数理解析研究所講究録
第 1866 巻 2013 年 135-148

135

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

lResearch Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto, 606-8502, JAPAN
Email: tomotada@kurims.kyoto-u.ac.jp

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1

Super- $A$ -polynomials of knots(Hiroyuki Fuji)
In the generalized volume conjecture, the A-polynomial appears in the asymptotic
expansion of the colored Jones polynomial $J_{n}(K;q)$ :$J_{n}(K;q= e^{\hslash})\sim\exp(\frac{1}{\hslash}S_{0}(x)+\sum_{k=1}^{\infty}\hslash^{k-1}S_{k}(x)) (x:=q^{n})$,$\frac{\partial S_{0}(x)}{\partial x}=\log y(x) , A_{K}(x, y(x))=0.$

The higher order terms $S_{k}(x)$ in the above asymptotic expansion is found by solving
the -difference equation iteratively [11]:$q$$A_{K}(\hat{x},\hat{y};q)J_{n}(K;q)=0$$\hat{x}J_{n}(K;q)=q^{n}J_{n}(K;q) , \hat{x}J_{n}(K;q)=J_{n+1}(K;q)$.

reduces to the A-polynomial in $qarrow 1$ limit,
Such -difference operator
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 $H_{R}(K;a, q)$ for the completely symmetric representation $R=S^{r}$ . From-difference equation of the colored HOMFLY polynomial, one finds a 1-parameter
generalization of the A-polynomial $A_{K}^{Q-def}(x, y;a)$ that is named as the -deformed-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 superpolynomialwith $R=S^{r}$ , and the 2-parameter generalization of the A-polynomial
is found. This $(a, t)$ -deformed polynomial is named as super- -polynomial.
knots, the super- -polynomial are obtained as follows:
For
and$A_{K}(\hat{x},\hat{y};q)$

$q$

$q$

$Q$

$A$

$\mathcal{P}^{R}(K;a, q, t)$$A$$4_{1}$$3_{1}$$A$$A_{3_{1}}^{\sup er}(x, y;a, t)=a^{2}t^{4}(x-1)x^{3}+(1+at^{3}x)y^{2}$$-a(1-t^{2}x+2t^{2}(1+at)x^{2}+at^{5}x^{3}+a^{2}t^{6}x^{4})y,$$A_{4_{1}}^{\sup er}(x, y;a, t)=a^{2}t^{5}(x-1)^{2}x^{2}+at^{2}x^{2}(1+at^{3}x)^{2}y^{3}$$+at(x-1)(1+t(1-t)x+2at^{3}(t+1)x^{2}-2at^{4}(t+1)x^{3}+a^{2}t^{6}(1-t)x^{4}-a^{2}t^{8}x^{5})y$$-(1+at^{3}x)(1+at(1-t)x+2at^{2}(t+1)x^{2}+2a^{2}t^{4}(t+1)x^{3}+a^{2}t^{5}(t-1)x^{4}+a^{3}t^{7}x^{5})y^{2}$

Question 1.1 (H. Fuji). Is it possible to express the augmentation polynomial andthe super- -polynomial via some $(a, t)$ -deformed holonomy representation2 Can we
extract the non-trivial geometric information of
for $A_{K}^{\sup er}(x, y;a, t)$ ?
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 $C[12]$ :$A$$S^{3}\backslash K$$C= \{(x, y)\in \mathbb{C}\cross \mathbb{C}^{2}|y^{2}=M(x)^{2}S(x), S(x)=\prod_{i=1}^{2n}(x-x_{i})\}.$

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$+$

$g$

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

The topological recursion solves the meromorphic
onin an iterative manner:$(1, \cdots\rangle 1)$-forms $W^{(h,n)}(p_{1}, \cdots,p_{n})$$(p_{1)}\cdots,p_{n})\in C^{\otimes n}$$W^{(0,1)}(p)=y(p)dx(p)$ ,$W^{(0,2)}(p_{1},p_{2})=B(p_{1},p_{2})$,$K(p_{0},p)= \frac{-\frac{1}{2}\int_{\overline{p}}^{p}B(\cdot,p_{0})}{(y(p)-y(\overline{p}))dx(p)},$$W^{(h,n)}(p_{1}, \cdots,p_{n})=\sum_{a:Branchpts}{\rm Res}_{qarrow a}K(p_{0}, q)[W^{(h-1,n+1)}(q,\overline{q},p_{I})$$+ \sum_{h’,J}W^{(h’,|J|+1)}(q,p_{J})W^{(h-h’,n-|J|)}(\overline{q},p_{I\backslash J})],$

where $B(x, y)$ is the Bergman kernel on which is (1, 1)-form on
Using the theta function of the spectral curve and meromorphic forms
the tau function
is studied systematically in [4].
is known as the$C\cross C.$$C$$W^{(h,n)},$$C$$\mathcal{T}_{g_{s}}[ydx]$$\mathcal{T}_{g_{s}}[ydx]$

non-perturbative partition function, and the $(n|n)$ -Baker-Akhierzer kernel which isdenoted as
is defined as the Schlesinger transformation of
the non-perturbative partition function:$\psi_{g_{s}}^{[n|n]}$$(p_{1},0_{1};\cdots ; p_{n}, 0_{n})$$\psi_{g_{s}}^{[n|n]}(p_{1},0_{1};\cdots;p_{n}, 0_{n})=\frac{\mathcal{T}_{g_{s}}[ydx+\sum_{k=1}^{n}dS_{O_{k,Pk}}]}{\mathcal{T}_{g_{s}}[ydx]}$

where the meromorphic 1-form
is given by
In [10, 4], the topological recursion on the character variety$dS_{O_{k,p_{k}}}$$dS_{o_{k},p_{k}}= \int_{0_{k}}^{p_{k}}B(\cdot,p)$.

$C_{K}=\{(x, y)\in \mathbb{C}^{*}\cross \mathbb{C}^{*}|A_{K}(x, y)=0\}$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$|$$\mathcal{T}_{g_{S}}[4]$Conjecture 1.2 ([4]). As formal power series, the colored Jones polynomial expanded around the exponential growth point coincides with the $(2|2)$ -Baker-Akhierzer
kemel on the character variety $C_{K}=\{(x, y)\in \mathbb{C}^{*}\cross \mathbb{C}^{*}|A_{K}(x, y)=0\},$$J_{n}(K;q=e^{\hslash/2})\simeq\psi_{\hslash}^{[2|2]}(p, 0;\iota(p), \iota(0))^{1/2},$

where $x(\iota(p))=x$ and the involution acts as $(x(\iota(p)), y(\iota(p)))=(x(\iota(p))^{-1}, y(\iota(p))^{-1})$ .
$\iota$3

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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 $(2|2)$ -kemel on the the superchamcter variety whose defining equation is given by $A_{K}^{\sup er}(x, t;a, t)$ related withthe asymptotic expansion of the colored superpolynomial?
$\mathcal{P}^{s^{r}}(K;a, q, t)$

In the case of $K=3_{1}$ , the super-character variety is expressed in$\mathbb{C}^{2}$variables as:$C=\{(x, y)\in \mathbb{C}^{2}|y^{2}=M(x)^{2}S(x)\},$$S(x)=x^{4}+2a^{-1}t^{-1}u^{3}+(a^{-2}t^{2}+2a^{-1}t^{-3}+4a^{-2}t^{-4})-2a^{-2}t-4+a^{-2}t^{-6},$$A(x)= \frac{1-t^{2}x+2t^{2}x^{2}+2at^{3}x^{2}+at^{5}x^{3}+a^{2}t^{6}x^{4}}{at^{3}(1+at^{3}x^{2})},$$M(x)= \frac{1}{2x\sqrt{S(x)}}\log\frac{A(x)+\sqrt{S(x)}}{A(x)-\sqrt{S(x)}}.$

andhas deformed reciprocal symmetry : $x\mapsto-1/(at^{3}x)$ . The
andterms are given by definition of the free energy as the Abel-Jacobi map anddiscriminant on
respectively [10]:$A_{3_{1}}^{\sup er}(x, y;a, t)$$\iota$$S_{0}$$S_{1}$$C_{0}$$S_{0}(x)=l^{x} \frac{dx}{x}\log y(x)$,$S_{1}(x)= \frac{1}{2}\log\frac{\gamma}{\sqrt{S(x)}},$

where is a constant. The next order[4], and the result is$\gamma$$S_{2}$

is computed via the topological recursion$S_{2}(x)=(1+2at+2t^{2}+4at^{3}+2a^{2}t^{4}+(3t^{2}-2t^{4}-4at^{5}-2a^{2}t^{6})x$$+(-16t^{2}-11at^{3}-21t^{4}+2a^{2}t^{4}-28at^{5}-2t^{6}-8a^{2}t^{6}-4at^{7}+2a^{3}t^{7}-2a^{2}t^{8})x^{2}$$+(-30at^{5}-7t^{6}-24a^{2}t^{6}-48at^{7}+2t^{8}-68a^{2}t^{8}+4at^{9}-28a^{3}t^{9}+2a^{2}t^{10})x^{3}$$+(16at^{5}+11a^{2}t^{6}+21at^{7}-2a^{3}t^{7}+28a^{2}t^{8}+2at^{9}+8a^{3}t^{9}$$+4a^{2}t^{10}-2a^{4}t^{10}+2a^{3}t^{11})x^{4}$$+(3a^{2}t^{8}-2a^{2}t^{10}-4a^{3}t^{11}-2a^{4}t^{12})x^{5}$$+(-a^{3}t^{9}-2a^{4}t^{10}-2a^{3}t^{11}-4a^{4}t^{12}-2a^{5}t^{13})x^{6})$$/(48(1+at)(1+t^{2}+at^{3})(1-2t^{2}x+4t^{2}x^{2}+2at^{3}x^{2}+t^{4}x^{2}+2at^{5}x^{3}+a^{2}t^{6}x^{4})^{3/2})$

The details of the computations are given in [15]. The above result and the asympbehave differently. This discrepancy would be
totic expansion of-dependenceofand . This point may be cured by taking into
related with the-mirror symmetry” proposed in [2] carefully.
account of the mirror map of the$\mathcal{P}^{S^{r}}(K;a, q, t)$

$q$

$x$

$a$

$D$

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Quasi-trivial quandles and link homotopy invariants

2(Ayumu Inoue)

A quandle $X$ is said to be quasi-trivial if it satisfies the condition $x*\varphi(x)=x$
for each $x\in X$ andInn(X) [25]. Here, Inn(X) denotes the inner automorphismgroup of $X$ . Modifying usual quandle homology slightly, we have homology of aquasi-trivial quandle [25]. We letdenote the n-th homology
andand cohomology groups of a quasi-trivial quandle $X$ respectively.
For each oriented and ordered link in , 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 componentof , we have the fundamental classwhich is invariant under link-homotopy [25, 26].
Conjecture 2.1 (A. Inoue). Two -component links $L=K_{1}\cup\cdots\cup K_{n}$ and$L’=K_{1}’\cup\cdots\cup K_{n}’$ are link-homotopic
if and only if there is an isomorphism: $RQ(L)arrow RQ(L’)$ such thatfor each $i(1\leq i\leq n)$ . Here,$\varphi_{\#}:H_{2}^{Q,qt}(RQ(L))arrow H_{2}^{Q,qt}(RQ(L’))$ denotes the isomorphism induced
fromThis conjecture is true in the case $n\leq 3$ . Prove the conjecture, or give a counterexample.
Suppose that $X$ is a quasi-trivial quandle. By definition, for each $x\in X$ andInn(X), the pair $(x, \varphi(x))$ is a 2-cycle.
Question 2.2 (A. Inoue). Let $L=K_{1}\cup\cdots\cup K_{n}$ be an -component link. For all$x\in RQ(L)$ andInn(X), is not the classin$H_{2}^{Q,qt}(RQ(L))$ ?
$\varphi\in$$H_{n}^{Q,qt}(X)$$H_{Q,qt}^{n}(X)$$S^{3}$$L$$L$$K_{i}$$[K_{i}]^{qt}\in H_{2}^{Q,qt}(L)$$n$$\varphi_{\#}([K_{i}]^{qt})=[K_{i}’]^{qt}$$\varphi$$\varphi.$$\varphi\in$$n$$[x, \varphi(x)]$$\varphi\in$$\langle[K_{1}]^{qt}\rangle\oplus\cdots\oplus\langle[K_{n}]^{qt}\rangle\triangleleft$

which is a certain quotient of
We have another homology group$H_{n}^{Q,qt}(RQ(L))$ .
. We can show that
By definition, $[x, \varphi(x)]=0$ in. If
for each and$H_{2}^{Q,qt}(RQ(L))$
is sufficient
Question 2.2 is true, considering
instead of$\tilde{H}_{n}^{Q,qt}(RQ(L))$$\tilde{H}_{n}^{Q,qt}(RQ(L))$$[K_{i}]^{qt}\in\tilde{H}_{n}^{Q,qt}(RQ(L))$$\tilde{H}_{n}^{Q,qt}(RQ(L))=\langle[K_{1}]^{qt}\rangle\oplus\cdots\oplus\langle[K_{n}]^{qt}\rangle$$i$$\tilde{H}_{2}^{Q,qt}(RQ(L))$

to classifying links up to link-homotopy.
Choose and fix a quasi-trivial quandle $X$ and its 2-cocycle with coefficientsin an abelian group . For each -component link $L=K_{1}\cup\cdots\cup K_{n}$ , consider
all homomorphisms : $RQ(L)arrow X$ . Then the multiset consisting of all elements
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 trivia13-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 -invariants[33],$\theta$$A$$n$$\varphi$$\langle\theta,$$\varphi_{\#}([K_{i}]^{qt})\rangle\in A$$\overline{\mu}$5140
Question 2.4 (A. Inoue). How are quandle cocycle invariants related with Milnor’s-invariants?

$\overline{\mu}$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 is called a lefl-ordering (resp. right-ordering)$g\in G.$ $<G$ is called a
if $a<Gb$ implies $ga<Ggb$ resp. $ag<Gbg)$ for allgroup is left-orderable (resp.
-ordering if it is both left- and right-ordering.
-ordemble) if 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.
$G$$a,$ $b,$

$($

$G$

$A$

$bi$

$G$

$bi$

Question 3.1 (T. Ito (Bi-ordering of knot groups)).
(1) Find an example of -ordering of non-fibered knot groups.
knot -orderable 2(2) Is the knot group of theknot -orderable?
(3) Is the knot group of the-orderable,
but virtually -orderable knot groups. (More(4) Find an example of not
genemlly, is a knot group virtually -ordemble?)$bi$$5_{2}$$bi$$6_{2}$

$bi$

$bi$

$bi$

$bi$

knot isknot is the simplest non-fibered knot. As for (3), thethe 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 ofa left-orderable group is called universally cofinal if for every left-ordering $<c$ ofleft orderingsuch that $a^{-N}<cg<c^{a^{N}}$ holds.
and $g\in G$ , there exists

As for (2), the$6_{2}$$5_{2}$

$a$

$G$

$G$

$<c$

$A$

$N\in \mathbb{Z}$of $G$ is called discrete if $<c$ admits $a<G$ -positive minimum element.

Question 3.2 (T. Ito (Left-ordering of knot groups)).
(1) Give an “explicit” example of lefl-ordering $<of$ knot groups.
(2) Can we genemlize the above mentioned orderings of torus knot groups to otherclass of knot groups, such as, 2-bridge knot gmups?
(3) Which knot gmup has (does not have) universally cofinal element?
(4) Do every knot group $<G$ admit a discrete ordering $<G$ such that its meridian is$the<c$ -minimum positive element?
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As for (1), explicit) means that we have a reasonable characterization of -positiveelements, or, one can check given element 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.
$<$

$<$

$x$

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 -space. Several supporting
evidences of this conjecture have been established by many researchers, but generalcase is widely open.
$L$

Question 3.3 (T. Ito ( $(Non-)left$ -orderable 3-manifold)).
(1) Is the double bmnched covering of a quasi-alternating link non-lefl-orderable?
(2) Chamcterize non-lefl-ordemble 3-manifold of Heegaard genus two.
Let $M$ be non-left-ordemble (hyperbolic) 3-manifold. Is there a finite covering$M$of with property $(a)$ and $(b)$ below 2(a) $M$ is left-ordemble.
(b) $M$ is a mtional homology 3-sphere.
$\underline{(3)}$$\underline{M}$As for (1), see [29] for related results. As for (2), see [31] for the corresponding
results for 3-manifolds admitting genus one open book decomposition. As for (3),by virtual fibered conjecture, if we drop the condition (b), we can always find such$M.$

Left-orderable surgeries and4$L$-space surgeries(Kimihiko Motegi)2$L$-spacesHeegaard Floer homology is a package of 3-manifold invariants introduced by Ozsv\’athand Szab\’o [37, 38]. In the following we consider the simplest one called the “hat”
gaard Floer homology $HF(Y)$ has a relative). Theversion, denoted by$\overline{HF}(Y)=HF_{0}(M)\oplus
HF_{1}(Y)$. For a rational homology 3-sphere $M,$
-grading
we have $\chi(\overline{HF}(Y))=|H_{1}(M;\mathbb{Z})|$ . In particular, rank $\hat{HF}(M)\geq|H_{1}(M;\mathbb{Z})|.$$\overline{HF}\underline{(Y}$$\underline{Hee}$$Z_{2}$

There are several kinds of definitions for -spaces. In particular, we often findthe following definitions in literatures.
$L$

Definition ( -space)rational homology 3-sphere $M$ is called an -space if the Heegaard Floer(1)
is a free abelian group of rankoefficients
homology with(2)
rational homology 3-sphere $M$ is called an -space if the Heegaard Floer
$L$

$L$

$A$

$\mathbb{Z}-c$$|H_{1}(Y;\mathbb{Z})|.$$\hat{HF}(Y;\mathbb{Z})$

$L$

$A$

$2I$
would like to thank Cameron Gordon, Hiroshi Matsuda, Yi Ni, Mot $oo$ Tange for private communications
concerning Question 4.1. would also like to thank Masakazu Teragaito for useful discussions.
$I$

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has the rank
homology with -coeffcients$M$
is called an -space if the Heegaard Floer(3)
rational homology 3-sphere
has the rank
homology with -coefficients$\overline{HF}(Y;\mathbb{Z})$$\mathbb{Z}$$|H_{1}(Y;\mathbb{Z})|.$

$L$

$A$

$\overline{HF}(Y;\mathbb{Z}_{2})$$\mathbb{Z}_{2}$$|H_{1}(Y;\mathbb{Z})|.$(1) is the original definition of Ozsv\’ath and Szab\’o [39], which does not allow “torsion”, but (2) allows “torsion” The difference between (2) and (3) is the choice of“coefficients” for $HF(Y)$ which we use.
Question 4.1 (K. Motegi). Are these three conditions equivalent? In particular,
have no torsion?
for any mtional homology sphere , does$\overline{HF}(Y;\mathbb{Z})$$Y$

has torFor rational homology 3-spheres, no examples are known where
moment,However,(3)
equivalent.
at
the
(2)
(1),
are
and
sion. So possibly definitions
it seems convenient to adopt the definition (3) for homogeneity; see [5, 1.1].
$\overline{HF}(Y;\mathbb{Z})$The set of left-orderable surgeries

We say that a nontrivial group is lefl-ordemble if there exists a strict total ordering
on its elements such that $g<h$ implies $fg<fh$ for all elements $f,$ $h\in G$ . The
left-orderability of fundamental groups of 3-manifolds has been studied by Boyer,
rreducible 3-manifold
Rolfsen and Wiest [6]. In particular, it is known that
$M$ with first Betti number
has the left-orderable fundamental group. Hencefor any knot $K$ in , its exterior $E(K)$ has the left-orderable fundamental group.
However if $r\neq 0$ , the result $K(r)$ of -Dehn surgery on $K$ , which is a rational
homology 3-sphere, may not have such a fundamental group. A Dehn surgery issaid to be lefl-ordemble 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
$G$

$<$

$g,$

$P^{2}arrow$$b_{1}\geq 1$$S^{3}$$r$$S_{LO}(K)=\{r\in \mathbb{Q}|\pi_{1}(K(r))$

is left-orderable$\}.$

For any nontrivial knot $K,$ $K(O)$ is irreducible [17, Corollary 8.3] andand hence $\{0\}\subset S_{LO}(K)$ [ , Theorem 1.1]. If $K$ is a trivial knot then $S_{LO}(K)=\{0\},$which has the smallest size. In the opposite direction, in [34], we demonstrate that
there are infinitely many hyperbolic knots $K$ with $S_{LO}(K)=\mathbb{Q}$ . It seems interesting
to ask:$H_{1}(K(O))\cong \mathbb{Z},$$6$

Question 4.2 (K. Motegi). If $K$ is a nontrivial knot in?
contain$S^{3}$, then does $S_{LO}(K)$$(-1,1)\cap \mathbb{Q}$

Recently Li and Roberts [30, Corollary 1.2] prove that for any hyperbolic knot $K,$
such that $\{\frac{1}{n}||n|>N_{K}\}\subset S_{LO}(K)$ . More strongly wethere exists a constantwould like to ask:
$N_{K}$

Question 4.3 (K. Motegi). If $K$ is a nontrivial knot in?
orcontain$(-\infty, 1)\cap \mathbb{Q}$$S^{3}$, then does $S_{LO}(K)$$(-1, \infty)\cap \mathbb{Q}$), the argument in the proof of [9,(resp.
For the simplest nontrivial knot
$S_{LO}(T_{3,2})=(-\infty,1)\cap\mathbb{Q}$resp. $S_{LO}(T_{-3,2})=(-1, \infty)\cap \mathbb{Q})$ .
Theorem 1.4] shows that
$T_{-3,2}$$T_{3,2}$

$($

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Question 4.4 (K. Motegi). If $S_{LO}(K)=(-\infty, 1)\cap \mathbb{Q}$ or, respectively 2then is $K$ a trefoil knotor$\mathcal{S}_{LO}(K)=(-1, \infty)\cap \mathbb{Q},$$T_{-3,2}$$T_{3,2}$

Question 4.5 (K. Motegi). Let $K$ be a nontrivial knot inhave no maximum and minimum?

$S^{3}$. Then does$\mathcal{S}_{LO}(K)$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 -spaces. For instance, lens spaces, spherical 3-manifolds are -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].
$L$

$L$

Conjecture 4.6 ([5]). An irreducible mtional homology 3-sphere is an
and only if its fundamental group is not left-ordemble.

$L$-space if

A Dehn surgery is called an -space surgery if the resulting manifold of thesurgery is an -space. Define the set of -space surgery slopes for $K$ as
$L$

$L$

$L$

$S_{L}(K)=\{r\in \mathbb{Q}|K(r)$is an -space
$L$

$\}.$

Now let us look at the shape of $S_{L}(K)$ , 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 and, then $S_{L}(K)=[2g(K)-1, \infty)\cap \mathbb{Q}$ or $\mathcal{S}_{L}(K)=(-\infty, -2g(K)+1]\cap \mathbb{Q}.$are complementary to each other
Since Conjecture 4.6 says that $S_{L}(K)$ and$K$
in if has no reducing surgery, we expect the following explicit form of $S_{LO}(K)$ .

$\mathcal{S}_{L}(K)\neq\emptyset$$\mathcal{S}_{LO}(K)$$\mathbb{Q}$

Conjecture 4.7. Let $K$ be a nontrivial knot in
which has no reducing surgery.
$(-\infty,
2g(K)-1)\cap\mathbb{Q}$and $(-2g(K)+1, \infty)\cap \mathbb{Q}.$Then $S_{LO}(K)$ coincides with one of$S^{3}$$\mathbb{Q},$Finally we give a comment on Question 4.4 in case of $S_{LO}(K)=(-\infty, 1)\cap \mathbb{Q}$ ;the other case follows by taking the mirror image. By the assumption $1\not\in S_{LO}(K)$ ,if Conjecture 4.6 is true, then $1\in S_{L}(K)$ or $K(1)$ is reducible. The latter possibility
is eliminated by [20, Corollary 3.1], and hence $K(1)$ is an -space. Further, it isis an -space, then $n=1$ (resp.
known [23] that, if $K$ is a nontrivial knot and$K$ is a trefoil knot$-1)$ and $K$ is a trefoil knot
Therefore,).
(resp.
$L$$K( \frac{1}{n})$$T_{3,2}$

5$T_{-3,2}$$L$$T_{3,2}.$Groups with addition(Rinat M. Kashaev)

A group with addition is a group with an additional commutative and associative binary operation $(x, y)\mapsto x+y$ with respect to which the group product is
distributive from both sides.
Examples (groups with addition)orwith the usual addition.
(1)$\mathbb{Q}_{>0}$$\mathbb{R}_{>0}$

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(2)
or with tropical addition $\max(x, y)$ .
(3) The group of upper triangular n-by-n matrices overelements and with the addition of matrices.
$\mathbb{Z},$$\mathbb{Q}$$\mathbb{R}$$\mathbb{Q}$or$\mathbb{R}$with positive diagonal

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 addition2(2) What are the conditions under which a given infinite gmup admits an addition2(3) Is there a systematic pmcedure 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 freely acted uponby a group , one can associate a connected groupoid
with the set of -orbits
in as the set of objects Ob, the -orbits in(with respect to the diagonal
action) as the set of morphisms Mor, the domain and the codomain maps
$S$

$G$

$G$

$\mathcal{G}s,c$

$S$

$G$

$\mathcal{G}s,c$$S\cross S$$\mathcal{G}s,c$dom, cod: Mor$\mathcal{G}s,carrow$Ob$\mathcal{G}s,c$,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]=$$dom[u, v]$ , i.e. $[y]=[u]$ with the composition (adopting the convention used for thecomposition in fundamental groupoids of topological spaces) $[x, y]0[u, v]=[x, gv],$
where is the unique group element such that $y=gu$ . The groupoid
isa connected groupoid with any vertex group being isomorphic to , so that any
representation of
restricts to a representation of $G.$Let
be a closed oriented surface of genus with punctures such that$(2g-2+s)s>0$ . An ideal triangulation of is a $CW$-decomposition of the closed
surface X where the set of vertices coincides with the set of punctures, and all cells
are simplices.
decorated ideal triangulation of is an ideal triangulation where
each triangle is provided with a distinguished corner and the set of all triangles islinearly ordered. Let
be the set of decorated ideal triangulations of . The
mapping class group of, freely acts on. The associated groupoidis called the groupoid of (decorated) ideal triangulations of$g$$\mathcal{G}s,c$$G$$\mathcal{G}_{S,G}$$\Sigma=\Sigma_{g,s}$

$g$

$s$

$\Sigma$$A$$\Sigma$$\Sigma$$\triangle_{\Sigma}$$\Sigma,$$\triangle_{\Sigma}$$\Gamma_{\Sigma}$$\mathcal{G}_{\Delta_{\Sigma},\Gamma_{\Sigma}}$$\Sigma.$A presentation of the groupoid of ideal triangulations. One can show that

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$\mathcal{G}_{\triangle_{\Sigma},\Gamma_{\Sigma}}$is generated by the following three types of morphisms:

Diagonal flips:

Corner changes:

Permutations:which satisfy the following relations:Pentagons:
Triple corner changes:
Double flips:
Permutation relations:$\omega_{ij}0\omega_{ik}0\omega_{jk}=\omega_{jk}0\omega_{ij}$$\rho_{i}\circ\rho_{i}=\rho_{i}^{-1}$$\omega_{ij}0\rho_{i}o(ij)\circ\omega_{ij}=\rho_{i}\circ\rho_{j}$$(ij)o(jk)\circ(ij)=(jk)\circ(ij)\circ(jk)$etc.

Semisymmetric -matrices.
-matrix in a symmetric monodical category$C=(C, \otimes, s)$ is an automorphism $T\in$ Aut, with $V\in$ Ob , which satisfies the equation $T_{12}T_{13}T_{23}=T_{23}T_{12}$ in Aut. $AT$-matrix $T\in$ Aut
issemisymmetric if there exists a symmetry $A\in$ Aut(V) such that $A^{3}=id_{V}$ and
$T$

$A$

$T$

$C$

$(V^{\otimes 2})$$(V^{\otimes 3})$$(V^{\otimes 2})$

$T(A\otimes id_{V})s_{V},{}_{V}T=A\otimes A.$

Theorem. Let

be an oriented surface of genus with punctures such that $(2g-$$2+s)s>0$ . Then, for any semisymmetric -matrix $T\in$ Aut
there exists
a unique representation$n_{g,s}:=4g-4+2s$ , such that$\Sigma$

$s$

$g$

$T$

$(V^{\otimes 2})$$\pi_{T}:\mathcal{G}_{\Delta_{\Sigma},\Gamma_{\Sigma}}arrow Aut(V^{\otimes n_{g,s}}),$$\pi_{T}(\omega_{ij})=T_{ij},$$\pi_{T}(\rho_{i})=A_{i},$$\pi_{T}((1,2))=s_{V,V}\otimes id_{V^{\otimes(n_{g,s}-2)}}.$

Theorem. Let be a group with addition, $c\in G$ a centml element, and $X=G\cross G.$
Then, there exists a semisymmetric set-theoretical -matrix $t:X^{2}\ni(x, y)\mapsto(x\cdot$$x*y)\in X^{2}$ with symmetry $a:X\ni(x_{1}, x_{2})\mapsto(cx_{1}^{-1}x_{2}, x_{1}^{-1})\in X,$ $x\cdot y=$, and $x*y=a^{-1}(a(y)\cdot a^{-1}(x))$ .
$G$

$T$

$y,$

$(x_{1}y_{1}, x_{1}y_{2}+x_{2})$6Periodic orbits of pseudo-Anosov flows(S\’ergio R. Fenley)

A pseudo-Anosov flow 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.
$\Phi$

Question 6.1 (S. R. Fenley). Do the conjugacy classes of the periodic orbits of thepseudo-Anosov flow genemte $\pi_{1}(M)$ ?
$\Phi$

This is not always true. For example if is a suspension pseudo-Anosov flow thenthis does not happen. Is that the only situation where this does not happen?
$\Phi$

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Question 6.2 (S. R. Fenley). Suppose that is a pseudo-Anosov flow tmnsverseto an -covered foliation and that is regulating for . Does this imply that the?
periodic orbits do not generate$\Phi$$\Phi$$\mathbb{R}$$\mathcal{F}$$\pi_{1}(M)$

Here -covered means that in the universal cover, the leaf space of the lifted foliation is homeomorphic to the set of real numbers . 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 is -covered,transversely orientable, and $M$ is aspherical and atoroidal, there is such a flow. Thiswas proved in [7] and [14].
An Anosov flow is said to be -covered if its stable (or unstable) foliation is-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 is freely homotopic to infinitely many other closed
orbits.
$\mathbb{R}$$\mathbb{R}$$\mathcal{F}$$\mathbb{R}$$\mathbb{R}$$\Phi$$\mathbb{R}$$\Phi$

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 is an -covered Anosov flow so that its stable foliationis transversely orientable then the following happens: given a periodic orbit, it isfreely homotopic to infinitely many other closed orbits. In [3] it is proved that is
in fact isotopic to each of these other closed orbits. That is, they represent the sameknot in $M.$$\Phi$$\mathbb{R}$$\gamma$$\gamma$

Question 6.4 (S. R. Fenley). Suppose that and are closed orbits of a pseudoAnosov flow which are freely homotopic. Does it follow that and are isotopic,
as is the case when is an -covered Anosov flow? When are they isotopic?
$\alpha$$\beta$$\Phi$$\alpha$$\Phi$$\beta$$\mathbb{R}$The proof in [3] uses the universal circle of Thurston [43] in an essential way. In thecase that the stable foliation of is not an -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.
$\Phi$$\mathbb{R}$

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