$ is shown in Table 2. Because of the symmetry
of $m125$, we obtain a manifold in two ways. Both manifolds admit an isometry
that interchanges the components and thus we also have the analogous result
for the second cusp.

###### Proof.

For the proof we use a result of Futer–Kalfagianni–Purcell
[FuterKalfagianniPurcell] giving a bound on the possible length of slopes
yielding manifolds of volume at most $3.07$. In [data] we explicitly compute
these finite sets of slopes and identify all filled manifolds. The results are
summarised in Table 2. ∎

Table 2: The parenthood diagram of the low volume $1$-cusped manifolds.  Manifold | Surgery on $m129$ | Surgery on $m125$
---|---|---
$m003$ | $m129(3,1)(0,0)$ |  $m125(-2,1)(0,0)$ and $m125(1,2)(0,0)$
$m004$ | $m129(-3,1)(0,0)$ | none
$m006$ | $m129(1,2)(0,0)$ |  $m125(3,1)(0,0)$ and $m125(-1,3)(0,0)$
$m007$ | $m129(-1,2)(0,0)$ | none
$m009$ | $m129(-4,1)(0,0)$ | none
$m010$ | $m129(4,1)(0,0)$ | none
$m011$ | none |  $m125(-3,1)(0,0)$ and $m125(1,3)(0,0)$
$m015$ | $m129(-3,2)(0,0)$ | none
$m016$ | none |  $m125(3,2)(0,0)$ and $m125(-2,3)(0,0)$
$m017$ | $m129(3,2)(0,0)$ | none
$m019$ | none |  $m125(-3,2)(0,0)$ and $m125(2,3)(0,0)$
$m022$ | $m129(5,1)(0,0)$ | none
$m023$ | $m129(-5,1)(0,0)$ | none
$m026$ | none |  $m125(4,1)(0,0)$ and $m125(-1,4)(0,0)$

###  4.3 Homology classes of minimisers

######  Example 4.4.

A minimiser does not need to be nullhomologous. Indeed, the unique minimiser
for $L(10,3)$ is given by $m003$; see LABEL:tab:complete. Since $m003(1,0)$ is
diffeomorphic to $L(10,3)$, the minimiser $m003$ represents the complement of
a knot $K$ in $L(10,3)$. We claim that $K$ is not nullhomologous. For that we
observe that the first homology of $m003$ is isomorphic to
$\mathbb{Z}\oplus\mathbb{Z}_{5}$ and thus $K$ must represent an element of
order two in $H^{1}(L(10,3))$. More concretely, we can describe $K$ as the
blue knot in the surgery diagram of $L(10,3)$ along the red unknot $U$ shown
in Figure 2. From that diagram we compute the linking number of $K$ and $U$ to
be $\pm 5$ and thus $K$ represents $5$ in $\mathbb{Z}_{10}\cong
H_{1}(L(10,3))$.

Figure 2: The surgery on the red unknot yields $L(10,3)$. We can use SnapPy to
verify that the complement of the blue knot is isometric to $m003$ and thus
represents the minimiser of $L(10,3)$.

In the proof of Proposition 4.1 we have listed only those knots that minimise
the volume. However, among the $1$-cusped manifolds with volume at most $3.07$
there are more exceptional fillings (that do not realise)
$\operatorname{volt}$. These are printed in Table 3. Since the homologies of
some of these knot complements differ from the homologies of the exteriors of
the minimisers from LABEL:tab:complete, we also obtain examples of minimisers
in different homology classes.

Table 3: All other hyperbolic knots in non-hyperbolic manifolds that have volume less than $3.07$. None of these knots is a minimiser since a knot of smaller volume is already presented in LABEL:tab:complete. knot | ambient manifold | realisation
---|---|---
$m015$ | $S3$ | $m015(1,0)$
$m016$ | $S3$ | $m016(1,0)$
$m023$ | $L(3,1)$ | $m023(1,0)$
$m019$ | $L(6,1)$ | $m019(1,0)$
$m022$ | $L(7,2)$ | $m022(1,1)$
$m007$ | $SFS[S2:(2,1)(3,2)(3,-1)]$ | $m007(2,1)$
$m016$ | $SFS[S2:(2,1)(3,2)(5,-3)]$ | $m016(1,1)$
$m009$ | $SFS[S2:(2,1)(4,1)(5,-4)]$ | $m009(2,1)$
$m023$ | $SFS[S2:(3,1)(3,1)(4,-3)]$ | $m023(-2,1)$
$m026$ | $SFS[S2:(2,1)(3,1)(9,-7)]$ | $m026(-2,1)$
$m016$ | $SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,2)]$ | $m016(-2,1)$
$m017$ | $SFS[D:(2,1)(3,1)]\cup_{(-1,1|0,1)}SFS[D:(2,1)(3,1)]$ | $m017(-1,2)$
$m019$ | $SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,1)]$ | $m019(1,2)$

######  Example 4.5.

In Table 4 we display the isomorphism types of the homologies of the knot
exteriors from Table 3 whenever the homology differs from the homology of the
minimiser. For example, the minimiser $m010$ of $L(6,1)$ is nullhomologous in
$L(6,1)$, while the hyperbolic knot with second smallest volume $m019$
represents a generator in the homology of $L(6,1)$.

######  Example 4.6.

We also have examples of manifolds where the knot with smallest and the knot
with second smallest hyperbolic volumes represent different non-trivial
torsion elements in homology. For example, let $M$ be
$SFS[S2:(2,1)(3,2)(3,-1)]$. Then the volume of $M$ is realised by $m003$ and
the hyperbolic knot with second smallest volume is given by $m007$. Since
their homologies are different they represent different non-trivial torsion
elements in $H_{1}(M)\cong\mathbb{Z}_{15}$.

Table 4: Homologies of the minimisers compared with the homologies of the knots from Table 3. ambient manifold | homology | minimiser | homology | knot | homology
---|---|---|---|---|---
$L(6,1)$ | $\mathbb{Z}_{6}$ | $m010$ | $\mathbb{Z}_{6}+\mathbb{Z}$ | $m019$ | $\mathbb{Z}$
$SFS[S2:(2,1)(3,2)(3,-1)]$ | $\mathbb{Z}_{15}$ | $m003$ | $\mathbb{Z}_{5}+\mathbb{Z}$ | $m007$ | $\mathbb{Z}_{3}+\mathbb{Z}$
$SFS[S2:(2,1)(4,1)(5,-4)]$ | $\mathbb{Z}_{2}$ | $m004$ | $\mathbb{Z}$ | $m009$ | $\mathbb{Z}_{2}+\mathbb{Z}$
$SFS[S2:(3,1)(3,1)(4,-3)]$ | $\mathbb{Z}_{3}$ | $m004$ | $\mathbb{Z}$ | $m023$ | $\mathbb{Z}_{3}+\mathbb{Z}$
$SFS[S2:(2,1)(3,1)(9,-7)]$ | $\mathbb{Z}_{3}$ | $m007$ | $\mathbb{Z}_{3}+Z$ | $m026$ | $\mathbb{Z}$
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,2)]$ | $\mathbb{Z}_{20}$ | $m003$ | $\mathbb{Z}_{5}+\mathbb{Z}$ | $m016$ | $\mathbb{Z}$
$SFS[D:(2,1)(3,1)]\cup_{(-1,1|0,1)}SFS[D:(2,1)(3,1)]$ | $\mathbb{Z}_{35}$ | $m006$ | $\mathbb{Z}_{5}+\mathbb{Z}$ | $m017$ | $\mathbb{Z}_{7}+\mathbb{Z}$
$SFS[D:(2,1)(2,1)]\cup_{(0,1|1,1)}SFS[D:(2,1)(3,1)]$ | $\mathbb{Z}_{28}$ | $m017$ | $\mathbb{Z}_{7}+\mathbb{Z}$ | $m019$ | $\mathbb{Z}$

###  4.4 Connected sums

One natural question to ask is how topological volume behaves under connected
sum. There is only one connected sum for which we can compute the topological
volume exactly in LABEL:tab:complete. Namely, we determine the topological
volume for $L(2,1)\\#L(3,1)$ to be realised by $m010(-1,1)$. Note that

$\displaystyle\operatorname{volt}(\;L(2,1)\\#L(3,1)\;)=\operatorname{vol_{\mathbb{H}}}(m010)=$
$\displaystyle\operatorname{vol_{\mathbb{H}}}(m009)$ $\displaystyle<$
$\displaystyle\operatorname{vol_{\mathbb{H}}}(m009)+\operatorname{vol_{\mathbb{H}}}(m007)$
$\displaystyle=$
$\displaystyle\operatorname{volt}(\;L(2,1)\;)+\operatorname{volt}(\;L(3,1)\;).$

A surgery diagram of $m010$ that represents that manifold as the complement of
a knot in $L(2,1)\\#L(3,1)$ is shown in Figure 3.

######  Question 4.7.

Let $M$ and $N$ be closed, orientable 3–manifolds. Is
$\operatorname{volt}(\;M\\#N\;)\leq\operatorname{volt}(M)+\operatorname{volt}(N)$?

Note that Statement (1) in Corollary 2.13 provides a similar, but weaker,
result.

Figure 3: The manifold $m010$ seen as a knot in $L(2,1)\\#L(3,1)$ minimises
the volume.

######  Question 4.8.

Let $M$ be a closed, orientable 3–manifold not diffeomorphic to $S^{3}$. What
is the limit for $n\rightarrow\infty$ of

$\frac{\operatorname{volt}(\\#_{n}M)}{n\operatorname{volt}(M)}\;?$

##  5 Examples obtained from the Whitehead link and its sister

Let $W$ denote the complement of the right-handed Whitehead link with Seifert
framing. This is $m129$ and L5a1 in the census, where the latter denotes the
left-handed Whitehead link. We also let $P$ denote the $(-2,3,8)$ pretzel link
complement, again, with Seifert framing. This is often called the _sister
manifold_ of $W$ and appears as $m125$ and L13n5885 in the census of manifolds
and links respectively. In the following, we use $W$ and $P$ to denote the
complements of Whitehead link and sister manifold with the Seifert framing,
and $m129$ and $m125$ for the Snappy census manifolds with geometric framing.

Figure 4: The Whitehead link $L5a1$ (left) and its sister link $L13n5885$
(right). The unknotted blue component is the first cusp (which we fill).

In SnapPy [snappy], the framing change (under filling the first cusp) is as
follows:

$\displaystyle L5a1(p,q)(0,0)$ $\displaystyle\cong m129(p+2q,-q)(0,0)$
$\displaystyle L13n5885(p,q)(0,0)$ $\displaystyle\cong m125(p-4q,p-3q)(0,0)$

In this section we show that, for $p^{2}+q^{2}$ sufficiently large, $L(p,q)$
has as minimiser a filling of the Whitehead or its sister link, cf. Theorem
1.3.

Our computation of the smallest volume lens spaces given in LABEL:tab:complete
shows that the minimisers of all lens spaces of topological volume at most
3.07 – with $p^{2}+q^{2}$ quite small – also arise as fillings of the
Whitehead or its sister link. However, there are lens spaces for which the
minimiser is not realised as a filling on either of these. One example is
$L(4,1)$ (see Corollary 5.4). A surgery diagram of $m034$ that displays this
manifold as a knot complement of a knot in $L(4,1)$ is shown in Figure 5.

######  Question 5.1.

What is the complete list of all lens spaces that do not have minimiser a
filling of the Whitehead or its sister link?

Figure 5: A surgery diagram of $m034$

###  5.1 Volume functions on Dehn surgery space

Hodgson and Masai [Hodgson_Masai, §2] give an excellent exposition of results
of Neumann and Zagier [Neumann-volumes-1985] relating the decrease in volume
during hyperbolic Dehn filling to the length of the surgery slope. Moreover,
they analyse the corresponding functions for $P$ and $W$ in [Hodgson_Masai,
§4-5]. We only state the main results required here and refer the reader to
[Hodgson_Masai] for a justification and further details.

For a cusped hyperbolic 3–manifold $M,$ let
$\Delta\operatorname{vol}_{M}(p,q)=\operatorname{vol_{\mathbb{H}}}(M)-\operatorname{vol_{\mathbb{H}}}(M(p,q)),$
where it is understood that a fixed cusp of $M$ is filled.

We note that $(p,q)$–surgery on the first cusp of $M\in\\{P,W\\}$ for
$p^{2}+q^{2}$ sufficiently large gives a one-cusped hyperbolic 3–manifold
$M((p,q),\infty)$. Since $M((p,q),(1,0))\cong L(p,q),$ this gives
$\operatorname{volt}(L(p,q))\leq\operatorname{vol_{\mathbb{H}}}M((p,q),\infty)$
(note that other filling slopes may result in a homeomorphic lens space, which
must be addressed in the proof of Theorem 1.3). Since $P$ and $W$ are of equal
volume, we have

$\operatorname{vol_{\mathbb{H}}}P((p,q),\infty))-\operatorname{vol_{\mathbb{H}}}W((p,q),\infty))=\Delta\operatorname{vol}_{W}(p,q)-\Delta\operatorname{vol}_{P}(p,q)$

With respect to the topological framing on component $0$ on $W$,
[Hodgson_Masai, (5.14)] gives:

$\displaystyle\Delta\operatorname{vol}_{W}(p,q)=\frac{2\pi^{2}}{p^{2}+4pq+8q^{2}}$
$\displaystyle-\frac{\pi^{4}(p^{2}-8q^{2})(p^{2}+8pq+8q^{2})}{3(p^{2}+4pq+8q^{2})^{4}}$
$\displaystyle+O\Bigg{(}\frac{1}{(p^{2}+4pq+8q^{2})^{3}}\Bigg{)}$ (5.1)

With respect to the topological framing on component $0$ on $P$,
[Hodgson_Masai, (5.11)] gives:

$\displaystyle\Delta\operatorname{vol}_{P}(p,q)=\frac{\pi^{2}}{(p-4q)^{2}+(p-3q)^{2}}$
$\displaystyle+\frac{\pi^{4}(4p^{4}-80p^{3}q+540p^{2}q^{2}-1520pq^{3}+1535q^{4})}{24((p-4q)^{2}+(p-3q)^{2})^{4}}$
$\displaystyle+O\Bigg{(}\frac{1}{((p-4q)^{2}+(p-3q)^{2})^{3}}\Bigg{)}$ (5.2)

###  5.2 The topological volume of all but finitely many lens spaces

Agol [agol-2cusped] showed that $W$ and $P$ are the minimal volume orientable
hyperbolic 3-manifolds with two cusps. This property is key for the following:

######  Theorem 1.3.

For all but finitely many homeomorphism types of lens spaces $L(p,q)$
($p>q\geq 1$ coprime), the volume minimising link in $L(p,q)$ has complement
one of $W(\frac{p^{\prime}}{q^{\prime}},\infty)$ or
$P(\frac{p^{\prime\prime}}{q^{\prime\prime}},\infty)$, where the filling
slopes $\frac{p^{\prime}}{q^{\prime}}$ and
$\frac{p^{\prime\prime}}{q^{\prime\prime}}$ can be determined from an
efficiently computable finite set. For $P$ this means that we fill along the
unknotted component.

Throughout, the surgery coefficient $\infty$ indicates that a cusp is not
filled, whilst $\frac{1}{0}$ is the filling with the infinity slope.

###### Proof.

First note that, by Proposition 4.3, we can assume without loss of generality
that we fill along the first cusp of both $W$ and $P$. Since the first cusp is
an unknot (in $S^{3}$), it follows from the classification of lens spaces,
that all Dehn fillings of $W$ or $P$ yielding a knot in $L(p,q)$ must have
filling coefficients obtained from the following operations

$\displaystyle(p,q)$ $\displaystyle\mapsto(p,q+np),$ $\displaystyle\forall
n\in\mathbb{Z}$ (5.3) $\displaystyle(p,q)$ $\displaystyle\mapsto(p,q^{*}),$
$\displaystyle q^{*}q\equiv 1\mod p$ (5.4) $\displaystyle(p,q)$
$\displaystyle\mapsto(-p,q),(p,-q)$ (5.5)

The volume functions derived by Hodgson and Masai in [Hodgson_Masai] exclude
all but finitely many fillings slopes satisfying Equations 5.3, 5.4, and 5.5
from being volume minimising. Since these functions are strictly monotonically
increasing with respect to their near-elliptical level sets (cf. Section 5.1),
this is a straightforward calculation.

With this setup, suppose to the contrary that there are infinitely many
homeomorphism types of lens spaces $L(p_{i},q_{i})$, $p_{i}>q_{i}\geq 1$, with
the property that some volume minimising link complement $M_{i}\subset
L(p_{i},q_{i})$ is not homeomorphic with a Dehn filling on $P$ or $W.$ We may
assume that $p_{i}^{2}+q_{i}^{2}$ is sufficiently large, such that $W$ and $P$
have filling slopes resulting in hyperbolic knots in $L(p,q)$.

Hence for all $\frac{p^{\prime}_{i}}{q^{\prime}_{i}}$ obtained from
$\frac{p_{i}}{q_{i}}$ by Equations 5.3, 5.4, and 5.5 and with
$W(\frac{p^{\prime}_{i}}{q^{\prime}_{i}})$,
$P(\frac{p^{\prime}_{i}}{q^{\prime}_{i}})$ hyperbolic, we have
$\operatorname{vol_{\mathbb{H}}}(M_{i})\leq\operatorname{vol_{\mathbb{H}}}(W(\frac{p^{\prime}_{i}}{q^{\prime}_{i}})),\operatorname{vol_{\mathbb{H}}}(P(\frac{p^{\prime}_{i}}{q^{\prime}_{i}}))<\operatorname{vol_{\mathbb{H}}}(W)=\operatorname{vol_{\mathbb{H}}}(P)$.
According to [agol-2cusped], $W$ and $P$ are the minimal volume orientable
hyperbolic 3-manifolds with two cusps. Hence each $M_{i}$ has exactly one
cusp. Then $\\{\operatorname{vol_{\mathbb{H}}}(M_{i})\\}$ is a set of volumes
of hyperbolic manifolds with one cusp and bounded above by
$\operatorname{vol_{\mathbb{H}}}(W)=\operatorname{vol_{\mathbb{H}}}(P).$

If this is a finite set, then there is
$v_{0}\in\\{\operatorname{vol_{\mathbb{H}}}(M_{i})\\}$ with the property that
infinitely many of the manifolds $M_{i}$ have volume equal to $v_{0}.$ But
then infinitely many lens spaces have the same volume, contradicting
Proposition 2.6.

Hence $\\{\operatorname{vol_{\mathbb{H}}}(M_{i})\\}$ is an infinite set. But
the first accumulation point for volumes of hyperbolic manifolds with one cusp
is $\operatorname{vol_{\mathbb{H}}}(W).$ However, a manifold with volume
arbitrarily close to this value is obtained by Dehn surgery on either $P$ or
$W$, a contradiction to our assumption.

This proves that all but finitely many lens spaces $L(p,q)$ have volume
minimiser a Dehn filling on $P$ or $W.$ ∎

######  Corollary 5.2.

The sequence $(\;\operatorname{vol}(L(p,q))\;)_{(p,q)}$ converges, as
$p^{2}+q^{2}\to\infty$, to the volume of the Whitehead link complement.

######  Corollary 5.3.

For $p^{2}+q^{2}$ sufficiently large, the topological volume of the lens space
$L(p,q)$ is realised by a knot. Moreover, if $q=1,$ then the minimiser is
unique and the complement is a tunnel number one once-punctured torus bundle.

###### Proof.

The first statement directly follows from Theorem 1.3.

For the second statement, observe that $q$ must be within $\pm 1$ of a
multiple of $p$ (see Equations 5.3, 5.4, and 5.5). Inserting this into the
first terms of Equations 5.1 and 5.2, we can see that – for $p$ sufficiently
large – the minimiser must be obtained by filling slope $(p,-1)$ on $W$
(setting $p\geq 0$ without loss): For $(p,\pm 1)$, the function
$\Delta\operatorname{vol}_{P}(p,q)$ converges to $\frac{\pi^{2}}{2p^{2}}$ plus
lower order terms, while $\Delta\operatorname{vol}_{W}(p,q)$ converges to
$\frac{2\pi^{2}}{p^{2}}$ plus lower order terms; and we have
$\Delta\operatorname{vol}_{W}(p,-1)=\frac{2\pi^{2}}{p^{2}-4p+1}$ while
$\Delta\operatorname{vol}_{W}(p,1)=\frac{2\pi^{2}}{p^{2}+4p+1}$. And thus, the
unique minimiser is given by an integer filling of the Whitehead link. These
are known to be once-punctured torus bundles with tunnel number $1$
[torus_bundles, Theorem 1.3]. ∎

######  Corollary 5.4.

The lens space $L(4,1)$ has as a minimiser a one-cusped manifold of volume at
most $\operatorname{vol_{\mathbb{H}}}(m034)\approx 3.166$ and which is not
obtained by Dehn filling $W$ or $P.$

###### Proof.

The filling coefficients for $W$ and $P$ to obtain a knot in lens space
$L(4,1)$ are precisely $(\pm 4,2m+1)$, for $m\in\mathbb{Z}$. Of those, only
$W((4,-1))$ and $P((4,1))$ are not hyperbolic: The former contains a splitting
Klein bottle, the latter is a solid torus. The other cases can be verified to
be hyperbolic using the $6$-theorem [agol-bounds, lackenby-6] together with
Snappy.

Of the hyperbolic filling slopes, the smallest volume ones are $W((4,1))$ and
$W((4,-3))$ with a volume of $\approx 3.17729$. On the other hand, we have
$\operatorname{volt}(L(4,1))\leq\operatorname{vol_{\mathbb{H}}}(m034)\approx
3.166.$ ∎

######  Remark 5.5.

It is important to note that we require both the Whitehead link $W$ and its
sister $P$ for Theorem 1.3 to hold: The volume minimiser for $L(p,1)$, $p$
sufficiently large, is uniquely realised as filling $W(p,-1)$, as shown in
Corollary 5.3. On the other hand, it can be deduced from LABEL:tab:complete
that $L(13,3)$ has minimiser $m011$ and $L(19,7)$ and $L(18,5)$ both have
minimiser $m016$. But according to Table 2, these can only be obtained by Dehn
fillings of $P$, but not of $W$.

We conclude this section with a question that is analogous to the third
question in [Hodgson_Masai, §9].

######  Question 5.6.

What is the largest volume less than $\operatorname{vol_{\mathbb{H}}}(W)$ of a
one-cusped hyperbolic 3–manifold that does not arise from Dehn filling on
$W=m129$ or $P=m125$?

An answer to 5.6 would allow us to give a quantitative version of Theorem 1.3.

###  5.3 Determining the minimiser

We are interested in an answer to the following question.

######  Question 5.7.

Suppose $L(p,q)$ is known to have minimiser a Dehn filling of $P$ or $W.$ Can
we decide which of the two it is?

Since we do not have a lower bound on $(p,q)$ to ensure that the volume
minimising link for $L(p,q)$ is in fact a filling of $W$ or $P$, we answer 5.7
for $(p,q)$ sufficiently large, and only provide a sketch on how to obtain
explicit bounds.

5.7 is hardest to answer when
$\Delta\operatorname{vol}_{W}(p,q)-\Delta\operatorname{vol}_{P}(p,q)\approx 0$
for at least one (possibly minimising) filling slope yielding a knot in a
given homeomorphism type of lens space. Using the approximations from above,
we have
$\Delta\operatorname{vol}_{W}(p,q)\approx\Delta\operatorname{vol}_{P}(p,q)$
for $\frac{2\pi^{2}}{p^{2}+4pq+8q^{2}}=\frac{\pi^{2}}{(p-4q)^{2}+(p-3q)^{2}}$,
which is the case for lines

$\ell_{1,2}(p)=\frac{p}{42}\left(16\pm\sqrt{130}\right)$

The curve defined by
$\Delta\operatorname{vol}_{W}(p,q)-\Delta\operatorname{vol}_{P}(p,q)=0$
converges to these lines as we move $(p,q)$ away from the origin. Since we
only evaluate $\Delta\operatorname{vol}_{W}(p,q)$ and
$\Delta\operatorname{vol}_{P}(p,q)$ on integer lattice points, we are
interested in integer lattice points closest to this line, with respect to the
magnitude of $q$ (and $p$).

For this reason, we briefly review some facts about continued fractions and
refer the reader to [ContinuedFractions, Chapter 7] for details. Let
$x\in\mathbb{R}$, and let $x=\langle a;a_{1},\ldots,a_{n},\ldots\rangle$ be
its continued fraction expansion. We call the sequence of rational numbers
$(\frac{p_{i}}{q_{i}})_{i\in\mathbb{N}}$ with $\frac{p_{i}}{q_{i}}=\langle
a;a_{1},\ldots,a_{i}\rangle$ the sequence of convergents for $x$. We have
$|xq_{i}-p_{i}|<|xq_{i+1}-p_{i+1}|$ for all $i\geq 0$ (see
[ContinuedFractions, Theorem 7.12]), and
$\left|\frac{1}{q_{i+2}}\right|<|xq_{i}-p_{i}|<\left|\frac{1}{q_{i+1}}\right|$
(see [ContinuedFractions, Theorems 7.11 and the proof of 7.12]). Moreover, we
have the following statement.

######  Theorem 5.8 (Theorem 7.14 from [ContinuedFractions]).

Let $\frac{p}{q}\in\mathbb{Q}$, $p$, $q$ co-prime, $q>0$, and let
$x\in\mathbb{R}$ with convergents $(\frac{p_{i}}{q_{i}})_{i\in\mathbb{N}}$. If
$|qx-p|<|q_{i}x-p_{i}|$ for some $i$, then $q\geq q_{i+1}$.

A corollary of Theorem 5.8 is that, if we can correctly evaluate the statement

$\Delta\operatorname{vol}_{W}(p,q)>\Delta\operatorname{vol}_{P}(p,q)$

for the convergents $(\frac{p_{i}}{q_{i}})_{i\in\mathbb{N}}$ of
$x_{1}=\frac{1}{42}\left(16+\sqrt{130}\right)$ and
$(\frac{a_{i}}{b_{i}})_{i\in\mathbb{N}}$ of
$x_{2}=\frac{1}{42}\left(16-\sqrt{130}\right)$ using only the first $k$ terms
of the Taylor series expansion of the two functions and the asymptotics of the
error terms, then we can correctly evaluate this statement for all other pairs
of co-prime integers $(p,q)$.

We have the following continued fraction expansions:

$x_{1}=\frac{1}{42}\left(16+\sqrt{130}\right)=\langle
0;1,1,1,7,7,2,7,7,2,\ldots\rangle$

and

$x_{2}=\frac{1}{42}\left(16-\sqrt{130}\right)=\langle
0;9,7,2,7,7,2,7,\ldots\rangle$

From these expansions it is straightforward to compute that we have
$q_{i}\in\Theta(q_{i+1})$ and $b_{i}\in\Theta(b_{i+1})$. More precisely,
excluding small values of $i$, $q_{i+1}$ (resp. $b_{i+1}$) is at least $2$ and
at most $8$ times larger than $q_{i}$ (resp. $b_{i}$). It follows that - in
the worst case - we have integer points $(p,q)$ near lines $\ell_{1,2}$ at
distance $\Theta\left(\frac{1}{q}\right)$, with the difference
$D(p,q)=\frac{2\pi^{2}}{p^{2}+4pq+8q^{2}}-\frac{\pi^{2}}{(p-4q)^{2}+(p-3q)^{2}}$
at unit distance from $\ell_{1,2}$ being of magnitude
$\Theta\left(\frac{1}{q^{2}}\right)$. Note that, for $(p,q)$ large enough and
near $\ell_{1,2}$ we have $p\in\Theta(q)$. A short calculation reveals that,
at distance $\Theta\left(\frac{1}{q}\right)$ from $\ell_{1,2}$, this amounts
to a difference $D(p,q)\in\Theta\left(\frac{1}{q^{4}}\right)$, which can be
counteracted by the second term of the Taylor series expansion (which is also
of order $\Theta\left(\frac{1}{q^{4}}\right)$ near $\ell_{1,2}$), but not by
the third term. In particular, we have the following statement:

######  Proposition 5.9.

Let $(p,q)$ be sufficiently large. Then
$\Delta\operatorname{vol}_{W}(p,q)>\Delta\operatorname{vol}_{P}(p,q)$ if and
only if

$\displaystyle\frac{2\pi^{2}}{p^{2}+4pq+8q^{2}}-\frac{\pi^{4}(p^{2}-8q^{2})(p^{2}+8pq+8q^{2})}{3(p^{2}+4pq+8q^{2})^{4}}$
$\displaystyle>\qquad$
$\displaystyle\frac{\pi^{2}}{(p-4q)^{2}+(p-3q)^{2}}+\frac{\pi^{4}(4p^{4}-80p^{3}q+540p^{2}q^{2}-1520pq^{3}+1535q^{4})}{24((p-4q)^{2}+(p-3q)^{2})^{4}}$

If an oracle tells us $L(p,q)$ has as volume minimiser a Dehn filling of
either $W$ or $P$, we have the following procedure to determine which one it
is: we simply evaluate the first two terms of
$\Delta\operatorname{vol}_{W}(p,q)$ and $\Delta\operatorname{vol}_{P}(p,q)$
for the finite set of filling slopes given in (the proof of) Theorem 1.3.

Overall, Proposition 5.9 provides us with a deterministic, exact and efficient
algorithm to check if $P$ or $W$ provides the minimiser, provided that one of
them does.

## References

##  Appendix A Using tetrahedron shapes to prove equivalence of volumes

In this appendix we verify our assertions regarding the equality of volumes of
certain 3–manifolds, as claimed in the proof of Proposition 4.1. SnapPy
[snappy] gives gluing information for triangulations of each manifold that
determine the unique shapes of tetrahedra corresponding to the complete
hyperbolic structure. In each of the two cases below, we show that the
manifolds hypothesised to have the same volume are built from geometrically
equivalent tetrahedra, therefore proving the claimed equality of volumes.

###  A.1 Equivalence of volumes: m006 and m007

SnapPy gives the following output for

A=Manifold(‘m006’) and B=Manifold(‘m007’):

    In: A.gluing_equations(form=‘rect’)
    Out:
    [([-1, 1, 1], [2, -2, -2], 1),
     ([0, -1, -1], [-2, 1, 1], 1),
     ([1, 0, 0], [0, 1, 1], 1),
     ([-1, 0, 0], [0, -2, 0], 1),
     ([0, 1, -1], [0, 0, 0], 1)]

    In: B.gluing_equations(form=‘rect’)
    Out:
    [([1, 2, 1], [0, 0, 0], 1),
     ([-1, -2, -1], [-1, 1, -1], 1),
     ([0, 0, 0], [1, -1, 1], 1),
     ([0, -1, -1], [-1, 0, 1], -1),
     ([1, 1, 0], [1, 0, -1], -1)]

Note that $(\left[a,b,c\right],\left[d,e,f\right],g)$ translates to the
equation

$z_{1}^{a}(1-z_{1})^{d}z_{2}^{b}(1-z_{2})^{e}z_{3}^{c}(1-z_{3})^{f}=g.$

For clarity, we use $z$ variables for m006 and $w$ variables for m007.

Using Sage [sage], we find that both systems of equations have multiple
solutions, but only one solution consisting of complex numbers with positive
imaginary part, which is the one we are interested in. The approximate
solutions are $z_{1}=0.7733+1.4677i,z_{2}=z_{3}=0.3352+0.4011i$, and
$w_{1}=w_{3}=-0.1027+0.6654i,w_{2}=0.2266+1.4677i$. As such, we expect the
exact values to satisfy $w_{2}=z_{1}-1,w_{1}=z_{2}/(z_{2}-1)$ and
$w_{3}=z_{3}/(z_{3}-1)$. To confirm this, we compute a Groebner basis from the
union of the two sets of gluing equations along with the conjectured relations
between $w$ and $z$ variables. The output is

$\displaystyle\\{z_{1}^{3}-$ $\displaystyle 3z_{1}^{2}+5z_{1}-4,$
$\displaystyle-$ $\displaystyle z_{1}^{2}+z_{1}+2z_{2}-3,$ $\displaystyle-$
$\displaystyle z_{1}^{2}+z_{1}+2z_{3}-3,$ $\displaystyle
w_{3}-z_{1}^{2}+2z_{1}-3,$ $\displaystyle w_{2}+z_{1}-1,$ $\displaystyle
w_{1}-z_{1}^{2}+2z_{1}-3\\}$

The fact that the Groebner basis is not empty confirms that a solution exists.
In particular, the first polynomial in the Groebner basis has $z_{1}\approx
0.7733+1.4677i$ as a root, and the remaining polynomials allow us to express
the other variables in terms of $z_{1}$.

This confirms that the geometry of the tetrahedra used to triangulate the
manifolds $m006$ and $m007$ are identical, and the manifolds do indeed have
the same hyperbolic volume.

###  A.2 Equivalence of volumes: m015, m016 and m017

SnapPy gives the following output for

C=Manifold(‘m015’), D=Manifold(‘m016’) and E=Manifold(‘m017’):

    In: C.gluing_equations(form=‘rect’)
    Out:
    [([1, 1, 1], [-1, 1, -1], -1),
     ([0, 0, 0], [1, -2, 1], 1),
     ([-1, -1, -1], [0, 1, 0], -1),
     ([0, 0, 0], [-1, 1, 0], 1),
     ([3, 1, -1], [0, -1, 0], -1)]

    In: D.gluing_equations(form=‘rect’)
    Out:
    [([1, 0, 1], [-1, 1, 0], -1),
     ([-1, -1, 0], [2, -1, -2], -1),
     ([0, 1, -1], [-1, 0, 2], 1),
     ([-2, 0, -1], [2, 0, 0], 1),
     ([0, 0, 0], [-2, 2, 2], 1)]

    In: E.gluing_equations(form=‘rect’)
    Out:
    [([1, 0, -2], [-1, 1, 1], -1),
     ([0, -1, 2], [1, 0, 0], 1),
     ([-1, 1, 0], [0, -1, -1], -1),
     ([-1, 0, -2], [-1, 1, 0], -1),
     ([0, -1, 0], [-1, 0, 0], 1)]

Using $u,v,w$ as variables for each system, respectively, we use Sage to find
the unique solution consisting of complex numbers with positive imaginary part
for each system. In approximate values, these solutions are

$u_{1}=u_{2}=u_{3}=0.662359+0.56228i$

$v_{1}=0.78492+1.30714i,v_{2}=v_{3}=0.122561+0.744862i$

$w_{1}=0.662359+0.56228i,w_{2}=w_{3}=0.78492+1.30714i$

We therefore expect to find that the exact values for all of
$u_{1},u_{2},u_{3}$ and $w_{1}$ are identical. Additionally, we expect
$v_{1}=w_{2}=w_{3}=\frac{1}{1-u_{1}}$ and $v_{2}=v_{3}=\frac{u_{1}-1}{u_{1}}$.

Again, we compute a Groebner basis for all three sets of gluing equations
along with the hypothesised relations between shapes. The output is the
following:

$\displaystyle\\{1-$ $\displaystyle z1+z1^{3},$ $\displaystyle-$
$\displaystyle z1+z2,-z1+z3,$ $\displaystyle w3-z1^{2},$ $\displaystyle
w2-z1^{2},$ $\displaystyle w1-z1-z1^{2},$ $\displaystyle v3-z1-z1^{2},$
$\displaystyle v2-z1-z1^{2},$ $\displaystyle v1-z1\\}$

This is confirmation that the geometry of the tetrahedra used to triangulate
the manifolds $m015$, $m016$ and $m017$ are identical, and the manifolds do
indeed have the same hyperbolic volume.