Problems on Low-dimensional Topology, 2013
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 22–24, 2013.

Contents
1 Super-A-polynomials of knots

22 Quasi-trivial quandles and link homotopy invariants

5

3 Ordering of knot and 3-manifold groups

6

4 Left-orderable surgeries and L–space surgeries

7

5 Groups with addition

9

6 Periodic orbits of pseudo-Anosov flows

11

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

Email: tomotada@kurims.kyoto-u.ac.jp

11Super-A-polynomials of knots(Hiroyuki Fuji)
In the generalized volume conjecture, the A-polynomial appears in the asymptotic
expansion of the colored Jones polynomial Jn (K; q):∞(1)∑
Jn (K; q = eℏ ) ∼ exp
S0 (x) +ℏk−1 Sk (x)(x := q n ),
ℏ
k=1

(
)
∂S0 (x)= log y(x), AK x, y(x) = 0.
∂x
The higher order terms Sk (x) in the above asymptotic expansion is found by solving
the q-diﬀerence equation iteratively [11]:ÂK (x̂, ŷ; q) Jn (K; q) = 0
x̂Jn (K; q) = q n Jn (K; q),x̂Jn (K; q) = Jn+1 (K; q).

Such q-diﬀerence operator ÂK (x̂, ŷ; q) reduces to the A-polynomial in q → 1 limit,
and its existence is conjectured in the quantum volume conjecture/AJ conjecture[21, 18].
In [2, 1], the quantum volume conjecture is generalized to the colored HOMFLY
polynomial HR (K; a, q) for the completely symmetric representation R = S r . Fromq-diﬀerence equation of the colored HOMFLY polynomial, one finds a 1-parameter
generalization of the A-polynomial AQ−def(x, y; a) that is named as the Q-deformed
K
A-polynomial. In [2], this polynomial is proposed to be equivalent to the augmentation polynomial of the knot contact homology [36]. In [16] we proposed the further
generalization of the quantum volume conjecture for the colored superpolynomial
P R (K; a, q, t) with R = S r , and the 2-parameter generalization of the A-polynomial
is found. This (a, t)-deformed polynomial is named as super-A-polynomial.
For 31 and 41 knots, the super-A-polynomial are obtained as follows:
2 4
3
3
2
Asuper31 (x, y; a, t) = a t (x − 1)x + (1 + at x)y()− a 1 − t2 x + 2t2 (1 + at)x2 + at5 x3 + a2 t6 x4 y,
2 5
2 2
2 2
3 2 3
Asuper41 (x, y; a, t) = a t (x − 1) x + at x (1 + at x) y

+ at(x−1)(1 + t(1−t)x + 2at3 (t+1)x2 − 2at4 (t+1)x3 + a2 t6 (1−t)x4 − a2 t8 x5 )y− (1 +at3 x)(1 +at(1−t)x +2at2 (t+1)x2 +2a2 t4 (t+1)x3 +a2 t5 (t−1)x4 +a3 t7 x5 )y 2 .
Question 1.1 (H. Fuji). Is it possible to express the augmentation polynomial andthe super-A-polynomial via some (a, t)-deformed holonomy representation? Can we
extract the non-trivial geometric information of S 3 \K for Asuper(x, y; a, t)?
K
In the context of the matrix models and topological strings, we find a similar
integrable structure as the quantum volume conjecture from the Eynard-Orantin’s
topological recursion on the spectral curve C [12]:
2n
}
{
∏
2
2
2
(x − xi ) .
C = (x, y) ∈ C × C y = M (x) S(x), S(x) =
i=1

2

p

p1

ph

p1

p

=

ph

q

q

+ Σ

q

g

p
pij pk1

pi1

lJ

g 1

pk

hj

q

g l

l

Figure 1: Diagrammatic representation of the Eynard-Orantin’s topological recursion relation.

The topological recursion solves the meromorphic (1, · · · , 1)-forms W (h,n) (p1 , · · · , pn )on (p1 , · · · , pn ) ∈ C ⊗n in an iterative manner:∫p− 21 p̄ B(·, p0 )
(0,1)
(0,2)
W
(p) = y(p)dx(p), W(p1 , p2 ) = B(p1 , p2 ), K(p0 , p) =,(y(p) − y(p̄))dx(p)[∑W (h,n) (p1 , · · · , pn ) =Res K(p0 , q) W (h−1,n+1) (q, q̄, pI )q→aa:Branch pts.

+

∑

W

(h′ ,|J|+1)(q, pJ )W(h−h′ ,n−|J|)](q̄, pI\J ) ,h′ ,Jwhere B(x, y) is the Bergman kernel on C which is (1, 1)-form on C × C.
Using the theta function of the spectral curve C and meromorphic forms W (h,n) ,the tau function Tgs [ydx] is studied systematically in [4]. Tgs [ydx] is known as thenon-perturbative partition function, and the (n|n)-Baker-Akhierzer kernel which is[n|n]denoted as ψgs (p1 , o1 ; · · · ; pn , on ) is defined as the Schlesinger transformation of
the non-perturbative partition function:∑
Tgs [ydx + nk=1 dSok ,pk ][n|n]ψgs (p1 , o1 ; · · · ; pn , on ) =Tgs [ydx]∫pwhere the meromorphic 1-form dSok ,pk is given by dSok ,pk = okk B(·, p).
In [10, 4], the topological recursion on the character variety{}
CK = (x, y) ∈ C∗ × C∗ AK (x, y) = 0
is studied for the figure eight knot and the SnapPea census manifold m009. More precisely speaking, the formal power series for the colored Jones polynomial around theexponential growth point is conjectured to be realized as the (2|2) Baker-Akhierzer
kernel Tgs [4]:Conjecture 1.2 ([4]). As formal power series, the colored Jones polynomial expanded around the exponential growth point coincides with the (2|2)-Baker-Akhierzer
kernel on the character variety CK = {(x, y) ∈ C∗ × C∗ | AK (x, y) = 0},)1/2[2|2] (,Jn (K; q = eℏ/2 ) ≃ ψℏ p, o; ι(p), ι(o)where x(ι(p)) = x and the involution ι acts as (x(ι(p)), y(ι(p))) = (x(ι(p))−1 , y(ι(p))−1 ).
3Here we consider the possibility for the generalization of this conjecture to thecolored HOMFLY polynomial and superpolynomial.
Question 1.3 (H. Fuji). Is the Baker-Akhierzer (2|2)-kernel on the the supercharacter variety whose defining equation is given by AsuperK r (x, t; a, t) related withthe asymptotic expansion of the colored superpolynomial P S (K; a, q, t)?
In the case of K = 31 , the super-character variety is expressed in C2 variables as:{}
C = (x, y) ∈ C2 y 2 = M (x)2 S(x) ,S(x) = x4 + 2a−1 t−1 u3 + (a−2 t2 + 2a−1 t−3 + 4a−2 t−4 ) − 2a−2 t−4 + a−2 t−6 ,1 − t2 x + 2t2 x2 + 2at3 x2 + at5 x3 + a2 t6 x4
A(x) =,
at3 (1 + at3 x2 )√A(x) + S(x)1√
M (x) = √log.
2x S(x)A(x) − S(x)3and Asuper
31 (x, y; a, t) has deformed reciprocal symmetry ι : x 7→ −1/(at x). The S0
and S1 terms are given by definition of the free energy as the Abel-Jacobi map anddiscriminant on C0 respectively [10]:∫ xdx
log y(x),S0 (x) =xo1γ
S1 (x) = log √
,
2
S(x)

where γ is a constant. The next order S2 is computed via the topological recursion[4], and the result is(()
S2 (x) = 1 + 2at + 2t2 + 4at3 + 2a2 t4 + 3t2 − 2t4 − 4at5 − 2a2 t6 x()
+ − 16t2 − 11at3 − 21t4 + 2a2 t4 − 28at5 − 2t6 − 8a2 t6 − 4at7 + 2a3 t7 − 2a2 t8 x2()+ − 30at5 − 7t6 − 24a2 t6 − 48at7 + 2t8 − 68a2 t8 + 4at9 − 28a3 t9 + 2a2 t10 x3(+ 16at5 + 11a2 t6 + 21at7 − 2a3 t7 + 28a2 t8 + 2at9 + 8a3 t9)+ 4a2 t10 − 2a4 t10 + 2a3 t11 x4()+ 3a2 t8 − 2a2 t10 − 4a3 t11 − 2a4 t12 x5
(
) 6)
3 9
4 10
3 11
4 12
5 13
+ − a t − 2a t − 2a t − 4a t − 2a t x)/(
48(1 + at)(1 + t2 + at3 )(1 − 2t2 x + 4t2 x2 + 2at3 x2 + t4 x2 + 2at5 x3 + a2 t6 x4 )3/2 .
The details of the computations are given in [15]. The above result and the asympr
totic expansion of P S (K; a, q, t) behave diﬀerently. This discrepancy would be
related with the q-dependence of a and x. This point may be cured by taking into
account of the mirror map of the “D-mirror symmetry” proposed in [2] carefully.

4

2

Quasi-trivial quandles and link homotopy invariants(Ayumu Inoue)
A quandle X is said to be quasi-trivial if it satisfies the condition x ∗ φ(x) = x
for each x ∈ X and φ ∈ Inn(X) [25]. Here, Inn(X) denotes the inner automorphismgroup of X. Modifying usual quandle homology slightly, we have homology of an(X) denote the n-th homology
quasi-trivial quandle [25]. We let HnQ,qt (X) and HQ,qt
and cohomology groups of a quasi-trivial quandle X respectively.
For each oriented and ordered link L in S 3 , we have the reduced knot quandle
RQ(L) which is invariant under link-homotopy [24]. Here, given two links are said
to be link-homotopic if they are related to each other by a finite sequence of ambient
isotopies and self-crossing changes [32]. The reduced knot quandle RQ(L) is quasitrivial. Associated with each component Ki of L, we have the fundamental class[Ki ]qt ∈ H2Q,qt (L) which is invariant under link-homotopy [25, 26].
Conjecture 2.1 (A. Inoue). Two n-component links L = K1 ∪ · · · ∪ Kn andL′ = K1′ ∪ · · · ∪ Kn′ are link-homotopic if and only if there is an isomorphism
φ : RQ(L) → RQ(L′ ) such that φ♯ ([Ki ]qt ) = [Ki′ ]qt for each i (1 ≤ i ≤ n). Here,φ♯ : H2Q,qt (RQ(L)) → H2Q,qt (RQ(L′ )) denotes the isomorphism induced from φ.
This conjecture is true in the case n ≤ 3. Prove the conjecture, or give a counterexample.
Suppose that X is a quasi-trivial quandle. By definition, for each x ∈ X andφ ∈ Inn(X), the pair (x, φ(x)) is a 2-cycle.
Question 2.2 (A. Inoue). Let L = K1 ∪ · · · ∪ Kn be an n-component link. For allx ∈ RQ(L) and φ ∈ Inn(X), is not the class [x, φ(x)] in ⟨[K1 ]qt ⟩ ⊕ · · · ⊕ ⟨[Kn ]qt ⟩ ◁H2Q,qt (RQ(L))?
e nQ,qt (RQ(L)) which is a certain quotient of
We have another homology group H
e Q,qt (RQ(L)). We can show that
HnQ,qt (RQ(L)). By definition, [x, φ(x)] = 0 in Hne Q,qt (RQ(L)) for each i and H
e Q,qt (RQ(L)) = ⟨[K1 ]qt ⟩ ⊕ · · · ⊕ ⟨[Kn ]qt ⟩. If[Ki ]qt ∈ Hnne 2Q,qt (RQ(L)) instead of H2Q,qt (RQ(L)) is suﬃcient
Question 2.2 is true, considering H
to classifying links up to link-homotopy.
Choose and fix a quasi-trivial quandle X and its 2-cocycle θ with coeﬃcientsin an abelian group A. For each n-component link L = K1 ∪ · · · ∪ Kn , consider
all homomorphisms φ : RQ(L) → X. Then the multiset consisting of all elements⟨θ, φ♯ ([Ki ]qt )⟩ ∈ A obviously gives rise to a link-homotopy invariant. We call thisnumerical invariant a quandle cocycle invariant.
Problem 2.3 (A. Inoue). Classify links up to link-homotopy using quandle cocycle
invariants.
For example, we can distinguish the trivial 3-component link and the Borromeanrings using a certain quandle cocycle invariant [25]. Can we distinguish the ncomponent trivial link and Brunnian links using quandle cocycle invariants?
We have other numerical link-homotopy invariants called Milnor’s µ-invariants[33].
5

Question 2.4 (A. Inoue). How are quandle cocycle invariants related with Milnor’sµ-invariants?
We hope that (quasi-trivial) quandles and their homology would be useful toinvestigate other areas in knot theory, e.g., tunnel number, sliceness, etc.

3

Ordering of knot and 3-manifold groups(Tetsuya Ito)
A total ordering <G of a group G is called a left-ordering (resp. right-ordering)if a <G b implies ga <G gb (resp. ag <G bg) for all a, b, g ∈ G. <G is called abi-ordering if it is both left- and right-ordering. A group G is left-orderable (resp.
bi-orderable) if G has at least one left-ordering (resp. bi-ordering).
The problem that which knot group or 3-manifold group admits a left- or biordering recently gathered much attention, but many basic questions remain unsolved. Moreover, we have few (non-trivial) examples of orderings. For backgrounds
on orderability and 3-dimensional topology, see [5, 6]. Here we list several problems
concerning orderings and orderability which may not be too hard.
For bi-orderings of knot groups, our knowledge is limited to fibered knot groups[8, 27, 41, 42]. Thus we pose the following questions.
Question 3.1 (T. Ito (Bi-ordering of knot groups)).
(1) Find an example of bi-ordering of non-fibered knot groups.
(2) Is the knot group of the 52 knot bi-orderable ?
(3) Is the knot group of the 62 knot bi-orderable ?
(4) Find an example of not bi-orderable, but virtually bi-orderable knot groups. (Moregenerally, is a knot group virtually bi-orderable ?)As for (2), the 52 knot is the simplest non-fibered knot. As for (3), the 62 knot is
the first example of fibered knot whose bi-orderability is unknown.
Contrary, it is known that all knot groups are left-orderable, although still wehave few concrete examples. We have several interesting and highly non-trivial leftorderings (such as, isolated orderings) of torus knot groups [28, 35]. An element a ofa left-orderable group G is called universally cofinal if for every left-ordering <G of
G and g ∈ G, there exists N ∈ Z such that a−N <G g <G aN holds. A left ordering<G of G is called discrete if <G admits a <G -positive minimum element.
Question 3.2 (T. Ito (Left-ordering of knot groups)).
(1) Give an “explicit” example of left-ordering < of knot groups.
(2) Can we generalize the above mentioned orderings of torus knot groups to otherclass of knot groups, such as, 2-bridge knot groups ?
(3) Which knot group has (does not have) universally cofinal element ?
(4) Do every knot group <G admit a discrete ordering <G such that its meridian is
the <G -minimum positive element ?
6

As for (1), “explicit” means that we have a reasonable characterization of <-positiveelements, or, one can check given element x of the knot group is <-positive or notin an reasonable way. As for (3), it is easy to see that a torus knot group admits
universally cofinal element. As for (4), of course, constructing discrete ordering forknot group is already interesting.
It is conjectured [5] that 3-manifold M is non-left-orderable (that is, M has nonleft-orderable fundamental group) if and only if M is an L-space. Several supporting
evidences of this conjecture have been established by many researchers, but generalcase is widely open.
Question 3.3 (T. Ito ((Non-)left-orderable 3-manifold)).
(1) Is the double branched covering of a quasi-alternating link non-left-orderable ?
(2) Characterize non-left-orderable 3-manifold of Heegaard genus two.
(3) Let M be non-left-orderable (hyperbolic) 3-manifold. Is there a finite covering
f of M with property (a) and (b) below ?
M
f is left-orderable.
(a) M
f is a rational homology 3-sphere.
(b) M
As for (1), see [29] for related results. As for (2), see [31] for the corresponding
results for 3-manifolds admitting genus one open book decomposition. As for (3),by virtual fibered conjecture, if we drop the condition (b), we can always find suchf.
M

4

Left-orderable surgeries and L–space surgeries(Kimihiko Motegi)2
L–spaces
Heegaard Floer homology is a package of 3–manifold invariants introduced by Ozsváthand Szabó [37, 38]. In the following we consider the simplest one called the “hat”
d (Y ). The Heegaard Floer homology HF
d (Y ) has a relative
version, denoted by HF
d (Y ) = HF
d 0 (M ) ⊕ HF
d 1 (Y ). For a rational homology 3–sphere M ,Z2 –grading HF
d (Y )) = |H1 (M ; Z)|. In particular, rankHF
d (M ) ≥ |H1 (M ; Z)|.
we have χ(HF
There are several kinds of definitions for L–spaces. In particular, we often findthe following definitions in literatures.
Definition (L–space)(1) A rational homology 3–sphere M is called an L–space if the Heegaard Floerd (Y ; Z) is a free abelian group of rank |H1 (Y ; Z)|.
homology with Z–coeﬃcients HF(2) A rational homology 3–sphere M is called an L–space if the Heegaard Floer
2 I would like to thank Cameron Gordon, Hiroshi Matsuda, Yi Ni, Motoo Tange for private communications
concerning Question 4.1. I would also like to thank Masakazu Teragaito for useful discussions.

7d (Y ; Z) has the rank |H1 (Y ; Z)|.
homology with Z–coeﬃcients HF(3) A rational homology 3–sphere M is called an L–space if the Heegaard Floerd (Y ; Z2 ) has the rank |H1 (Y ; Z)|.
homology with Z2 –coeﬃcients HF(1) is the original definition of Ozsváth and Szabó [39], which does not allow “torsion”, but (2) allows “torsion”. The diﬀerence between (2) and (3) is the choice ofd (Y ) which we use.
“coeﬃcients” for HF
Question 4.1 (K. Motegi). Are these three conditions equivalent? In particular,d (Y ; Z) have no torsion?
for any rational homology sphere Y , does HF
d (Y ; Z) has torFor rational homology 3–spheres, no examples are known where HF
sion. So possibly definitions (1), (2) and (3) are equivalent. However, at the moment,
it seems convenient to adopt the definition (3) for homogeneity; see [5, 1.1].
The set of left-orderable surgeries
We say that a nontrivial group G is left-orderable if there exists a strict total ordering< on its elements such that g < h implies f g < f h for all elements f, g, h ∈ G. The
left-orderability of fundamental groups of 3–manifolds has been studied by Boyer,
Rolfsen and Wiest [6]. In particular, it is known that P 2 –irreducible 3–manifold
M with first Betti number b1 ≥ 1 has the left-orderable fundamental group. Hencefor any knot K in S 3 , its exterior E(K) has the left-orderable fundamental group.
However if r ̸= 0, the result K(r) of r–Dehn surgery on K, which is a rational
homology 3–sphere, may not have such a fundamental group. A Dehn surgery issaid to be left-orderable if the resulting manifold of the surgery has the left-orderable
fundamental group, and the set of left-orderable surgery slopes for K is defined as{()}
SLO (K) = r ∈ Q π1 K(r) is left-orderable .
For any nontrivial knot K, K(0) is irreducible [17, Corollary 8.3] and H1 (K(0)) ∼= Z,
and hence {0} ⊂ SLO (K) [6, Theorem 1.1]. If K is a trivial knot then SLO (K) = {0},which has the smallest size. In the opposite direction, in [34], we demonstrate that
there are infinitely many hyperbolic knots K with SLO (K) = Q. It seems interesting
to ask:Question 4.2 (K. Motegi). If K is a nontrivial knot in S 3 , then does SLO (K)contain (−1, 1) ∩ Q?
Recently Li and Roberts [30, Corollary 1.2] prove that for any hyperbolic knot K,
there exists a constant NK such that { n1 | |n| > NK } ⊂ SLO (K). More strongly wewould like to ask:
Question 4.3 (K. Motegi). If K is a nontrivial knot in S 3 , then does SLO (K)contain (−∞, 1) ∩ Q or (−1, ∞) ∩ Q?
For the simplest nontrivial knot T3,2 (resp. T−3,2 ), the argument in the proof of [9,Theorem 1.4] shows that SLO (T3,2 ) = (−∞, 1)∩Q (resp. SLO (T−3,2 ) = (−1, ∞)∩Q).
8

Question 4.4 (K. Motegi). If SLO (K) = (−∞, 1) ∩ Q or SLO (K) = (−1, ∞) ∩ Q,
then is K a trefoil knot T3,2 or T−3,2 , respectively?
Question 4.5 (K. Motegi). Let K be a nontrivial knot in S 3 . Then does SLO (K)have no maximum and minimum?
Boyer-Gordon-Watson conjecture
Results in [5] and previously known examples suggest that there exists a strange
correspondence between 3–manifolds whose fundamental groups are left-orderable
and L–spaces. For instance, lens spaces, spherical 3–manifolds are L–spaces, on
the other hand their fundamental groups are not left-orderable (because they havetorsions). The following conjecture is formulated by Boyer, Gordon and Watson [5].
Conjecture 4.6 ([5]). An irreducible rational homology 3–sphere is an L–space if
and only if its fundamental group is not left-orderable.
A Dehn surgery is called an L–space surgery if the resulting manifold of thesurgery is an L–space. Define the set of L–space surgery slopes for K as{}
SL (K) = r ∈ Q K(r) is an L–space .
Now let us look at the shape of SL (K), which is well-understood by Proposition 9.6
in [40] ([22, Lemma 2.13]). It is known [40, 22] that, if K is a nontrivial knot andSL (K) ̸= ∅, then SL (K) = [2g(K) − 1, ∞) ∩ Q or SL (K) = (−∞, −2g(K) + 1] ∩ Q.
Since Conjecture 4.6 says that SL (K) and SLO (K) are complementary to each otherin Q if K has no reducing surgery, we expect the following explicit form of SLO (K).
Conjecture 4.7. Let K be a nontrivial knot in S 3 which has no reducing surgery.
Then SLO (K) coincides with one of Q, (−∞, 2g(K)−1)∩Q and (−2g(K)+1, ∞)∩Q.
Finally we give a comment on Question 4.4 in case of SLO (K) = (−∞, 1) ∩ Q;
the other case follows by taking the mirror image. By the assumption 1 ̸∈ SLO (K),if Conjecture 4.6 is true, then 1 ∈ SL (K) or K(1) is reducible. The latter possibility
is eliminated by [20, Corollary 3.1], and hence K(1) is an L–space. Further, it is
known [23] that, if K is a nontrivial knot and K( n1 ) is an L–space, then n = 1 (resp.
−1) and K is a trefoil knot T3,2 (resp. T−3,2 ). Therefore, K is a trefoil knot T3,2 .

5Groups with addition(Rinat M. Kashaev)
A group with addition is a group with an additional commutative and associative binary operation (x, y) 7→ x + y with respect to which the group product is
distributive from both sides.
Examples (groups with addition)(1) Q>0 or R>0 with the usual addition.
9

(2) Z, Q or R with tropical addition max(x, y).
(3) The group of upper triangular n-by-n matrices over Q or R with positive diagonal
elements and with the addition of matrices.
One can easily show that no finite group can be a group with addition.
Question 5.1.
(1) Are there other examples of groups with addition?
(2) What are the conditions under which a given infinite group admits an addition?
(3) Is there a systematic procedure for constructing groups with addition?
Motivation for groups with addition
Groups with addition can be used for construction of representations of mappingclass groups of punctured surfaces as follows.
Groupoid of ideal triangulations. Recall that for any set S freely acted uponby a group G, one can associate a connected groupoid GS,G with the set of G-orbits
in S as the set of objects Ob GS,G , the G-orbits in S ×S (with respect to the diagonal
action) as the set of morphisms Mor GS,G , the domain and the codomain maps
dom, cod : Mor GS,G → Ob GS,G ,dom[x, y] = [x], cod[x, y] = [y],so that two morphisms [x, y] and [u, v] are composable if and only if cod[x, y] =dom[u, v], i.e. [y] = [u] with the composition (adopting the convention used for thecomposition in fundamental groupoids of topological spaces) [x, y] ◦ [u, v] = [x, gv],
where g is the unique group element such that y = gu. The groupoid GS,G isa connected groupoid with any vertex group being isomorphic to G, so that any
representation of GS,G restricts to a representation of G.
Let Σ = Σg,s be a closed oriented surface of genus g with s punctures such that(2g − 2 + s)s > 0. An ideal triangulation of Σ is a CW-decomposition of the closed
surface Σ where the set of vertices coincides with the set of punctures, and all cells
are simplices. A decorated ideal triangulation of Σ is an ideal triangulation where
each triangle is provided with a distinguished corner and the set of all triangles islinearly ordered. Let ∆Σ be the set of decorated ideal triangulations of Σ. The
mapping class group of Σ, ΓΣ , freely acts on ∆Σ . The associated groupoid G∆Σ ,ΓΣis called the groupoid of (decorated) ideal triangulations of Σ.
A presentation of the groupoid of ideal triangulations. One can show that10G∆Σ ,ΓΣ is generated by the following three types of morphisms:
Diagonal flips:

ωij

j

i

−→

i
j

ρi

Corner changes:i−→Permutations:

σ(i)

−→

σ

i
i

which satisfy the following relations:Pentagons:

ωij ◦ ωik ◦ ωjk = ωjk ◦ ωij
ρi ◦ ρi = ρ−1iTriple corner changes:
Double flips:

ωij ◦ ρi ◦ (ij) ◦ ωij = ρi ◦ ρj

Permutation relations: (ij) ◦ (jk) ◦ (ij) = (jk) ◦ (ij) ◦ (jk) etc.
Semisymmetric T -matrices. A T -matrix in a symmetric monodical category
C = (C, ⊗, s) is an automorphism T ∈ Aut(V ⊗2 ), with V ∈ Ob C, which satisfies the equation T12 T13 T23 = T23 T12 in Aut(V ⊗3 ). A T -matrix T ∈ Aut(V ⊗2 ) issemisymmetric if there exists a symmetry A ∈ Aut(V ) such that A3 = idV and
T (A ⊗ idV )sV,V T = A ⊗ A.
Theorem. Let Σ be an oriented surface of genus g with s punctures such that (2g −2 + s)s > 0. Then, for any semisymmetric T -matrix T ∈ Aut(V ⊗2 ) there exists
a unique representation πT : G∆Σ ,ΓΣ → Aut(V ⊗ng,s ), ng,s := 4g − 4 + 2s, such that
πT (ωij ) = Tij , πT (ρi ) = Ai , πT ((1, 2)) = sV,V ⊗ idV ⊗(ng,s −2) .
Theorem. Let G be a group with addition, c ∈ G a central element, and X = G×G.
Then, there exists a semisymmetric set-theoretical T -matrix t : X 2 ∋ (x, y) 7→ (x ·−1y, x ∗ y) ∈ X 2 with symmetry a : X ∋ (x1 , x2 ) 7→ (cx−1
1 x2 , x1 ) ∈ X, x · y =−1−1(x1 y1 , x1 y2 + x2 ), and x ∗ y = a (a(y) · a (x)).

6Periodic orbits of pseudo-Anosov flows(Sérgio R. Fenley)
A pseudo-Anosov flow Φ has a countable number of periodic orbits. Each orbit
defines a free homotopy class, or a conjugacy class in the fundamental group of M .
The orbit is traversed in the positive flow direction.
Question 6.1 (S. R. Fenley). Do the conjugacy classes of the periodic orbits of thepseudo-Anosov flow Φ generate π1 (M )?
This is not always true. For example if Φ is a suspension pseudo-Anosov flow thenthis does not happen. Is that the only situation where this does not happen?
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Question 6.2 (S. R. Fenley). Suppose that Φ is a pseudo-Anosov flow transverseto an R-covered foliation and that Φ is regulating for F. Does this imply that theperiodic orbits do not generate π1 (M )?
Here R-covered means that in the universal cover, the leaf space of the lifted foliation is homeomorphic to the set of real numbers R. Regulating means that in theuniversal cover, every orbit of the lifted flow intersects every leaf of the lifted foliation and vice versa. This is very common: whenever the foliation F is R-covered,transversely orientable, and M is aspherical and atoroidal, there is such a flow. Thiswas proved in [7] and [14].
An Anosov flow Φ is said to be R-covered if its stable (or unstable) foliation is
R-covered. They are very common and there are infinitely many examples where
M is hyperbolic as proved in [13]. Suppose that M is atoroidal. It was proved in[13] that every closed orbit of Φ is freely homotopic to infinitely many other closed
orbits.
Question 6.3 (S. R. Fenley). Are there other examples where some closed orbits
are freely homotopic to infinitely many other closed orbits of the flow? Does it imply
that M has to be atoroidal?
In [3] it was proved that if Φ is an R-covered Anosov flow so that its stable foliationis transversely orientable then the following happens: given γ a periodic orbit, it isfreely homotopic to infinitely many other closed orbits. In [3] it is proved that γ isin fact isotopic to each of these other closed orbits. That is, they represent the sameknot in M .
Question 6.4 (S. R. Fenley). Suppose that α and β are closed orbits of a pseudoAnosov flow Φ which are freely homotopic. Does it follow that α and β are isotopic,
as is the case when Φ is an R-covered Anosov flow? When are they isotopic?
The proof in [3] uses the universal circle of Thurston [43] in an essential way. In thecase that the stable foliation of Φ is not an R-covered foliation, the universal circle
is much more complicated and the analysis of Question 6.4 is bound to be more
complicated too. There may be some simple examples or counterexamples whichshould be analysed first.

References[1] M. Aganagic, T. Ekholm, L. Ng, C. Vafa, Topological strings, D-Model, and knot contact
homology, arXiv:1304.5778.
[2] M. Aganagic, C. Vafa, Large N duality, mirror symmetry, and a Q-deformed A-polynomial forknots, arXiv:1204.4709.
[3] T. Barthelme, S. Fenley, Knot theory of R-covered Anosov flows: homotopy versus isotopy ofclosed orbits, math.Arxiv.DS.1208.6487.
[4] G. Borot, B. Eynard, All-order asymptotics of hyperbolic knot invariants from non-perturbative
topological recursion of A-polynomials, arXiv:1205.2261.

12

[5] S. Boyer, C. Gordon, L. Watson, On L-spaces and left-orderable fundamental groups, Math.
Ann. to appear.
[6] S. Boyer, D. Rolfsen, B. Wiest, Orderable 3-manifold groups, Ann. Inst. Fourier (Grenoble),55 (2005) 243-288.
[7] D. Calegari, The geometry of R-covered foliations, Geometry and Topology 4 (2000) 457–515.
[8] A. Clay, D. Rolfsen, Ordered groups, eigenvalues, knots, surgery and L-spaces, Math. Proc.
Cambridge Philos. Soc. 152 (2012) 115–129.
[9] A. Clay, L. Watson, On cabled knots, Dehn surgery, and left-orderable fundamental groups,
Math. Res. Lett. 18 (2011) 1085–1095.
[10] R. Dijkgraaf, H. Fuji, M. Manabe, The volume conjecture, perturbative knot invariants, andrecursion relations for topological strings, Nucl. Phys. B849 (2011) 166–211.
[11] T. Dimofte, S. Gukov, J. Lenells, D. Zagier, Exact results for perturbative Chern-Simons theory
with complex gauge group, Commun. Number Th. Phys. 3 (2009) 363–443.
[12] B. Eynard, N. Orantin, Algebraic methods in random matrices and enumerative geometry,
arXiv:0811.3531.
[13] S. Fenley, Anosov flows in 3-manifolds, Ann. of Math. 139 (1994) 79–115.
[14], Foliations and the topology of 3-manifolds: R-covered foliations and transverse
pseudo-Anosov flows, Comm. Math. Helv. 77 (2002) 415–490.

[15] H. Fuji, Colored HOMFLY homology and super-A-polynonial, Proceedings of the conference“Intelligence of Low-dimensional Topology”, RIMS Kōkyūroku (the same volume as thismanuscript).
[16] H. Fuji, S. Gukov, P. Sulkowski, Super-A-polynomial for knots and BPS states, Nucl. Phys.
B867 (2013) 506–546.
[17] D. Gabai, Foliations and the topology of 3–manifolds. III, J. Diﬀ. Geom. 26 (1987) 479–536[18] S. Garoufalidis, On the characteristic and deformation varieties of a knot, Geometry andTopology Monographs 7 (2004) 291–304.
[19] F. González-Acuña, H. Short, Knot surgery and primeness, Math. Proc. Camb. Phil. Soc. 99(1986) 89–102.
[20] C. McA. Gordon, J. Luecke, Knots are determined by their complements, J. Amer. Math. Soc.
2 (1989) 371–415.
[21] S. Gukov, Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial,
Commun. Math. Phys. 255 (2005) 577–627.
[22] M. Hedden, On knot Floer homology and cabling. II, Int. Math. Res. Not. IMRN, (12):2248–
2274, 2009.
[23] M. Hedden, L. Watson, Does Khovanov homology detects the unknot?, Amer. J. Math. 132(2010) 1339–1345.
[24] J. R. Hughes, Link homotopy invariant quandles, J. Knot Theory Ramifications 20 (2011)
763–773.

13

[25] A. Inoue, Quasi-triviality of quandles for link-homotopy, to appear in J. Knot Theory Ramifications.
[26], Toward obtaining a table of link-homotopy classes: The second homology of a reduced
knot quandle, to appear in RIMS Kōkyūroku.

[27] T. Ito, A remark on the Alexander polynomial criterion for the bi-orderability of fibered 3manifold groups, Int. Math. Res. Not. IMRN. 2013 (2013) 156–169.
[28], Dehornoy-like left orderings and isolated left orderings, J. Algebra, 374 (2013) 42–58.

[29], Non-left-orderable double branched coverings, Algebr. Geom. Topol., to appear.

[30] T. Li, R. Roberts, Taut foliations in knot complements, arXiv:1211.3066.
[31] Y. Li, L. Watson, Genus one open books with non-left-orderable fundamental group, Proc.
Amer. Math. Soc., to appear.
[32] J. Milnor, Link groups, Ann. of Math. 59 (1954) 177–195.
[33], Isotopy of links, Algebraic geometry and topology (A symposium in honor of S.
Lefschetz), Princeton Univ. Press, Princeton, N. J., 1957, 280–306.

[34] K. Motegi, M. Teragaito, Left-orderable, non–L–space surgeries on knots, arXiv:1301.5729.
[35] A. Navas, A remarkable family of left-ordered groups: Central extensions of Hecke groups, J.
Algebra, 328 (2011) 31–42.
[36] L. Ng, Framed knot contact homology, Duke Math. J. 141 (2008) 365–406.
[37] P. Ozsváth, Z. Szabó, Holomorphic disks and topological invariants for closed three-manifolds,
Ann. of Math. 159 (2004) 1027–1158.
[38], Holomorphic disks and three-manifold invariants: properties and applications, Ann.
of Math. 159 (2004) 1159–1245.

[39], On knot Floer homology and lens space surgeries, Topology 44 (2005) 1281–1300.

[40], Knot Floer homology and rational surgeries, Algebr. Geom. Topol. 11 (2011) 1–68.

[41] B. Perron, D. Rolfsen, On orderability of fibred knot groups, Math. Proc. Cambridge Philos.
Soc. 135 (2003) 147–153.
[42], Invariant ordering of surface groups and 3-manifolds which fibre over S 1 , Math. Proc.
Cambridge Philos. Soc. 141 (2006) 273–280.

[43] W. Thurston, 3-manifolds, foliations and circles, I, arXiv:math/9712268.

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