# How Quantum is the Resonance Behavior in Vibrational Polariton Chemistry?

Marit R. Fiechter  [  Johan E. Runeson  [  Joseph E. Lawrence  [  Jeremy O.
Richardson<EMAIL_ADDRESS>[

###### Abstract

Recent experiments in polariton chemistry have demonstrated that reaction
rates can be modified by vibrational strong coupling to an optical cavity
mode. Importantly, this modification only occurs when the frequency of the
cavity mode is tuned to closely match a molecular vibrational frequency. This
sharp resonance behavior has proved difficult to capture theoretically. Only
recently, Lindoy et al. reported the first instance of a sharp resonant effect
in the cavity-modified rate simulated in a model system using exact quantum
dynamics. We investigate the same model system with a different method, ring-
polymer molecular dynamics (RPMD), which captures quantum statistics but
treats dynamics classically. We find that RPMD does not reproduce this sharp
resonant feature at the well frequency, and we discuss the implications of
this finding for future studies in vibrational polariton chemistry.

ETH] Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland
ox] Department of Chemistry, University of Oxford, Physical and Theoretical
Chemistry Laboratory, South Parks Road, Oxford, OX1 3QZ, United Kingdom ETH]
Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland ETH]
Laboratory of Physical Chemistry, ETH Zürich, 8093 Zürich, Switzerland

A recent series of experiments has revealed the surprising result that one can
alter chemical reaction rates just by placing the reaction mixture in an
optical cavity 1, 2, 3, 4, 5, 6, 7, 8, 9, _i.e._ between a pair of carefully
spaced mirrors which support standing waves of light at specific frequencies.
In particular, when strongly coupling a cavity mode to molecular vibrations
(called vibrational strong coupling, or VSC), 10, 11 the rate constant of
ground-state reactions can be modified even without external driving, _i.e._
without explicitly adding photons into the cavity. As the cavity mode can be
treated as a harmonic oscillator coupled to the molecular system under study,
it is relatively straightforward to incorporate into standard theoretical
chemistry methods. However, in spite of the plethora of theoretical studies
conducted on the topic, recently reviewed in _e.g._ Refs. 12, 13, 14, the
mechanism behind this cavity effect on the chemical reaction rate is not yet
well understood.

One of the features observed in experiments that has proven hard to reproduce
theoretically is the resonance behavior: the rate-constant modification is
only significant when the cavity length is tuned such that one of the cavity
modes is in resonance with a vibrational mode in the reaction mixture (either
of a reactant 2, 3, 7, 8, 4, 6, 9 or a solvent molecule 5, 4, 9). The width of
this resonant feature in the plot of rate vs. cavity frequency is comparable
to the linewidth of the molecular resonance in the infrared spectrum 2, 5, 3,
7, typically on the order of tens of wavenumbers. Another feature of
experiments complicating a theoretical analysis is the fact that, in
experiments, a large number of molecules is coupled to a single cavity mode.
This induces collective effects, so that spectral characteristics such as the
Rabi splitting depend on the number of molecules coupled to the cavity 15, 16.
The question remains by what mechanism these collective effects influence the
rate. However, for now we will constrain ourselves to the single-molecule
regime, as the focus of this work is to compare an approximate theory to a
fully quantum-mechanical benchmark, for which the extension to multiple
molecules quickly becomes prohibitively expensive.

In this work, we focus on the resonance behavior and further investigate to
what extent quantum effects could play a role for these sharp resonances in
the rate around a reactant vibrational frequency. Although analytical rate
theories such as Grote–Hynes theory17, Eyring theory 18 or
Pollak–Grabert–Hänggi theory 19 do predict a cavity effect, these effects tend
to be spread over a broad range of cavity frequencies. Additionally, the
largest effect observed in these theories does not necessarily occur when a
cavity mode is on resonance with a vibrational mode, contrary to experimental
observations. Only recently has a sharp resonance behavior, more in line with
experiment, been reported theoretically by Lindoy _et al._20 In this study
they used a fully quantum-dynamical approach (hierarchical equations of
motion, HEOM) 21 for a specific low-friction parameter regime of a model
double-well system coupled to a cavity mode. These quantum simulations give
peaks in the rate modification centered at the reactant frequency with a full-
width half-maximum (FWHM) as small as $80\text{\,}{\mathrm{cm}}^{-1}$, which
is significantly narrower than the FWHM seen in earlier classical simulations
22, 23 (for example $\sim$$350\text{\,}{\mathrm{cm}}^{-1}$ for the
isomerization of HONO on an _ab initio_ PES 22). Importantly, the quantum
results are comparable to the resonance widths observed in experiment 2, 3, 5,
9, although it should be noted that Lindoy _et al._ considered only a single
molecule in the cavity, making up for the lack of collective enhancement of
the effect by choosing an extremely large light–matter coupling strength.

We assess the importance of quantum effects in forming these sharp peaks by
comparing the HEOM results with ring-polymer molecular dynamics (RPMD) rate
theory 24, 25, 26. RPMD is based on imaginary-time path integrals and is thus
able to capture quantum statistics (including tunneling and zero-point energy
effects), but treats dynamics classically, meaning it cannot reproduce effects
due to quantum coherence.27 Hence, whether or not RPMD succeeds in reproducing
the HEOM results tells us whether the quantum effects responsible for the
sharp resonant feature are statistical or dynamical in nature. This will again
have important implications for future studies in this field: as quantum-
dynamics calculations on full-dimensional realistic systems are prohibitively
expensive, it is natural to resort to more affordable methods that capture
quantum effects only approximately. 28, 22, 29 The results in this study show
that, if the single-molecule low-friction models studied in Ref. 20 are indeed
representative of experiment, one needs quantum dynamics to describe the
physics correctly and reproduce sharp resonances.

_Model._ The model system we study here is equivalent to the model of Ref. 20.
For completeness, in the following we give a full description of the form and
parameters of this model.

The model Hamiltonian can be expressed as

$\hat{H}=\hat{H}_{\text{mol}}+\hat{H}_{\text{solv}}+\hat{H}_{\text{cav}}+\hat{H}_{\text{cav-
loss}},$ (1)

where $\hat{H}_{\text{mol}}$ is the molecular Hamiltonian,
$\hat{H}_{\text{solv}}$ describes the solvent and its coupling to the
molecule, $\hat{H}_{\text{cav}}$ describes the cavity mode and its interaction
with the molecule, and $\hat{H}_{\text{cav-loss}}$ represents cavity loss
through interaction with electromagnetic modes outside the cavity. A
diagrammatic representation of the degrees of freedom in this Hamiltonian and
their coupling is shown in Figure 1a. All calculations are performed at a
temperature of $300\text{\,}\mathrm{K}$.

The molecular Hamiltonian $\hat{H}_{\text{mol}}$ is chosen to be a one-
dimensional symmetric double well, so that for the molecular coordinate $R$
(with unit mass) we have $\hat{H}_{\text{mol}}=\hat{P}_{R}^{2}/2+V(\hat{R})$
with the potential given by

$V(R)=\frac{\omega_{\mathrm{b}}^{4}}{16E_{\mathrm{b}}}R^{4}-\tfrac{1}{2}\omega_{\mathrm{b}}^{2}R^{2}+E_{\mathrm{b}},$
(2)

where $\omega_{\mathrm{b}}$ is the imaginary part of the barrier frequency and
$E_{\mathrm{b}}$ is the barrier height of the double well. Note that, as
illustrated in Figure 1b, the vibrational states of such a double-well system
are split by a very small energy, the tunneling splitting. We define the
(anharmonic) well frequency as the difference between the first and second
pair of these tunneling-split states, _i.e._
$\hbar\omega_{0}=\tfrac{1}{2}(E_{2}+E_{3})-\tfrac{1}{2}(E_{0}+E_{1})$, with
$E_{n}$ denoting the energy of the $n$th eigenstate of $\hat{H}_{\text{mol}}$.
In this work we will study two of these double-well systems: I, a relatively
shallow double-well system (studied in the main text of Ref. 20), with
$\omega_{\mathrm{b}}=$1000\text{\,}{\mathrm{cm}}^{-1}$$ and
$E_{\mathrm{b}}=$2250\text{\,}{\mathrm{cm}}^{-1}$$, so that
$\omega_{0}\approx$1190\text{\,}{\mathrm{cm}}^{-1}$$; and II, a deeper double-
well system (studied in the Supplementary Information of Ref. 20) with
$\omega_{\mathrm{b}}=$500\text{\,}{\mathrm{cm}}^{-1}$$ and
$E_{\mathrm{b}}=$2000\text{\,}{\mathrm{cm}}^{-1}$$, yielding
$\omega_{0}\approx$650\text{\,}{\mathrm{cm}}^{-1}$$.

The interaction of the molecule to the solvent is taken to be

$\displaystyle\hat{H}_{\text{solv}}$
$\displaystyle=\sum_{i}\bigg{[}\frac{\hat{P}_{i}^{2}}{2}+\tfrac{1}{2}\Omega_{i}^{2}\bigg{(}\hat{Q}_{i}+\frac{C_{i}\hat{R}}{\Omega_{i}^{2}}\bigg{)}^{2}\bigg{]},$
(3)

which couples the molecular coordinate to a harmonic bath described by the
canonical operators $\hat{P}_{i}$ and $\hat{Q}_{i}$. Here we take this bath to
be characterized by a Debye spectral density
$J_{R}(\omega)=\tfrac{\pi}{2}\sum_{i}\frac{C_{i}^{2}}{\Omega_{i}}\delta(\omega-\Omega_{i})=\eta_{\mathrm{s}}\omega\Gamma^{2}/(\omega^{2}+\Gamma^{2})$,
where we set $\Gamma=$200\text{\,}{\mathrm{cm}}^{-1}$$; $\eta_{\mathrm{s}}$
can be varied to change the solvent friction.

Within the dipole approximation, the coupling of the molecular coordinate to
the cavity mode is given by the Pauli–Fierz Hamiltonian 13, 12,

$\hat{H}_{\text{cav}}=\frac{1}{2}\hat{p}_{\mathrm{c}}^{2}+\frac{1}{2}\omega_{\mathrm{c}}^{2}\left(\hat{q}_{\mathrm{c}}+\sqrt{\frac{2}{\hbar\omega_{\mathrm{c}}}}\eta\hat{\mu}\right)^{2}.$
(4)

Here, $\hat{p}_{\mathrm{c}}$ and $\hat{q}_{\mathrm{c}}$ are the canonical
displacement field operators, $\omega_{\mathrm{c}}$ is the cavity frequency,
and $\eta=\sqrt{\frac{\hbar}{2\omega_{\mathrm{c}}\epsilon_{0}V}}$ is a measure
of the coupling strength, with $\epsilon_{0}$ the permittivity of vacuum and
$V$ the quantization volume. $\hat{\mu}$ is dipole moment operator projected
onto the electronic ground state and along the cavity polarization,
$\hat{\mu}=\hat{\mathbf{e}}\cdot\hat{\bm{\mu}}$. We follow Ref. 20 and choose
the dipole moment to be linear in the molecular coordinate,
$\hat{\mu}=\hat{R}$.

If the cavity mirrors are not perfectly reflective, the electromagnetic mode
inside the cavity can interact with the continuum of modes outside the cavity,
enabling for example the escape of a photon from the cavity. This can
effectively be described by

$\hat{H}_{\text{cav-
loss}}=\sum_{j}\bigg{[}\frac{\hat{p}_{j}^{2}}{2}+\frac{1}{2}{\omega}_{j}^{2}\bigg{(}\hat{q}_{j}+\frac{c_{j}\hat{q}_{\mathrm{c}}}{{\omega}_{j}^{2}}\bigg{)}^{2}\bigg{]},$
(5)

where $\hat{{p}}_{j}$ and $\hat{{q}}_{j}$ are the canonical operators
associated to field modes outside the cavity. Making the assumption that this
bath of external light modes is Markovian, one can relate the parameters in
this Hamiltonian, _i.e._ the set of frequencies ${\omega}_{j}$ and couplings
$c_{j}$ described by a spectral density
$J_{\mathrm{L}}(\omega)=\frac{\pi}{2}\sum_{j}\frac{c_{j}^{2}}{{\omega}_{j}}\delta(\omega-{\omega}_{j})$,
to the cavity lifetime $\tau_{\mathrm{c}}$. One possible such relation is
given in Ref. 20:

$\tau_{\mathrm{c}}=\omega_{\mathrm{c}}\frac{1-\mathrm{e}^{-\beta\hbar\omega_{\mathrm{c}}}}{2J_{\mathrm{L}}(\omega_{\mathrm{c}})},$
(6)

which is the definition we will use throughout this work for consistency.111We
note that alternatively, one could take
$\tau_{\mathrm{c}}=\omega_{\mathrm{c}}/J_{\mathrm{L}}(\omega_{\mathrm{c}})$;
this relation can be obtained by calculating the expectation value of the
photon number operator for an empty cavity (no matter) with the bath at zero
temperature, giving an exponential decay that allows for extraction of a
lifetime:
$\langle\hat{n}(t)\rangle=\langle\hat{n}(0)\rangle\mathrm{e}^{-J(\omega_{\mathrm{c}})t/\omega_{\mathrm{c}}}$
(Ref. 30, Eq. 1.80).  In Ref. 20, $J_{\mathrm{L}}(\omega)$ was chosen to be a
Debye spectral density
$J_{\mathrm{L}}(\omega)=\eta_{\mathrm{L}}\omega\Gamma_{\mathrm{L}}^{2}/(\omega^{2}+\Gamma_{\mathrm{L}}^{2})$
with $\Gamma_{\mathrm{L}}=$1000\text{\,}{\mathrm{cm}}^{-1}$$. Debye baths are
the natural choice for HEOM calculations, whereas for RPMD, there is a
significant advantage in using an Ohmic spectral density instead. Therefore,
to make the cost of our RPMD calculations tractable for system I, we replaced
this Debye spectral density by an Ohmic spectral density
$J_{\mathrm{L}}(\omega)=\gamma_{\mathrm{L}}\omega$ with $\gamma_{\mathrm{L}}$
defined by $\tau_{\mathrm{c}}$ and Eq. 6. As the spectral density near
$\omega_{\mathrm{c}}$ is unaltered, this is not expected to significantly
change the dynamics. Details on our treatment of the spectral densities and
results supporting the validity of this exchange are given in the Supporting
Information.

Figure 1: Outline of the model. (a) Diagram representing the degrees of
freedom in the model and their coupling. The molecule-cavity coupling strength
is given by $\eta$, the magnitude of the solvent friction determined by
$\eta_{\mathrm{s}}$ (blue) and the strength of the coupling between cavity
mode and external modes by $\tau_{\mathrm{c}}$ (red). (b) Double well I and
its vibrational eigenstates, with the blue arrow indicating the well frequency
$\omega_{0}$. (c) Rate constant $k_{0}$ as a function of solvent friction
$\eta_{\mathrm{s}}$ outside the cavity (_i.e._ $\eta=0$) for system I, with
the gray dashed line indicating the friction at which the results of Figure 2
were obtained.

_Theory._ We perform the calculations in this study with RPMD 31, 24, 25, 32.
Here we summarize the general idea behind this method; for details of our
implementation, we refer the reader to the Supporting Information.

In short, RPMD is based on discretized closed paths in imaginary time called
ring polymers. These ring polymers emerge for example in path-integral
molecular dynamics, 33, 34 where, by sampling over ring-polymer
configurations, one can compute static equilibrium properties of a quantum
system exactly. It is however not feasible to rigorously extend this to real-
time dynamics because of the infamous sign problem. RPMD circumvents this by
instead propagating the ring polymer classically in time to obtain an
approximation to real-time quantum correlation functions such as the flux–side
correlation function, from which one can extract the rate constant. RPMD rate
theory can be shown to capture static quantum effects such as zero-point
energy effects and tunneling 24, 25, 35, 36, 26; on the other hand it is not
able to capture truly dynamical quantum effects, such as interference. 27

_System I: shallow wells._ We start by investigating the behavior of the
chemical reaction rate without the cavity, $k_{0}$, with increasing solvent
friction, $\eta_{\mathrm{s}}$, in Figure 1c. For frictions larger than
$\eta_{\mathrm{s}}\approx 0.2\omega_{\mathrm{b}}$, this figure shows typical
Kramers turnover behavior 37, 24, i.e. the rate increases with increasing
friction until it reaches a maximum (“turnover”), and then decreases with
increasing friction after that. For very low friction however, the exact rate
in this system shoots up again due to coherent nuclear tunneling between
states localized in the left and in the right well; this is a dynamic quantum
effect not captured by RPMD. For higher frictions, the RPMD rate more
accurately approximates the quantum rate, being within a few percent after the
Kramer’s turnover. The classical rate on the other hand underestimates the
quantum benchmark rate by an order of magnitude.

Figure 2: Cavity modification of the rate of system I as a function of cavity
frequency $\omega_{\mathrm{c}}$ for a set of cavity lifetimes
$\tau_{\mathrm{c}}$, with $\eta=0.00125$ a.u. and
$\eta_{\mathrm{s}}=0.1\omega_{\mathrm{b}}$. Neither classical dynamics nor
RPMD is able to reproduce the sharp peak in rate enhancement at the well
frequency $\omega_{0}$; instead they give a broad peak centered at the barrier
frequency $\omega_{\mathrm{b}}$. The error bars indicate a 68% confidence
interval.

In Figure 2 we investigate the change in rate, $k/k_{0}$, when coupling the
molecular coordinate to the cavity mode for a low value of the solvent
friction, $\eta_{\mathrm{s}}=0.1\omega_{\mathrm{b}}$. In particular, in Ref.
20 it was found that the effect of the cavity is negligible when the cavity is
lossless ($\tau_{\mathrm{c}}\rightarrow\infty$): in that case energy transfer
from the molecular mode to the cavity mode may be possible, but from there the
energy cannot dissipate. The cavity mode will therefore not efficiently dampen
the motion in the reaction coordinate. When increasing the coupling between
the cavity mode and the environment (_i.e._ reducing $\tau_{\mathrm{c}}$), a
dramatic increase in the exact rate is observed. Moreover, the resulting rate
profile as a function of cavity frequency features a single sharp peak (with a
full-width half-maximum (FWHM) of about $80\text{\,}{\mathrm{cm}}^{-1}$ for
$\tau_{\mathrm{c}}=$100\text{\,}\mathrm{fs}$$ cavity lifetime), and it is
centered around the well frequency. At least in this respect it is in
agreement with experimental observations (although the experiments involve
many molecules in a cavity, and it is unclear whether the current model is
physically realistic for a reaction in solution).

Both classical and RPMD simulations display a markedly different behavior from
the HEOM results: the peak in the cavity-induced rate enhancement is centered
around the barrier frequency $\omega_{\mathrm{b}}$ rather than the well
frequency, and it is also a much broader feature, its FWHM spanning hundreds
of wavenumbers. Interestingly, the effect of cavity loss ($\tau_{\mathrm{c}}$)
on the rate is much smaller here: the rate modification is not entirely
suppressed for lossless cavities, as it was in the quantum case, and the
results do not change dramatically when introducing a finite cavity lifetime
(from a rate enhancement by a factor of 1.2 for a lossless cavity to a factor
of 1.4 for a cavity with lifetime
$\tau_{\mathrm{c}}=$100\text{\,}\mathrm{fs}$$).

Even though the absolute rate without cavity, $k_{0}$, predicted by classical
and RPMD simulations differs by a factor of about $8$, they predict similar
rate modifications, $k/k_{0}$, which agree with each other within the error
bars. This implies that quantum statistics does not play an important role in
the effect the cavity has on the relative rate $k/k_{0}$ in this case. On the
other hand, the cavity modification of the RPMD rate constant does not even
qualitatively agree with the full quantum rate constant, implying the quantum
dynamics is crucial in this model system for capturing the right behavior.

_System II: deep wells._ We now move on to a double-well system with a higher
barrier, so that there are three (instead of two) tunneling-split eigenstates
below the barrier (see Figure 3a, _c.f._ Figure 1b). The cavityless rate
constant as a function of solvent friction $\eta_{\mathrm{s}}$, shown in
Figure 3b, reveals that this system loosely speaking behaves less quantum
mechanically; there is for example no coherent-tunneling regime at very low
frictions. The classical rate in this case only underestimates the full
quantum rate constant by a factor of 2, whereas RPMD yields a spot-on
prediction of the rate.

Figure 3: System II and cavity modification of its rate. (a) Double well II
and its eigenstates, with the blue arrow indicating the well frequency
$\omega_{0}$. (b) Rate constant $k_{0}$ as a function of solvent friction
$\eta_{\mathrm{s}}$ outside the cavity ($\eta=0$). (c) Rate-constant
modification $k/k_{0}$ as a function of cavity frequency for
$\eta_{\mathrm{s}}=0.1\omega_{\mathrm{b}}$, $\eta=0.005$ a.u., and
$\tau_{\mathrm{c}}=1000$ fs.

Coupling the cavity mode to this deeper double well yields quite a different
cavity-frequency-dependence of the rate, as displayed in Figure 3c. Firstly,
the rate profile is not composed of a single sharp peak; rather, it is made up
of a much broader peak (FWHM $>$300\text{\,}{\mathrm{cm}}^{-1}$$) and a
somewhat lower broad shoulder. Moreover, the main peak is no longer centered
at the molecular vibrational frequency; at
$\omega_{\mathrm{c}}\approx$560\text{\,}{\mathrm{cm}}^{-1}$$ it is somewhere
in between the barrier frequency
($\omega_{\mathrm{b}}=$500\text{\,}{\mathrm{cm}}^{-1}$$) and the well
frequency ($\omega_{0}\approx$650\text{\,}{\mathrm{cm}}^{-1}$$).

Interestingly, in this case the rate-enhancement profile is captured
qualitatively by the classical simulations, and adding in quantum statistics
via RPMD has little additional effect. This suggests that the origin of the
cavity-induced rate modification in this system can be understood classically.
Note, however, that by going from system I to system II, we have also lost the
sharp peak centered at $\omega_{0}$, and therefore this system does not
exhibit one of the key features of the experimental results.

It should be noted, however, that although RPMD agrees very well with HEOM in
the cavityless case, it overestimates the magnitude of the rate modification
by the cavity. This is in line with the saturation of the HEOM rate
modification for larger light–matter coupling strengths, $\eta$, observed in
the Supplementary Information of Ref. 20; this saturation effect is much less
pronounced in the classical case. RPMD does not improve upon classical
dynamics in capturing this, suggesting the saturation effect is due to a
mechanism based on coherent exchange of energy between the cavity mode and the
double-well system.

_Discussion._ In summary, we have investigated two variations of the model
system studied in Ref. 20. In system I, where the molecular coordinate is a
relatively shallow double well, full quantum dynamics predicts that the rate
modification as a function of cavity frequency is peaked sharply around the
well frequency. We show that this feature cannot be reproduced with classical
dynamics, and that even adding quantum statistics with RPMD does not improve
upon this. This indicates that the resonance in this system requires quantum
dynamical effects. In system II we consider a deeper double-well system, where
intuitively quantum effects should play a less prominent role. Indeed, even
classical dynamics qualitatively captures the correct frequency-dependent rate
modification $k/k_{0}$ in this case (even though one needs statistical quantum
effects in RPMD to correctly predict the absolute rate constant). This system
however lacks the provocatively sharp resonance feature that made system I so
intriguing. Also note that, surprisingly, introducing coupling to the cavity
mode causes RPMD to overestimate the rate by $\sim$14%, even though it was
spot-on in the cavityless case.

The implications of these findings are as follows. We have seen that the only
sharp peak in the rate modification predicted in theory so far can exclusively
be simulated by quantum dynamics. However, this has only been observed for a
specific model, and it is not at all clear whether this model is
representative of the reactions done in experiment. Firstly, in this model,
the molecular coordinate is only very weakly coupled to its bath, so that the
reaction takes place in the low-friction regime. Chemical reactions typical of
this regime are unimolecular dissociation reactions in low-pressure gases 37;
in this light, it seems unlikely that the model studied here is a good
presentation of the (solution-phase) reactions performed in experiment.
Secondly, our findings seem to suggest that only systems for which the rate
exhibits quantum-dynamical behavior (even before coupling to the cavity) will
exhibit sharp resonances. This is rather atypical for solution-phase
reactions, where coherences are expected to wash out rapidly 32. In fact, this
is the reason that RPMD normally performs so well for predicting reaction
rates in solution; the very fact that RPMD fails to correctly predict the rate
of system I at the friction of interest without cavity coupling in itself
already suggests that this system may not be representative of reactions in
solution.

Nevertheless, it cannot be ruled out that somehow, in the reactions studied in
experiment, a critical part of the motion leading to products is only weakly
damped, effectively introducing a low friction locally and leading to a
situation in which by some means coherences can survive longer than expected.
Even if this is true, and quantum dynamics is vital for capturing the
resonance behavior correctly, not all is lost; one may for example still be
able to cut computational costs by only treating important molecular and
cavity degrees of freedom quantum mechanically in a mixed quantum–classical
scheme 38, 39, 40, 41, 42, 43, 44, 45.

Another possibility is that the sharp peak in rate modification in system I is
more of an exotic feature, the double well being very shallow, and the
resonance prominent at a friction where the rate is enhanced by coherent
nuclear tunneling. The sharp peaks observed in experiment, however, may be of
a different kind. That they have not been observed in simulations could be
because our models or methods in the high friction regime just still lack
essential features.

Finally, it is also possible that by some unknown mechanism one can only get
these resonances to be sharp in the high-friction regime by accounting for
collective effects. This has previously been suggested in for example Refs.
29, 46, 9, and is an interesting avenue for further exploration.

## Acknowledgements

The authors would like to thank Lachlan Lindoy for helpful discussions. J.E.R.
would like to thank David Manolopoulos for advice on evolving the explicit
bath. M.R.F. was supported by an ETH Zurich Research Grant, J.E.L. was
supported by an ETH Zurich Postdoctoral Fellowship, and J.E.R. by a mobility
fellowship from the Swiss National Science Foundation.

## Supporting Information Available

The following files are available free of charge:

  * •

SupportingInformation.pdf: details on the computational method used and
results supporting the validity of the replacement of the Debye spectral
density by an Ohmic spectral density in the light-mode bath.

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See pages 1-5 of SI.pdf


Let $M$ be a $\bbQ$-homology solid torus. In this paper, we give a cohomological criterion for the existence of an interval of left-orderable Dehn surgeries on $M$. We apply this criterion to prove that the two-bridge knot that corresponds to the continued fraction $[1,1,2,2,2j]$ for $j\geq 1$ admits an interval of left-orderable Dehn surgeries. This family of two-bridge knots gives some positive evidence for a question of Xinghua Gao.

§ INTRODUCTION

A group $G$ is left orderable if it admits a strict total ordering on the group elements such that $g < h $ implies that $fg < fg$ for all elements $f,g,h \in G$. Left-orderability arises naturally in the study of low-dimensional topology, foliation theory, and group theory. Well-known examples of left-orderable groups include torsion-free abelian groups, free (non-abelian) groups, surface groups and the group of orientation preserving homeomorphisms of the real line. In 3-manifold topology, left-orderability is an important concept due to its role in the L-space conjecture.

For an irreducible $\bbQ$-homology 3-sphere $M$, the following are equivalent

* $\pi_1(M)$ is left-orderable.

* $M$ is not an L-space.

* $M$ admits a coorientable taut foliation.

An L-space is a $\bbQ$-homology 3-sphere with $\dim \hat{HF}(M) = |H_1(M;\bbZ)|$ where $\hat{HF}(M)$ is the Heegaard Floer homology of $M$ <cit.>. There has been a substantial amount of evidence in favor of this conjecture. For example, the L-space conjecture holds for all graph manifolds <cit.> and <cit.>.

In view of <ref>, there have been a lot of ideas developed to study left-orderability of 3-manifold groups. It is a well-known fact that a countable group is left-orderable if and only if it embeds in the group of orientation-preserving homeomorphisms of the real line <cit.>. In the case of an irreducible compact 3-manifold, its fundamental group is left-orderable if and only if it admits a non-trivial homomorphism onto a left-orderable group <cit.>. In particular, all manifolds with positive first Betti number are left-orderable. Therefore, it is interesting to construct left orders on $\bbQ$-homology spheres, for example those coming from doing Dehn filling and from taking cyclic branched covering of $\bbQ$-homology solid torus.

A fruitful way to build left-orderings on $\bbQ$-homology spheres is by lifting $\PSL_2(\bbR)$ representations to $\widetilde{\PSL}_2(\bbR)$. This strategy has been employed with a lot of success, for example see <cit.>. Recently, Dunfield, Culler and independently Gao have introduced the idea of using the extension locus of a compact 3-manifold with torus boundary $M$ to order families of $\bbQ$-homology spheres arising by doing Dehn filling on $M$ <cit.>. Furthermore, they gave several criteria implying the existence of intervals of left-orderable Dehn fillings on $M$. To state their results, we need the following definition:

A compact 3-manifold $Y$ has few characters if each positive-dimensional component of the $\PSL_2(\bbC)$-character variety $X(Y)$ consists entirely of characters of reducible representations. An irreducible $\bbQ$-homology solid torus $M$ is called longitudinally rigid when $M(0)$ has few characters where $M(0)$ is the closed manifold obtained from $M$ by doing Dehn filling along the homological longitude.

We summarize their results in the following:

Suppose that $M$ is longitudinally rigid irreducible $\bbZ$-homology solid torus. Then the following are true:

* If the Alexander polynomial of $M$ has a simple root $\xi \neq 1$ on the unit circle, then there exists $a >0$ such that for every rational $r \in (-a,0) \cup (0,a)$ the Dehn filling $M(r)$ is orderable.

* If the Alexander polynomial of $M$ has a simple positive real root $\xi \neq  1$, then there exists a nonempty interval $(-a,0]$ or $[0,a)$ such that for every rational $r$ in the interval, the Dehn filling $M(r)$ is orderable.

In fact, their techniques also apply to the case where $M$ is a $\bbQ$-homology solid torus with some further hypothesis on $\xi$ in the first statement. The full version of the second statement is stated below in <ref>.

Culler and Dunfield also proved the following criterion for left-orderability:

Suppose that $M$ is a hyperbolic $\bbZ$-homology solid torus, whose trace field has a real embedding, then there exists $a >0$ such that for every rational $r\in(-a,0) \cup (0,a)$ the Dehn filling $M(r)$ is orderable.

In view of <ref>, it is natural to ask when a $\bbQ$-homology solid torus is longitudinally rigid. Since the character variety is notoriously hard to compute, see for example <cit.>, longitudinal rigidity is difficult to study in a general setting. Culler and Dunfield gave a topological condition which implies longitudinal rigidity. In particular, they introduced the following concept:

Let $M$ be a knot exterior. We say that $M$ is lean if the longitudinal Dehn filling $M(0)$ is prime and every closed essential surface in $M(0)$ is a fiber in a fibration over $S^1$.

For example, the $(-2,3,2s+1)$-pretzel knots were shown to be lean for $s\geq 3$, so there is an interval of about $0$ of left-orderable Dehn surgeries on these knot complements <cit.>. However as remarked in <cit.>, this leanness condition is rather restrictive. In particular, for a knot complement $K$ in $S^3$ being lean implies that $K$ fibers. Nevertheless, the first statement of <ref> was proved to be true without the condition of longitudinal rigidity by Herald and Zhang <cit.>. Motivated by this result, Xinghua Gao asked:

Can the longitudinal rigidity condition be dropped from the second statement of <ref>? Is it possible to prove $H^1(\pi_1(M(0));\liesl_2(\bbR)_{\rho}) = 0$ where $\rho$ is the non-abelian reducible representation of $\pi_1(M)$ coming from the root of the Alexander polynomial?

Following the suggestion in this question, we say that $\bbQ$-homology solid torus is locally longitudinal rigid at a root of the Alexander polynomial if $H^1(\pi_1(M(0));\liesl_2(\bbC)_{\rho}) = 0$ where $\rho$ is a non-abelian reducible representation coming from this root, see <ref> for a precise definition. In fact, we prove that the second item of <ref> still holds true under this weakened hypothesis.

Suppose that $M$ is an irreducible $\bbQ$-homology solid torus and that the Alexander polynomial of $M$ has a simple positive real root $\xi \neq 1$. Furthermore, suppose that $M$ locally longitudinally rigid at $\xi$. Then there exists a nonempty interval $(-a,0]$ or $[0,a)$ such that for every rational $r$ in the interval, the Dehn filling $M(r)$ is orderable.

As an application, we apply this result to produce an interval of left-orderable Dehn surgeries on an infinite family of two-bridge knots complement.

For every two-bridge knot $K_j$ corresponding to the contiued fraction $[1,1,2,2,2j]$ where $j\geq 1$, there exists a nonempty interval $(-a,0]$ or $[0,a)$ such that for every rational $r$ in the interval, the Dehn filling $M(r)$ is left-orderable.

As we will see in <ref>, the Alexander polynomial of $K_j$ has all simple positive real roots that are not 1, and is not monic for $j \geq 2$. In particular, the complement of $K_j$ is not lean for $j\geq 2$. Furthermore, the trace field of $K_j$ for $1 \leq j \leq 30 $ has no real places, and it is most likely that the trace fields of all knots in this family share this property. Therefore, <ref> is not a direct consequence of <ref> nor <ref>. The family of two-bridge knots $[1,1,2,2,2j]$ is a genuinely new family of knots with an interval left-orderable Dehn surgeries which cannot be obtained from prior techniques.

§.§ Outline

In <ref>, we review some background materials on group cohomology, $\PSL_2$-representation variety, formal deformation of representation and holonomy extension locus. At the end of this section, we will give a proof of <ref>. In <ref>, we will carry out the group cohomology calculation and prove that the complement of $K_j$ is locally longitudinally rigid at all roots of the Alexander polynomial. As a result, <ref> will follow from <ref>.

§ PRELIMINARIES

§.§ Group cohomology and $\PSL_2$-representation variety

Following the notation in <cit.> and <cit.>, we set $G = \PSL_2(\bbR)$ and $G_\bbC = \PSL_2(\bbC)$ throughout the paper. For a compact manifold $M$ and a group $H$, we let $R_H(M) = \Hom(\pi_1(M),H)$ be the representation variety. When $H = G_\bbC$, we denote $R(M) := R_{G_\bbC}(M)$. Since $\pi_1(M)$ is finitely generated, $R(M)$ can be identified with an algebraic subset in some affine space $\bbC^N$. The group $G$ acts on $R(M)$ by conjugation. Let us consider the minimal Hausdorff quotient $X(M) := R(M)//G_\bbC$ and the quotient map $\pi:R(M) \to X(M)$. Given a representation $\rho \in R(M)$, a character of $\rho$ is the map $\chi_\rho:\pi(M) \to \bbC$ defined by $\chi_\rho(\gamma) = \tr^2(\rho(\gamma))$. By <cit.>, there exists a bijection between between the points of $X(M)$ and the characters of representations in $R(M)$ such that the point $t(\rho) = [\rho]$ corresponds to $\chi_\rho$. Therefore, we refer to $X(M)$ as the $\PSL_2(\bbC)$-character variety of $M$.

Let $\Gamma$ be a group and $\rho:\Gamma \to G_\bbC$ be a representation. The Lie algebra of $G_\bbC$ can be identified with the set of trace-less 2-by-2 matrices over $\bbC$. Using the adjoint representation, the Lie algebra $\liesl_2(\bbC)$ becomes a $\Gamma$-module by

\begin{equation*}
\gamma\cdot v = \rho(\gamma) v \rho(\gamma)^{-1}.
\end{equation*}

We denote this $\Gamma$-module by $\liesl_2(\bbC)_\rho$. The space of 1-cocycles is

\begin{equation*}
\label{eq:SpaceCocycles}
Z^1(\Gamma;\liesl_2(\bbC)_\rho) = \{ z: \Gamma \to \liesl_2(\bbC) \mid z(\gamma\gamma') = z(\gamma)  + \gamma \cdot z(\gamma') \ \forall \gamma, \gamma' \in \Gamma\}.
\end{equation*}

Alternatively when $\Gamma$ is a finitely presented group, we can also describe the space of cocycles as maps $\Gamma \to \liesl_2(\bbC)$ satisfying the group relations of $\Gamma$. In particular, suppose that $\Gamma = \langle \gamma_1,\dots,\gamma_n \mid w_1(\gamma_i),\dots,w_k(\gamma_i)\rangle$ is a finite presentation and that $z(\gamma_i) = v_i \in \liesl_2(\bbC)$. Given any element $w \in\Gamma$, we can express $w$ as a word $w(\gamma_i)$ in the generators $\gamma_i$'s of $\Gamma$. The equation

\begin{equation}
\label{eq:FoxDerivatives}
z(\gamma\gamma') = z(\gamma)  + \gamma \cdot z(\gamma') \quad \forall \gamma, \gamma' \in \Gamma
\end{equation}

determines the image of $z(w)$. This gives us a well-defined cocycle on $\Gamma$ if and only if $z(w_j) = 0$ for all relations $w_j$ of $\Gamma$, see <cit.>. The space of 1-coboundaries is

\begin{equation}
\label{eq:Coboundary}
B^1(\Gamma; \liesl_2(\bbC)_\rho) = \{b: \Gamma \to \liesl_2(\bbC) \mid \exists \ v \in \liesl_2(\bbC), \ b(\gamma) = (\gamma - 1_\Gamma)\cdot v\}.
\end{equation}

Finally, the group cohomology is defined by

\begin{equation*}
\label{eq:Cohomology}
H^1(\Gamma;\liesl(\bbC)_\rho) =  Z^1(\Gamma;\liesl_2(\bbC)_\rho)/    B^1(\Gamma; \liesl_2(\bbC)_\rho).
\end{equation*}

Suppose that $V$ is an affine algebraic variety in $\bbC^n$. Let
\[I(V) = \{ f \in \bbC[x_1,\dots,x_n] \mid f(x) = 0 \quad \forall x \in V\}\]
be the vanishing ideal of $V$. Define the Zariski tangent space to $V$ at $p$ to be the vector space of derivatives of polynomials.

\begin{equation*}
T^{Zar}_p(V) = \left\{ \frac{d\gamma}{dt}\rvert_{t=0}\in \bbC^n \mid \gamma \in (\bbC[t])^n, \gamma(0) = p \text{ and } f\circ \gamma \in t^2 \bbC[t] \ \forall f \in I(V)\right\}.
\end{equation*}

It was observed by Weil in <cit.> that for any Lie group $H$ and $\rho \in R_H(M)$ the Zariski tangent space embeds in the space of 1-cocycles $Z^1(\pi_1(M);\mathfrak{h})$ where $\mathfrak{h}$ is the Lie algerba of $H$. In particular, we have the following inequalities

\[
\dim Z^1(\Gamma;\liesl_2(\bbC)_\rho) \geq \dim  T^{Zar}_\rho(R(\Gamma)).
\]

§.§ Formal deformation of representation

We will review some background materials on formal deformations of representations and integrability of cocycles. The concept of integrable cocycles will be important to building a certain path of representations required in the proof of <ref>, see also <ref>. For this discussion, let $\Gamma$ be a finitely presented group, $A_k := \bbR[[t]]/(t^{k+1})$ for $k\in \mathbb{N}$ and $A_\infty := \bbR[[t]]$. Consider the following groups $G_k := \PSL_2(A_k)$ and $G_\infty := \PSL_2(A_\infty)$.

Let $\rho:\Gamma \to G$ be a representation. A formal deformation of $\rho$ is a representation $\rho_\infty:\Gamma \to G_\infty$ such that $\rho = p_0 \circ \rho_\infty$ where $p_0: G_\infty \to G$ is the homomorphism induced by evaluating the formal power series at $t=0$.

For any formal deformation $\rho_\infty:\Gamma \to G_\infty$ of $\rho$, we can write

\begin{equation}
\label{eq:FormalDef}
\rho_\infty(\gamma) = \exp\left(\sum_{i=1}^\infty t^i u_i(\gamma)\right) \rho(\gamma)
\end{equation}

where $u_i:\Gamma \to \liesl_2(\bbR)_\rho \in C^1(\Gamma;\liesl_2(\bbR)_\rho)$. Since $\rho_\infty$ is a homomorphism, a calculation using the Taylor series for the exponential map implies that $u_1 \in Z^1(\Gamma;\liesl_2(\bbR)_\rho)$. Conversely, we have the following definition:

A cocycle $u_1\in Z^1(\Gamma;\liesl_2(\bbR)_\rho)$ is integrable if there exists a formal deformation $\rho_\infty$ of $\rho$ given by <ref>. In this case, we say that $\rho_\infty$ is a formal deformation of $\rho$ with leading term $u_1$.

Given a representation $\rho: \Gamma \to G$ and a cocycle  $u_1 \in Z^1(\Gamma;\liesl_2(\bbR)_\rho)$, the existence of a formal deformation of $\rho$ with leading term $u_1$ is equivalent to the vanishing of a series of obstruction classes in $H^2(\Gamma;\liesl_2(\bbR)_\rho)$ <cit.>. In particular, we have the following proposition from <cit.>:

Let $\rho \in R_G(\Gamma)$ and $u_i \in C^1(\Gamma;\liesl_2(\bbR)_\rho)$ for $1 \leq i \leq k$ be given. Suppose that we have constructed a representation $\rho_k:=\rho_k^{(\rho;u_1,\dots,u_k)}:\Gamma \to G_k$ given by

\[
\rho_k(\gamma) = \exp\left(\sum_{i=1}^kt^i u_i(\gamma)\right)\rho(\gamma) \mod t^{k+1}.
\]

There exists an obstruction class $\zeta_{k+1}:=\zeta_{k+1}^{(u_1,\dots,u_k)} \in H^2(\Gamma;\liesl_2(\bbR)_\rho)$ with the following properties:

* There is a cochain $u_{k+1}$ such that $\rho_{k+1}^{(\rho;u_1,\dots,u_{k+1})}: \Gamma \to G_{k+1}$ given by
\[\rho_{k+1}^{(\rho;u_1,\dots,u_{k+1})} = \exp\left(\sum_{i=1}^{k+1}t^i u_i(\gamma)\right)\rho(\gamma) \mod t^{k+2}\]
is a homomorphism if and only if $\zeta_{k+1} = 0$.

* The obstruction $\zeta_{k+1}$ is natural in the following sense: if $f:\Gamma' \to \Gamma$ is a homomorphism then

\[
f^*\rho_{k}^{(\rho;u_1,\dots,u_k)} = \rho_{k}^{(f^*\rho;f^*u_1,\dots,f^*u_k)}
\]

is a homomorphism and $f^*\zeta_{k+1}^{(u_1,\dots,u_k)} = \zeta_{k+1}^{(f^*u_1,\dots,f^*u_k)}$.

Consequently, an infinite sequence $\{u_i\}_{i=1}^\infty \subset C^1(\Gamma;\liesl_2(\bbR))_\rho$ defines a formal deformation of $\rho$, $\rho_\infty: \Gamma\to G_\infty$ via <ref> if and only if $u_1$ is a cocycle and $\zeta_{k+1}^{(u_1,\dots,u_k)} = 0$ for all $k \geq 1$.

<ref> was stated over $\bbC$ in <cit.>. Since the construction of the obstruction $\zeta_{k+1}$, see <cit.>, and the proof of <ref> is purely homological, it remains true over $\bbR$.

§.§ Holonomy extension locus

Now we recall some definitions and results about the holonomy extension locus from <cit.>. The group $G$ acts on $P^1_\bbC$ by Mobius transformation leaving $P^1_\bbR$ invariant. Any nontrivial abelian subgroup of $G$ either contains only parabolic elements and has one fixed point in $P^1_\bbC$ or contains only hyperbolic or elliptic elements and has two fixed points in $P^1_\bbC$. Let $\widetilde{G} = \widetilde{\PSL}_2(\bbR)$ be the universal covering group of $G$. The group $\widetilde{G}$ also acts on $P^1_\bbC$ by pulling back the action of $G$. We say that an element $\widetilde{g}\in\widetilde{G}$ is hyperbolic, parabolic, elliptic, or central if the image of $\widetilde{g}$ in $G$ is hyperbolic, parabolic, elliptic, or trivial, respectively.

We denote by $M$ a compact 3-manifold with a single torus boundary component and define the augmented representation $R_G^{\aug}(M)$ as follows. Since abelian subgroups of $G$ act with global fixed points, we define the augmented representation variety $R_G^{\aug}(M)$ to be the subvariety of $R_G(M) \times P^1_\bbC$ consisting of pairs $(\rho,z)$ where $z$ is a fixed point of $\rho(\pi_1(\partial M))$. Since the action of $\widetilde{G}$ on $P^1_{\bbC}$ comes from pulling back the action of $G$, we can also define $R^{\aug}_{\widetilde{G}}(M)$ to be the real analytic subvariety of $R_{\widetilde{G}}(M)\times P^1_\bbC$ consisting of pairs $(\rho,z)$ where $z$ is a fixed point of $\rho(\pi_1(\partial M))$. Similarly, we define $R^{\aug}_{\widetilde{G}}(\partial M)$ to be the real analytic subvariety of $R_{\widetilde{G}}(\partial M)\times P^1_\bbC$ consisting of pairs $(\rho,z)$ where $z$ is a fixed point of $\rho(\pi_1(\partial M))$.

Given a hyperbolic, parabolic or central element $\tilde{g} \in \widetilde{G}$ with a fixed point $v \in P^1_\bbC$, let $g\in G$ be the image of $\widetilde{g}$ and $a$ be a square root of the derivative of $g$ at $v$. We define
\[\ev(\widetilde{g},v) := (\ln(|a|),\trans(\widetilde{g}))\]
where $\trans:\widetilde{G} \to \bbR$ is the translation number given by

\[
\trans(\widetilde{g})=\lim_{n\to \infty} \frac{\widetilde{g}^n(0)}{n}.
\]

for some $x \in\bbR$. This limit exists for all $\widetilde{g} \in \widetilde{G}$, see <cit.>. It is shown in <cit.> that $\ev(-,v)$ is a homomorphism when restricted to hyperbolic or parabolic abelian subgroups of $\widetilde{G}$ fixing $v$. We get a group homomorphism

\[
\ev(\widetilde{\rho}(-),v) : \pi_1(\partial M) \to \bbR \times \bbZ
\]

for $\widetilde{\rho} \in R^{\aug}_{\widetilde{G}}(\partial M)$ whose image in $\widetilde{G}$ is hyperbolic, parabolic or central. In other words, we can view $\ev(\widetilde{\rho}(-),v)$ as an element of $\Hom(\pi_1(\partial M),\bbR \times \bbZ)$. We are now ready to define the holonomy extension locus.

Let $PH_{\widetilde{G}}(M)$ be the subset of $R_{\widetilde{G}}^{\aug}(M)$ whose restriction to $\pi_1(\partial M)$ is either hyperbolic, parabolic or central. Consider the restriction map $i^*: R^{\aug}_{\widetilde{G}}(M)  \to R^{\aug}_{\widetilde{G}}(\partial M)$ induced by the inclusion $i : \partial M \to M$. Define $\EV: i^*(PH_{\widetilde{G}}(M)) \to H^1(\partial M;\bbR) \times H^1(\partial M;\bbZ) $ by

\begin{equation*}
(\widetilde{\rho},v) \mapsto \ev((\widetilde{\rho}(-),v)).
\end{equation*}

Consider the composition

\begin{equation*}
PH_{\widetilde{G}}(M) \subset R_{\widetilde{G}}^{\aug}(M) \xrightarrow{i^*} R_{\widetilde{G}}^{\aug}(\partial M) \xrightarrow{\EV} H^1(\partial M; \bbR) \times H^1(\partial M;\bbZ)
\end{equation*}

The closure of $\text{EV} \circ i^*( PH_{\widetilde{G}}(M))$ in $H^1(\partial M; \bbR) \times H^1(\partial M;\bbZ)$ is called the holonomy extension locus of $M$ and denoted $HL_{\widetilde{G}}(M).$

We call a point in $HL_{\widetilde{G}}(M)$ a hyperbolic/parabolic/central point if it comes from a representation $\widetilde{\rho}\in PH_{\widetilde{G}}(M)$ such that $i^*(\widetilde{\rho})$ is hyperbolic/parabolic/central. We call points in $HL_{\widetilde{G}}(M)$ but not in $\text{EV} \circ i^*( PH_{\widetilde{G}}(M))$ ideal points.

To get concrete coordinates on the holonomy extension locus as well as the Dehn surgery space, let us pick a basis $(\mu, \lambda)$ for $H_1(\partial M;\bbR)$ where $\lambda$ is the homological longitude of $M$. We identify $H^1(\partial M;\bbR)$ with $\bbR^2$ using the dual basis $(\mu^*,\lambda^*)$. Let $L_r$ be the line through the origin in $\bbR^2$ of slope $-r$ where $r \in \bbQ \cup \{\infty\}$. In terms of the dual basis $(\mu^*,\lambda^*)$, the line $L_r$ consists of linear functions that vanish on the primitive element $\gamma$ representing the slope $r$ in $\pi_1(\partial M)$ with respect to the basis $(\mu,\lambda)$. The structure of the holonomy extension locus is summarized as follows:

The holonomy extension locus
\[HL_{\tilde{G}}(M) = \bigsqcup_{i,j\in \bbZ} H_{i,j}(M)\]
is a locally finite union of analytic arcs and isolated points. Each component $H_{i,j}(M)$ contains at most one parabolic point and has finitely many ideal points locally. The locus $H_{0,0}$ contains the horizontal axis $L_0$, which comes from representations to $\tilde{G}$ with abelian image.

The holonomy extension locus gives a tool to detect left-orderable Dehn surgies. We have the following lemma:

If $L_r$ intersects the component $H_{0,0}(M)$ of $HL_{\widetilde{G}}(M)$ at non-parabolic and non-ideal points, and assume that $M(r)$ is irreducible, then $M(r)$ is left-orderable.

Using the previous lemma, Xinghua Gao gives a criterion in terms of the $\PSL_2(\bbR)$-character variety to produce an interval of left-orderable Dehn surgery around the $0$-filling.

Suppose that $M$ is a longitudinally rigid irreducible $\bbQ$-homology solid torus and that the Alexander polynomial of $M$ has a simple positive real root $\xi \neq 1$. Then there exists a nonempty interval $(-a,0]$ or $[0,a)$ such that for every rational $r$ in the interval, the Dehn filling $M(r)$ is orderable.

For completeness, we include the proof of this theorem. The key to the proof of <ref> is to produce an arc in $H_{0,0}(M)$ transverse to the horizontal axis. By construction, this arc does not contain any parabolic or ideal points. <ref> then follows from <ref>. To construct an arc in $H_{0,0}(M)$, we start by deforming abelian representations coming from the roots of the Alexander polynomial into irreducible representations. In particular, let $\xi$ be a simple positive real root of the Alexander polynomial and $\alpha: \pi_1(M) \to \bbR_+$, the multiplicative group of the real numbers, such that $\alpha$ factors through $H_1(M;\bbZ)_{\text{free}} \cong \bbZ$ and takes a generator of $H_1(M;\bbZ)_{\text{free}}$ to $\xi$. We let $\rho_\alpha:\pi_1(M) \to  G_\bbC$ be the associated diagonal representation given by

\begin{equation}
\label{eq:AlexanderAbelian}
\rho_\alpha(\gamma) = \pm \begin{pmatrix}
\alpha(\gamma)^{1/2} & 0 \\
0 &  \alpha(\gamma)^{-1/2}
\end{pmatrix}
\end{equation}

where $\alpha(\gamma)^{1/2}$ is either square root. The condition on the root of the Alexander polynomial allows one to deform $\rho_\alpha =: \rho_0$ into an analytic path of representations $\rho_t :\pi_1(M) \to G$ where $t\in [-1,1]$, see <cit.>. Furthermore this path of representations has the following properties.

The path $\rho_t:[-1,1] \to R_G(M)$ constructed above satisfies:

* The representations $\rho_t$ are irreducible over $G_\bbC$ for $t\neq 0$.

* The corresponding path $[\rho_t]$ of characters in $X_G(M)$ is also a non-constant analytic path.

* The function $\tr^2(\gamma)$ is nonconstant in $t$ for some $\gamma \in \pi_1(\partial M)$.

Let $\rho_t$ be the path of representations from <ref>. Using this path, we can produce an arc in $H_{0,0}(M)$ as follows. Since $\rho_0$ factors through $H_1(M;\bbZ)_{\text{free}} \cong \bbZ$, we can lift this representation to $\tilde{\rho}_0:\pi_1(M) \to \widetilde{G}$. As the obstruction of lifting a representation from $G$ to $\widetilde{G}$ has discrete values and is continuous on $R_G(M)$, we can lift the path $\rho_t$ to a path $\widetilde{\rho}_t$ in $R_{\widetilde{G}}(M)$. Adjusting $\widetilde{\rho}_0$ by the appropriate central element of $\widetilde{G}$, we can assume that $\trans(\widetilde{\rho}_0(\mu)) = 0$. The image $\rho_0(\lambda)$ is trivial implies that $\trans(\widetilde{\rho}_0(\lambda)) = 0$. Therefore,  $\widetilde{\rho}_0$ is mapped to a point on the horizontal axis of $H_{0,0}(M)$. Since $\xi \neq 1$, the $x$-coordinate of $\widetilde{\rho}_0$, $\ln(|\xi|)$, is nonzero.

Let $k$ be the index of $[\mu]$ in $H_1(M;\bbZ)_{\text{free}}$ and $\xi \neq 1$ be the positive real root of the Alexander polynomial that corresponds to $\rho_0$. We have $\tr^2(\rho_0(\mu)) = \xi^k + 2 + \xi^{-k} > 4$. Therefore, $\rho_0(\mu)$ is hyperbolic and so is $\widetilde{\rho}_0(\mu)$. This implies that the image of the abelian subgroup $\langle \mu,\lambda\rangle$ under $\widetilde{\rho}_0$ contains only hyperbolic elements and that $\widetilde{\rho}_0\in PH_{\widetilde{G}}(M)$. Since being hyperbolic is an open condition, there exists $\varepsilon>0$ such that $\widetilde{\rho}_t \in PH_{\widetilde{G}}(M)$ for all $t \in[-\varepsilon,\varepsilon]$. The path $\widetilde{\rho}_t$ restricted to $[-\varepsilon,\varepsilon]$ projects to a path $A$ in $H_{0,0}(M)$. The projection $A$ must be non-constant since $[\rho_t]$ is a non-constant path on the character variety $X_G(M)$.

Finally, we use the hypothesis that $M$ is longitudinally rigid to argue that $A$ is not contained in the horizontal axis of $H_{0,0}(M)$. If $A$ is contained in $L_0$, then $\ln|a| = 0$ where $a$ is the eigenvalue of $\rho_t(\lambda)$. Since $\rho_t(\lambda)$ is either hyperbolic or trivial, we must have $\rho_t(\lambda)$ is trivial in $G$. Therefore, the representation $\rho_t$ factors through $M(0)$. We get a non-constant path $[\rho_t]$ in $X(M(0))$ of irreducible characters when $t\neq 0$. However, this contradicts the assumption that $M$ is longitudinally rigid.

The condition that $M$ is longitudinally rigid ensures that the representation $\rho_t$ obtained by deforming the abelian representation $\rho_0$ does not factor through the longitudinal filling. We can weaken this hypothesis by a local condition at the non-abelian reducible representation $\rho^+_\xi$ that corresponds to a root $\xi$ of the Alexander polynomial.

Recall that, we have the following theorem of Burde and de Rham:

Let $\alpha:\pi_1(M) \to \bbC^*$ be a representation and define $\rho_\alpha$ as in <ref>. Then there exists a reducible, non-abelian representation $\rho^+_\xi:\pi_1(M) \to \PSL_2(\bbC)$ such that $[\rho^+_\xi] = [\rho_\alpha]$ in $X(M)$ if and only if $\alpha$ factors through $H_1(M;\bbZ)_{\text{free}} \cong \bbZ$ sending a generator to the root $\xi$ of the Alexander polynomial of $M$.

Suppose that $M$ be an irreducible $\bbQ$-homology solid torus. Let $\xi$ be a root of the Alexander polynomial of $M$ and $\rho^+_\xi$ be a non-abelian reducible representation associated to $\xi$. We say that $M$ is locally longitudinally rigid at $\xi$ if

\begin{equation*}
H^1(M(0);\mathfrak{sl}_2(\bbC)_{\rho^+_\xi}) = 0.
\end{equation*}

Before proving <ref>, we need the following lemmas from <cit.> in the real setting. We include the proof of these lemmas for completeness.

Let $\xi$ be a simple positive real root of the Alexander polynomial that is not 1 and

\[\phi:=\rho^+_\xi: \pi_1(M) \to \PSL_2(\bbR)\]
be a non-abelian reducible representation that corresponds to $\xi$. Then the map
\[H^2(\pi_1(M);\liesl_2(\bbR)_\phi) \to H^2(\pi_1(\partial M);\liesl_2(\bbR)_\phi)\]
induced by the inclusion $\pi_1(\partial M) \hookrightarrow \pi_1(M)$ is injective.

We have $\phi|_{\pi_1(\partial M)}$ is nontrivial since $\tr^2(\phi(\mu)) = \xi^k + 2 + \xi^{-k} > 4$ where $k$ is the index $\langle[\mu]\rangle$ in $H_1(M;\bbZ)_{\text{free}}$. Since $\partial M$ is aspherical, we have $H^*(\partial M;\liesl_2(\bbR)_\phi) \cong H^*(\pi_1(\partial M);\liesl_2(\bbR)_\phi)$. Since $\phi|_{\pi_1(\partial M)}$ is non-trivial, we have

\[H^0(\partial M;\liesl_2(\bbR)_\phi) \cong \liesl_2(\bbR)^{\phi(\pi_1(\partial M))}\cong \bbR.\]

By duality and Euler characteristic, we have

\[
H^2(\partial M;\liesl_2(\bbR)_\phi) \cong \bbR
\quad \text{and} \quad
H^1(\partial M;\liesl_2(\bbR)_\phi) \cong \bbR^2.
\]

Since $\xi$ is a simple root of the Alexander polynomial, <cit.> gives that
\[H^1(M;\liesl_2(\bbR)_\phi)\cong H^1(\pi_1(M);\liesl_2(\bbR)_\phi) \cong \bbR.\]
By duality, we have

\[
H^2(M,\partial M;\liesl_2(\bbR)_\phi) \cong H^1(M;\liesl_2(\bbR)_\phi) \cong \bbR.
\]
Therefore, the following segment of the long exact sequence of pair for $(M,\partial M)$

\[
H^1(M;\liesl_2(\bbR)_\phi) \to H^1(\partial M;\liesl_2(\bbR)_\phi) \to
H^2(M,\partial M;\liesl_2(\bbR)_\phi)
\]

is short exact. Therefore from the long exact sequence of pair for $(M,\partial M)$ we see that the map

\[
H^2(M;\liesl_2(\bbR)_\phi) \to H^2(\partial M;\liesl_2(\bbR)_\phi)
\]

is injective. The conclusion of the lemma follows from the following commutative diagram

H^2(M;_2()_ϕ) [r,hook]     H^2(∂M;_2()_ϕ)

H^2(π_1(M);_2()_ϕ) [r][u]     H^2(π_1(∂M);_2()_ϕ)[u, "≅"]

and the fact that $H^2(\pi_1(M);\liesl_2(\bbR)_\phi) \to H^2(M;\liesl_2(\bbR)_\phi)$ is injective, see <cit.>.

Let $\xi$ be a simple positive real root of the Alexander polynomial that is not 1 and

\[\phi:=\rho^+_\xi: \pi_1(M) \to \PSL_2(\bbR)\]
be a non-abelian reducible representation that corresponds to $\xi$. All cocycles in $Z^1(\pi_1(M);\liesl_2(\bbR)_\phi)$ are integrable.

As noted in the proof of <ref>, $\phi|_{\pi_1(\partial M)}$ is non-trivial. Since $\phi(\pi_1(\partial M)) \subset \PSL_2(\bbR)$, the image $\phi(\pi_1(\partial M))$ cannot be the Klein 4-group. By <cit.>, $\phi|_{\pi_1(\partial M)}$ is a smooth point of an irreducible component of $R_G(\bbZ^2)$ with local dimension four.

Let $i:\pi_1(\partial M) \to \pi_1(M)$ be an inclusion map and $u_1:\pi_1(M) \to \liesl_2(\bbR)$ be a cocycle. Suppose we have cochains $u_2,\dots,u_k:\pi_1(M)\to \liesl_2(\bbR)$ such that

\[
\phi_k(\gamma) = \exp\left(\sum_{i=1}^{k}t^i u_i(\gamma)\right) \phi(\gamma)
\]

is a homomorphism modulo $t^{k+1}$. From <ref>, we get an obstruction class

\[
\zeta_{k+1}^{(u_1,\dots,u_k)} \in H^2(\pi_1(M);\liesl(\bbR)_\phi),
\]

which vanishes if and only if $\phi_k$ can be extended to a homomorphism modulo $t^{k+2}$.

The restriction $\phi_k\circ i$ is a homomorphism modulo $t^{k+1}$. Since $\phi \circ i$ is a smooth point of $R_G(\bbZ^2)$, $\phi_k \circ i$ extends to a homomorphism modulo $t^{k+2}$, see <cit.>. Therefore, the order $k+1$ obstruction vanishes on the boundary:

\[
i^*\zeta_{k+1}^{(u_1,\dots,u_k)} = \zeta_{k+1}^{(i^*u_1,\dots,i^*u_k)} = 0.
\]

By <ref>, $i^*$ is injective, and so the obstruction $\zeta_{k+1}^{(u_1,\dots,u_k)}$ vanishes for $\pi_1(M)$ as well. Iterating this process starting with $u_1$, we get an infinite sequence of cochains $\{u_i\}_{i=1}^\infty$ such that $u_1$ is a cocycle and the obstruction

\[
\zeta_{k+1}^{(u_1,\dots,u_k)} =0
\]

for all $k\geq 1$. By <ref>, we get a representation $\phi_\infty: \pi_1(M) \to \PSL_2(\bbR[[t]])$

\[
\phi_\infty(\gamma) = \exp\left(\sum_{i=1}^{\infty}t^iu_i(\gamma)\right)\phi(\gamma)
\]

for all cocycle $u_1$. Therefore, all cocycles of $Z^1(\pi_1(M);\liesl_2(\bbR))$ are integrable.

This strategy of proving that all cocycles are integrable was carried out over $\bbC$ in <cit.>. The key tool is <cit.> which uses the formal implicit function theorem. Since the formal implicit function theorem holds over $\bbR$, we can also carry out this strategy over $\bbR$. The same strategy to prove that certain cocyles are integrable over $\bbR$ was also carried out in the proof of <cit.>.

Following the proof of <ref>, it suffices to prove that the arc $A$ constructed in the proof of <ref> is not contained in $L_0$. Arguing by contradiction, suppose this arc is contained in $L_0$. As in the proof of <ref>, this would imply that the path of representation $\rho_t$ factors through $M(0)$. Since $\rho_t$ is irreducible for all $t \neq 0$, we obtain an arc in $X(M(0))$ that contains $[\rho_\alpha] = [\rho_\xi^+]$.

On the other hand, we claim that there exists a path $\phi_t:[-1,1] \to R_G(M)$ such that $\phi_0 = \rho^+_\xi$, the non-abelian reducible representation that corresponds to $\xi$. For convenience, we let $\phi:= \rho^+_\xi$. We have the following isomorphism of cohomology groups

\[
H^1(\pi_1(M);\liesl_2(\bbC)_\phi)  = H^1(\pi_1(M);\liesl_2(\bbR)_\phi) \otimes_\bbR \bbC.
\]

Since $\xi$ is a simple root of the Alexander polynomial, <cit.> gives that $H^1(\pi_1(M);\liesl_2(\bbR)_\phi)$ is one-dimensional. Therefore, $Z^1(\pi_1(M);\liesl_2(\bbR)_\phi)$ is four-dimensional. By <ref>, all cocycles in $Z^1(\pi_1(M);\liesl_2(\bbR)_\phi)$ are integrable. Therefore, $\phi$ is a smooth point of $R_G(M)$ with local dimension 4. Integrating a cocycle that generates $H^1(\pi_1(M);\liesl_2(\bbR)_\phi)$, we obtain a path $\phi_t:[-1,1] \to R_G(M)$ which has $\phi_0 = \phi$ and is transverse to the orbit of $\phi$ at $t = 0$.

We note that since $\phi$ is a smooth point of $R_G(M)$, it is contained in a unique irreducible component of $R_G(M)$. Since the abelian representations of $\pi_1(M)$ form an irreducible component of dimension 3, $\phi$ is locally four-dimensional implies that the path $\phi_t$ cannot contain any abelian representation. Consider the path $[\phi_t]$ in $X_G(M)$ which contains the character $[\phi_0]=[\rho_0]$ coming from an abelian representation. By <cit.>, $[\rho_0]$ is contained in precisely two real curves of characters. One of the curves is associated with abelian representations, and the other one with irreducible representations. Since $\phi_t$ is non-abelian representation for all $t$, the path $[\phi_t]$ is contained in the curve of irreducible characters. Therefore, for $t\neq 0$, the character $[\phi_t]$ is the character of some irreducible representation. Up to shrinking either $\rho_t$ or $\phi_t$, we may assume that $[\phi_t] = [\rho_t]$ for all $t$. Since $\phi_t$ has the same character as an irreducible representation $\rho_t$ for $t\neq 0$, the representation $\phi_t$ is conjugate to $\rho_t$ for all $t \neq 0$ by <cit.>. Since $\rho_t$ factors through $M(0)$ for all $t$, we also get that $\phi_t$ factors through $M(0)$ for all $t$. We obtain a path $[\phi_t]$ in $R_G(M(0))$ going through $\phi = \rho^+_\xi$ that is transverse to the orbit of $\rho^+_\xi$. The existence of this path implies that

\[
\dim_\bbR Z^1(\Gamma(0),\liesl_2(\bbR)_{\rho_\xi^+}) \geq \dim_\bbR T^{Zar}_{\rho_\xi^+}(R(\Gamma(0))) \geq 1 + \dim_\bbR B^1(\Gamma(0),\liesl_2(\bbR)_{\rho_\xi^+}) = 4.
\]

This would imply that $\dim_\bbR H^1(\Gamma(0),\liesl_2(\bbR)_{\rho_\xi^+}) \geq 1$. Since $\liesl_2(\bbC) = \liesl_2(\bbR) \otimes_\bbR \bbC $, we have

\[
H^1(\Gamma(0),\liesl_2(\bbC)_{\rho_\xi^+}) = H^1(\Gamma(0),\liesl_2(\bbR)_{\rho_\xi^+}) \otimes_\bbR \bbC.
\]

Therefore, the dimension of $H^1(\Gamma(0),\liesl_2(\bbC)_{\rho_\xi^+})$ is at least 1. This gives a desired contradiction to the condition that $M$ is locally longitudinally rigid at $\xi$.

§ THE [1,1,2,2,2J] TWO-BRIDGE KNOTS

In this section, we apply <ref> to study left-orderability on the family of two-bridge knots $K_j$ associated to the continued fraction $[1,1,2,2,2j]$ for $j\geq 1$ and prove <ref>. We first make some remarks about this family of two-bridge knot complements.

These knot complements are obtained by doing $1/j$ Dehn filling on the unknot component of the link $L^2_{25}$, see <ref>. The first two members of the family are the knots $8_{12}$ and $10_{13}$ in Rolfsen's table. As we will see in <ref>, the Alexander polynomial of $K_j$ has all simple positive real roots, that are not 1, and is not monic for $j \geq 2$. In particular, the complement of $K_j$ is not lean for $j\geq 2$. Furthermore, the trace field of $K_j$ for $1 \leq j \leq 30 $ has no real places, and it is most likely that the trace fields of all knots in this family share this property. Therefore, <ref> is not a direct consequence of <ref> nor <ref>. The family of two-bridge knots $[1,1,2,2,2j]$ is a new family of knots with an interval left-orderable Dehn surgeries which cannot be obtained from prior techniques.

The link complement $L^2_{25}$

§.§ Group presentation

We will denote by $\Gamma$ the fundamental group of the complement of the knot $K_j$. The knot corresponding to the continued fraction $[1,1,2,2,2j]$ has the associated fraction

\[
[1,1,2,2,2j] = 1\frac{1}{1+\frac{1}{2+\frac{1}{2+\frac{1}{2j}}}} = \frac{24j + 5}{14j + 3}.
\]

By <cit.>, the knot group $\Gamma$ has the presentation $ \Gamma = \langle x,y \mid xw = wy\rangle$. The word $w$ is given by

\begin{equation}
\label{eq:wxy}
w = y^{e_1}x^{e_2}\dots y^{e_{24j+3}}x^{e_{24j+4}}
\end{equation}

where $e_i = (-1)^{\lfloor i(14j+3)/(24j+5)\rfloor}$. Also by <cit.>, the homological longitude of $K_j$ that commutes with $x$ is given by $\ell = wv$ where

\begin{equation}
\label{eq:vxy}
v = x^{e_{24j+4}} y^{e_{24j+3}} \dots x^{e_2}y^{e_1}.
\end{equation}

We first give an explicit description of $w$ in terms of $x$ and $y$ by giving a formula for the right-hand sides of <ref> and <ref>. We have the following lemma.

In the terms of the generators $x,y$ of $\Gamma$, the word $w$ has the form

\begin{equation}
\label{eq:wj}
w = (yx^{-1}y^{-1}x)u^j \quad \text{and} \quad v = s^j(xy^{-1}x^{-1}y)
\end{equation}

\begin{equation*}
\label{eq:u}
u = (yx^{-1}yx)(y^{-1}x^{-1}yx^{-1})(y^{-1}xyx^{-1})(y^{-1}xy^{-1}x^{-1})(yxy^{-1}x)(yx^{-1}y^{-1}x)
\end{equation*}

and $s$ is $u$ spelled backwards.

Since $v$ is $w$ spelled backwards, it suffices to prove the lemma for $w$. Let us consider

\[k_{i,j} = \frac{i(14j+3)}{24j + 5}\]

for $1 \leq i \leq 24j + 4$. We first claim that

\begin{equation}
\label{eq:kijPieceWiseConstant}
\lfloor k_{i,j} \rfloor = \lfloor k_{i,m} \rfloor = \left\lfloor \frac{7i}{12}\right\rfloor
\end{equation}

for all $j\geq m$ and $\varepsilon_m \leq  i \leq 24m + 4$ where $\varepsilon_m = \max\{1,24(m-1) + 5\} $. Fixing $i$, we can view $k_{i,j}$ as a continuous function in the variable $j$. Since $i\geq 1$, the derivative of $k_{i,j}$ with respect to $j$ is

\begin{equation*}
\frac{d k_{i,j}}{dj} = -\frac{2i}{(24j+5)^2} < 0.
\end{equation*}

The function $k_{i,j}$ is strictly decreasing with and has a horizontal asymptote at $7i/12$ as $j\to +\infty$. Therefore, we have the following chain of inequalities

\[\frac{7i}{12} < k_{i,j} < k_{i,m} = \frac{(14 m + 3) i }{24 m + 5} \]

for all $j\geq m$ and $\varepsilon_m \leq i \leq 24 m + 4 $. We have

\[
0< k_{i,m} - \left\lfloor\frac{7i}{12}\right\rfloor \leq k_{i,m} - \frac{7i}{12} +\frac{11}{12} =\frac{264m + 55 + i}{288m + 60} < 1
\]

for all $ \varepsilon_m\leq i\leq 24m+4$. It follows that $k_{i,j}$ is contained in the interval $(\left\lfloor\frac{7i}{12}\right\rfloor,\left\lfloor\frac{7i}{12}\right\rfloor+1]$ for all $j\geq m$ and $\varepsilon_m \leq i \leq 24 m + 4 $. To verify <ref>, it remains to show that $k_{i,m}$ is not an integer for all $\varepsilon_m \leq i \leq 24m + 4$. Since $14m+3$ and $24m+5$ are relatively prime, $k_{i,m}$ is an integer if and only if $24m+5$ divides $i$. But this is not possible since $1\leq i < 24m+5$.

By a direct computation, we can verify <ref> when $j=1$. From <ref>, we see that the right-hand side of <ref> has prefix $w_1 = yx^{-1}y^{-1}xu$ for all $j \geq 1$. We write $w = w_1 w_j' = (yx^{-1}y^{-1}x)u w_j'$. It remains to show that $w_j' = u^{j-1}$. Using <ref>, we have

\[
\lfloor k_{i+24n,j} \rfloor = \left\lfloor \frac{7i}{12} + 14n \right\rfloor =\left\lfloor \frac{7i}{12}  \right\rfloor + 14n =   \lfloor k_{i,j} \rfloor + 14n
\]

for all $5 \leq i \leq 28$ and $5 \leq i + 24n \leq 24j + 4$. We have

\begin{equation}
\label{eq:kijParity}
\lfloor k_{i,j} \rfloor \equiv \lfloor k_{i + 24n,j} \rfloor \mod 2
\end{equation}

for all $5 \leq i \leq 28$ and $5 \leq i + 24n \leq 24j + 4$. <ref> implies that the parity of $\lfloor k_{i,j} \rfloor$ repeats with period 24 when $i\geq 5$. Since the word for $w$ in $x$ and $y$ only depends on this parity, the word $w$ is given by <ref> as claimed. This completes the proof of the lemma.

§.§ The Alexander polynomial of $K_j$

Now we will compute the Alexander polynomial of $K_j$ using non-abelian reducible representations. Let $\rho:\Gamma \to \SL_2(\bbC)$ be a non-abelian reducible representation of $\Gamma$. Since $\Gamma$ is generated by two conjugate meridians $x$ and $y$, the representation $\rho$ can be conjugated to have the form

\begin{equation}
\label{eq:NonAbelianReducible}
x \mapsto \begin{pmatrix}
t & 0 \\ 0 & t^{-1}
\end{pmatrix},
\quad \text{and} \quad
y \mapsto \begin{pmatrix}
t & 1 \\ 0 & t^{-1}
\end{pmatrix}
\end{equation}

where $t\neq \pm 1$. By <ref>, for a knot group $\Gamma$ the assignment in <ref> defines a representation of $\Gamma$ if and only if $t^2$ is a root of the Alexander polynomial $\Delta(\tau) \in \bbZ[\tau^{\pm 1}]$. Consequently, we can use this fact to compute the Alexander polynomial of the knot $K_j$ as follows.

Let $F_2$ be the free group on two letters $X$ and $Y$. Consider the representation $P:F_2 \to \SL_2(\bbZ[\tau^{\pm 1}])$

\[
X \mapsto
\begin{pmatrix}
\tau & 0 \\ 0 & \tau^{-1}
\end{pmatrix},
\quad \text{and} \quad
Y \mapsto
\begin{pmatrix}
\tau & 1 \\ 0 & \tau^{-1}
\end{pmatrix}.
\]

Let $W$ be the word in $X$ and $Y$ given by <ref>. A direct calculation shows that

\[
P(W) = \begin{pmatrix}
1 & -j \tau^3 + (5j+ 1) \tau - (5j+1)\tau^{-1} +j \tau^{-3}\\ 0 & 1
\end{pmatrix}
\]

The representation $P$ factors through the natural projection $F_2 \to \Gamma$ if and only if $P(XW)  = P(WY)$. Or equivalently, we have

\[
j\tau^4 - (6j+1)\tau^2 + (10j+3) - (6j+1) \tau^{-2} + j \tau^{-4} = 0.
\]

The expression above is the Alexander polynomial of $K_j$ evaluated at $\tau^2$. As a convention, we will normalize the Alexander polynomial so that the lowest term of $\Delta(\tau)$ is a non-zero constant term. We have the following lemma.

The Alexander polynomial of $K_j$ has the form

\begin{equation}
\label{eq:AlexanderPoly}
\Delta(\tau) = j \tau^4 -(6j+1) \tau^3 + (10j+3)\tau^2 -(6j+1)\tau +j.
\end{equation}

Furthermore, $\Delta(\tau)$ has exactly 4 simple real roots.

The discussion prior to the lemma implies that

\[
\Delta(\tau^2) = j\tau^8 - (6j+1)\tau^6 + (10j+3)\tau^4 - (6j+1) \tau^{2} + j.
\]

This gives us <ref> as claimed. For the claim about the roots of $\Delta$, we consider $\delta(\tau) = \Delta(\tau)/j$. We note that

\begin{equation*}
\delta_j(0) = 1, \
\delta_j(1/2) = \frac{-3j+2}{16j}, \
\delta_j(1) = \frac{1}{j}, \
\delta_j(2) = \frac{-3j+2}{j}, \
\delta_j(5) = \frac{96j -55}{j}.
\end{equation*}

For all $j \geq 1$, we see that $\delta_j$ changes signs 4 times in the interval $[0,5]$. By continuity, $\delta_j(\tau)$ has 4 distinct real roots in the interval $[0,5]$. Therefore, $\Delta(\tau)$ has at least 4 positive real roots. Since $\Delta$ has degree 4, $\Delta$ has precisely 4 simple positive real roots for all $j \geq 1$.

§.§ The group cohomology $H^1(\Gamma(0);\liesl_2(\bbC)_\rho)$

In this section, we will prove that the knots $K_j$ are locally longitudinal rigid by directly computing the group cohomology with coefficients in $\liesl_2(\bbC)$. We first identify $\liesl_2(\bbC)$ with $\bbC^3$ by choosing the following basis

\begin{equation*}
v_+ = \begin{pmatrix}
0 & 1 \\ 0 & 0
\end{pmatrix}, \quad  v_0 = \begin{pmatrix}
1 & 0 \\ 0 & -1
\end{pmatrix}, \quad \text{and} \quad
v_- = \begin{pmatrix}
0 & 0 \\ 1 & 0
\end{pmatrix}.
\end{equation*}

With respect to this basis, the adjoint representation $\Ad: \SL_2(\bbC) \to \SL_3(\bbC)$ becomes

\begin{equation*}
% \label{eq:Adjoint}
\begin{pmatrix}
a & b \\ c & d
\end{pmatrix}
\mapsto
\begin{pmatrix}
a^2 & -2ab & -b^2\\
-ac & ad + bc & bd \\
-c^2 & 2cd & d^2\\
\end{pmatrix}.
\end{equation*}

By <ref>, we can choose $t \in \bbR$ such that $t^2$ is a simple root of the Alexander polynomial $\Delta(\tau)$. Since the longitude $\ell$ belongs to the second commutator subgroup of $\Gamma$, any non-abelian reducible representation on $\Gamma$ factors through $\Gamma(0)$. We get a non-abelian reducible representation $\rho:\Gamma(0)\to \SL_2(\bbC)$ given by <ref>. For convenience, we will write

\begin{equation*}
\rho(w) = \begin{pmatrix} 1 & f \\ 0 & 1
\end{pmatrix}
\end{equation*}

where $f = -j t^3 +(5j+1)t - (5j+1)t^{-1} + jt^{-3}$. The action of $\Gamma(0)$ on $\liesl_2(\bbC)$ is given by

\begin{equation*}
% \label{eq:Gamma(0)Action}
\begin{pmatrix}
t^2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & t^{-2}
\end{pmatrix}
\quad \text{and} \quad
\begin{pmatrix}
t^2 & -2t & -1 \\ 0 & 1 & t^{-1} \\ 0 & 0 & t^{-2}
\end{pmatrix}.
\end{equation*}

Using <ref>, we see that the space of coboundaries can be parametrized by $d: \Gamma(0) \to  \liesl_2(\bbC)_{\rho}$ such that

\begin{equation}
\label{eq:ParametrizedCoboundary}
d(x) =
\begin{pmatrix}(t^2-1) a \\ 0 \\ (t^{-2}-1) c \end{pmatrix}
\quad \text{and} \quad
d(y) =
\begin{pmatrix}
(t^2-1) a   -2tb - c \\ t^{-1}c \\ (t^{-2}-1)c
\end{pmatrix}.
\end{equation}

Any cohomology class in $H^1(\Gamma(0);\liesl_2(\bbC)_{\rho})$ can be represented by a 1-cocycle $z\in Z^1(\Gamma(0);\liesl_2(\bbC)_\rho)$ such that

\begin{equation}
\label{eq:NormalizedCocycle}
z(x) = \begin{pmatrix}
0 \\ \alpha \\ \beta
\end{pmatrix}
\quad \text{and} \quad
z(y) = \begin{pmatrix}
0 \\ \alpha \\ 0
\end{pmatrix}.
\end{equation}

Let $z\in Z^1(\Gamma(0);\liesl_2(\bbC)_\rho)$ be a 1-cocycle. Since $t^2 \neq 1$, by an appropriate choice of $a,b,c \in \bbC$ for a coboundary $d$ in <ref>, we can assume that

\begin{equation*}
z(x) = \begin{pmatrix}
0 \\ \alpha \\ \beta
\end{pmatrix}
\quad \text{and} \quad
z(y) = \begin{pmatrix}
0 \\ \delta \\ 0
\end{pmatrix}.
\end{equation*}

The relation $z(xw) = z(wy)$ implies that

\[z(x) + (x - 1)\cdot z(w) - w \cdot z(y) = 0.\]

Or equivalently, we have

\begin{equation*}
\begin{pmatrix}
0 \\ \alpha \\ \beta
\end{pmatrix} +
\begin{pmatrix}
t^2 - 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & t^{-2}- 1
\end{pmatrix}
\begin{pmatrix}
\omega_1 \\ \omega_2 \\ \omega_3
\end{pmatrix} -
\begin{pmatrix}
1 & -2 f & -f^2 \\ 0 & 1 &  f \\ 0 & 0 & 1
\end{pmatrix}
\begin{pmatrix}
0 \\ \delta \\ 0
\end{pmatrix} =
\begin{pmatrix}
0 \\ 0 \\ 0
\end{pmatrix}.
\end{equation*}

The second coordinate of the previous equation implies that $\delta = \alpha$.

We will need the following lemma

Let $z\in Z^1(\Gamma(0);\liesl_2(\bbC)_\rho)$ be given by <ref>. Suppose that

\begin{align*}
z(w) = \omega_1 v_+ +  \omega_2 v_0 +  \omega_3v_- \quad \text{and}\quad  z(v) = \nu_1 v_+ +  \nu_2 v_0 +  \nu_3v_-.
\end{align*}

\begin{align*}
\omega_1 = &\alpha(-4jt^3 +(10j+2)t-2jt^{-3}) + \beta h \\
\omega_2 =  &\left(\frac{1}{2}j(j+1)t^7 -5j(j+1)t^5 +\frac{1}{2}(35j^2+31j + 2)t^3 -\frac{1}{2}(52j^2+28j+4)t \right. \\
&\left.+ \frac{1}{2}j(35j-3)t^{-1}-\frac{1}{2}j(10j-6)t^{-3} + \frac{1}{2}j(j-1)t^{-5}\right)\beta\\
\nu_2 =  &\left(\frac{1}{2}j(j+1)t^7 -5j(j+1)t^5 +\frac{1}{2}(35j^2+27j + 2)t^3 -\frac{1}{2}(52j^2+12j)t \right. \\
&\left.+ \frac{1}{2}j(35j-7)t^{-1}-\frac{1}{2}j(10j-6)t^{-3} + \frac{1}{2}j(j-1)t^{-5}\right)\beta\\
\omega_3 &= -\nu_3 = tf \beta
\end{align*}

where $h \in \bbC$.

The proof of this lemma is a direct calculation. By a repeat application of the cocycle relation in <ref>, we have

\begin{align*}
z(w) &= z(yx^{-1}y^{-1}x) + (yx^{-1}y^{-1}x)\cdot\sum_{i=0}^{j-1} u^i \cdot z(u),\\
z(v) &=  s^j \cdot z(xy^{-1}x^{-1}y) + \sum_{i=0}^{j-1} s^i \cdot z(s).
\end{align*}

We also have

\begin{align*}
\sum_{i=0}^{j-1} (\Ad\circ \rho) (u^i) &= \begin{pmatrix}
j & (j^2-j)(t^3-5t+5t^{-1} - t^{-3}) & -\frac{1}{6}(2j^3-3j^2 +j)(t^3-5t+5t^{-1} - t^{-3})^2 \\
0 & j & -\frac{1}{2}(j^2-j)(t^3-5t+5t^{-1} - t^{-3}) \\
0 & 0 & j
\end{pmatrix}, \\
\sum_{i=0}^{j-1} (\Ad\circ \rho) (s^i) &= \begin{pmatrix}
j & -(j^2-j)(t^3-5t+5t^{-1} - t^{-3}) & -\frac{1}{6}(2j^3-3j^2 +j)(t^3-5t+5t^{-1} - t^{-3})^2 \\
0 & j & \frac{1}{2}(j^2-j)(t^3-5t+5t^{-1} - t^{-3}) \\
0 & 0 & j
\end{pmatrix},
\end{align*}

\begin{equation*}
z(u) =
\begin{pmatrix}
(-4t^3 + 10t -2t^{-3})\alpha + h'\beta\\ (t^7 - 9t^5 + 27t^3 - 30t +10t^{-1} -t^{-3})\beta \\ (-t^4 +5t^2-5 +t^{-2})\beta
\end{pmatrix} \quad \text{and} \quad
z(s) =
\begin{pmatrix}
(4t^3 - 10t +2t^{-3})\alpha + h''\beta\\ (t^7 - 9t^5 + 25t^3 - 22t +8t^{-1} -t^{-3})\beta \\ (t^4 - 5t^2 +5 -t^{-2})\beta
\end{pmatrix}
\end{equation*}

for some $h',h'' \in \bbC$. The lemma will follow once we note that

\[
z(yx^{-1}y^{-1}x)  = \begin{pmatrix}
2t\alpha - (t^4-3t^2+1)\beta \\ (t^3-2t)\beta \\ (t^2-1)\beta
\end{pmatrix} \quad \text{and} \quad
z(xy^{-1}x^{-1}y) = \begin{pmatrix}
(-6t + 2t^{-1})\alpha + t^4\beta \\
t^3\beta \\
(-t^2 + 1)\beta
\end{pmatrix}.
\]

Now we are ready to show that $H^1(\Gamma(0);\liesl_2(\bbC)_\rho) = 0$.

Now we will show that $K_j$ is locally longitudinally rigid for all $j\geq 1$ at any root of the Alexander polynomial. Let $[z] \in H^1(\Gamma(0);\liesl_2(\bbC)_\rho)$. By <ref>, we can assume that $z$ satisfies <ref>. Since $z(\ell) = 0$, we must have

\[
z(w) + w\cdot z(v) = 0.
\]

Let us write $z(w) =\omega_1 v_+ +\omega_2v_0 + \omega_3v_-$ and $z(v) =\nu_1 v_+ +\nu_2v_0 + \nu_3v_-$. The second coordinate of this equation implies that $\omega_2 + \nu_2 + f \nu_3 = 0$. Using <ref>, we have

\[
\frac{(t^4 - 1)(t^4 + j (t^4 -4t^2 + 1)^2)}{t^5}\beta = 0.
\]

Suppose that $\beta \neq 0$. Since $t \in \bbR - \{\pm 1\}$, we have

\begin{align*}
t^4 + j (t^4 -4t^2 + 1)^2 = 0.
\end{align*}

Since $t\in \bbR$ and $j \geq 1$, the above equation holds if and only if

\[
t = 0 \quad \text{and} \quad t^4 -4t^2 + 1 = 0.
\]

This is the desired contradiction. Therefore, we must have $\beta = 0$.

From the first coordinate of the relation $z(xw) = z(wy)$, we have $(t^2-1) \omega_1 + 2f \alpha = 0$. Using $\beta = 0$ and <ref>, this equation is equivalent to

\begin{equation*}
(t^4- 1)(2jt^4-(6j+1)t^2+2j))\alpha = 0.
\end{equation*}

Similarly, if $\alpha \neq 0$, we must have $(2jt^4-(6j+1)t^2+2j)) = 0 $. It follows that $t^2$ is a root of both

\[\Delta(\tau) \quad \text{and} \quad h(\tau) :=(2j\tau^2-(6j+1)\tau+2j).\]

Note that the roots of $h(\tau)$ are reciprocal of each other. Since $\Delta(\tau)$ is a reciprocal polynomial, the roots of $\Delta(\tau)$ come in reciprocal pairs. Therefore, $h(\tau)$ divides $\Delta(\tau)$. By Gauss's lemma, we can write $\Delta(\tau) = h(\tau)k(\tau)$ for $k(\tau)\in \bbZ[\tau]$. This implies that $\Delta(0) = h(0)k(0)$ or $j = 2j k(0)$. This contradicts the fact that $j \geq 1$ and $k(0) \in \bbZ$. Therefore, $\alpha = 0$ and $z$ can only be the zero cocycle. Consequently, $H^1(\Gamma(0);\liesl_2(\bbC)_\rho) = 0$ where $\rho$ is any non-abelian reducible representation of $\Gamma(0)$. In other words, the knot $K_j$ is locally longitudinally rigid at any root of the Alexander polynomial. By <ref>, there exists an interval of left-orderable Dehn surgeries near 0.

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