Left-ordering
L-space conjecture
Our Result

Left-orderable fundamental groups
and Dehn surgery on two-bridge knots
Masakazu Teragaito
Department of Mathematics Education
Hiroshima University

Intelligence of Low-dimensional Topology
RIMS, May 23, 2013

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Outline1

Left-ordering
Definition and Examples2

L-space conjecture
L-space
Previous Works

3

Our Result
Main Result
Basic Ideas for Proof

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Definition and Examples

Left-ordering

A non-trivial group G is said to be left-orderable (LO)if G admits a strict total ordering "<" which is invariant underleft-multiplication.
g < h ⇒ fg < fh

Masakazu Teragaito

for any f , g, h ∈ G

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Definition and Examples

Left-ordering

A non-trivial group G is said to be left-orderable (LO)if G admits a strict total ordering "<" which is invariant underleft-multiplication.
g < h ⇒ fg < fh

Masakazu Teragaito

for any f , g, h ∈ G

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Definition and Examples

ExamplesMany groups, which appear in Topology, are known to be LO.
free groups, free abelian groups
fundamental groups of surface 6= P 2(classical) knot/link groups
braid groups

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Definition and Examples

ExamplesMany groups, which appear in Topology, are known to be LO.
free groups, free abelian groups
fundamental groups of surface 6= P 2(classical) knot/link groups
braid groups

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

L-space
A rational homology sphere Y is an L-space if Heegaard–Floerd(Y ) is a free abelian group with rank equal tohomlogy HF|H1 (Y ; Z)|.
lens spaces
elliptic manifolds
double branched covers over non-split alternatingknots/linksd(Y ) ≥ |H1 (Y ; Z)| for any rational homology* In general, rank HF
sphere Y .

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

L-space
A rational homology sphere Y is an L-space if Heegaard–Floerd(Y ) is a free abelian group with rank equal tohomlogy HF|H1 (Y ; Z)|.
lens spaces
elliptic manifolds
double branched covers over non-split alternatingknots/linksd(Y ) ≥ |H1 (Y ; Z)| for any rational homology* In general, rank HF
sphere Y .

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

L-space conjecture

It is an open problem to find a non-Heegaard–Floer
characterization of L-spaces.
Conjecture (Boyer-Gordon-Watson 2011)
Let Y be an irreducible rational homology sphere.
Then Y is an L-space if and only if π1 (Y ) 6= LO.
They confirmed the conjecture for Seifert fibered manifolds,
Sol-manifolds, etc.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

L-space conjecture

It is an open problem to find a non-Heegaard–Floer
characterization of L-spaces.
Conjecture (Boyer-Gordon-Watson 2011)
Let Y be an irreducible rational homology sphere.
Then Y is an L-space if and only if π1 (Y ) 6= LO.
They confirmed the conjecture for Seifert fibered manifolds,
Sol-manifolds, etc.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

L-space conjecture

It is an open problem to find a non-Heegaard–Floer
characterization of L-spaces.
Conjecture (Boyer-Gordon-Watson 2011)
Let Y be an irreducible rational homology sphere.
Then Y is an L-space if and only if π1 (Y ) 6= LO.
They confirmed the conjecture for Seifert fibered manifolds,
Sol-manifolds, etc.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Rational homology spheres are obtained by Dehn surgery on
knots in huge quantities.
Let K be a knot in S 3 . If K admits Dehn surgery yielding an
L-space (L-space surgery), then
K is fibered (Yi Ni),
the Alexander polynomial has a special form(Ozsváth-Szabó).

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Rational homology spheres are obtained by Dehn surgery on
knots in huge quantities.
Let K be a knot in S 3 . If K admits Dehn surgery yielding an
L-space (L-space surgery), then
K is fibered (Yi Ni),
the Alexander polynomial has a special form(Ozsváth-Szabó).

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Rational homology spheres are obtained by Dehn surgery on
knots in huge quantities.
Let K be a knot in S 3 . If K admits Dehn surgery yielding an
L-space (L-space surgery), then
K is fibered (Yi Ni),
the Alexander polynomial has a special form(Ozsváth-Szabó).

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Motivation

Hence we can say that “most” knots have no L-space surgery.
If we support L-space conjecture, then we can expect
any non-trivial Dehn surgery on “most” knots yields
a 3-manifold whose π1 = LO.
A slope r is said to be left-orderable (LO) if π1 K (r ) is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Motivation

Hence we can say that “most” knots have no L-space surgery.
If we support L-space conjecture, then we can expect
any non-trivial Dehn surgery on “most” knots yields
a 3-manifold whose π1 = LO.
A slope r is said to be left-orderable (LO) if π1 K (r ) is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Motivation

Hence we can say that “most” knots have no L-space surgery.
If we support L-space conjecture, then we can expect
any non-trivial Dehn surgery on “most” knots yields
a 3-manifold whose π1 = LO.
A slope r is said to be left-orderable (LO) if π1 K (r ) is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Motivation

Hence we can say that “most” knots have no L-space surgery.
If we support L-space conjecture, then we can expect
any non-trivial Dehn surgery on “most” knots yields
a 3-manifold whose π1 = LO.
A slope r is said to be left-orderable (LO) if π1 K (r ) is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Previous works on Dehn surgery vs LO
Boyer-Gordon-Watson 2011
For the figure-eight knot, any slope r in (−4, 4) is LO.
Idea: Use a continuous family of representations of π1 (S 3 − K )to SL2 (R).
Clay-Lidman-Watson 2011
For the figure-eight knot, the slopes ±4 are also LO.
Idea: Use the graph manifold structure and gluing technique ofleft-orderings.
* Any integral slope is LO for the figure-eight knot (Fenley1994).
Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Previous works on Dehn surgery vs LO
Boyer-Gordon-Watson 2011
For the figure-eight knot, any slope r in (−4, 4) is LO.
Idea: Use a continuous family of representations of π1 (S 3 − K )to SL2 (R).
Clay-Lidman-Watson 2011
For the figure-eight knot, the slopes ±4 are also LO.
Idea: Use the graph manifold structure and gluing technique ofleft-orderings.
* Any integral slope is LO for the figure-eight knot (Fenley1994).
Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Previous works on Dehn surgery vs LO
Boyer-Gordon-Watson 2011
For the figure-eight knot, any slope r in (−4, 4) is LO.
Idea: Use a continuous family of representations of π1 (S 3 − K )to SL2 (R).
Clay-Lidman-Watson 2011
For the figure-eight knot, the slopes ±4 are also LO.
Idea: Use the graph manifold structure and gluing technique ofleft-orderings.
* Any integral slope is LO for the figure-eight knot (Fenley1994).
Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Previous works on Dehn surgery vs LO
Boyer-Gordon-Watson 2011
For the figure-eight knot, any slope r in (−4, 4) is LO.
Idea: Use a continuous family of representations of π1 (S 3 − K )to SL2 (R).
Clay-Lidman-Watson 2011
For the figure-eight knot, the slopes ±4 are also LO.
Idea: Use the graph manifold structure and gluing technique ofleft-orderings.
* Any integral slope is LO for the figure-eight knot (Fenley1994).
Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Previous works on Dehn surgery vs LO
Boyer-Gordon-Watson 2011
For the figure-eight knot, any slope r in (−4, 4) is LO.
Idea: Use a continuous family of representations of π1 (S 3 − K )to SL2 (R).
Clay-Lidman-Watson 2011
For the figure-eight knot, the slopes ±4 are also LO.
Idea: Use the graph manifold structure and gluing technique ofleft-orderings.
* Any integral slope is LO for the figure-eight knot (Fenley1994).
Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Previous works on Dehn surgery vs LO
Boyer-Gordon-Watson 2011
For the figure-eight knot, any slope r in (−4, 4) is LO.
Idea: Use a continuous family of representations of π1 (S 3 − K )to SL2 (R).
Clay-Lidman-Watson 2011
For the figure-eight knot, the slopes ±4 are also LO.
Idea: Use the graph manifold structure and gluing technique ofleft-orderings.
* Any integral slope is LO for the figure-eight knot (Fenley1994).
Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Teragaito 2011
For any hyperbolic twist knot, the slope 4 is LO.
Idea: Generalize Clay-Lidman-Watson’s argument by using
Navas’s ordering.
Clay-Teragaito 2011
For a hyperbolic 2-bridge knot, any exceptional slope is LO.
Idea: Use the graph manifold structure with 3 pieces.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Teragaito 2011
For any hyperbolic twist knot, the slope 4 is LO.
Idea: Generalize Clay-Lidman-Watson’s argument by using
Navas’s ordering.
Clay-Teragaito 2011
For a hyperbolic 2-bridge knot, any exceptional slope is LO.
Idea: Use the graph manifold structure with 3 pieces.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Teragaito 2011
For any hyperbolic twist knot, the slope 4 is LO.
Idea: Generalize Clay-Lidman-Watson’s argument by using
Navas’s ordering.
Clay-Teragaito 2011
For a hyperbolic 2-bridge knot, any exceptional slope is LO.
Idea: Use the graph manifold structure with 3 pieces.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Teragaito 2011
For any hyperbolic twist knot, the slope 4 is LO.
Idea: Generalize Clay-Lidman-Watson’s argument by using
Navas’s ordering.
Clay-Teragaito 2011
For a hyperbolic 2-bridge knot, any exceptional slope is LO.
Idea: Use the graph manifold structure with 3 pieces.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

Dehn surgery

Teragaito 2011
For any hyperbolic twist knot, the slope 4 is LO.
Idea: Generalize Clay-Lidman-Watson’s argument by using
Navas’s ordering.
Clay-Teragaito 2011
For a hyperbolic 2-bridge knot, any exceptional slope is LO.
Idea: Use the graph manifold structure with 3 pieces.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous WorksTwist knots

Hakamata-Teragaito, Tran 2012For a hyperbolic twist knot, any slope in [0, 4] is LO.
The argument works for the figure-eight knot, too. It is simpler
than BGW’s one which involves character varieties.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous WorksTwist knots

Hakamata-Teragaito, Tran 2012For a hyperbolic twist knot, any slope in [0, 4] is LO.
The argument works for the figure-eight knot, too. It is simpler
than BGW’s one which involves character varieties.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous WorksTwist knots

Hakamata-Teragaito, Tran 2012For a hyperbolic twist knot, any slope in [0, 4] is LO.
The argument works for the figure-eight knot, too. It is simpler
than BGW’s one which involves character varieties.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

L-space
Previous Works

All LO

Theorem (Motegi-Teragaito, 2013)
There exist infinitely many hyperbolic knots such that all
non-trivial slopes are LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Genus one 2-bridge knots
Let K = K (m, n) be a genus one 2-bridge knot.
m–full twists

Vertical twists areleft-handed if m > 0.
Horizontal twists areright-handed if n > 0.
K (m, n) = K (−n, −m).
K (−m, −n) is the mirror
image of K (m, n).

n-full twists* We may assume m > 0.
* Except the trefoil K (1, −1), K is hyperbolic.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Genus one 2-bridge knots
Let K = K (m, n) be a genus one 2-bridge knot.
m–full twists

Vertical twists areleft-handed if m > 0.
Horizontal twists areright-handed if n > 0.
K (m, n) = K (−n, −m).
K (−m, −n) is the mirror
image of K (m, n).

n-full twists* We may assume m > 0.
* Except the trefoil K (1, −1), K is hyperbolic.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Genus one 2-bridge knots
Let K = K (m, n) be a genus one 2-bridge knot.
m–full twists

Vertical twists areleft-handed if m > 0.
Horizontal twists areright-handed if n > 0.
K (m, n) = K (−n, −m).
K (−m, −n) is the mirror
image of K (m, n).

n-full twists* We may assume m > 0.
* Except the trefoil K (1, −1), K is hyperbolic.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Main result

Theorem (Hakamata-Teragaito, Tran 2013)
Let K = K (m, n) be a hyperbolic genus one 2-bridge knot.
Let I be the interval defined by
if n > 0,(−4n, 4m)
I = [0, max{4m, −4n}) if m > 1 and n < −1,
[0, 4]
otherwise.
Then any slope in I is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Main result

Theorem (Hakamata-Teragaito, Tran 2013)
Let K = K (m, n) be a hyperbolic genus one 2-bridge knot.
Let I be the interval defined by
if n > 0,(−4n, 4m)
I = [0, max{4m, −4n}) if m > 1 and n < −1,
[0, 4]
otherwise.
Then any slope in I is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

For twist knots

For positive twist knots, we obtain a wider range ofleft-orderable slopes.
Corollary
Let K (1, n) be the n-twist knot with n > 0. Then any slope in(−4n, 4] is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

For twist knots

For positive twist knots, we obtain a wider range ofleft-orderable slopes.
Corollary
Let K (1, n) be the n-twist knot with n > 0. Then any slope in(−4n, 4] is LO.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

SchemeLike BRW and BGW, we use a continuous family ofrepresentaions of knot group G = π1 (S 3 − K ) to SL2 (R).
G/hhr ii
^
SL
2 (R)
ρ̃s

G

ρs

Masakazu TeragaitoχSL2 (R)LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

PointsG/hhr ii
^
SL
2 (R)

G

^
SL
2 (R) is LO.

ρ̃s

χ

ρs

SL2 (R)

Any non-trivial subgroupof LO group is LO.

Boyer-Rolfsen-Wiest 2005
Let M be a prime, connected, compact 3-manifold.
Then π1 M is LO if and only if there is a surjection from π1 M
onto a LO group.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

PointsG/hhr ii
^
SL
2 (R)

G

^
SL
2 (R) is LO.

ρ̃s

χ

ρs

SL2 (R)

Any non-trivial subgroupof LO group is LO.

Boyer-Rolfsen-Wiest 2005
Let M be a prime, connected, compact 3-manifold.
Then π1 M is LO if and only if there is a surjection from π1 M
onto a LO group.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Presentation of knot group

Let G = π1 (S 3 − K (m, n)).
It is well known that G has a presentation
G = hx, y : w n x = yw n i,where x and y are meridians, and w = (xy −1 )m (x −1 y )m .

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Representations

For real numbers s > 0 and t > 1, defineρs : G → SL2 (R)by√ρs (x) =

t
0

0
1
√
t



!
,



t−s−1
√
1
 t− √√ s−1( t− √1 )2


−s



ρs (y ) = tts+1− 1t√t− √1t

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Representations

For real numbers s > 0 and t > 1, defineρs : G → SL2 (R)by√ρs (x) =

t
0

0
1
√
t



!
,



t−s−1
√
1
 t− √√ s−1( t− √1 )2


−s



ρs (y ) = tts+1− 1t√t− √1t

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Riley Polynomials
The map ρs : G → SL2 (R) gives an irreducible non-abelian
representation if and only if s and t satisfy Riley’s equation
φK (s, t) = 0.

φK (s, t) = (τn+1 − τn ) + (s + 2 − t − 1/t)fm−1 gm−1 τn ,λk −λkwhere τk = λ++ −λ−, λ± are eigenvalues of ρs (w),−fi and gi are certain polynomials of variable s.
22* trρs (w) = λ+ + λ− = fm2 + fm−1
− s(t + 1/t)gm−1.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Riley Polynomials
The map ρs : G → SL2 (R) gives an irreducible non-abelian
representation if and only if s and t satisfy Riley’s equation
φK (s, t) = 0.

φK (s, t) = (τn+1 − τn ) + (s + 2 − t − 1/t)fm−1 gm−1 τn ,λk −λkwhere τk = λ++ −λ−, λ± are eigenvalues of ρs (w),−fi and gi are certain polynomials of variable s.
22* trρs (w) = λ+ + λ− = fm2 + fm−1
− s(t + 1/t)gm−1.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Riley Polynomials
The map ρs : G → SL2 (R) gives an irreducible non-abelian
representation if and only if s and t satisfy Riley’s equation
φK (s, t) = 0.

φK (s, t) = (τn+1 − τn ) + (s + 2 − t − 1/t)fm−1 gm−1 τn ,λk −λkwhere τk = λ++ −λ−, λ± are eigenvalues of ρs (w),−fi and gi are certain polynomials of variable s.
22* trρs (w) = λ+ + λ− = fm2 + fm−1
− s(t + 1/t)gm−1.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Existence of Representaions
In general, we cannot solve the equation φK (s, t) = 0. But wecan guarantee the existence of real solutions.
Existence of solutions
If |n| > 1, then Riley’s equation φK (s, t) = 0 has a real solutiont > 1 for any s > 0 such that
s+2+

c
2
sgm−1

<t+

d
1
<s+2+ 2tsgm−1where c, d ∈ (0, 4) are constants depending only on n.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Existence of Representaions
In general, we cannot solve the equation φK (s, t) = 0. But wecan guarantee the existence of real solutions.
Existence of solutions
If |n| > 1, then Riley’s equation φK (s, t) = 0 has a real solutiont > 1 for any s > 0 such that
s+2+

c
2
sgm−1

<t+

d
1
<s+2+ 2tsgm−1where c, d ∈ (0, 4) are constants depending only on n.

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Slope
We need to annihilate the image ρs (r ) of the slope r = µp λq .
The images of meridian and longitude under ρs are
diagonal.
Let A and B be their (1, 1)-entries.
Hence,
ρs (r ) = I ⇐⇒ Ap B q = 1
p
log B
⇐⇒
=−
q
log A

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Slope
We need to annihilate the image ρs (r ) of the slope r = µp λq .
The images of meridian and longitude under ρs are
diagonal.
Let A and B be their (1, 1)-entries.
Hence,
ρs (r ) = I ⇐⇒ Ap B q = 1
p
log B
⇐⇒
=−
q
log A

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Slope
We need to annihilate the image ρs (r ) of the slope r = µp λq .
The images of meridian and longitude under ρs are
diagonal.
Let A and B be their (1, 1)-entries.
Hence,
ρs (r ) = I ⇐⇒ Ap B q = 1
p
log B
⇐⇒
=−
q
log A

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Range of Slopes

Lemma
If p/q ∈ (0, 4m), then there exists s > 0 such that ρs (r ) = I.
This is established by examining the image of a functiong : (0, ∞) → R
defined byg(s) = −Masakazu Teragaito

log B
.
log A

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Range of Slopes

Lemma
If p/q ∈ (0, 4m), then there exists s > 0 such that ρs (r ) = I.
This is established by examining the image of a functiong : (0, ∞) → R
defined byg(s) = −Masakazu Teragaito

log B
.
log A

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

The image of g

By definition, A =√t. On the other hand,B=−fm + tfm−1.
−fm−1 + tfm

Lemma
The image of g contains (0, 4m).

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

The image of g

By definition, A =√t. On the other hand,B=−fm + tfm−1.
−fm−1 + tfm

Lemma
The image of g contains (0, 4m).

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for ProofControl of Lift
Thus we have a representation ρs : G → SL2 (R) with ρs (r ) = I.
^
In general, it lifts to ρ̃s : G → SL
2 (R), but ρ̃s (r ) may not be theidentity element.
G/hhr ii
^
SL
2 (R)
ρ̃s

χ

ρs

SL2 (R)So, we need furthermore tomodify the lift so that ρ̃s (r ) =^(0, 0) ∈ SL
2 (R). This partneeds some argument.

G

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for ProofControl of Lift
Thus we have a representation ρs : G → SL2 (R) with ρs (r ) = I.
^
In general, it lifts to ρ̃s : G → SL
2 (R), but ρ̃s (r ) may not be theidentity element.
G/hhr ii
^
SL
2 (R)
ρ̃s

χ

ρs

SL2 (R)So, we need furthermore tomodify the lift so that ρ̃s (r ) =^(0, 0) ∈ SL
2 (R). This partneeds some argument.

G

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for ProofControl of Lift
Thus we have a representation ρs : G → SL2 (R) with ρs (r ) = I.
^
In general, it lifts to ρ̃s : G → SL
2 (R), but ρ̃s (r ) may not be theidentity element.
G/hhr ii
^
SL
2 (R)
ρ̃s

χ

ρs

SL2 (R)So, we need furthermore tomodify the lift so that ρ̃s (r ) =^(0, 0) ∈ SL
2 (R). This partneeds some argument.

G

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Proof of Theorem

From the argument so far, any slope in (0, 4m) is LO.
Assume n > 0. Apply the argument for K (n, m), which is themirror of K (m, n). Then we obtain (0, 4n).
Thus any slope in (−4n, 4m) is LO for K (m, n).

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Proof of Theorem

From the argument so far, any slope in (0, 4m) is LO.
Assume n > 0. Apply the argument for K (n, m), which is themirror of K (m, n). Then we obtain (0, 4n).
Thus any slope in (−4n, 4m) is LO for K (m, n).

Masakazu Teragaito

LO for Two-bridge Knots

Left-ordering
L-space conjecture
Our Result

Main Result
Basic Ideas for Proof

Proof of Theorem

From the argument so far, any slope in (0, 4m) is LO.
Assume n > 0. Apply the argument for K (n, m), which is themirror of K (m, n). Then we obtain (0, 4n).
Thus any slope in (−4n, 4m) is LO for K (m, n).

Masakazu Teragaito

LO for Two-bridge Knots

Appendix

References

R. Hakamata and M. Teragaito,
Left-orderable fundamental groups and Dehn surgery on
genus one 2-bridge knots, preprint.

Masakazu Teragaito

LO for Two-bridge Knots