# Experimental quantum speed-up in reinforcement learning agents

V. Saggio1∗ B. E. Asenbeck1 A. Hamann2 T. Strömberg1 P. Schiansky1 V. Dunjko3
N. Friis4 N. C. Harris5,6 M. Hochberg7 D. Englund6 S. Wölk2,8 H. J. Briegel2,9
P. Walther1,10 1University of Vienna, Faculty of Physics, Vienna Center for
Quantum Science and Technology (VCQ), Boltzmanngasse 5, Vienna A-1090, Austria
2Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21a,
6020 Innsbruck, Austria  3LIACS, Leiden University, Niels Bohrweg 1, 2333 CA
Leiden, Netherlands  4Institute for Quantum Optics and Quantum Information -
IQOQI Vienna, Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna,
Austria  5Lightmatter, 60 State Street, Floor 10, Boston, MA, 02130, USA
6Research Laboratory of Electronics, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  7NOKIA Corporation, New York, NY, USA  8Deutsches
Zentrum für Luft- und Raumfahrt e.V. (DLR), Institut für Quantentechnologien,
Söflingerstr. 100, 89077 Ulm, Germany  9Fachbereich Philosophie, Universität
Konstanz, Fach 17, 78457 Konstanz, Germany  10Christian Doppler Laboratory for
Photonic Quantum Computer, Faculty of Physics, University of Vienna, Vienna,
Austria.

###### Abstract

Increasing demand for algorithms that can learn quickly and efficiently has
led to a surge of development within the field of artificial intelligence
(AI). An important paradigm within AI is reinforcement learning (RL), where
agents interact with environments by exchanging signals via a communication
channel. Agents can learn by updating their behaviour based on obtained
feedback. The crucial question for practical applications is how fast agents
can learn to respond correctly. An essential figure of merit is therefore the
learning time. While various works have made use of quantum mechanics to speed
up the agent’s decision-making process, a reduction in learning time has not
been demonstrated yet. Here we present a RL experiment where the learning of
an agent is boosted by utilizing a quantum communication channel with the
environment. We further show that the combination with classical communication
enables the evaluation of such an improvement, and additionally allows for
optimal control of the learning progress. This novel scenario is therefore
demonstrated by considering hybrid agents, that alternate between rounds of
quantum and classical communication. We implement this learning protocol on a
compact and fully tunable integrated nanophotonic processor. The device
interfaces with telecom-wavelength photons and features a fast active feedback
mechanism, allowing us to demonstrate the agent’s systematic quantum advantage
in a setup that could be readily integrated within future large-scale quantum
communication networks.

## INTRODUCTION

Rapid advances in the field of machine learning (ML) and in general artificial
intelligence (AI) have been paving the way towards intelligent algorithms and
automation. An important paradigm within AI is reinforcement learning (RL),
where agents interact with an environment and ‘learning’ is facilitated by
feedback exchanges sutton . RL has applications in many sectors, from robotics
(e.g. robot control johannink , text and speech recognition tjandra ) to the
healthcare domain (e.g. finding optimal treatment policies komorowski ) to
simulation of brain-like computing thakur  and neural networks shen ;
steinbrecher . RL is also at the heart of the celebrated AlphaGo algorithm
silver , able to beat even the most skilled human players at the game of Go.

At the same time, thanks to the capability of quantum physics to outperform
classical algorithms, the development of quantum technologies has been
experiencing remarkable progress arute . Quantum physics offers improved
performance in a variety of applications, from quantum-enhanced sensing
giovannetti  to secure communication sergienko  and quantum information
processing monroe . Quantum mechanics also inspires new advantageous
algorithms in ML, and in particular RL dong ; dunjko . In fact, RL has been
successfully implemented to aid in problems encountered in quantum information
processing itself, e.g. decoding of errors baireuther ; breuckmann ;
chamberland , quantum feedback fosel , adaptive code-design poulsen , and even
the design of new quantum experiments krenn ; melnikov . Conversely, quantum
technology has been employed to enable a quadratically faster decision-making
process for RL agents via the quantization of their internal hardware paparo ;
dunjko2 ; jerbi ; sriarunothai .

In all of these RL applications, the interaction between the agent and the
environment is performed via entirely classical communication. Here we
consider a novel RL setting where the agent and the environment can also
communicate via a quantum channel dunjko3 . In particular, we introduce a
hybrid agent that enables quantum as well as classical information transfer,
and thus combines quantum amplitude amplification with a classical update
policy via a feedback loop. Within such a scenario, it is possible for the
first time to quantify and achieve a quantum speed-up in the agent’s learning
time with respect to RL schemes based on classical communication only.

Figure 1: Schematic of a learning agent. (a) An agent interacts with an
environment by receiving perceptual input $s_{i}$ and outputting consequent
actions $a_{i}$. In the case of the correct $a_{i}$ being performed, the
environment issues a reward $r$ that the agent uses to enhance its performance
in the next round of interaction. (b) Agent and environment interacting via a
classical channel, where communication is only possible via a fixed preferred
basis (e.g. ‘vertical’ or ‘horizontal’). (c) Agent and environment interacting
via a quantum channel, where arbitrary superposition states are exchanged.

The learning time is defined as the average number of interactions until the
agent receives a reward (‘accomplishes a specific task’). Such improvement
represents a remarkable advantage over previously implemented protocols, and
may prove crucial in the development of increasingly complex learning devices
thakur ; shen ; steinbrecher .

We demonstrate this protocol using a fully-programmable nanophotonic processor
interfaced with photons at telecom wavelengths. The setup enables the
implementation of an active feedback mechanism, thus proving suitable for
demonstrations of RL algorithms. Moreover, such a photonic platform holds the
potential of being fully interfaceable with future quantum communication
networks thanks to the photons’ telecom wavelengths. In fact, a long-standing
goal of the current development of quantum communication technologies lies in
establishing a form of ‘quantum internet’ kimble ; cacciapuoti  — a highly
interconnected network able to distribute and manipulate complex quantum
states via fibres and optical links (ground-based or even via satellites). We
therefore envisage AI and RL to play an important role in future large-scale
quantum communication networks, including a potential quantum internet, much
in the same way that AI forms integral parts of the internet today. Our
results thus additionally demonstrate the feasibility of integrating quantum
mechanical RL speed-ups in future complex quantum networks.

## QUANTUM ENHANCEMENT IN REINFORCEMENT LEARNING

The conceptual idea of RL is shown in Fig. 1(a). Here an agent (e.g. physical
or web-based) interacts with an environment by receiving perceptual input,
called ‘percepts’ $s_{i}$, and outputting specific ‘actions’ $a_{i}$
accordingly. ‘Rewards’ $r$ issued by the environment for correct combinations
of percepts and actions incentivize agents to improve their decision-making,
and thus to learn sutton .

Although RL has already been shown amenable to quantum enhancements, in
practical demonstrations the interaction between agents and environments has
so far been restricted exclusively to classical communication, meaning that
agents are limited to exchanges in a fixed preferred basis, e.g. ‘vertical’ or
‘horizontal’ as shown in Fig. 1(b). In general, it has been shown that
granting an agent access to quantum hardware (while still restricting it to
classical exchanges with the environment) does not reduce the average number
of interactions that the agent needs to accomplish its task, although it
allows it to output actions quadratically faster paparo ; sriarunothai . To
achieve a reduction in learning times, and thus in sample complexity, fully
quantum-mechanical interactions between agent and environment must be
considered.

Figure 2: Experimental setup. (a) Single programmable unit consisting of a
Mach-Zehnder interferometer (MZI) equipped with two fully tunable phase
shifters $\theta$ and $\phi$. (b) Real-life picture of a single MZI in the
processor. The third phase shifter in the bottom arm of the interferometer is
not used. (c) Overview of the setup. A single-photon source generates single-
photon pairs at telecom wavelength. One photon is sent straight to a single-
photon detector D0, while the other one is coupled into the processor and
undergoes the desired computation. It is then detected, in coincidence with
the photon in D0, either in detector D1 or D2/D3 after the agent plays the
classical/quantum strategy (refer to Fig. 3 for more details). The coincidence
events are recorded with a custom-made Time Tagging Module (TTM). Different
areas of the processor are assigned to either the agent or the environment.
The former one also comprises a classical control updating its policy. They
alternately play a Grover-like search to look for the rewarded sequence of
actions.

Our goal is to experimentally demonstrate such a quantum-enhanced RL scenario.
We start with considering an agent and environment with access to internal
quantum (as well as classical) hardware that can interact by coherently
exchanging quantum states ${\left|{a_{i}}\right\rangle}$,
${\left|{s_{i}}\right\rangle}$, and ${\left|{r}\right\rangle}$, representing
actions $a_{i}$, percepts $s_{i}$, and rewards $r$, respectively, via a joint
quantum channel, as illustrated in Fig. 1(c). In this case, communication is
no longer limited to a fixed preferred basis, but allows for an exchange of
arbitrary superpositions. Agents react to (sequences of) percepts
${\left|{s_{i-1}}\right\rangle}$ with (sequences of) actions
${\left|{a_{i}}\right\rangle}$ according to a policy $\pi(a_{i}|s_{i-1})$ that
is updated during the learning process. In particular, this quantum framework
includes so-called deterministic strictly epochal (DSE) learning scenarios,
which we focus on. Here, ‘epochs’ consist of strings of percepts
$\vec{s}=(s_{0},\cdots,s_{L-1})$ and actions $\vec{a}=(a_{1},\cdots,a_{L})$ of
fixed length $L$, and a final reward $r$, and both $\vec{s}=\vec{s}(\vec{a})$
and $r=r(\vec{a})$ are completely determined by $\vec{a}$ (see Methods, Sec.
I). Many interesting environments are epochal, e.g. in applications of RL to
quantum physics denil ; melnikov ; bukov ; poulsen2  or popular problems such
as playing Go silver . A non-trivial feature of the DSE scenario is that the
effective behaviour of the environment can be modelled via a unitary $U_{E}$
on the action and reward quantum registers (indicated with subscripts
$\mathrm{A}$ and $\mathrm{R}$, respectively):

$U_{E}{\left|{\vec{a}}\right\rangle}_{\mathrm{A}}{\left|{0}\right\rangle}_{\mathrm{R}}=\left\\{\begin{array}[]{cc}{\left|{\vec{a}}\right\rangle}_{\mathrm{A}}{\left|{1}\right\rangle}_{\mathrm{R}}&\text{
if }r(\vec{a})>0\\\
{\left|{\vec{a}}\right\rangle}_{\mathrm{A}}{\left|{0}\right\rangle}_{\mathrm{R}}&\text{else}\end{array}\right.,$
(1)

which identifies rewarded sequences of actions. The quantum-enhanced agent can
use $U_{E}$ to perform a quantum search for rewarded action sequences and
teach the found sequences to a classical agent dunjko3 . We push this idea
further and grant the quantum and the classical part of the agent complete
access to each other. In this way, it is possible to create a hybrid agent
with a feedback loop between quantum search and classical policy update. This
approach allows us to quantify the speed-up in learning time, which is not
possible in the general setting discussed in dunjko3  (see Methods, Sec. I).

It is known that near-term quantum devices are generally subject to limited
coherence times, restricting the number of quantum gates that can be performed
before essential quantum features are lost. However, already a few or even a
single coherent query to $U_{E}$ are enough to construct a quantum-enhanced RL
framework that can outperform a classical one.

In a classical epoch, the agent prepares the state
${\left|{\vec{a}}\right\rangle}_{\mathrm{A}}{\left|{0}\right\rangle}_{\mathrm{R}}$
where $\vec{a}$ is determined by directly sampling from a classical
probability distribution $p(\vec{a})$. From classically interacting with the
environment, the agent determines the corresponding reward $r(\vec{a})$ and
percepts $\vec{s}(\vec{a})$, and uses the obtained information to update its
policy $\pi$, thus learning.

In a quantum epoch, the agent prepares the state
${\left|{\psi}\right\rangle}_{\mathrm{A}}{\left|{-}\right\rangle}_{\mathrm{R}}$,
with
${\left|{\psi}\right\rangle}=\sum_{\vec{a}}\sqrt{p(\vec{a})}\,{\left|{\vec{a}}\right\rangle}$
being determined according to $p(\vec{a})$ dictated by its current classical
policy $\pi$, and
${\left|{-}\right\rangle}=({\left|{0}\right\rangle}-{\left|{1}\right\rangle})/\sqrt{2}$.
Both states are sent to the environment, which acts on them via $U_{E}$
inducing a phase of $-1$ for all rewarded sequences of actions. The states are
sent back to the agent which performs a reflection
$U_{Ref}=(2{\left|{\psi}\right\rangle}{\left\langle{\psi}\right|}_{\mathrm{A}}-\mathds{1}_{\mathrm{A}})$
over the initial superposition state. This leads to an amplitude amplification
of rewarded action sequences similar to the Grover’s algorithm grover .
Parametrising the classical probability to find a reward via $\sin^{2}(\xi)$
with $\xi\in[0,2\pi]$, the steps above followed by a final measurement on the
action register result with an increased probability $\sin^{2}(3\xi)$ in a
rewarded action sequence. However, this procedure does not reveal the reward
or the corresponding sequence of percepts. These can be determined only during
classical test epochs, where the measured sequence of actions is used as input
state.

The quantum-enhanced agent plays in a hybrid way by alternating between these
quantum and classical test epochs. Such agents find rewarded sequences of
actions faster, and hence usually learn faster than entirely classical agents.
The learning speed-up manifests in a reduced average quantum learning time
$\langle{T}\rangle_{Q}$, that is, the average number of epochs necessary to
achieve a predefined winning probability $Q_{L}$. In general, a quadratic
improvement in the learning time given by
$\langle{T}\rangle_{Q}\leq\alpha\sqrt{J\langle{T}\rangle_{C}}$ for the hybrid
agent can be achieved if the maximal number of coherent interactions between
the agent and the environment scales with the problem size. Here, $J$
quantifies the number of rewards the agent needs to find in order to learn and
$\langle{T}\rangle_{C}$ the average classical learning time. However, already
a single (coherent) query to $U_{E}$ is enough to improve the learning time
albeit by a constant factor, as demonstrated in the Methods, Sec. I.

## EXPERIMENTAL IMPLEMENTATION

Quantized RL protocols can be compactly realized using state-of-the-art
photonic technology flamini . In particular the path towards miniaturization
of photonic platforms holds the advantage of providing scalable architectures
where many elementary components can be accommodated on small devices. Here we
use a programmable nanophotonic processor with dimensions of $4.9$ x $2.4$ mm,
comprising $26$ waveguides fabricated to form $88$ Mach-Zehnder
interferometers (MZIs) in a trapezoidal configuration. A quantum gate is
implemented by a MZI equipped with two thermo-optic phase shifters, one
internal to allow for a scan of the output distribution over
$\theta\in[0,2\pi]$ and one external dictating the relative phase
$\phi\in[0,2\pi]$ between the two output modes, as shown in Fig. 2(a). This
makes each MZI act as a fully-tunable beam splitter and allows for coherent
implementation of sequences of quantum gates. Information is spatially encoded
onto two orthogonal modes ${\left|{0}\right\rangle}=(1,0)^{\mathrm{T}}$ and
${\left|{1}\right\rangle}=(0,1)^{\mathrm{T}}$, which constitute the
computational basis. Fig. 2(b) shows how a MZI looks like in our integrated
device.

Pairs of single photons are generated in the telecom wavelength band from a
single-photon source pumped by laser light at 789.5 nm. One photon is coupled
into one waveguide in the processor to perform a particular computation, while
the other one is sent straight to a single-photon detector D0 for heralding.
The detectors are superconducting nanowires that combine extremely high
efficiency (up to $\sim$90%) with a very short dead time ($<$ 100 ns), thus
proving the best candidates for fast feedback at telecom wavelengths.
Detection events recorded in D0 and at the output of the processor falling
into a temporal window of 1.3 ns are registered with a time tagging module
(TTM) as coincidence events (see Methods, Sec. II for more details). Different
areas of the processor are assigned to the agent and the environment, who
alternately perform the aforementioned steps of the Grover-like amplitude
amplification. The agent is further equipped with a classical control
mechanism that updates its learning policy. All of this can be seen in Fig.
2(c).

In our experiment, we represent all possible sequences of actions by a single
qubit (${\left|{1}\right\rangle}_{\mathrm{A}}\widehat{=}$ rewarded,
${\left|{0}\right\rangle}_{\mathrm{A}}\widehat{=}$ not rewarded) and use
another qubit to encode the reward (${\left|{1}\right\rangle}_{\mathrm{R}}$,
${\left|{0}\right\rangle}_{\mathrm{R}}$). This results in a four-level system
where each level is a waveguide path in our processor, as shown in Fig. 3
(which illustrates only the part of the processor needed for our computation).

The probability for the agent to choose a correct sequence of actions is
initially set to $\sin^{2}(\xi)=\varepsilon=0.01$, representing, for instance,
a single rewarded sequence of actions out of 100. After a single photon is
coupled into the mode ${\left|{0_{\mathrm{A}}0_{\mathrm{R}}}\right\rangle}$,
the agent puts it into the superposition state
${\left|{\psi}\right\rangle}_{\mathrm{A}}=(\cos{\xi}{\left|{0}\right\rangle}_{\mathrm{A}}+\sin{\xi}{\left|{1}\right\rangle}_{\mathrm{A}}){\left|{0}\right\rangle}_{\mathrm{R}}$
via the unitary $U_{p}$. Next, it can decide to play classically or quantum-
mechanically.

Figure 3: Circuit implementation. One photon is coupled into the
${\left|{0_{\mathrm{A}}0_{\mathrm{R}}}\right\rangle}$ spatial mode and
undergoes different unitaries depending on whether a classical (a) or a
quantum (b) epoch is implemented. The waveguides highlighted in yellow show
the photon’s possible paths. Identity gates are represented with straight
waveguides. Figure 4: Behaviour of the average reward $\eta$ for different
learning strategies. The solid line represents the theoretical data simulated
with $n=10,000$ agents, while the dots represent the experimental data
measured with $n=165$ agents. The shaded regions indicate the errors
associated to each single data point. (a) $\eta$ of agents playing a quantum
(blue) or classical (orange) strategy. (b) $\eta$ accounting, in the quantum
case, for rewards obtained only every second epoch, compared to (c) the case
where the reward is distributed over the two epochs an agent needs to acquire
it. (d) Comparison between the classical and combined quantum-classical case,
where an advantage over the classical case is visible. Here, the agents stop
the quantum strategy at their best performance (at $\varepsilon=0.396$) and
continue playing classically. The inset shows the point at which the agent
playing the quantum strategy reaches the predefined winning probability
$Q_{L}=0.37$, after $\langle{T}\rangle_{Q}=100$ epochs.

Classical strategy. The environment flips the reward qubit only if the action
qubit is in the correct state via the unitary $U_{E}$, as shown in Fig. 3(a).
Next, the photon is coupled out and detected in either D1 or D2 with
probability $\cos^{2}(\xi)$ and $\sin^{2}(\xi)$, respectively. Only if D2 is
triggered (i.e. the agent has been rewarded), a classical feedback mechanism
updates the policy $\pi$, and thus $p(\vec{a})$ and $\varepsilon$, via rules
given by the projective simulation framework (see Methods, Sec. I).

Quantum strategy. Here, the theoretical framework described above is exploited
to speed up the learning process. As illustrated in Fig. 3(b), after the
reward qubit is rotated to ${\left|{-}\right\rangle}_{\mathrm{R}}$ via the
unitaries $U_{H0}$ and $U_{H1}$, the environment acts as an oracle via
$U_{E}$. Consecutively, the agent reverses the effect of $U_{H0}$ and
$U_{H1}$, and then it performs the reflection $U_{Ref}$. Next, a measurement
in the computational basis of the action register is performed, leading to
detection of a rewarded sequence of actions in D3 with increased probability
$\sin^{2}(3\xi)$. The classical test epoch needed to determine the reward is
not experimentally implemented for practical reasons, since the circuit
visibility is bigger than 0.99.

In general, any Grover-like algorithm faces a drop in the amplitude
amplification after the optimal point is reached. Keeping in mind that each
agent will reach this optimal point at different epochs, one can identify the
probability $\varepsilon=0.396$ (see Methods, Sec. I) until which it is
beneficial for all agents to use a quantum strategy, as on average they will
observe more rewards than with the classical strategy. As soon as this
probability is surpassed, it is advantageous for the agents to switch to an
entirely classical strategy. This combined quantum-classical strategy thus
avoids the winning probability drop without introducing any additional
overheads in terms of experimental resources.

## RESULTS

Here, we show the experimental comparison between quantum and classical
strategies.

At the end of each classical epoch, we record outcomes $1$ and $0$ for the
rewarded and not rewarded behaviour, obtaining a binary sequence whose length
equals the number of played epochs for the classical learning strategy, and
half of the number of epochs for the quantum strategy. To enable a fair
comparison between the quantum and classical scenarios, the reward in the
quantum case is distributed — i.e. averaged — over the two epochs needed to
determine a test sequence of actions $\vec{a}$ and its reward. The reward is
then averaged over many different agents playing independently of one another.
Fig. 4 shows the average reward $\eta$ for the different learning strategies.

The theoretical data is simulated for $n=10,000$ agents and the experimental
data obtained from $n=165$. Fig. 4(a) visualizes the quantum improvement
originating from the use of amplitude amplification in comparison with a
purely classical strategy. For completeness, we also show in Fig. 4(b) and (c)
the comparison between not distributing and distributing the reward over two
epochs in the quantum case.

As soon as $\varepsilon=0.396$ is reached, $\eta$ starts decreasing. Our setup
allows the agents to always choose the favorable strategy by switching from
quantum to classical as soon as the second becomes more advantageous. In this
way, such combined strategy always outperforms the purely classical scenario,
as shown in Fig. 4(d).

We define $Q_{L}=0.37$ as the winning probability. Note however that any
probability below $\varepsilon=0.396$ can be defined as the winning
probability $Q_{L}$. The learning time $\langle{T}\rangle$ for $Q_{L}$
decreases from $\langle{T}\rangle_{C}=270$ in the classical case to
$\langle{T}\rangle_{Q}=100$ in the combined quantum-classical case. This
implies a reduction of $63\%$, which fits well to the theoretical values
$T_{C}^{theory}=293$ and $T_{Q}^{theory}=97$, taking into account small
experimental imperfections.

As the combined configuration prevents the average reward from dropping, it is
particularly relevant for real-life applications when a Grover search (with
its intrinsic overshooting drawback) is implemented. An average reward that
saturates at a high niveau without a subsequent drop can also be achieved by
employing other algorithms like the fixed-point algorithm yoder . However,
they show a less favorable speed-up than Grover-like amplitude amplification,
especially considering the limited size of our integrated processor.

In general, an agent can experience a quadratic speed-up in its learning time
if it can perform an arbitrary number of coherent Grover iterations hamann
even if the number of actual rewarded sequences is unknown boyer .

## CONCLUSIONS

We have demonstrated a novel RL protocol where information is alternately
communicated via a quantum and a classical channel. This makes it possible to
evaluate the agent’s performance resulting in a speed-up in the learning time,
and gain optimal control of the learning process. Learning agents relying on
purely classical communication are therefore outperformed. At the same time,
emerging photonic circuit technology provides the advantages of compactness,
full tunability and low-loss communication, thus proving suitable for RL
algorithms where active feedback mechanisms, even over long distances, need to
be implemented.

We highlight that although photonic architectures prove particularly suitable
for these types of learning algorithms, the presented demonstration is based
on a novel theoretical background that is general and applicable to any
quantum platform. Moreover, as the field of integrated optics is moving
towards the fabrication of increasingly large devices, this demonstration
could be extended to more complex quantum circuits allowing for processing of
high-dimensional states, and thus paving the way for achieving superior
performance in increasingly complex learning devices.

∗Corresponding author<EMAIL_ADDRESS>
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Acknowledgments: The authors thank Lee A. Rozema, Irati Alonso Calafell and
Philipp Jenke for help with the detectors. A.H. acknowledges support from the
Austrian Science Fund (FWF) through the project P 30937-N27. V.D. acknowledges
support from the Dutch Research Council (NWO/ OCW), as part of the Quantum
Software Consortium programme (project number 024.003.037). N.F. acknowledges
support from the Austrian Science Fund (FWF) through the project P 31339-N27.
H.J.B. acknowledges support from the Austrian Science Fund (FWF) through SFB
BeyondC F7102, the Ministerium für Wissenschaft, Forschung, und Kunst Baden-
Württemberg (Az. 33-7533-30-10/41/1) and the Volkswagen Foundation (Az.
97721). P.W. acknowledges support from the research platform TURIS, the
European Commission through ErBeStA (no. 800942), HiPhoP (no. 731473), UNIQORN
(no. 820474), EPIQUS (no. 899368), and AppQInfo (no. 956071), from the
Austrian Science Fund (FWF) through CoQuS (W1210-N25), BeyondC (F 7113) and
Research Group (FG 5), and Red Bull GmbH. The MIT portion of the work was
supported in part by AFOSR award FA9550-16-1-0391 and NTT Research. Author
contributions: V.S. and B.E.A. implemented the experiment and performed data
analysis. A.H., V.D., N.F., S.W. and H.J.B. developed the theoretical idea.
T.S. and P.S. provided help with the experimental implementation. N.C.H.,
M.H., and D.E. designed the nanophotonic processor. V.S., S.W., and P.W.
supervised the project. All the authors contributed to writing the paper.
Competing interests: The authors declare they have no competing interests.
Data and material availability: Data related to this work may be available
upon request.

## METHODS

###  I Quantum-enhanced learning agents

In this section, we develop an explicit method for combining a classical
reinforcement learning agent with quantum amplitude amplification. Our
approach for such hybrid agents goes beyond the ideas of Ref. dunjko3  by
introducing a feedback loop between classical policy update and quantum
amplitude amplification. The developed model allows us to determine achievable
improvements in sample complexity, and thus learning time. In addition, the
final policy of our agent has similar properties as the policy of the
underlying classical agent, leading to a comparable behavior as discussed in
more detail in hamann  (a paper dedicated to discuss the theoretical
background of the hybrid agent presented here more specifically).

In the following, we concentrate on simple deterministic, strictly epochal
(DSE) environments. That is, the interaction between the agent and the
environment is structured into epochs where each epoch starts with the same
percept $s_{0}$, and at each time step $i$ an action-percept pair
$(a_{i},s_{i})$ is exchanged. At the end of each epoch, after $L$ action-
percept pairs are communicated, a reward $r\in\\{0,1\\}$ is given to the
agent. The rules of the game are deterministic and time independent, such that
performing a specific action $a_{i}$ after receiving a percept $s_{i-1}$
always leads to the same following percept $s_{i}$.

The behaviour of an agent is determined by its policy described by the
probability $\pi(a_{i}|s_{i-1})$ to perform the action $a_{i}$ given the
percept $s_{i-1}$. In deterministic settings, the percept $s_{i}$ is
completely determined by all previous performed actions $a_{1},\cdots,a_{i}$
such that $\pi(a_{i}|s_{i-1})=\pi(a_{i}|a_{1},\cdots,a_{i-1})$. Thus, the
behaviour of the agent within one epoch is described by sequences of actions
$\vec{a}=(a_{1},\cdots,a_{L})$ and their corresponding probabilities

$\displaystyle
p(\vec{a})=\prod\limits_{i=1}^{L}\pi(a_{i}|a_{1},\cdots,a_{i-1}).$ (A.1)

The learning agent implemented here uses a policy based on projective
simulation briegel , where each sequence of actions $\vec{a}$ is associated
with a weight factor $h(\vec{a})$ initialized to $h=1$. Its policy is defined
via the probability distribution

$\displaystyle p(\vec{a})$ $\displaystyle=$
$\displaystyle\frac{h(\vec{a})}{\sum\limits_{\vec{a}^{\prime}}h(\vec{a}^{\prime})}.$
(A.2)

If the agent has played the sequence $\vec{a}$, it updates the corresponding
weight factor via

$\displaystyle h(\vec{a})$ $\displaystyle\rightarrow$ $\displaystyle
h(\vec{a})+\lambda r(\vec{a}),$ (A.3)

where $\lambda=2$ in our experiment. In general, the update method for
quantum-enhanced agents is not limited to projective simulation and can be
used to enhance any classical learning scenario, provided that $p(\vec{a})$
exists and that the update rule is solely based on the observed rewards.

We generalize the given learning problem to the quantum domain by encoding
different sequences of actions $\vec{a}$ into orthogonal quantum states
${\left|{\vec{a}}\right\rangle}$ defining our computational basis. In
addition, we create a fair unitary oracular variant $\tilde{U}_{E}$ of the
environment dunjko3 , whose behaviour is described by

$\displaystyle\tilde{U}_{E}{\left|{\vec{a}}\right\rangle}=\left\\{\begin{array}[]{cc}{\left|{\vec{a}}\right\rangle}&\text{
if }r(\vec{a})=0\\\ -{\left|{\vec{a}}\right\rangle}&\text{ else
}\end{array}\right..$ (A.6)

The unitary oracle $\tilde{U}_{E}$ can be used to perform, for instance, a
Grover search or amplitude amplification for rewarded sequences of actions by
performing Grover iterations

$U_{G}=\Big{(}2{\left|{\psi}\right\rangle}{\left\langle{\psi}\right|}-\mathds{1}\Big{)}\tilde{U}_{E}$
(A.7)

on an initial state ${\left|{\psi}\right\rangle}$. A quantum-enhanced agent
with access to $\tilde{U}_{E}$ can thus find rewarded sequences of actions
faster than a corresponding classical agent, defined by the same initial
policy $\pi(a_{i}|s_{i-1})$ and update rules, without access to
$\tilde{U}_{E}$.

In general, the optimal number $k$ of Grover iterations
$U_{G}^{k}{\left|{\psi}\right\rangle}$ depends on the winning probability

$\displaystyle q=\sum\limits_{\\{\vec{a}|r(\vec{a})\neq 0\\}}p(\vec{a})$ (A.8)

via $k\sim 1/\sqrt{q}$ grover . In the following, we assume that $q$ is known
at least to a good approximation. This is for instance possible if the number
of winning sequences ${\left|{\vec{a}}\right\rangle}$ is known. However, a
similar quantum-enhanced learning agent can be also developed if $q$ is
unknown by adapting methods from boyer  as described in hamann .

####  I.1 Description of the agent

A hybrid agent that learns faster than a classical one can be constructed by
alternating between quantum amplitude amplification and classical policy
update by repeatedly performing the following steps:

  1. 1.

Given the classical probability distribution $p(\vec{a})$, determine the
success probability $q$ based on the current policy and prepare the quantum
state

$\displaystyle{\left|{\psi}\right\rangle}=\sum\limits_{\\{\vec{a}\\}}\sqrt{p(\vec{a})}{\left|{\vec{a}}\right\rangle}.$
(A.9)

  2. 2.

Apply the optimal number of Grover iteration $k(\sqrt{q})$ leading to

$\displaystyle\centering{\left|{\psi^{\prime}}\right\rangle}=U_{G}^{k}{\left|{\psi}\right\rangle}\@add@centering$
(A.10)

and perform a measurement on ${\left|{\psi^{\prime}}\right\rangle}$ in the
computational basis to determine a test sequence of actions $\vec{a}$.

  3. 3.

Play one classical epoch by using the test sequence $\vec{a}$ determined in
step 2 and record the corresponding sequence of percepts $\vec{s}(\vec{a})$
and the reward $r(\vec{a})$.

  4. 4.

Update the classical policy $\pi(a_{k}|s_{k-1})$ and the resulting probability
distribution $p(\vec{a})$ according to A.3.

There exists a limit $Q$ on $q$ determining whether it is more advantageous
for the agent to perform a Grover iteration with $k(\sqrt{q})\geq 1$ or sample
directly from $p(\vec{a})$ (therefore $k(\sqrt{q})=0$) to determine $\vec{a}$.
In the latter case the agent would only interact classically (as in step 3)
with the environment.

After each epoch, a classical agent receives a reward with probability
$q=\varepsilon=\sin^{2}(\xi)$. We assume that the agent can use one epoch to
either perform one Grover iteration (step 2) or to determine the reward of a
given test sequence $\vec{a}$ (step 3). Thus, for $k=1$ it receives after
every second epoch a reward with probability $\sin^{2}(3\xi)$. As a
consequence, we define the expected average reward of an agent playing a
classical strategy as $\eta_{C}=2\sin^{2}(\xi)$ and of an agent playing a
quantum strategy with $k=1$ as $\eta_{Q}=\sin^{2}(3\xi)$. For $q<Q$,
$\eta_{Q}>\eta_{C}$, meaning that the quantum strategy proves advantageous
over the classical case. However, as soon as $\eta_{C}=\eta_{Q}$ (at
$Q=0.396$), a classical agent starts outperforming a corresponding quantum
agent which still performs Grover iterations.

Determining the winning probability $q$ exactly such that $q=\varepsilon$ as
in the example presented here is not always possible. In general, additional
information like the number of possible solutions and model building helps to
perform this task. Note that a $Q$ smaller than $0.396$ should be chosen if
$q$ can only be estimated up to some range. To circumvent this problem,
methods like Grover search with unknown reward probability boyer , or fixed-
point search yoder , can be used to determine if and how many steps of
amplitude amplification should be performed hamann .

####  I.2 Learning time

We define the learning time $T$ as the number of epochs an agent needs on
average to reach a predefined winning probability $Q_{L}$. The above described
quantum-enhanced agent can reach the probability $Q$ on average with less
epochs than its classical counterpart. However, once both reach $Q$, they need
on average the same number of epochs to reach $Q+\Delta_{Q}$ with
$0\leq\Delta_{Q}<1-Q$. Therefore, we choose $Q_{L}\leq Q$ in order to quantify
the achievable improvement of a hybrid agent compared to its classical
counterpart. In our experiment, we choose $Q_{L}=0.37$ to define the learning
time.

Let $\ell_{J}=\\{\vec{a}_{1},\cdots,\vec{a}_{J}\\}$ be a time-ordered list of
all the rewarded sequences of actions an agent has found until it reaches
$Q_{L}$. Note that the actual policy $\pi_{j}$, and thus $p_{j}$, of our
agents depend only on the list $\ell_{j}$ of observed rewarded sequences of
actions, and this is independent of whether it has found them via classical
sampling or quantum amplitude amplification. As a result, a classical agent
and its quantum-enhanced version are described by the same policy
$\pi(\ell_{j})$ and behave similar if they have found the same rewarded
sequences of actions. However, the quantum-enhanced agent finds them faster.

In general, the actual policy and overall success probability might depend on
the found rewarded action sequence. Thus the number $J$ of observed rewarded
action sequences necessary to learn might vary. However, this is not the case
for the here reported experiment. The learning time in this case can be
determined via

$T(J)=\sum\limits_{j=1}^{J}t_{j}$ (A.11)

where $t_{j}$ determines the number of epochs necessary for the agent to find
the next rewarded sequence $\vec{a}_{j}$ after it has observed $j-1$ rewards.
For a purely classical agent, the average time is given by

$\langle t_{j}\rangle_{C}=\frac{1}{q_{j}}$ (A.12)

where $q_{j}$ is the actual success probability. This time is quadratically
reduced to

$\langle t_{j}\rangle_{Q}=\frac{\alpha}{\sqrt{q_{j}}}$ (A.13)

for the quantum-enhanced agent. Here, $\alpha$ is a parameter depending only
on the number of epochs needed to create one oracle query $\tilde{U}_{E}$
dunjko3  and on whether $q_{j}$ is known. In the case considered here, we find
$\alpha=\pi/4$. As a consequence, the average learning time for the quantum-
enhanced learning agent is given by

$\displaystyle\langle T(J)\rangle_{Q}$ $\displaystyle=$
$\displaystyle\sum\limits_{j=1}^{J}\frac{\alpha}{\sqrt{q_{j}}}$ (A.14)
$\displaystyle\leq$
$\displaystyle\alpha\sqrt{J}\sqrt{\sum\limits_{j=1}^{J}\frac{1}{q_{j}}}$
(A.15) $\displaystyle\leq$ $\displaystyle\alpha\sqrt{J}\sqrt{\langle
T(J)\rangle_{C}}$ (A.16)

where we used the Cauchy-Schwarz inequality in the second step. The classical
learning time typically scales with $\langle T\rangle_{C}\sim A^{K}$ for a
learning problem with episode length $K$ and the choice between $A$ different
actions in each step. The number $J$ of observed rewarded sequences in order
to learn depends on the specific policy update and sometimes also on the list
$\ell_{J}$ of observed rewarded sequences of actions. For an agent sticking
with the first rewarded action sequence, we would find $J=1$. However, typical
learning agents are more explorative, and common scalings are $J\sim K$ such
that we find for these cases

$\langle T(J)\rangle_{Q}\sim\sqrt{\log(\langle T(J)\rangle_{C})}\sqrt{\langle
T(J)\rangle_{C}}.$ (A.17)

This is equivalent to a quasi-quadratic speed-up in the learning time if it is
possible to perform arbitrary numbers of Grover iterations.

In more general settings, there exist several possible $\ell_{J}$ with
different length $J$ such that the learning time $\langle T(J)\rangle$ needs
to be averaged over all possible $\ell_{J}$, which leads again to a quadratic
speed-up in learning hamann .

####  I.3 Limited coherence times

In general, all near-term quantum devices allow for coherent evolution only
for a limited time and thus a maximal number $n$ of Grover iterations. For
winning probabilities $q=\sin^{2}\xi$ with $(2n+1)\xi\leq\pi/2$, performing
$n$ Grover iterations leads to the highest probability of finding a rewarded
action.

Again, we assume that the actual policy of an agent only depends on the number
of observed rewards an agent has found. As a consequence, the average time a
quantum-enhanced agent limited to $n$ Grover iterations needs to achieve the
success probability $Q<\sin^{2}(\pi/(4n+2))$ is given by

$\langle
T(J,n)\rangle_{Q}=\sum\limits_{j=1}^{J}\frac{\alpha_{0}n+1}{\sin[(2n+1)\xi_{j}]^{2}}$
(A.18)

with $\sin^{2}\xi_{j}=q_{j}$ and $\alpha_{0}$ determining the number of epochs
necessary to create one oracle query $\tilde{U}_{E}$. For $\alpha_{0}=1$,
$n>>1$ and $(2n+1)\xi_{J}\ll\pi/2$ we can approximate the learning time for
the quantum-enhanced agent via

$\langle
T(J,n)\rangle_{Q}\approx\sum\limits_{j=1}^{J}\frac{1}{4nq_{j}}=\frac{\langle
T(J)\rangle_{C}}{4n}$ (A.19)

where we used $\sin x\approx x$ for $x\ll 1$. In general, it can be shown
hamann  that the success probability $Q_{n}=\sin^{2}(\pi/(4n+2))$ can be
reached by a quantum-enhanced agent limited to $n$ Grover iterations in a time

$\langle T(n)\rangle_{Q}\leq\gamma\frac{\langle T\rangle_{C}}{n}$ (A.20)

where $\gamma$ is a factor depending on the specific setting.
In our case Eq. A.18 can be used to compute the lower bound for the average
quantum learning time, with $\alpha_{0}=n=1$. For the classical strategy Eq.
A.12 is used.

###  II Experimental details

A continuous wave laser (Coherent Mira HP) is used to pump a single-photon
source producing photon pairs in the telecom wavelength band. The laser light
has a central wavelength of 789.5 nm and pumps the single photon source at a
power of approximately 100 mW. The source is a periodically poled KTiOPO4 non-
linear crystal placed in a Sagnac interferometer kim ; saggio , where the
emission of single photons occurs via a type II Spontaneous Parametric Down-
Conversion (SPDC) process. The crystal (produced by Raicol) is 30 mm long, set
to a temperature of 25 ∘C, has a poling period of 46.15 $\mu$m and is quasi-
phase matched for degenerate emission of photons at 1570 nm when pumping with
coherent laser light at 785 nm. As the processor is calibrated for a
wavelength of 1580 nm, we shift the wavelength of the laser light to 789.75 nm
in order to produce one photon at 1580 nm (that is then coupled into the
processor) and another one at 1579 nm (the heralding photon).

The processor used for the experiment is a silicon-on-insulator (SOI) type,
designed by the Quantum Photonics Laboratory at MIT (Massachusetts Institute
of Technology) harris . Each programmable unit on the device acts as a tunable
beam splitter implementing the unitary

$U_{\theta,\phi}=\begin{pmatrix}e^{i\phi}\sin\frac{\theta}{2}&e^{i\phi}\cos\frac{\theta}{2}\\\
\cos\frac{\theta}{2}&-\sin\frac{\theta}{2}\end{pmatrix}$ (A.21)

where $\theta$ and $\phi$ are the internal and external phases (as in Fig.
2(b)) set via thermo-optical phase shifters controlled by a voltage supply.
The achievable precision for phase settings is higher than $250$ $\mu$rad. The
bandwidth of the phase shifters is around 130 kHz. The waveguides, spatially
separated from one another by 25.4 $\mu$m, are designed to admit one linear
polarization only. The high contrast in refractive index between the silicon
and silica (the insulator) allows for waveguides with very small bend radius
(less than 15 $\mu$m), thus enabling a high component density (in our case 88
MZIs) on small areas (in our case $4.9$ x $2.4$ mm). Given the small
dimensions, the in- (and out-)coupling is realized with the help of
$Si_{3}N_{4}-SiO_{2}$ waveguide arrays (produced by Lionix International),
that shrink (and enlarge) the 10 $\mu$m optical fibers’ mode to match the 2
$\mu$m mode size of the waveguides in the processor. The total input-output
loss is around 7 dB. The processor is stabilized to a temperature of 28 ∘C and
calibrated at 1580 nm for optimal performance. To reduce the black-body
radiation emission due to the heating of the phase shifters when voltage is
applied, wavelength division multiplexers with a transmission peak centered at
1571 nm and bandwidth of 13 nm are used before the photons are sent to the
detection apparatus. In our processor, two external phase shifters in the
implemented circuits were not responding to the supplied voltage. This defects
were accounted for by deploying an optimization procedure.

The single-photon detectors are multi-element superconducting nanowires
(produced by photonSpot) with efficiencies up to 90% in the telecom wavelength
band. They have a dark count rate of $\backsim 100$ c.p.s, low timing jitter
(hundreds of ps) and a reset time $<$ 100 ns marsili .


# RIBBONNESS OF A STABLE-RIBBON SURFACE-LINK, I. A STABLY TRIVIAL LINK

Akio KAWAUCHI
Osaka City University Advanced Mathematical Institute
Sugimoto, Sumiyoshi-ku, Osaka 558-8585, Japan
<EMAIL_ADDRESS>
( )

###### ABSTRACT

There is a question asking whether a handle-irreducible summand of every
stable-ribbon surface-link is a unique ribbon surface-link. This question for
the case of a trivial surface-link is affirmatively answered. That is, a
handle-irreducible summand of every stably trivial surface-link is only a
trivial 2-link. By combining this result with an old result of F. Hosowaka and
the author that every surface-knot with infinite cyclic fundamental group is a
stably trivial surface-knot, it is concluded that every surface-knot with
infinite cyclic fundamental group is a trivial (i.e., an unknotted) surface-
knot.

x

Keywords: Trivial surface-link, Stably trivial surface-link, Orthogonal
2-handle pair.

Mathematics Subject Classification 2010: Primary 57Q45; Secondary 57M15, 57N13

1\. Introduction

A surface-link is a closed oriented (possibly disconnected) surface $F$
embedded in the 4-space ${\mathbf{R}}^{4}$ by a smooth (or a piecewise-linear
locally flat) embedding. When a (posibly disconnected) closed surface
$\mathbf{F}$ is fixed, it is also called an $\mathbf{F}$-link. If $\mathbf{F}$
is the disjoint union of some copies of the 2-sphere $S^{2}$, then it is also
called a 2-link. When $\mathbf{F}$ is connected, it is also called a surface-
knot, and a 2-knot for ${\mathbf{F}}=S^{2}$.

Two surface-links $F$ and $F^{\prime}$ are equivalent by an equivalence $f$ if
$F$ is sent to $F^{\prime}$ orientation-preservingly by an orientation-
preserving diffeomorphism (or piecewise-linear homeomorphism)
$f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$. The notation $F\cong F^{\prime}$ is
used for equivalent surface-links $F$, $F^{\prime}$. A trivial  surface-link
is a surface-link $F$ which is the boundary of the union of mutually disjoint
handlebodies smoothly embedded in ${\mathbf{R}}^{4}$, where a handlebody is a
3-manifold which is a 3-ball, a solid torus or a boundary-disk sum of some
number of solid tori. A trivial surface-knot is also called an unknotted
surface-knot. A trivial disconnected surface-link is also called an unknotted
and unlinked surface-link. For any given closed oriented (possibly
disconnected) surface $F$, a trivial $F$-link exists uniquely up to
equivalences (see [6]). A ribbon surface-link is a surface-link $F$ which is
obtained from a trivial $2$-link $O$ by the surgery along an embedded 1-handle
system (see [10, 11, 12, 13], [16, II]). A stabilization of a surface-link $F$
is a connected sum $\bar{F}=F\\#_{k=1}^{s}T_{k}$ of $F$ and a system $T$ of
trivial torus-knots $T_{k}\,(k=1,2,\dots,s)$. By granting $s=0$, we understand
that a surface-link $F$ itself is a stabilization of $F$. The trivial torus-
knot system $T$ is called the stabilizer on the stabilization $\bar{F}$ of $F$
with stabilizer components $T_{k}\,(k=1,2,...,s)$.

A stable-ribbon surface-link is a surface-link $F$ such that a stabilization
$\bar{F}$ of $F$ is a ribbon surface-link. For every surface-link $F$, there
is a surface-link $F^{*}$ with minimal total genus such that $F$ is equivalent
to a stabilization of $F^{*}$. The surface-link $F^{*}$ is called a handle-
irreducible summand of $F$. The following question is a central question.

x

Question 1.0. A handle-irreducible summand of every stable-ribbon surface-link
is a ribbon surface-link which is unique up to equivalences ?

x

A stably trivial surface-link is a surface-link $F$ such that a stabilization
$\bar{F}$ of $F$ is a trivial surface-link.

In this paper, the following theorem is shown answering affirmatively this
question for the case of a stably trivial surface-link. This question in the
general case will be answered affirmatively in [15].

x

Theorem 1.1. Any handle-irreducible summand of every stably trivial surface-
link is a trivial 2-link.

x

The following corollary is directly obtained from Theorem 1.1:

x

Corollary 1.2. Every stably trivial surface-link is a trivial surface-link.

x

If a surface-knot $F$ has an infinite cyclic fundamental group, then $F$ is a
TOP-trivial surface-knot, which was shown by Freedman for a 2-knot and by [3,
9] for a higher genus surface-knot. In the case of a piecewise linear surface-
knot(equivalent to a smooth surface-knot), it is known by [6] that a
stabilization $\bar{F}$ of the surface-knot $F$ is a trivial surface-knot,
namely the surface-knot $F$ is a stably trivial surface-knot. Hence the
following corollary is directly obtained from Corollary 1.2 answering the
problem [17, Problem 1.55(A)] on unknotting of a 2-knot positively (see [14]
for the surface-link version):

x

Corollary 1.3. A surface-knot $F$ is a trivial surface-knot if the fundamental
group $\pi_{1}({\mathbf{R}}^{4}\setminus F)$ is an infinite cyclic group.

x

The exterior of a surface-knot $F$ is the 4-manifold
$E=\mbox{cl}({\mathbf{R}}^{4}\setminus N(F))$ for a tubular neighborhood
$N(F)$ of $F$ in ${\mathbf{R}}^{4}$. Then the boundary $\partial E$ is a
trivial circle bundle over $F$. A surface-knot $F$ is of Dehn’s type if there
is a section $F^{\prime}$ of $F$ in the bundle $\partial E$ such that the
inclusion $F^{\prime}\to E$ is homotopic to a constant map. By [3, Corollary
4.2], the fundamental group $\pi_{1}({\mathbf{R}}^{4}\setminus F)$ of a
surface-knot $F$ of Dehn’s type is an infinite cyclic group. Thus, we have the
following corollary(answering the problem [17, Problem 1.51)] on unknotting of
a 2-knot of Dehn’s type positively):

x

Corollary 1.4. A surface-knot of Dehn’s type is a trivial surface-knot.

x

Unknotting Conjecture asks whether an $n$-knot $K^{n}$(i.e., a smooth
embedding image of the $n$-sphere $S^{n}$ in the(n+2)-sphere $S^{n+2}$) is
unknotted (i.e., bounds a smooth $(n+1)$-ball in $S^{n+2}$) if and only if the
complement $S^{n+2}\backslash K^{n}$ is homotopy equivalent to $S^{1}$ (see
[8] for example). This conjecture was previously known to be true for $n>3$ by
[18], for $n=3$ by [20] and for $n=1$ by [5, 19]. The conjecture for $n=2$ was
known only in the TOP category by [1](see also [2]). Corollary 1.3 answers
this finally remained smooth unknotting conjecture affirmatively and hence
Unknotting Conjecture can be changed into the following:

x

Unknotting Theorem. A smooth $S^{n}$-knot $K^{n}$ in $S^{n+2}$ is unknotted if
and only if the complement $S^{n+2}\backslash K^{n}$ is homotopy equivalent to
$S^{1}$ for every $n\geq 1$.

x

A main idea in our argument is to use the surgery of a surface-link on an
orthogonal 2-handle pair, which is much different from the surgery of a
surface-link on a single 2-handle. It is known that every surface-link $F$ in
${\mathbf{R}}^{4}$ is obtained from a higher genus trivial surface-knot
$F^{\prime}$ by the surgery of $F^{\prime}$ on a system of mutually disjoint
2-handles, because a handlebody in ${\mathbf{R}}^{4}$ is obtained from a
connected Seifert hypersurface of $F$ by removing mutually disjoint 1-handles
(see [6]). Thus, for example, every 2-twist spun 2-bridge knot in [21] is
obtained from a trivial torus-knot $T$ in ${\mathbf{R}}^{4}$ by the surgery of
$T$ on a single 2-handle, because it bounds a once-punctured lens space as a
Seifert hypersurface.

In Section 2, it is shown that every stably trivial surface-link is a trivial
surface-link if and only if the uniqueness of an orthogonal 2-handle pair on
every trivial surface-link holds. In Section 3, the uniqueness of every
orthogonal 2-handle pair on every surface-link is shown, by which Theorem 1.1
is obtained.

x

2\. A triviality condition of a stably trivial surface-link

A 2-handle on a surface-link $F$ in ${\mathbf{R}}^{4}$ is an embedded 2-handle
$D\times I$ on $F$ with $D$ a core disk such that $(D\times I)\cap F=(\partial
D)\times I$, where $I$ denotes a closed interval containing $0$ and $D\times
0$ is identified with $D$. If $D$ is an immersed disk, then call it an
immersed 2-handle. An orthogonal 2-handle pair (or simply, an O2-handle pair)
on $F$ is a pair $(D\times I,D^{\prime}\times I)$ of 2-handles $D\times I$,
$D^{\prime}\times I$ on $F$ such that

$(D\times I)\cap(D^{\prime}\times I)=(\partial D)\times I\cap(\partial
D^{\prime})\times I$

and $(\partial D)\times I$ and $(\partial D^{\prime})\times I$ meet
orthogonally on $F$ (that is, $\partial D$ and $\partial D^{\prime}$ meet
transversely at one point $p$ and the intersection $(\partial D)\times
I\cap(\partial D^{\prime})\times I$ is homeomorphic to the square $Q=p\times
I\times I$) (see Fig. 1).

Figure 1: An orthogonal 2-handle pair(=: an O2-handle pair)

Let $(D\times I,D^{\prime}\times I)$ be an O2-handle pair on a surface-link
$F$. Let $F(D\times I)$ and $F(D^{\prime}\times I)$ be the surface-links
obtained from $F$ by the surgeries along $D\times I$ and $D^{\prime}\times I$,
respectively. Let $F(D\times I,D^{\prime}\times I)$ be the surface-link which
is the union of the once punctured surface

$F^{c}=\mbox{cl}(F\setminus((\partial D)\times I\cup(\partial
D^{\prime})\times I))$

of $F$ and the plumbed disk

$\delta_{D}=D\times(\partial I)\cup Q\cup D^{\prime}\times(\partial I).$

A compact once-punctured torus of a torus $T$ is simply called a punctured
torus and denoted by $T^{o}$. A punctured torus $T^{o}$ in a 3-ball $B$ is
trivial if $T^{o}$ is smoothly and properly embedded in $B$ and there is a
solid torus $V$ in $B$ with $\partial V=T^{o}\cup\delta$ for a disk $\delta$
in $\partial B$. A bump of a surface-link $F$ is a 3-ball $B$ in
${\mathbf{R}}^{4}$ with $F\cap B=T^{o}$ a trivial punctured torus in $B$. Let
$F(B)$ be a surface-link $F^{c}\cup\delta_{B}$ for the surface
$F^{c}=\mbox{cl}(F\setminus T^{o})$ and a disk $\delta_{B}$ in $\partial B$
with $\partial\delta_{B}=\partial T^{o}$, where note that $F(B)$ is uniquely
determined up to cellular moves on $\delta_{B}$ keeping $F^{c}$ fixed. For an
O2-handle pair $(D\times I,D^{\prime}\times I)$ on a surface-link $F$, let
$\Delta=D\times I\cup D^{\prime}\times I$ is a 3-ball in ${\mathbf{R}}^{4}$
called the 2-handle union. By adding a boundary collar to the 2-handle union
$\Delta$, we have a bump $B_{D}$ of $F$, which we call the associated bump of
the O2-handle pair $(D\times I,D^{\prime}\times I)$ (see Fig. 2). The
following lemma shows that giving an O2-handle unordered pair on a surface-
link $F$ is the same as giving a bump of $F$.

Figure 2: A 2-handle union whose boundary collar is an associated bump

x

Lemma 2.1. An O2-handle unordered pair $(D\times I,D^{\prime}\times I)$ on a
surface-link $F$ is uniquely constructed from any given bump $B$ of $F$ in
${\mathbf{R}}^{4}$ with $F(D\times I,D^{\prime}\times I)\cong F(B)$.

x

Proof of Lemma 2.1. For a bump $B$ of $F$, the set of two solid tori bounded
by $T^{o}=F\cap B$ is unique, whose meridian-longitude disk pair is an
O2-handle pair. $\square$

x

The following lemma represents the unique nature of the surgery of a surface-
link $F$ on an O2-handle pair.

x

Lemma 2.2. For any O2-handle pair $(D\times I,D^{\prime}\times I)$ on any
surface-link $F$ and the associated bump $B$, there are equivalences

$F(B)\cong F(D\times I,D^{\prime}\times I)\cong F(D\times I)\cong
F(D^{\prime}\times I).$

Further, these equivalences are attained by cellular moves on the 2-handles
$D\times I$ and $D^{\prime}\times I$ keeping $F^{c}$ fixed.

x

Proof of Lemma 2.2. By definition, we have $F(B)\cong F(D\times
I,D^{\prime}\times I)$. The surface-link $F(D\times I,D^{\prime}\times I)$ is
equivalent to $F(D\times I)$ and $F(D^{\prime}\times I)$ by cellular moves on
the 3-balls $D\times I$ and $D^{\prime}\times I$ from the disks
$D\times\partial I\cup Q$ and $D^{\prime}\times\partial I\cup Q$ to the disks
$\mbox{cl}((\partial D)\times I\setminus Q)$ and $\mbox{cl}((\partial
D^{\prime})\times I\setminus Q)$, respectively. $\square$

x

The following lemma shows how an O2-handle pair on $F$ is obtained from an
immersed 2-handle and a 2-handle on a surface-link $F$ whose attaching annuli
meet orthogonally on $F$.

x

Lemma 2.3. Let $D\times I$ be an immersed 2-handle on a surface-link $F$. Let
$D^{\prime}\times I$ be a 2-handle on $F$ such that the annuli $(\partial
D)\times I$ and $(\partial D^{\prime})\times I$ meet orthogonally on $F$. Then
there is a 2-handle $D^{*}\times I$ on $F$ with $(\partial D^{*})\times
I=(\partial D)\times I$ such that $(D^{*}\times I,D^{\prime}\times I)$ is an
O2-handle pair on $F$.

x

Proof of Lemma 2.3. Assume that the immersed disk $D$ has only transversely
intersecting double points. Let $a$ be an arc obtained from $D^{\prime}\times
I$ by shrinking $D^{\prime}$ into a point such that $\partial D=a\cup(D\cap
F(D^{\prime}\times I))$. Let $N(a)$ be a thin regular neighborhood of $a$ in
$D^{\prime}\times I$. Add a small embedded disk $A$ to $D$ along a part of the
arc $a$. Then take mutually disjoint simple arcs from the double points of $D$
to the interior of $A$. Slide the double points of $D$ into $A$ along the arcs
by finger moves. Forget this additional disk $A$ to obtain an embedded disk
$D^{\prime\prime}$ with $\partial D^{\prime\prime}=\partial D$. The free arc
$a$ can be replaced by $N(a)$ without meeting the interior of
$D^{\prime\prime}$. By an ambient isotopic deformation, the pair
$(D^{\prime\prime}\times I,N(a))$ constructs an O2-handle pair $(D^{*}\times
I,D^{\prime}\times I)$ on $F$. $\square$

x

Lemma 2.3 is applied for the following result useful to our argument.

x

Lemma 2.4. Let $D\times I$ be an immersed 2-handle on a surface-link $F$. Let
$D^{\prime}_{i}\times I\,(i=1,2)$ be any 2-handles on $F$ such that $(\partial
D^{\prime}_{1})\times I=(\partial D^{\prime}_{2})\times I$, and this annulus
and $(\partial D)\times I$ meet orthogonally on $F$. Then there is a 2-handle
$D^{*}\times I$ on $F$ with $(\partial D^{*})\times I=(\partial D)\times I$
such that the pairs $(D^{*}\times I,D^{\prime}_{i}\times I)\,(i=1,2)$ are
O2-handle pairs on $F$.

x

Proof of Lemma 2.4. For each $i\,(i=1,2)$, let $a_{i}$ be an arc obtained from
$D^{\prime}_{i}\times I$ by shrinking $D^{\prime}_{i}$ into a point such that
$\partial D=a_{i}\cup(D\cap F(D^{\prime}_{i}\times I))$. Then $a_{1}=a_{2}$
ince $(\partial D^{\prime}_{1})\times I=(\partial D^{\prime}_{2})\times I$.
Let $N(a_{i})$ be a thin regular neighborhood of $a_{i}$ in
$D^{\prime}_{i}\times I$. By assuming that a boundary-collar of $\partial
D^{\prime}_{1}$ in the immersed disk $D^{\prime}_{1}$ coincides with a
boundary-collar of $\partial D^{\prime}_{2}$ in the immersed disk
$D^{\prime}_{2}$, consider that $N(a_{1})=N(a_{2})$. By a cell-move on
$D^{\prime}_{i}\times I$, let $r_{i}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ be a
diffeomorphism sending $F\cup D^{\prime}_{i}\times I$ into
$F(D^{\prime}_{i}\times)\cup N(a_{i})$ such that the restriction of $r_{i}$ to
a neighborhood of $D\times I$ is the identity map and the restriction of
$r_{2}r_{1}^{-1}$ to $N(a_{1})=N(a_{2})$ is the identity map. By Lemma 2.3,
there is a 2-handle $D^{\prime\prime}\times I$ on $r_{i}(F)$ with $(\partial
D^{\prime\prime})\times I=(\partial D)\times I$ such that
$(D^{\prime\prime}\times I,N(a_{i}))$ is an O2-handle pair on $r_{i}(F)$ for
each $i$. Since $N(a_{i})=r_{i}(D^{\prime}_{i}\times I)$, the pair
$(r_{i}^{-1}(D^{\prime\prime})\times I,D^{\prime}_{i}\times I)$ is an
O2-handle pair on $F$ for each $i$. Since the restriction of $r_{2}r_{1}^{-1}$
to $N(a_{1})=N(a_{2})$ is the identity map, we can also assume that the
restriction of $r_{2}r_{1}^{-1}$ to a neighborhood of $N(a_{1})=N(a_{2})$ is
the identity map, so that
$r_{2}r_{1}^{-1}(D^{\prime\prime})=D^{\prime\prime}$. Then
$r_{1}^{-1}(D^{\prime\prime})=r_{2}^{-1}(D^{\prime\prime})$ which we denote by
$D^{*}$. The pairs $(D^{*}\times I,D^{\prime}_{i}\times I)\,(i=1,2)$ are
desired O2-handle pairs on $F$. $\square$

x

A surface-link $F$ has only unique O2-handle pair if for any O2-handle pairs
$(D\times I,D^{\prime}\times I)$ and $(E\times I,E^{\prime}\times I)$ on $F$
with $(\partial D)\times I=(\partial E)\times I$ and $(\partial
D^{\prime})\times I=(\partial E^{\prime})\times I$, there is an equivalence
$f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ sending $F$ to $F$ such that
$f(D\times I)=E\times I$ and $f(D^{\prime}\times I)=E^{\prime}\times I$. A
surface-link not admitting any O2-handle pair is understood as a surface-link
with only unique O2-handle pair. We have the following characterization on a
stably trivial surface-link.

x

Lemma 2.5. The following (1)-(3) are mutually equivalent.

(1) If a connected sum $F\\#T$ of a surface-link $F$ and a trivial torus-knot
$T$ is a trivial surface-link, then $F$ is a trivial surface-link.

(2) If $F$ is a trivial surface-link and $(D\times I,D^{\prime}\times I)$ is
an O2-handle pair on $F$, then $F(D\times I,D^{\prime}\times I)$ is a trivial
surface-link.

(3) Any trivial surface-link has only unique O2-handle pair.

x

Proof of Lemma 2.5. $(1)\Rightarrow(2)$: Let $B$ be the associated bump of the
O2-handle pair $(D\times I,D^{\prime}\times I)$. A 4-ball $A$ obtained by
taking a bi-collar $c(B\times[-1,1])$ of $B$ in ${\mathbf{R}}^{4}$ with
$c(B\times 0)=B$ gives a connected sum decomposition $F\cong F(D\times
I,D^{\prime}\times I)\\#T$. By (1), $F(D\times I,D^{\prime}\times I)$ is a
trivial surface-link.

$(2)\Rightarrow(3)$: Let $(D\times I,D^{\prime}\times I)$ and $(E\times
I,E^{\prime}\times I)$ be O2-handle pairs with $(\partial D)\times I=(\partial
E)\times I$ and $(\partial D^{\prime})\times I=(\partial E^{\prime})\times I$.
Let $F(D\times I,D^{\prime}\times I)=F^{c}\cup\delta_{D}$ and $F(E\times
I,E^{\prime}\times I)=F^{c}\cup\delta_{E}$ be trivial surface-links for disks
$\delta_{D}$ and $\delta_{E}$ in the boundaries $\partial B_{D}$ and $\partial
B_{E}$ of the associated bumps $B_{D}$ and $B_{E}$ the O2-handle pairs
$(D\times I,D^{\prime}\times I)$ and $(E\times I,E^{\prime}\times I)$,
respectively. Let $F_{D}$ and $F_{E}$ be the components of $F(D\times
I,D^{\prime}\times I)$ and $F(E\times I,E^{\prime}\times I)$ containing the
loop $\partial\delta_{D}=\partial\delta_{E}$, respectively, which are made
split from the other components in ${\mathbf{R}}^{4}$ because all the
components of every trivial surface-link are split in ${\mathbf{R}}^{4}$.
Since $F_{D}$ and $F_{E}$ are trivial surface-knots of the same genus, there
is an equivalence $f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ from $F_{D}$ to
$F_{E}$ orientation-preservingly. By a cellular move of $\delta_{D}$ in
$F_{D}$, this map $f$ is modified to have $f(\delta_{D})=\delta_{E}$. Further,
this map $f$ is modified to send $F^{c}\cup\delta_{D}$ to
$F^{c}\cup\delta_{E}$ by sending all the components except for $F_{D}$ and
$F_{E}$ identically. Thus, we have an equivalence $f$ with $f(F^{c})=F^{c}$
and $f(\delta_{D})=\delta_{E}$. The map $f$ is isotopic to a diffeomorphism
$f^{\prime}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ sending the associated bump
$B_{D}$ of $(D\times I,D^{\prime}\times I)$ to the associated bump $B_{E}$ of
$(E\times I,E^{\prime}\times I)$ by regarding $B_{D}$ and $B_{E}$ as collars
of $\delta_{D}$ and $\delta_{E}$, respectively. The diffeomorphism
$f^{\prime}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ is modified into an
equivalence $f^{\prime\prime}:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ from $F$ to
$F$ such that $f(D\times I)=E\times I$ and $f(D^{\prime}\times
I)=E^{\prime}\times I$.

$(3)\Rightarrow(1)$: Let $F_{i}\,(i=0,1,\dots,r)$ be the components of $F$,
and $F\\#T=F_{0}\\#T\cup F_{1}\cup\dots\cup F_{r}$ a trivial surface-link. Let
$V$ be the split union of handlebodies $V_{i}\,(i=0,1,\dots,r)$ in
${\mathbf{R}}^{4}$ such that $\partial V_{0}=F_{0}\\#T$ and $\partial
V_{i}=F_{i}\,(i=1,2,\dots,r)$.

A loop basis of $F_{0}\\#T$ of genus $g+1$ is a system of oriented simple loop
pairs $(e_{j},e^{\prime}_{j})\,(j=0,1,2,\dots,g)$ on $F_{0}\\#T$ representing
a basis for $H_{1}(F_{0}\\#T;{\mathbb{Z}})$ such that $e_{j}\cap
e_{j^{\prime}}=e^{\prime}_{j}\cap e^{\prime}_{j^{\prime}}=e_{j}\cap
e^{\prime}_{j^{\prime}}=\emptyset$ for all distinct $j,j^{\prime}$ and
$e_{j}\cap e^{\prime}_{j}$ is one point with the intersection number
$\mbox{Int}(e_{j},e^{\prime}_{j})=+1$ in $F_{0}\\#T$ for all $j$. The
following result is given in [3, Lemma 2.1].

x

(2.5.1) For a surface-knot $F_{0}\\#T$ of genus $g+1$ in ${\mathbf{R}}^{4}$,
there is a loop basis $(e_{j},e^{\prime}_{j})\,(j=0,1,2,\dots,g)$ on
$F_{0}\\#T$ which is spin (namely, with $q(e_{j})=q(e^{\prime}_{j})=0$ for all
$j$ for the ${\mathbb{Z}}_{2}$-quadratic function
$q:H_{1}(F_{0}\\#T;{\mathbb{Z}}_{2})\to{\mathbb{Z}}_{2}$ associated with the
surface-knot $F_{0}\\#T$). In particular, for a trivial surface-knot
$F_{0}\\#T$ bounded by a handlebody $V_{0}$ in ${\mathbf{R}}^{4}$, every loop
basis $(e_{j},e^{\prime}_{j})\,(j=0,1,2,\dots,g)$ on $\partial V_{0}$ with
$e^{\prime}_{j}\,(j=0,1,2,\dots,g)$ a meridian loop system of $V_{0}$ has
$q(e^{\prime}_{j})=0$ and either $q(e_{j})=0$ or $q(e_{j}+e^{\prime}_{j})=0$
for all $j$, where $e_{j}+e^{\prime}_{j}$ denotes a Dehn twist of $e_{j}$
along $e^{\prime}_{j}$.

x

The following result is obtained from [4]:

x

(2.5.2) For any two loop bases $(e_{j},e^{\prime}_{j})\,(j=0,1,2,\dots,g)$ and
$(\tilde{e}_{j},\tilde{e}^{\prime}_{j})\,(j=0,1,2,\dots,g)$ on a trivial genus
$g$ surface-knot $F_{0}\\#T$ with $q(e_{j})=q(\tilde{e}_{j})$ and
$q(e^{\prime}_{j})=q(\tilde{e}^{\prime}_{j})$ for all $j$, there is an
orientation-preserving diffeomorphism $f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$
with $f(F_{0}\\#T)=F_{0}\\#T$ such that $f(e_{j})=\tilde{e}_{j}$ and
$f(e^{\prime}_{j})=\tilde{e}^{\prime}_{j}$ for all $j$.

x

Let $(D\times I,D^{\prime}\times I)$ be an O2-handle pair on $F\\#T$ in
${\mathbf{R}}^{4}$ attached to $T^{o}$ such that $(F\\#T)(D\times
I,D^{\prime}\times I)\cong F$. By (2.5.1), there is a spin loop basis for
$F_{0}\\#T$ containing the pair $(\partial D,\partial D^{\prime})$. Also, let
$(e_{i},e^{\prime}_{i})\,(i=0,1,2,\dots,g)$ be a spin loop basis for $F_{0}$
such that $e_{0}$ bounds a disk $d$ in ${\mathbf{R}}^{4}$ with $d\cap V=e_{0}$
and $e^{\prime}_{0}$ bounds a meridian disk $d^{\prime}$ of $V_{0}$. By
(2.5.2), there is an orientation-preserving diffeomorphism
$f:{\mathbf{R}}^{4}\to{\mathbf{R}}^{4}$ with $f(F_{0}\\#T)=F_{0}\\#T$ and
$f|_{V_{i}}=1\,(i=1,2,\dots,r)$ such that $f(\partial D)=e_{0}$ and
$f(\partial D^{\prime})=e^{\prime}_{0}$. A thickening pair $(d\times
I,d^{\prime}\times I)$ of the disk pair $(d,d^{\prime})$ is an O2-handle pair
with $(F\\#T)(d\times I,d^{\prime}\times I)$ is a trivial surface-knot. Since
$(f(D)\times I,f(D^{\prime})\times I)$ is an O2-handle pair on $F\\#T$, we
obtain from (3) that

$\displaystyle F$ $\displaystyle\cong$ $\displaystyle(F\\#T)(D\times
I,D^{\prime}\times I)$ $\displaystyle\cong$ $\displaystyle(F\\#T)(f(D)\times
I,f(D^{\prime})\times I)$ $\displaystyle\cong$ $\displaystyle(F\\#T)(d\times
I,d^{\prime}\times I).$

Thus, $F$ is a trivial surface-link. $\square$

x

3\. Uniqueness of an orthogonal 2-handle pair

The following theorem is our main result.

x

Theorem 3.1. Any two O2-handle pairs $(D\times I,D^{\prime}\times I)$ and
$(E\times I,E^{\prime}\times I)$ on any (not necessarily trivial) surface-link
$F\\#T$ attached to $T^{o}\subset T$ with $(\partial D)\times I=(\partial
E)\times I$ and $(\partial D^{\prime})\times I=(\partial E^{\prime})\times I$
are moved into each other by a finite number of cellular moves on $D\times I$
and $D^{\prime}\times I$ keeping $F^{c}=\mbox{cl}(F\setminus T^{o})$ fixed. In
particular, any (not necessarily trivial) surface-link has only unique
O2-handle pair.

x

Theorem 3.1 and Lemma 2.5 implies Theorem 1.1.

x

Proof of Theorem 1.1. Let $F$ be a stably trivial link. That is, assume that a
stabilization $\bar{F}=F\\#_{k=1}^{s}T_{k}$ of $F$ is a trivial link for some
$s\geq 1$. By Theorem 3.1 and Lemma 2.5, $F\\#_{k=1}^{s-1}T_{k}$ is a trivial
surface-link. Inductively, $F$ is a surface-link, so that any handle-
irreducible summand $F^{*}$ of $F$ is a trivial $S^{2}$-link. $\square$

x

For a subset $A$ of ${\mathbf{R}}^{3}$ and an interval $J$, the subset
$\\{(x,t)|\,x\in A,t\in I\\}$ of ${\mathbf{R}}^{4}$ is denoted by the notation
$AJ$.

Let $B$ be a bump of a surface-link $F$ such that the punctured torus
$T^{o}=F\cap B$ is in ${\mathbf{R}}^{3}[0]$. Let $V_{B}$ be a solid torus in
$B$ with $\partial V_{B}=T^{o}\cup\delta_{B}$ for a disk $\delta_{B}$ in
$\partial B$, and $V_{\mathbf{R}}$ a solid torus in ${\mathbf{R}}^{3}[0]$ with
$\partial V_{\mathbf{R}}=T^{o}\cup\delta_{\mathbf{R}}$ for a disk
$\delta_{\mathbf{R}}$. Assume that a boundary collar of $T^{o}$ in $V_{B}$
overlaps with a boundary collar of $T^{o}$ in $V_{\mathbf{R}}$. Let
$(d_{B}^{\ell},d_{B}^{m})$ be a longitude-meridian disk pair of $V_{B}$ in $B$
with intersection point $p$, and $d_{\mathbf{R}}^{m}$ a meridian disk of
$V_{\mathbf{R}}$. For a boundary collar $n_{B}^{m}$ of $d_{B}^{m}$ and a
boundary collar $n_{\mathbf{R}}^{m}$ of $d_{\mathbf{R}}^{m}$, assume that
$n_{B}^{m}=n_{R}^{m}$. Let $q$ be an interior point of this overlapped
boundary collar. The following lemma is needed for a deformation on a
O2-handle pair.

x

Lemma 3.2. There is a 2-sphere $S$ in ${\mathbf{R}}^{4}$ such that

(1) $S\cap F=\emptyset$,

(2) $S\cap d_{B}^{m}=S\cap d_{R}^{m}=\\{q\\}$,

(3) $S$ meets $d_{B}^{\ell}$ transversely in finitely many points.

x

Figure 3: A twisted disk

Proof of Lemma 3.2. Regard the 2-handle union $\Delta=d_{B}^{\ell}\times I\cup
d_{B}^{m}\times I$ as a line bundle of the disk $d_{B}^{*}=d_{B}^{\ell}\cup
b^{\tau}\cup d_{B}^{m}$ in $B$ for a half twist band $b^{\tau}$ by lending a
part of $d_{B}^{\ell}\times I$ (see Fig. 3). Since $B$ is obtained from
$\Delta$ by adding a boundary collar, the bump $B$ is the total space $|\xi|$
of a line bundle $\xi$ over a disk $\bar{d}_{B}^{*}$ obtained from $d_{B}^{*}$
by adding a boundary collar. For a subspace $X$ of $\bar{d}_{B}^{*}$, let
$\xi(X)$ denote the line bundle over $X$ induced from $\xi$, and $|\xi(X)|$
the total space of $\xi(X)$. Also, denote the associated $S^{0}$-bundle by
$\dot{\xi}(X)$, and the total space by $|\dot{\xi}(X)|$. Let
$E_{B}^{m}=\mbox{cl}(d_{B}^{m}\setminus n_{B}^{m})$ and
$E_{R}^{m}=\mbox{cl}(d_{R}^{m}\setminus n_{R}^{m})$. Assume that
$d_{B}^{\ell}\cap n_{R}^{m}=\\{p\\}$. Since we can assume that

$\displaystyle(\mbox{int}E_{R}^{m})\cap(d_{B}^{\ell}\times I)$
$\displaystyle=$ $\displaystyle((\mbox{int}d_{R}^{m})\cap d_{B}^{\ell})\times
I\quad{\mbox{and}}$ $\displaystyle(\mbox{int}E_{R}^{m})\cap(d_{B}^{m}\times
I)$ $\displaystyle=$ $\displaystyle((\mbox{int}d_{R}^{m})\cap d_{B}^{m})\times
I,$

the open disk $\mbox{int}E_{R}^{m}$ is considered to meet $B$ as finitely many
fibers of the line bundle $\xi$ disjoint from $T^{o}$. Let $u$ be a trivial
loop in ${\mathbf{R}}^{3}$ such that

(a) $u[0]$ is in the overlapping part of the boundary collars of $V_{B}$ and
$V_{\mathbf{R}}$,

(b) $u[0]$ meets $n_{B}^{m}=n_{R}^{m}$ transversely at the point $q$, and

(c) $u[0]$ does not meet $T^{o}$, $d_{B}^{\ell}$, $E_{B}^{m}$ and $E_{R}^{m}$
(see Fig. 4).

x

Figure 4: A situation on a bump

Let $U=u(-\infty,+\infty)$ be an infinite cylinder in ${\mathbf{R}}^{4}$.
Since $T^{o}$ is a connected summand of $F$, we can assume that $U\cap
F=\emptyset$. By construction,

$U\cap d_{R}^{m}=U\cap n_{B}^{m}=\\{q\\}.$

Also, assume that $U$ meets $d_{B}^{\ell}$ and $E_{B}^{m}$ transversely in
finite points.

Let

$\displaystyle f$ $\displaystyle=$ $\displaystyle(U\setminus\\{q\\})\cap
d_{B}^{m}=\\{x_{i}|\,i=1,2,\dots,s\\},$ $\displaystyle f^{\prime}$
$\displaystyle=$ $\displaystyle U\cap d_{B}^{\ell},$ $\displaystyle
f^{\prime\prime}$ $\displaystyle=$ $\displaystyle
d_{B}^{\ell}\cap\mbox{int}E_{R}^{m},$ $\displaystyle f^{\prime\prime\prime}$
$\displaystyle=$ $\displaystyle\mbox{int}E_{B}^{m}\cap\mbox{int}E_{R}^{m}$

be finite sets of transversely meeting points.

Since we can assume that

$\displaystyle(U\setminus u)\cap(d_{B}^{\ell}\times I)$ $\displaystyle=$
$\displaystyle((U\setminus u)\cap d_{B}^{\ell})\times I\quad{\mbox{and}}$
$\displaystyle(U\setminus u)\cap(d_{B}^{m}\times I)$ $\displaystyle=$
$\displaystyle((U\setminus u)\cap d_{B}^{m})\times I,$

the subspace $U\setminus u$ of $U$ is considered to meet $B=|\xi|$ as finitely
many fibers of the line bundle $\xi$ disjoint from $T^{o}$. In particular,
$U\setminus u$ and $\mbox{int}E_{R}^{m}$ meet $B=|\xi|$ in different fibers of
the line bundle $\xi$ disjoint from $T^{o}$.

For interior points $y_{i}\,(i=1,2,\dots,s)$ in $d_{B}^{\ell}\setminus f$, let
$\gamma_{i}\,(i=1,2,\dots,s)$ a system of mutually disjoint simple arcs in
$d_{B}^{*}$ such that $\gamma_{i}$ joins $x_{i}$ with $y_{i}$ and avoids
passing through the finite sets $f\setminus\\{x_{i}\\}$, $f^{\prime}$,
$f^{\prime\prime}$, $f^{\prime\prime\prime}$ and meets $T^{o}$ in just one
point.

Let $|\xi(\gamma_{i})|$ be the total space of the induced line bundle
$\xi(\gamma_{i})$ over $\gamma_{i}$ which meets $T^{o}$ as a proper arc as in
Fig. 5.

x

Figure 5: An intersecting arc of $T^{o}$ in the total space
$|\xi(\gamma_{i})|$

Let $c(B\times[-1,1])$ be a bi-collar of $B$ in ${\mathbf{R}}^{4}$ with
$c(B\times 0)=B$. By a collaring technique, the infinite cylinder $U$ is
assumed so that

$c(B\times[-1,1])\cap(U\setminus u)=c((B\cap(U\setminus u)\times[-1,1]).$

Let $\xi^{\prime}(\gamma_{i})$ be a line bundle over the arc $\gamma_{i}$ with
fibers deformed from the fibers of $\xi(\gamma_{i})$ so that the total space
$|\xi^{\prime}(\gamma_{i})|$ is obtained from the total space
$|\xi(\gamma_{i})|$ by truncating two triangles as in Fig. 6.

Figure 6: A deformed line bundle $\xi^{\prime}(\gamma_{i})$ of the line bundle
$\xi(\gamma_{i})$

Deform a disk neighborhood of $x_{i}$ in $U$ into a proper disk

$D(x_{i})=c(|\xi^{\prime}(x_{i})|\times[-1,1])$

of the 4-ball $c(B\times[-1,1])$. For this purpose, note that the fiber line
$|\xi(x_{i})|$ is the component of the intersection $U\cap B$ containing the
point $x_{i}$ because $U\setminus u$ meets $B$ in fibers of the line bundle
$\xi$. Deform $U$ so that $|\xi(x_{i})|$ moves into $|\xi^{\prime}(x_{i})|$
through the total space $|\xi(\gamma_{i})|$. Then deform the resulting $U$ so
that $D(x_{i})=c(|\xi^{\prime}(x_{i})|\times[-1,1])$ is a part of $U$. The
disks $D(x_{i})\,(i=1,2,\dots,s)$ are mutually disjoint. Further, note that
$D(x_{i})\cap d_{B}^{m}=\\{x_{i}\\}$, $D(x_{i})\cap d_{R}^{m}=\emptyset$, and
$D(x_{i})$ meets $T^{o}$ with two points. Let $U^{\prime}$ be the infinite
cylinder obtained from $U$ in this way, which meets $F$ with the $2s$ points
of $D(x_{i})\,(i=1,2,\dots,s)$.

Let $D(y_{i})=c(|\xi^{\prime}(y_{i})|\times[-1,1])$ be a proper disk of the
4-ball $c(B\times[-1,1])$. The disks $D(y_{i})\,(i=1,2,\dots,s)$ are mutually
disjoint. Let

${\mathcal{A}}(\gamma_{i})=c(|\xi^{\prime}(\gamma_{i})|\times(-1)\cup
c(|\dot{\xi}^{\prime}(\gamma_{i})|\times[-1,1])\cup
c(|\xi^{\prime}(\gamma_{i})|\times 1)$

be a cylinder in the spherical shell $c((\partial B)\times[-1,1])$. The
cylinders ${\mathcal{A}}(\gamma_{i})\,(i=1,2,\dots,s)$ are mutually disjoint.
Note that $D(y_{i})$ and $A(\gamma_{i})$ are disjoint from $F$, $d_{R}^{m}$
and $d_{B}^{m}$.

Let $U^{\prime\prime}$ be the infinite cylinder obtained from $U^{\prime}$ by
replacing $D(x_{i})$ with the disk $D(y_{i})\cup{\mathcal{A}}(\gamma_{i})$ for
all $i$. Let $d_{u}$ be a disk in ${\mathbf{R}}^{3}$ bounded by $u$. The
2-sphere

$S=d_{u}[-t]\cup(U^{\prime\prime}\cap{\mathbf{R}}^{3}[-t,t])\cup d_{u}[t]$

for a large number $t>0$ is a desired one. $\square$

x

We have a position to prove Theorem 3.1.

x

Figure 7: The replacement operation on the sphere $S$ replacing the square in
the figure with the complementary region in $S$

Proof of Theorem 3.1. By taking an associated bump of $(E\times
I,E^{\prime}\times I)$ in ${\mathbf{R}}^{3}[0]$, let $V_{\mathbf{R}}$ be a
standard solid torus in ${\mathbf{R}}^{3}[0]$ with $\partial
V_{\mathbf{R}}=T^{o}\cup\delta_{\mathbf{R}}$ for the connected summand $T^{o}$
of $F\\#T$ and a disk $\delta_{\mathbf{R}}$ in ${\mathbf{R}}^{3}[0]$. Further,
let the disk pair $(E,E^{\prime})$ be a standard longitude-meridian disk pair
$(d_{\mathbf{R}}^{\ell},d_{\mathbf{R}}^{m})$ of $V_{\mathbf{R}}$, whose
longitude-meridian pair $(\partial d_{\mathbf{R}}^{\ell},\partial
d_{\mathbf{R}}^{m})$ with $\partial d_{\mathbf{R}}^{\ell}\cap\partial
d_{\mathbf{R}}^{m}=p$ for a point $p\in T^{o}$ coincides with the pair
$(\partial D,\partial D^{\prime})$.

Assume that a boundary collar $n_{\mathbf{R}}^{m}$ of a meridian disk
$d_{\mathbf{R}}^{m}$ completely overlaps with a boundary collar of
$D^{\prime}$.

We show that the O2-handle pair $(D\times I,D^{\prime}\times I)$ is moved into
the O2-handle pair $(E\times I,E^{\prime}\times I)$ by a finite number of
cellular moves on the 2-handles $D\times I$ and $D^{\prime}\times I$ keeping
$F^{c}$ fixed.

By Lemmas 2.1 and 3.2, there is a 2-sphere $S$ in ${\mathbf{R}}^{4}$ such that
$S\cap(F\\#T)=S\cap(D^{\prime}\setminus n_{\mathbf{R}}^{m})=\emptyset$ and $S$
meets $n_{\mathbf{R}}^{m}\times I$ transversely in just one collared arc
$q\times I$ of a point $q$ in $n_{R}^{m}$. We apply the replacement operation
on $S$ in Fig. 7. Let

$L=D\cap(d_{\mathbf{R}}^{m}\times I)=(D\cap d_{\mathbf{R}}^{m})\times I.$

Apply the replacement operation on a parallel of $S$ by moving $L$ near
$q\times I$. Then an O2-handle pair $(\tilde{D}\times
I,d_{\mathbf{R}}^{m}\times I)$ with $\tilde{D}$ an immersed disk disjoint from
$(\mbox{int}d_{\mathbf{R}}^{m})\times I$ is obtained from the pair $(D\times
I,d_{\mathbf{R}}^{m}\times I)$ without changing the attaching part on $T^{o}$.
Since $(\tilde{D}\times I,D^{\prime}\times I)$ is also an O2-handle pair with
$\tilde{D}$ an immersed disk, we see from Lemma 2.4 that there is an embedded
2-handle $D^{*}\times I$ on $F\\#T$ such that the pairs $(D^{*}\times
I,d_{\mathbf{R}}^{m}\times I)$ and $(D^{*}\times I,D^{\prime}\times I)$ are
O2-handle pairs on $F\\#T$.

By Lemma 2.2, we have

$\displaystyle(F\\#T)(D\times I,D^{\prime}\times I)$ $\displaystyle\cong$
$\displaystyle(F\\#T)(D^{\prime}\times I)$ $\displaystyle\cong$
$\displaystyle(F\\#T)(D^{*}\times I,D^{\prime}\times I)$ $\displaystyle\cong$
$\displaystyle(F\\#T)(D^{*}\times I)$ $\displaystyle\cong$
$\displaystyle(F\\#T)(D^{*}\times I,d_{\mathbf{R}}^{m}\times I)$
$\displaystyle\cong$ $\displaystyle(F\\#T)(d_{\mathbf{R}}^{m}\times I)$
$\displaystyle\cong$ $\displaystyle(F\\#T)(d_{\mathbf{R}}^{\ell}\times
I,d_{\mathbf{R}}^{m}\times I)$ $\displaystyle=$ $\displaystyle(F\\#T)(E\times
I,E^{\prime}\times I).$

Each equivalence is attained by a finite number of cellular moves keeping
$F^{c}$ fixed, as it is observed in Lemma 2.2. $\square$

x

Acknowledgements. This work was in part supported by Osaka City University
Advanced Mathematical Institute (MEXT Joint Usage/Research Center on
Mathematics and Theoretical Physics). An idea in this paper together with an
idea in [15] was presented at the meeting “Differential Topology 19” held at
Ritsumeikan Tokyo Campus on March 12, 2019, organized by Tetsuya Abe and Motoo
Tange. The author would like to thank them for giving him a talk chance and
the other participants for making some discussions. The first trial to this
research project was done during his stay at Pusan National University in the
season of cherry blossoms in full bloom of 2018 spring where the author would
like to thank Sang Youl Lee and Jieon Kim for kind hospitalities during his
stay. Since then several improvements of this paper were done and a nearly
last improvement was done during his stay at Novosibirsk State University for
“VI Russian-Chinese Conference on Knot Theory and Related Topics”held on June
17-21, 2019 where he would like to thank Nikolay Abrosimov and Andrei Vesnin
for kind hospitalities during his stay. The author also would like to thank
Seiichi Kamada for taking up a topic on a stabilization of a ribbon surface-
knot in his lecture [7], which gave him a motivation to consider a stable
surface-knot and Maggie Miller for giving sharp observations on earlier
versions of this paper, and Takao Matumoto for telling him where it is
difficult to read.

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