Problems on Low-dimensional Topology, 2015
Edited by T. Ohtsuki1

This is a list of open problems on low-dimensional topology with expositions of
their history, background, significance, or importance. This list was made by editing
manuscripts written by contributors of open problems to the problem session of theconference “Intelligence of Low-dimensional Topology” held at Research Institutefor Mathematical Sciences, Kyoto University in May 20–22, 2015.

Contents
1 The slice-ribbon conjecture and Akbulut-Kirby’s conjecture for 0surgeries along knots
2
2 Random links

3

3 Braid ordering

44 Relation between knots in S 3 and knots in RP 3

65 Surgery presentations of genus two 3-manifolds

6

6 Tight contact structures on 3-manifolds with open book decompositions
6
7 Random 3-manifolds and orderability

7

8 Stable complexity and stable presentation length for 3-manifolds

8

9 Invariants of knots and 3-manifolds with representations of theirfundamental groups
9

1 Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, JAPAN

Email: tomotada@kurims.kyoto-u.ac.jp
The editor is partially supported by JSPS KAKENHI Grant Number 24340012.

1

1

The slice-ribbon conjecture and Akbulut-Kirby’s conjecture for 0-surgeries along knots(Tetsuya Abe, Keiji Tagami)
The slice-ribbon conjecture asks whether any slice knot in S 3 bounds a ribbon disk
in the standard 4-ball B 4 ([12]). On the other hand, Akbulut and Kirby conjectured
the following:
Conjecture 1.1 ([17, Problem 1.19]). If 0-surgeries on two knots give the same
3-manifold, then the knots with relevant orientations are concordant.
In our paper [1], we proved that if the slice-ribbon conjecture is true, Conjecture 1.1
is false. More precisely, we proved the following: Let K0 and K1 be the knots
depicted in Figure 1. Orient K0 and K1 arbitrarily. Then, we see• K0 ♯K1 is not a ribbon knot, and• K0 and K1 admit the same 0-surgery,where K1 is the mirror image of K1 . If the slice-ribbon conjecture is true, K0 ♯K1
is not a slice knot. Namely, K0 and K1 are not concordant. In particular, Conjecture 1.1 is not true.

Figure 1: K0 and K1 . Each rectangle labeled 1 implies a right-handed full twist.

Here, we give the following question.
Question 1.2 (T. Abe, K. Tagami). Is K0 ♯K1 slice?
If the answer of Question 1.2 is “YES”, we see that K0 ♯K1 is a counterexample ofthe slice-ribbon conjecture. If the answer is “NO”, we see that K0 ♯K1 is a counterexample of Conjecture 1.1.
Remark. Recently, in private communications, Kouichi Yasui showed us infinitely
many counterexamples of Conjecture 1.1 by utilizing cork twists and satellite maps.

2

2

Random links(Kazuhiro Ichihara)2
In Knot theory, and larger, in the low-dimensional topology, there have been
several studies on random links (manifolds) recently. For example, in [19, 20], Jiming
Ma introduced some models of random links, the random braid model and therandom bridge presentation model. In the following of this section, as in [19], arandom link via the random braid model (resp. the random bridge presentation
model) is a link obtained as the closure (resp. the plat closure) of the braid induced
from a random walk of length k on the braid group (resp. the mapping class groupon a punctured sphere), and we consider the behavior of an invariant (or a property)of such a link at k → ∞.
Based on his results, the following problems can be considered.
Problem 2.1 (K. Ichihara). Does the probability of hyperbolic random links go to
1? This was already answered in [20] for the random braid model. How about forthe bridge model? The author and Jiming Ma are considering this as an ongoing
joint work.
Problem 2.2 (K. Ichihara). For some N, does the probability of random links of
N components go to 1? Or, for any N, does the probability go to 0? The expectedvalues of the number of components goes to the infinity, as shown in [19], withrespect to the standard generators. How about the other generating sets (i.e., theother probability distribution)?
Problem 2.3 (K. Ichihara). Does the expected value of the (simplicial or hyperbolic)volume for random link diverge? If this is true, what is the growth late? For any
fixed V , does the probability of the ones with the volume at most V go to 0?
Problem 2.4 (K. Ichihara). Does the probability of the sufficiently large random
links (i.e., containing closed essential surfaces in the exterior) go to 1 or 0 ?
Problem 2.5 (K. Ichihara). Can we consider all the problems above for the random
links in general 3-manifolds? For the random braid model, in [15], Ito introduced angeneralization to the links in general 3-manifolds by considering open book decompositions.
Problem 2.6 (K. Ichihara). What is the behavior of knot invariants for random
links? For example, can we say something Alexander polynomials for random links?

2 The author would like to thank Jiming Ma for useful discussions and advises to make this note.

33

Braid ordering(Patrick Dehornoy)
Termination of handle reduction
Let Bn be Artin’s n-strand braid group. An n-strand braid word is any word in the±1alphabet {σ1±1 , ..., σn−1}. For 1 6 i < n, define a σi -handle to be an n-strand braid−eeword of the form σi vσi with e = ±1 and v an n-strand braid word that contains
no letter σj±1 with j 6 i.
v
i+1i
For instance, σ2 σ3−1 σ2−1 is a σ2 -handle, but neither σ2 σ3−1 nor σ2 σ1−1 σ2−1 is.
−1Call a σi -handle σie vσi−e reducible if at most one of σi+1 , σi+1
occurs in v and, in−eethis case, define the reduct of σi vσi to be the word obtained from v by replacing±1−e ±1 eevery occurrence of σi+1
with σi+1
σi σi+1 , as in the diagram
7→
Finally, say that a braid word w reduces to another braid word w ′ if w ′ is obtained
from w by replacing a reducible handle by its reduct, that is, we have w = w1 w2 w3
and w ′ = w1 w2′ w3 with w2 a reducible σi -handle and w2′ the corresponding reduct.
Conjecture 3.1 (P. Dehornoy). For every n > 2, there exists a constant Cn suchthat no more than Cn ℓ2 reduction steps may be performed starting from an n-strand
braid word of length ℓ.
Conjecture 3.1 is known to be true for n 6 3 [8, Chap. V]. It is also known that,for every n, no infinite sequence of reductions may start from an n-strand word [5],but, for n > 4, the only upper bound on the number of reductions proved so far is
4
exponential in ℓ, namely O(2n ℓ ). Computer experiments provide a strong evidence
for Conjecture 3.1, but the only currently known proof, which combines algebraic andorder arguments in a nontrivial way, seems hard to improve. This suggests that the“true” proof is still missing. Note the connection with the Word Problem for Bn :an n-strand braid word w represents 1 in Bn if some/any sequence of reductions
starting from w leads to the empty word. Conjecture 3.1 is also connected with thestandard braid ordering (see next Subsection), since, in an irreducible braid word,
the generator σi with least index appears only positively, or only negatively.
The µn function
For β, β ′ in the n-strand braid group Bn , declare β <D β ′ if β −1 β ′ admits at least
one expression in terms of the generators σi±1 in which the generator σi with least4index occurs only positively (no σi−1 ). It is known [8] that the relation <D is a linear
order on Bn that is compatible with multiplication on the left (β <D β ′ implies
γβ <D γβ ′ for every γ). A deep result of R. Laver [18] states that, for every n, therestriction of <D to the submonoid Bn+ of Bn generated by σ1 , ..., σn−1 (“positive
braids”) is a well-ordering, that is, every nonempty subset of Bn+ has a <D -smallest
element. In particular, if β is a positive braid, there exists a <D -smallest positive
braid conjugate to β.
Question 3.2 (P. Dehornoy). Can one effectively compute the function µn : Bn+ →Bn+ defined by µn (β) = min{β ′ ∈ Bn+ | β ′ is conjugate to β}?
Question 3.2 is open for n > 3. Note that any method for computing µn would leadto a solution of the Conjugacy Problem for the monoid Bn+ , which, by using the factthat multiplying an arbitrary braid by a sufficient power of the central braid ∆2nyields a positive braid, would in turn lead to a solution of the Conjugacy Problem
for the group Bn . The alternating normal form of [6] or the rotating normal formof [14] might be useful for answering Question 3.2. A natural step toward a solution
could be the following computational formula, extensively tested on examples.
Conjecture 3.3 (Dehornoy, Fromentin, Gebhardt). For every β in B3+ , one hasµ3 (β∆23 ) = σ1 σ22 σ1 · µ3 (β) · σ22 .
Exotic orders and presentations
The Artin presentation of the braid group Bn in terms of Artin’s generators σ1 , ..., σn−1
is standard.
It is equally well-known that Bn can be presented simply in terms of−1−1
the n2 band generators ai,j = σj−1
· · · σI+1
σi σi+1 · · · σj−1 with 1 6 i < j 6 n [2].
Other natural generating families can be considered.
Question 3.4 (P. Dehornoy). Put τi = σi σi−1 · · · σ1 for 1 6 i < n. Does Bn admita finite presentation in terms of the generators τ1 , ..., τn−1 ?
The answer is positive for n = 3: writing a for τ1 and b for τ2 , one easily checks that
B3 admits the finite presentation ha, b | aba = b2 i, which provides an alternative“Garside structure” on B3 . The general case is open (and seemingly not easy; anegative answer seems likely).
Similar questions arise in connection with the Dubrovina–Dubrovin ordering of
braids [10].
iQuestion 3.5 (P. Dehornoy). Put ρi = (σ1 σ2 · · · σi )(−1) for 1 6 i < n. Does Bnadmit a finite presentation in terms of the generators ρ1 , ..., ρn−1 ?
Again the answer is positive for n = 3. Writing a for ρ1 and b for ρ2 , one seesthat B3 admits the presentation ha, b | b = ab2 ai. The submonoid of Bn generatedby ρ1 , ..., ρn−1 is the positive cone of the Dubrovina-Dubrovin order on Bn , a variantof the left-invariant ordering <D with the interesting property that it is an isolated
point in the space of orderings on Bn .
5(−1)i

Similar questions may be raised for the signed version ai,jof the band genera-(−1)itors, or for that ∆iof Garside fundamental braids, see [7] for further results andcomments on these (strange and interesting) questions.

4

Relation between knots in S 3 and knots in RP 3

(Sergei Matveev)
A knot in RP 3 is the image of a smooth embedding of S 1 into RP 3 . When aknot K in RP 3 is not null-homotopic in RP 3 , the preimage of K with respect tothe natural projection S 3 → RP 3 is a knot in S 3 .
Question 4.1 (S. Matveev). Do there exist non-equivalent knots in RP 3 such that
their lifts to S 3 are equivalent knots?
Motivation: The answer would clarify the relation between the theory of knots in
S 3 and the one in RP 3 .

5Surgery presentations of genus two 3-manifolds(Sergei Matveev, Vladimir Tarkaev)
A genus g Heegaard splitting of a closed orientable 3-manifold M is a decomposition of M into a union of two handlebodies of genus g. We say that a closed
3-manifold M is of genus g if M has a genus g Heegaard splitting.
Problem 5.1 (S. Matveev, V. Tarkaev). Prove that any closed 3-manifold of genus
two can be obtained by a rational surgery along a framed 4-component link in S 3 .
Motivation: Using 3-manifold Recognizer (an interactive computer tool), we havetested several hundreds of genus two 3-manifolds and for all of them have foundsuch a presentation.

6

Tight contact structures on 3-manifolds with open bookdecompositions(Keiko Kawamuro)
Let S be a compact surface with non-empty boundary. Let φ : S → S be adiffeomorphism that fixes the boundary ∂S of S pointwise. Let M(S,φ) be the 3manifold obtained by gluing the mapping torus of φ and solid tori in such a waythat a meridian of a solid torus is glued to {point} × S 1 ⊂ ∂S × S 1 . If a 3-manifold
M is diffeomorphic to M(S,φ) we say that M admits an open book decomposition(S, φ). Due to Giroux, there is a correspondence between (isotopy classes of) contact
structures on a closed oriented 3-manifold M and (certain equivalence classes of)open book decompositions of M.
6

Question 6.1 (T.Ito, K. Kawamuro). Let φ : S → S be a diffeomorphism of acompact surface S that fixes the boundary of the surface pointwise. If the fractional
Dehn twist coefficient c(φ, C) > 1 for all the boundary components C ⊂ ∂S thendoes the open book (S, φ) support a tight contact structure?
In [16], it is proven that the answer is “yes” for planar open books (that is, S has
genus 0).

7

Random 3-manifolds and orderability(Tetsuya Ito)
It is conjectured that for a compact 3-manifold M, the following three conditions
are equivalent (L-space conjecture, [3]):Non-LO: π1 (M) is not left-orderable, i.e., there is no total ordering < on π1 (M)that is invariant under the left action of π1 (M) itself.
L-space: M is an L-space. (Here we adopt the convention that if M is not a rational
homology sphere, it is automatically not an L-space)
No Taut: M admits no co-oriented taut foliation.
We consider a random walk on MCG(S), the mapping class group of a surface
with connected boundary S, with respect to a probability measure µ on MCG(S).
We denote by φk ∈ MCG(S) the position of the random walk at time k.
Recently it is shown that for a random open book (S, φ), the corresponding 3manifold M(S,φ) admits a co-oriented taut foliation, and is not an L-space [15] withasymptotic probability one. That is, the probability that M(S,φk ) has these properties
goes to one as k → ∞, under reasonable assumptions on µ. (In the current versionof [15], the proof of a key assertion on the Nielsen-Thurston ordering of φk (Theorem
1) has an error. Fortunately this assertion follows from the theorem of Malyutin[21] so the main results in [15] is unaffected).
Thus in the light of the aforementioned conjecture, we ask:
Question 7.1 (T. Ito). Is π1 (M(S,φ) ) left-orderable with asymptotic probability one?
One may ask a similar question for different models (e.g. random Heegaard
splitting) of random 3-manifolds:
Question 7.2 (T. Ito). For a random 3-manifold M (obtained by random Heegaard
splitting, or, other means – see [11] for precise formulation of random 3-manifold),
does M have the property Non-LO, L-space, No Taut (with probability one/zero)?
These questions can be seen as a “probabilistic approach” to disprove the Lspace conjecture. It should be noted that a (density model of) random group is notleft-orderable [4].

7

8Stable complexity and stable presentation length for 3manifolds(Ken’ichi Yoshida)
For a 3-manifold M, let kMk denote the simplicial volume of M, and let c(M)denote the complexity of M [22]. The complexity of a hyperbolic 3-manifold isthe minimal number of tetrahedra in an (ideal) triangulation of the manifold. The
stable complexity of M is defined as c∞ (M) = inf N c(N)/ deg(N → M), where theinfimum is taken for the finite coverings N of M [13].
Question 8.1 (S. Francaviglia, R. Frigerio, B. Martelli [13]). Does the equality
kMk = c∞ (M)hold for a 3-manifold M?
For a finitely presentable group G, let T (G) denote the presentation length of G[9]. The stable presentation length of G is defined as T∞ (G) = inf H T (H)/[G : H],
where the infimum is taken for the finite index subgroups H of G [27].
Question 8.2 (K. Yoshida [27]). Does the equality1
T∞ π1 (M) = c∞ (M)2hold for a 3-manifold M? In particular, does it hold that T∞ (π1 (M0 )) = 1 for the
figure-eight knot complement M0 ?
The above questions are reduced to considering for hyperbolic 3-manifolds, since
the simplicial volume, the stable complexity and the stable presentation length are
additive for the decomposition of geometrization.
The explicit values of stable complexity of hyperbolic 3-manifolds are known only
for the hyperbolic manifolds commensurable with the figure-eight knot complement.
No explicit value of stable presentation length is known for the fundamental groupof a hyperbolic 3-manifold.
Question 8.3 (K. Yoshida). Suppose that W is a hyperbolic 3-manifold obtained bygluing n ideal regular octahedra. Then does it hold that c(W ) = 4n?
If Question 8.3 is true for arbitrary W , it implies c∞ (W ) = 4n and a counterexampleof Question 8.1.
Question 8.4 (K. Yoshida). Suppose that a hyperbolic 3-manifold M ′ is obtainedby a Dehn filling from a hyperbolic 3-manifold M. Then does it holds that c∞ (M ′ ) <c∞ (M) and T∞ (π1 (M ′ )) < T∞ (π1 (M))?
It holds that c∞ (M ′ ) ≤ c(M) and T∞ (π1 (M ′ )) ≤ T (π1 (M)).
Ehrenpreis asked a problem whether any two closed Riemann surfaces of genus≥ 2 have finite coverings with arbitrarily small distance in an appropriate sense; it is8known that this problem was solved affirmatively. Francaviglia, Frigerio and Martelli[13] proposed to generalize this problem to any dimension, and call it an Ehrenpreis
problem. They observed that Question 8.1 would be proved from a positive answer
to an appropriate version of such a problem.
To formulate an Ehrenpreis problem for hyperbolic 3-manifolds, it is a problem
to give an appropriate “metric” on the set of hyperbolic 3-manifolds. Let H andC denote the isometry classes and the commensurability classes of finite volume
hyperbolic 3-manifolds, respectively. The set H admits the “geometric topology”,
which is the standard topology by Gromov-Hausdorff convergence. A metric on H
gives an Ehrenpreis problem, which asks if for M, N ∈ H, there are finite coverings
f, N
e of M, N such that the distance between M
f and Ne is arbitrarily small. If
M
an Ehrenpreis problem in the above sense holds, the quotient pseudometric on C is
identically zero. We expect a result that an Ehrenpreis problem does not hold, since
the claim of Question 8.3 is likely to hold.
Problem 8.5 (K. Yoshida). Find a natural metric on H whose quotient pseudometric on C is a metric.
We remark that the topology induced by a quotient pseudometric does not coincidewith the quotient topology in general. However, we have a following necessary
condition for Problem 8.5.
Question 8.6 (K. Yoshida). Is the quotient topology of C Hausdorff ?

9Invariants of knots and 3-manifolds with representationsof their fundamental groups(Toshie Takata)
In late 1980s, quantum invariants of closed oriented 3-manifolds have been introduced, motivated by the Chern–Simons path integral. This path integral itself
is a formal integral over G connections on a 3-manifold M in mathematical physics(where G is a compact Lie group), but there are two ways to obtain invariants of
3-manifolds from the path integral: the operator formalism and the perturbative
expansion. The operator formalism of the Chern–Simons path integral induces amathematically rigorous definition of quantum invariants of M. The perturbative
expansion of the Chern–Simons path integral gives a sum of contributions from flat
G connections, where each contribution gives a power series invariant of M. Byarithmetic expansion of quantum invariants, we obtain the perturbative invariant ofM, which can be regarded as a mathematical construction of the contribution fromthe trivial G connection. For details, see e.g. [23]. It would be a problem to givea mathematical construction of a contribution from a non-trivial flat G connection,noting that a flat G connection can be regarded as a representation π1 (M) → G.
The above observation suggests that there might exist many such invariants.
Problem 9.1 (T. Ohtsuki, T. Takata). Let G be a Lie group.
(1) Construct concrete invariants of a closed oriented 3-manifold M with a repre9

sentation π1 (M) → G.
(2) Construct concrete invariants of a knot K with a representation π1 (S 3 −K) → G.
So far, there have been some attempts to construct such invariants, such as “homotopy field theory” [25, 26]. This theory describes a formulation to construct suchinvariants in terms of G tensor categories, but it might be a problem to find concreteinteresting examples of such categories. We also note that, if G was a finite group,
Dijkgraaf–Witten invariant is such an invariant. In the same way as the construction of this invariant, we can also construct “Dijkgraaf–Witten invariant” for a Liegroup G by giving the discrete topology to G and finding a 3-cocycle of it, but it
might be a problem to find concrete interesting 3-cocycles of such G.
The volume conjecture suggests another aspect of asymptotic expansions of quantum invariants. The Kashaev invariant h L iN ∈ C of √a link L is defined for N =2, 3, · · · by using the quantum dilogarithm at q = e2π −1/N . Kashaev conjectured
that, for any hyperbolic link L, the hyperbolic volume of S 3 − L appears in theasymptotic behavior of h L iN at N → ∞ (Kashaev conjecture). H. Murakami andJ. Murakami showed that the Kashaev
invariant is equal to the N-colored Jones√
2π −1/N, where the N-colored Jones polynomial is thepolynomial evaluated at q = equantum invariant associated with the N-dimensional irreducible representation of
the quantum group Uq (sl2 ). They reformulated the Kashaev conjecture in terms of
the N-colored Jones polynomial (the volume conjecture). From the viewpoint ofmathematical physics, the asymptotic behavior of the problem can be regarded as
the asymptotic expansion of the SL2 C Chern–Simons path integral at the SL2 C flat
connection corresponding to the holonomy representation of the hyperbolic structure, noting that we can formally calculate the asymptotic behavior of the problem
by formally applying the saddle point method to the space of SU(2) connections inthe space of SL2 C connections. It is conjectured that, for a hyperbolic knot K, thesemi-classical part ω(K) of this expansion is equal to a scalar multiple of the squareroot of the twisted Reidemeister torsion τ (K) of K. For details, see [24]. In [24],for any hyperbolic two-bridge knot K, ω(K)2 and τ (K) are calculated in terms of“representations” Ψ and Φ of parameterized 3-braids, which are given by
!
!
1
1
xi
i+1 −1)1 2 xi+111− x(xi (xΨΦ
xi+1
i −1)xi+1
7−→ xi+1 0 −xi+1 −1 ,xi −xi+1xi
i −1− xxi+1
0
0
1
xi+1
−1
xi+1 11!
!
xi (xi+1 −1)
1
1
xi
1
0
0
1
Ψ
Φ
xi+1
(xi −1)xi+1
7−→ xi+1 −1 −xi+1 0 .
xi −xi+1
xi −1
xi
7−→
7−→

xi+1

xi+1 −1

1

xi+1

1

1

2 xi+11Here, we recall that the hyperbolic structure of a knot complement or, moregen
3erally, a knot K with a parabolic representation π1 (S − K) → SL2 C can be
described by a knot diagram with hyperbolicity parameters, which are solutions ofhyperbolicity equations.
Problem 9.2 (T. Ohtsuki, T. Takata).
(1) Construct concrete “representations” of parameterized braids (i.e., braids with
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hyperbolicity parameters).
(2) Construct concrete invariants of a knot K with a parabolic representation π1 (S 3 −K) → SL2 C.
As shown in [24], the above mentioned “representations” Ψ and Φ give the sameinvariant of hyperbolic 2-bridge knot, though they themselves do not look equivalent
in a usual sense of equivalent representations. It might be a problem to develop atheory to analyze “representations” of parameterized braids.

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