subfig

# Reinforcement Learning for Optimized Beam Training in Multi-Hop Terahertz
Communications

Arian Ahmadi and Omid Semiari This research was supported by the U.S. National
Science Foundation under Grants CNS-1941348 and CNS-2008646. Department of
Electrical and Computer Engineering, University of Colorado Colorado Springs,
Colorado Springs, CO
Email: {aahmadi<EMAIL_ADDRESS>
###### Abstract

Communication at terahertz (THz) frequency bands is a promising solution for
achieving extremely high data rates in next-generation wireless networks.
While the THz communication is conventionally envisioned for short-range
wireless applications due to the high atmospheric absorption at THz
frequencies, multi-hop directional transmissions can be enabled to extend the
communication range. However, to realize multi-hop THz communications,
conventional beam training schemes, such as exhaustive search or hierarchical
methods with a fixed number of training levels, can lead to a very large time
overhead. To address this challenge, in this paper, a novel _hierarchical beam
training scheme with dynamic training levels_ is proposed to optimize the
performance of multi-hop THz links. In fact, an optimization problem is
formulated to maximize the overall spectral efficiency of the multi-hop THz
link by dynamically and jointly selecting the number of beam training levels
across all the constituent single-hop links. To solve this problem in presence
of unknown channel state information, noise, and path loss, a new
reinforcement learning solution based on the multi-armed bandit (MAB) is
developed. Simulation results show the fast convergence of the proposed scheme
in presence of random channels and noise. The results also show that the
proposed scheme can yield up to 75% performance gain, in terms of spectral
efficiency, compared to the conventional hierarchical beam training with a
fixed number of training levels.

##  I Introduction

Future wireless networks are expected to support a new breed of wireless
technologies that not only require very high data rates, but also mandate very
low communications latency [1]. Among these emerging services include, but not
limited to, factory automation, autonomous vehicular platoon systems, swarm of
unmanned aerial vehicles (UAVs), and user interactions via wireless extended
reality applications [2]. Despite their unique service requirements, these
applications are similar in a number of key aspects: 1) they mainly rely on
direct device-to-device (D2D) or machine-to-machine (M2M) communications among
a group of user equipment (UE), 2) the communication network is formed over
_multi-hop_ D2D or M2M links, and 3) substantial traffic (e.g., sensing data)
must be managed within very short (sub-millisecond) time intervals.

These unique characteristics motivate leveraging the large available bandwidth
at very high-frequency bands, particularly over the terahertz (THz)
frequencies (collectively considered as $0.1-10$ THz)[3, 4]. In fact, compared
with the frequency bands considered in the fifth-generation (5G) new radio
specifications111In particular, frequency range 1 (sub-6 GHz) and frequency
range 2 (sub-100 GHz) millimeter wave (mmWave) bands., THz spectrum can offer
an order of magnitude larger bandwidth, suitable for managing large sensing
information required in autonomous systems. Additionally, deployment of
advanced phased arrays (composed of many antenna elements) with very small
form-factors is feasible at THz frequencies. This allows UEs to leverage large
array processing gains and form highly directional multi-hop links to cope
with the large atmospheric absorption at THz frequency bands, achieve
extremely high data rates, and extend the communication range to form larger
D2D or M2M networks (e.g., UAV swarm or vehicular platoons).

However, one of the key challenges for establishing directional links at high-
frequency bands is the lack of full or even partial knowledge of the channel
state information (CSI) at the transceivers during the initial access [5].
Therefore, prior to the actual data transmissions, UEs have to follow a
process, known as _beam training_ , during which the transceivers direct the
antenna array gain toward different directions to find the optimal spatial
path that maximizes the received power. As UEs move and the propagation
environment changes, the beam training process must be repeated to find the
best spatial path and maintain high data rates across the network. Moreover,
in multi-hop communications, the overall link performance (e.g., data rate)
depends on the performance of all the constituent single-hope links [6].
Hence, for multi-hop THz communications, the beam training process must be
completed jointly for multiple links at every transmission block, leading to
substantial time overhead.

Thus far, substantial work has been done to optimize the beam training
process, particularly for communications below 100 GHz [7, 8, 9, 10, 11, 12,
13, 14, 15]. The authors in [14] propose an exhaustive search beam training,
which sequentially tests all possible beam pairs in the angle domain between a
base station (BS) and a UE and chooses the precoding/combining codeword that
yields maximum received signal power. The main drawback of this approach is
that scanning the angular space via sequential search is very time consuming,
particularly in THz communications that require very high resolution angular
search to achieve the so-called pencil beams. To reduce the beam training time
overhead, the hierarchical beam training is developed in [15, 10, 13, 12, 9,
11, 7, 8]. This technique allows BSs/UEs to scan the angular space with wider
beams, and then, narrow down the search space and the beamwidth over multiple
training stages. While this approach has been widely adopted to decrease the
training time, its performance is highly dependent on the codebook design. In
[7], the authors present a fast discovery hierarchical search strategy to
decrease the delay of exhaustive search. The authors in [8] propose an analog
beamforming strategy using a hierarchical scheme for a single-hop transmission
scenario. In [9], an efficient hierarchical codebook is designed by jointly
exploiting sub-array and deactivation antenna processing techniques. The work
in [15] presents a hierarchical multi-resolution codebook based on hybrid
beamforming precoding in a single-UE mmWave system. In [10], the authors
introduce a Discrete Fourier Transform (DFT) based multi-level codebook design
that yields beam patterns with near-uniform gains at each training level. In
[16], the authors propose an online learning algorithm to solve the problem of
beam training in mmWave vehicular systems based on a contextual multi-armed
bandit (MAB) method. In [11], an online stochastic optimization problem is
solved as a unimodal MAB problem to improve beam training in mmWave networks.
In fact, the authors utilize the correlation and unimodality properties to
decrease the search space and maximize the received energy. The authors in
[12] consider a beam-training scheme based on Bayesian MAB to maximize the
throughput of the mmWave systems in a single-UE scenario. While the
hierarchical beam training schemes developed in [15, 10, 13, 12, 9, 11, 7, 8],
reduce the time overhead compared to the sequential search, they only focus on
single-UE, single-hop communications mainly at mmWave frequency bands. As we
will show in this paper, the hierarchical beam training schemes can lead to
large beam training overhead and performance degradation when applied directly
to multi-hop THz links. In fact, the time overhead of existing hierarchical
beam training schemes can scale linearly with the number of UEs, which makes
them inefficient for multi-hop THz communications.

The main contribution of this paper is a novel hierarchical beam training
scheme to reduce the time overhead of the beam training process and enhance
the performance for multi-hop THz communication links. To this end, we
formulate an optimization problem that aims to maximize the overall spectral
efficiency of the multi-hop THz link by finding the optimal number of training
levels in hierarchical beam search, jointly for all the constituent single-hop
links. In particular, instead of performing the hierarchical search for a pre-
defined number of training levels, the proposed scheme dynamically determines
the number of training levels for each single-hop link while considering the
performance of other links. Hence, the proposed scheme can effectively reduce
the beam training overhead and increase the available time for data
communication during each transmission block. To solve this problem, we
propose a new algorithm that builds on an MAB strategy to efficiently learn
the optimal values for the number of search levels during the hierarchical
beam training, without requiring prior knowledge about the CSI. The simulation
results show that compared with the conventional hierarchical beam training
with a fixed number of training levels, the proposed algorithm yields up to
75% performance gain in terms of spectral efficiency.

The rest of the paper is organized as follows. Section II presents the system
model. Section III describes the problem formulation based on the proposed
hierarchical beam training with dynamic training levels. The proposed
algorithm is presented in Section IV. Simulation results are provided in
Section V and conclusions are presented in Section VI.

Figure 1: An example of multi-hop THz communication composed of line-of-sight
(LOS) and non-LOS (NLOS) D2D/M2M links.

##  II System Model

Consider a network of $K$ UEs in a set $\mathcal{K}$ that communicate with one
another over a multi-hop THz link. Let $u_{1}\leftrightarrow
u_{2}\leftrightarrow\cdots\leftrightarrow u_{K}$ denote the multi-hop THz link
where $u_{k}\leftrightarrow u_{k+1}$ represents a single-hop bi-directional
THz link between UE $u_{k}$ to UE $u_{k+1}$. As an example, Fig. 1 shows a
two-hop THz communication $u_{1}\leftrightarrow u_{2}\leftrightarrow u_{3}$.
We note that more complex structures for the multi-hop network (e.g., mesh or
star networks) can be built based on the considered structure, i.e., a
connected polytree graph with each node having a maximum degree of two. To
form the directional links, each UE $u_{k}$ is equipped with a uniform linear
array composed of $N$ antenna elements.

###  II-A Channel model

Given an azimuth steering direction
$\psi\in\left[-\frac{\pi}{2},\frac{\pi}{2}\right]$, the transceiver’s response
vector of a UE is given by

$\displaystyle\boldsymbol{a}(\theta)=\left[1,e^{-j2\pi\frac{d}{\lambda}\theta},\cdots,e^{-j2\pi\frac{d}{\lambda}(N-1)\theta}\right]^{T},$
(1)

where $d$ is the antenna spacing, $\lambda$ is the wavelength, and
$\theta=\sin(\psi)$. For the channel model at high-frequency bands, it is
widely accepted to consider an LOS link or an NLOS link with a single spatial
path associated with a cluster of scatterers (as shown in Fig. 1)[10, 12]. In
fact, for an arbitrary single-hop THz link $u_{i}\leftrightarrow u_{j}$, the
multiple-input multiple-output (MIMO) channel matrix can be written as

$\displaystyle\mathbf{H}_{ij}=\beta\boldsymbol{a}(\theta_{i})\boldsymbol{a}^{H}(\theta_{j}),$
(2)

where $\theta_{i}$ and $\theta_{j}$ are, respectively, the angle-of-departure
(AoD) and the angle-of-arrival (AoA) associated with the spatial path for the
link between UEs $u_{i}$ and $u_{j}$. In addition, $\beta$ is the small-scale
fading channel gain which is modeled as a zero-mean, complex Gaussian random
variable with variance $\sigma_{\beta}^{2}$, i.e.,
$\beta\sim\mathcal{CN}(0,\sigma_{\beta}^{2})$.

###  II-B Beam training and data transmissions

We consider the transmission of data frames, each composed of $L$ time slots.
Denoting $T_{c}$ as the channel coherence time and $\tau$ as the duration of
each time slot, $L$ is selected such that $L\tau\ll T_{c}$. At the beginning
of each frame, $L^{\prime}$ time slots are allocated for the beam training
between the transmitter and the receiver of a THz link. Hence, $L-L^{\prime}$
time slots will be assigned for the transmission of data symbols. For an
arbitrary link from UE $u_{i}$ to UE $u_{j}$, the received signal over the
MIMO channel at a given time slot can be represented as

$\displaystyle
r_{j}=\sqrt{p_{j}}\boldsymbol{v}^{H}\mathbf{H}_{ij}\boldsymbol{w}x+\boldsymbol{v}^{H}\boldsymbol{n},$
(3)

where $x$ is the transmitted symbol with $\mathbbm{E}\\{|x|^{2}\\}=1$ and
$p_{j}$ is the omni-directional received power (i.e., transmit power after
impacted by the path loss) at the receiver $j$. The additive white Gaussian
noise (AWGN) vector $\boldsymbol{n}\in\mathbbm{C}^{N}$ has a zero mean with
$\mathbbm{E}\\{\boldsymbol{n}\boldsymbol{n}^{H}\\}=\sigma_{n}^{2}\mathbf{I}_{N}$.
Moreover, $\boldsymbol{w}\in\mathbbm{C}^{N}$ and
$\boldsymbol{v}\in\mathbbm{C}^{N}$ represent, respectively, the beamforming
and combining vectors. These vectors are selected from a pre-defined codebook
and satisfy $\lVert\boldsymbol{w}\rVert^{2}=\lVert\boldsymbol{v}\rVert^{2}=1$.
With this model, the received signal-to-noise ratio (SNR) is

$\displaystyle\gamma_{i,j}=\frac{p_{j}}{\sigma_{n}^{2}}|\boldsymbol{v}^{H}\mathbf{H}_{ij}\boldsymbol{w}|^{2}.$
(4)

Accordingly, the spectral efficiency of a single-hop link between UEs $u_{i}$
and $u_{j}$ is

$\displaystyle
R_{i,j}=\left[1-\mathbbm{P}(\gamma_{i,j}<\gamma_{\text{th}})\right]\left(1-\frac{L^{\prime}_{i,j}}{L}\right)\log_{2}(1+\gamma_{i,j}),$
(5)

where $\gamma_{\text{th}}$ denotes the minimum required SNR and
$\mathbbm{P}(\gamma_{i,j}<\gamma_{\text{th}})$ represents the outage
probability. Next, we describe why the time overhead of beam training in
multi-hop THz communications can severely impact the performance and we
justify the need for new solutions to optimize the beam training for multi-hop
THz links.

##  III Multi-Resolution Beam Training With Dynamic Training Levels

Here, we first focus on analyzing the beam training time overhead $L^{\prime}$
in multi-hop THz networks. As described in Sec. I, the sequential search will
result in a significant time overhead to establish bi-directional THz links.
Therefore, in this section, we first briefly overview the hierarchical (also
known as multi-resolution) search as a widely adopted beam training scheme and
analyze its time overhead for multi-hop THz communications. Then, we propose
an optimization problem to maximize the performance of multi-hop THz links by
effectively reducing the beam training time overhead.

###  III-A Time overhead of hierarchical beam training for multi-hop THz
communications

The hierarchical beam training is a technique which allows a UE to start the
search using codewords with wide beamwidths, and then, fine-tune the search
(i.e., narrow down the angular search space and the beamwidth) at each
subsequent level throughout the search process. While different multi-
resolution codebooks have been proposed in the literature, here, we build our
framework based on the phase-shifted DFT codebook introduced in [10] due to:
1) efficient DFT-based implementation, and 2) near-uniform antenna gain over
the beamwidths. As shown in Fig. 2, the hierarchical beam training process
requires $M$ training levels to complete the beam training at a transmitter
with
$\mathcal{W}^{(m)}=\\{\boldsymbol{w}_{1}^{(m)},\boldsymbol{w}_{2}^{(m)},\cdots,\boldsymbol{w}_{q_{m}}^{(m)}\\}$
representing the set of $q_{m}$ beamforming codewords at the $m$-th level.
Using the phase-shifted DFT codebook design, we can construct each codeword as

$\displaystyle\boldsymbol{w}_{i}^{(m)}=\frac{\sqrt{q_{m}}}{N}\sum_{k}a_{k}(\theta_{k})e^{j\omega_{m}k},$
(6)

where $(i-1)\frac{N}{q_{m}}+1\leq k\leq i\frac{N}{q_{m}}$ and
$\theta_{k}=-1+\frac{2k-1}{N}$. In addition, $\omega_{m}$ represents the phase
shift added to reduce the antenna gain fluctuations of the codeword over its
beamwidth of $\frac{2\pi}{q_{m}}$. To perform the beam training for a single
link, beam search can start at the transmitter while the receiver operates
with an omni-directional array gain and provides the index of best codeword to
the transmitter over a feedback channel. Then, the transmitter sends training
signals using the selected beam training codeword and the beam training can be
repeated at the receiver. To implement this process, we use an $s_{k}$-way
decision tree which allows a UE $k$ to only send $s_{k}$ training signals at
each level of the hierarchical search and complete the beam training at
$M_{k}=\log_{s_{k}}N$ levels. Considering that only one training signal is
sent per time slot, the time overhead of beam training for a single-hop link
$u_{k}\leftrightarrow u_{k+1}$ will be

$\displaystyle L^{\prime}_{k,k+1}$ $\displaystyle=s_{k}M_{k}+s_{k+1}M_{k+1}$
$\displaystyle=s_{k}\log_{s_{k}}N+s_{k+1}\log_{s_{k+1}}N.$ (7)

To establish a multi-hop communications with $K$ THz links, we consider a
time-division multiple access (TDMA) hierarchical beam training, i.e., the
beam training is performed sequentially across the single-hop links. With this
in mind, the time overhead increases with the order $O(Ks\log_{s}N)$, if
$s_{i}=s,\forall i$. For example, for the two-hop link in Fig. 1 with $N=64$,
$s=4$, the time overhead of the beam training will be $L^{\prime}=48$. To
better evaluate this time overhead within the total frame duration, let 5 Km/h
be the maximum UE velocity. Then, at 240 GHz carrier frequency and with 120
KHz subcarrier spacing, the total number of time slots within the channel
coherence time will be $L=108$. Thus, the beam training overhead for a two-hop
THz link will be close to $30\%$ which is significantly large. Here, we note
that simultaneous beam training schemes (e.g., in [17]) that are designed for
cellular downlink transmissions cannot be applied to distributed D2D/M2M
networks considered in this work. That is because such schemes require a
central node (e.g., a base station) to separate and group UEs in the spatial
domain and manage the interference resulting from concurrent beam trainings.
Next, we propose a new beam training approach to reduce this substantial delay
overhead in multi-hop THz communications.

Figure 2: Example of hierarchical beam training process at the transmitter
with $M=3$, $q_{m}\in[8,16,64]$, and $\omega_{m}=2.24$. At each stage, the
selected codeword is shown in red. The dashed line represents a
$45^{\circ}$AoD.

###  III-B Problem Formulation

The performance of the multi-hop link $u_{1}\leftrightarrow
u_{2}\leftrightarrow\cdots\leftrightarrow u_{K}$ depends on the spectral
efficiency of the involved single-hop links $u_{k}\leftrightarrow u_{k+1}$,
for $1\leq k<K$. More generally, with a decode-and-forward scheme, the overall
spectral efficiency of the multi-hop THz link between two arbitrary UEs
$u_{i},u_{j}\in\mathcal{K}$, with $1\leq i<K$ and $i<j\leq K$, is
$R_{i,j}=$

$\displaystyle\left[\prod_{i\leq
k<j}\mathbbm{P}(\gamma_{k,k+1}\\!\geq\\!\gamma_{\text{th}})\right]\\!\\!\left(1\\!-\\!\frac{L^{\prime}_{i,j}}{L}\right)\\!\\!\left(\frac{1}{j-i}\right)\log_{2}(1\\!+\\!\gamma_{\text{min}}),$
(8)

where $L^{\prime}_{i,j}=\sum_{k}L^{\prime}_{k,k+1}$, $j-i$ is the number of
hops, and $\gamma_{\text{min}}=\min\\{\gamma_{k,k+1}\\}$ for $i\leq k<j$. In
fact, (8) implies that the link with the smallest SNR, $\gamma_{\text{min}}$,
will limit the overall spectral efficiency of the multi-hop communication,
irrespective of the beam training and the resulting SNR value at other links
[6]. Hence, selecting larger $M$ values at other links (with
$\gamma_{k,k+1}>\gamma_{\text{min}}$) will only increase the beam training
overhead without increasing the overall spectral efficiency $R_{i,j}$ of the
multi-hop THz link. Thus, _we can increase the performance of a multi-hop THz
link by optimizing the number of search levels $M$ during the hierarchical
beam training_. In fact, given $\gamma_{\text{th}}$, we must find the optimal
value for the number of levels $1\leq m^{*}\leq M$ so as to maximize the
spectral efficiency of the multi-hop link.

For a single-hop link $u_{k}\leftrightarrow u_{k+1}$, let
$\gamma_{k,k+1}^{(m_{k})}$ denote the SNR when beam training is performed up
to the $m_{k}$-th level at the transceivers. This SNR can be calculated by
substituting the selected beamforming and combining vectors at the $m_{k}$-th
level in (4). With this in mind, we aim to find the optimal vector
$\boldsymbol{m}=[m_{i},m_{i+1},\cdots,m_{j-1}]$ for the multi-hop THz link,
such that

$\displaystyle\operatorname*{argmax}_{\boldsymbol{m}}\,\,\,$ $\displaystyle
R_{i,j}$ (9a) s.t.,
$\displaystyle\gamma_{\text{min}}=\min\left\\{\gamma^{(m_{k})}_{k,k+1}\right\\},\,\,\,i\leq
k<j,$ (9d) $\displaystyle
L^{\prime}_{i,j}<\Bigl{\lfloor}\frac{L}{K-1}\Bigr{\rfloor}(j-i),$
$\displaystyle m_{k}\in\\{1,2,\cdots,M_{k}\\},\,\,\,i\leq k<j.$

The first constraint in (9d) shows that $\gamma_{\text{min}}$ is calculated
based on the SNR at the selected search levels. Assuming that the total number
of time slots is uniformly allocated to each single-hop link, and let
$\lfloor.\rfloor$ denote the floor operation, then, the constraint in (9d)
guarantees that the time overhead of beam training is less than the allocated
time slots to the THz link with $j-i$ hops. In addition, the feasibility
constraint in (9d) ensures that $m_{k}$ at the transceivers of the $k-i+1$-th
single-hop link does not exceed the number of training levels in the
conventional multi-resolution beam training. Next, we develop a new approach
to solve the proposed problem for hierarchical beam training with _dynamic
training levels_.

##  IV Proposed Hierarchical Beam Training Algorithm for Multi-Hop THz
Communications

Clearly, the objective function in (9a) is not a monotonic function of
$\boldsymbol{m}$, since $\log_{2}(1+\gamma_{\text{min}})$ can be an increasing
function of $m_{k}$ parameters, while the pre-log factor,
$1-L^{\prime}_{i,j}/L$, is a decreasing function $m_{k}$ variables in
$\boldsymbol{m}$. To solve the proposed problem in (9a)-(9d), we note that it
is very challenging to derive the outage probability as a function of
$\boldsymbol{m}$. Moreover, the CSI of the MIMO channel for each link and
$\gamma_{\text{min}}$ are not known prior to the beam training phase. Hence,
it is not feasible to solve the proposed problem in (9a)-(9d) via standard
optimization techniques. To this end, we develop a new beam training approach,
based on reinforcement learning, to find the optimal $\boldsymbol{m}$ for a
multi-hop THz link while considering the joint performance of its constituent
single-hop links.

TABLE I: Proposed Hierarchical Beam Training Algorithm with Dynamic Training
Levels

Inputs: $\mathcal{K}$, $\gamma_{\text{th}}$, $\omega_{m}$, $\varepsilon_{0}$,
$s_{k}$, $t=0$. while  _(t $\leq$ T)_ do
---

Step 1: Optimize the number of training levels using the epsilon-decay
strategy: a. With probability $1-\varepsilon_{t}$, select the arm with the
current maximum average reward. Otherwise, select an arm
$\boldsymbol{l}^{t}\in\mathcal{L}$ randomly. Then, increase $t$ to $t+1$. b.
Update the value of $\varepsilon_{t}$ using $\varepsilon_{t}$ =
$\varepsilon_{t-1}(1000/(1000+t))$. Step 2: a. Using the selected arm, follow
the hierarchical beam training described in Section III. b. Calculate the
reward from (8) and update the average reward for the selected arm.

end while

Output: The arm with maximum average reward.

In fact, we can formulate the optimization problem in (9a)-(9d) as an MAB
problem, in which the transceivers of the multi-hop link (acting as the
agents) explore different choices for a vector $\boldsymbol{l}$ (analogous to
an arm in an MAB problem) with the $k$-th element being
$l_{k}=M_{k}-m_{k}=\log_{s_{k}}N-m_{k}$ for $i\leq k<j$. Here, the integer
variable $l_{k}$ ($0\leq l_{k}<M_{k}$) represents the number of reduced search
levels for beam training at the transmitter and receiver of the link $k$.
After playing an arm $\boldsymbol{l}$ from a set of all possible arms
$\mathcal{L}$, the UEs of the multi-hop link will receive a random reward
$P_{i,j}(\boldsymbol{l})$ which is equal to the spectral efficiency in (8). To
determine the size of the set $\mathcal{L}$, we note that each $l_{k}$ can
take $M_{k}$ integer values from $0$ to $M_{k}-1$. Therefore, for the multi-
hop link $u_{1}\leftrightarrow u_{2}\leftrightarrow\cdots\leftrightarrow
u_{K}$, the total number of arms will be equal to $\Pi_{k=1}^{K-1}M_{k}$.

The key advantage of the MAB-based solution is that it enables the
transceivers across the multi-hop THz link to jointly find the optimal
solution for (9a)-(9d), in presence of stochastic noise and channel
variations. The transceivers of the multi-hop link can try only one arm
$\boldsymbol{l}\in\mathcal{L}$ at each trial (i.e., block transmission of $L$
time slots). Within this MAB framework, we define the regret $\zeta(T)$ after
$T$ trial as

$\displaystyle\zeta(T)=\sum_{t=1}^{T}P_{i,j}(\boldsymbol{l}^{*})-P_{i,j}(\boldsymbol{l}^{t}),$
(10)

where $\boldsymbol{l}^{*}$ is the optimal arm with elements
$l_{k}^{*}=M_{k}-m_{k}^{*}$ where $m_{k},i\leq k<j$ is the solution of the
proposed problem in (9a)-(9d) and $\boldsymbol{l}^{t}$ is the played arm at
round $t$ with an associated reward $P_{i,j}(\boldsymbol{l}^{t})$. The
objective is to find a strategy that selects $\boldsymbol{l}^{t}$, for $1\leq
t\leq T$, such that $\lim_{T\rightarrow\infty}\zeta(T)=0$. Clearly, such
strategy will converge to the solution of the proposed problem in (9a)-(9d).

The proposed algorithm is summarized in Table I which builds on the epsilon-
decay strategy to solve the MAB problem. The reason for employing the epsilon-
decay strategy is that it can properly maintain the tradeoff between
exploration versus exploitation during the learning process, particularly if
the size of the set $\mathcal{L}$ is not too large. With this in mind, for a
given set of input parameters, the proposed algorithm follows a two-step
process during each trial (i.e., a transmission block of $L$ time slots). In
Step 1 of an arbitrary trial $t$, the transceivers of the multi-hop link
select an arm $\boldsymbol{l}^{t}\in\mathcal{L}$ based on the epsilon-decay
strategy. That is, the algorithm chooses a random arm with probability
$\varepsilon_{t}$ or selects the arm with the highest current average reward
with probability $1-\varepsilon_{t}$. Once an arm is selected, the
transceivers follow the hierarchical beam search according to the selected
arm. That is, the transceivers of the $k$-th single-hop link will follow the
beam training process explained in Sec. III for up to $m_{k}=M_{k}-l_{k}$
training levels. After receiving the instantaneous reward in (8) for the
$t$-th trial, the average reward for the selected arm will be updated and the
process is repeated until up to $T$ arms are played. The output of the
algorithm will be the arm $\boldsymbol{l}^{*}$ with the maximum average
reward.

##  V Simulation Results

TABLE II:  Simulation Parameters Notation |  Parameter |  Value
---|---|---
$f_{c}$ |  Carrier frequency |  $240$ GHz
$N$ |  Number of antennas |  $64$
$s_{k}$ |  Number of training signals |  $4$
$\omega_{m}$ |  Phase shift parameter of the DFT-based codebook |  $2.24$ rad/s
$v$ |  UE maximum speed |  $5$ km/h
$\sigma_{\beta}$ |  Channel gain standard deviation |  $1$
– |  Subcarrier spacing |  $120$ kHz [18]
$\gamma_{\text{th}}$ |  Minimum required SNR |  $-50$ dB
$N_{0}$ |  Noise power spectral density |  $-204$ dBm/Hz[19]
– |  Path loss exponent |  $2.02$ dB [18]
B |  Total system bandwidth |  4 GHz [19]

In this section, we present the simulation results and show the performance of
the proposed algorithm in terms of its convergence, the statistics of the
achievable spectral efficiency, and the probability of miss detection at the
receivers of the multi-hop THz link. For simulations, we consider three UEs
communicating with one another over a two-hop THz link, as shown in Fig. 1.
The distances between UEs 1 and 2 and UEs 2 and 3 are, respectively, 30 m and
5 m. The received SNR in (4) is calculated at the output of the matched filter
where the length of the filter is calculated based on sampling the training
signal at the Nyquist rate. Both links have the same transmit SNR, ranging
from 20 dB to 60 dB. Simulation parameters are summarized in Table II. We
compare the performance of the proposed beam training with dynamic training
levels (labeled as “Hierarchical, Dynamic”) with two other baseline schemes:
1) The conventional hierarchical beam training described in Sec. III-A
(labeled as “Hierarchical, Fixed”), and 2) The hierarchical beam training with
a random number of training levels (labeled as “Hierarchical, Random”). The
performance was evaluated by averaging the results over sufficiently large
Monte Carlo runs.

Figure. 3 shows the average regret resulting from the proposed MAB-based
algorithm versus the number of trials for different values of the transmit
SNR. Here, the average regret is computed by averaging the regret in (12) over
large independent runs. From Fig. 3, we observe that the regret decreases
rapidly, showing the fast convergence of the proposed learning approach in
presence of channel fading, random AoA/AoD, and the receiver noise. The
results show that the proposed beam training algorithm successfully converges
to the optimal solution within a reasonably small number of trials. As an
example, for 40 dB transmit SNR, the average regret will be less that
$10^{-5}$ after 100 trials.

In Fig. 4, we compare the average spectral efficiency of the two-hop THz link
versus the transmit SNR for the proposed approach and the two baseline
schemes. Clearly, the spectral efficiency increases with higher transmit SNR
values. The results in Fig. 4 also show that the proposed algorithm
outperforms the other two schemes, with up to $17\%$ and $75\%$ performance
gains compared to, respectively, the “Hierarchical, Random” and “Hierarchical,
Fixed” approaches at 60 dB transmit SNR.

Figure 3: Average regret versus the number of trials.

Figure 4: Average spectral efficiency of the two-hop THz link versus the
transmit SNR.

Figure 5: The CDF of spectral efficiency of the two-hop THz link at $50$ dB
transmit SNR.

Figure 6: Probability of miss detection versus transmit SNR.

Figure. 5 presents the cumulative distribution function (CDF) of the spectral
efficiency of the two-hop THz link resulting from the proposed algorithm and
the two baseline beam training schemes at 50 dB transmit SNR. The figure
indicates that although the “Hierarchical, Random” algorithm is more efficient
than the “Hierarchical, Fixed” scheme, its performance is not comparable to
the proposed approach. In fact, the figure shows that the proposed algorithm
achieves a spectral efficiency greater than 2 bps/Hz with about 0.9
probability. However, this probability is only $0.5$ for the ”Hierarchical,
Random” scheme while the ”Hierarchical, Fixed” cannot achieve the 2 bps/Hz
spectral efficiency.

While the proposed scheme can effectively optimize the performance of the
multi-hop THz link (as shown in Figs. 3-5), reducing the number of training
levels can impact the detection of training signals from noise. In fact, as
the proposed scheme reduces the number of training levels, the effective
antenna gain can be reduced and the receivers’ detectors may not be able to
detect the actual training signal from noise particularly at low transmit SNR
values. To study this effect, Fig. 6 compares the probability of miss
detection of the proposed algorithm with the two other baseline schemes. For
the signal detection, we use the Neyman-Pearson detector with a fixed
probability of false alarm ($P_{\text{FA}}$) of 0.01. Based on the results in
Fig. 6, the probability of miss detection decreases as we increase the
transmit SNR. Moreover, Fig. 6 shows that the performance degradation for the
proposed scheme is negligible compared to the “Hierarchical, Fixed” scheme.
Comparing the results of Figs. 4-6, we observe that the proposed scheme can
significantly improve the spectral efficiency of multi-hop THz communications
without any major impact on the detection of training signals.

##  VI Conclusions

In this paper, we have proposed a novel beam training approach, based on
reinforcement learning to optimize the performance of multi-hop THz
communication links. First, we have shown the substantial time overhead of
conventional hierarchical beam training schemes when applied to multi-hop THz
communications. Then, to address this challenge, we have introduced a new
hierarchical beam training scheme with dynamic training levels to effectively
reduce the time overhead of the beam training process and maximize the multi-
hop link’s performance. To find the optimal number of training levels across
the multi-hop link, we have formulated an optimization problem that maximizes
the spectral efficiency, while considering the beam training time constraints.
To solve the problem with no prior CSI knowledge of the links during the beam
training phase, we proposed an MAB-based algorithm that can effectively find
the optimal training levels with reasonably fast convergence. The simulation
results have shown that the proposed approach can yield up to 75% and 17%
performance gains in spectral efficiency, compared to, respectively, the
hierarchical beam training with a fixed and random number of training levels.

## References

  * [1] O. Semiari, W. Saad, M. Bennis, and M. Debbah,  “Integrated millimeter wave and sub-6 GHz wireless networks: A roadmap for joint mobile broadband and ultra-reliable low-latency communications,”  IEEE Wireless Communications, vol. 26, no. 2, pp. 109–115, 2019\.
  * [2] T. Rappaport, Y. Xing, O. Kanhere, S. Ju, A. Madanayake, S. Mandal, A. Alkhateeb, and G. Trichopoulos,  “Wireless communications and applications above 100 GHz: Opportunities and challenges for 6G and beyond,”  IEEE Access, vol. 7, pp. 78729–78757, 2019.
  * [3] N. Rajatheva, I. Atzeni, et al.,  “White paper on broadband connectivity in 6G,”  arXiv preprint arXiv:2004.14247, 2020.
  * [4] R. Barazideh, O. Semiari, S. Niknam, and B. Natarajan,  “Reinforcement learning for mitigating intermittent interference in terahertz communication networks,”  in 2020 IEEE International Conference on Communications Workshops (ICC Workshops), 2020, pp. 1–6.
  * [5] C. Barati, S. Hosseini, M. Mezzavilla, T. Korakis, S. Panwar, S. Rangan, and M. Zorzi,  “Initial access in millimeter wave cellular systems,”  IEEE Transactions on Wireless Communications, vol. 15, no. 12, pp. 7926–7940, 2016.
  * [6] M. Sikora, J. N. Laneman, M. Haenggi, D. J. Costello, and T. E. Fuja,  “Bandwidth- and power-efficient routing in linear wireless networks,”  IEEE Transactions on Information Theory, vol. 52, no. 6, pp. 2624–2633, 2006.
  * [7] V. Desai, L. Krzymien, P. Sartori, W. Xiao, A. Soong, and A. Alkhateeb,  “Initial beamforming for mmwave communications,”  in 2014 48th Asilomar Conference on Signals, Systems and Computers. IEEE, 2014, pp. 1926–1930.
  * [8] J. Wang, Z. Lan, et al.,  “Beam codebook based beamforming protocol for multi-Gbps millimeter-wave WPAN systems,”  IEEE Journal on Selected Areas in Communications, vol. 27, no. 8, pp. 1390–1399, 2009.
  * [9] Z. Xiao, T. He, P. Xia, and X. Xia,  “Hierarchical codebook design for beamforming training in millimeter-wave communication,”  IEEE Transactions on Wireless Communications, vol. 15, no. 5, pp. 3380–3392, 2016.
  * [10] S. Noh, M. Zoltowski, and D. Love,  “Multi-resolution codebook and adaptive beamforming sequence design for millimeter wave beam alignment,”  IEEE Transactions on Wireless Communications, vol. 16, no. 9, pp. 5689–5701, 2017.
  * [11] M. Hashemi, A. Sabharwal, C. Koksal, and N. Shroff,  “Efficient beam alignment in millimeter wave systems using contextual bandits,”  in IEEE INFOCOM 2018-IEEE Conference on Computer Communications. IEEE, 2018, pp. 2393–2401.
  * [12] M. Hussain and N. Michelusi,  “Second-best beam-alignment via bayesian multi-armed bandits,”  in 2019 IEEE Global Communications Conference (GLOBECOM). IEEE, 2019, pp. 1–6.
  * [13] S. Hur, T. Kim, D. Love, J. Krogmeier, T. Thomas, and A. Ghosh,  “Millimeter wave beamforming for wireless backhaul and access in small cell networks,”  IEEE transactions on communications, vol. 61, no. 10, pp. 4391–4403, 2013.
  * [14] C. Jeong, J. Park, and H. Yu,  “Random access in millimeter-wave beamforming cellular networks: issues and approaches,”  IEEE Communications Magazine, vol. 53, no. 1, pp. 180–185, 2015\.
  * [15] A. Alkhateeb, O. El Ayach, G. Leus, and R. Heath,  “Channel estimation and hybrid precoding for millimeter wave cellular systems,”  IEEE Journal of Selected Topics in Signal Processing, vol. 8, no. 5, pp. 831–846, 2014.
  * [16] A. Asadi, S. Müller, G. Sim, A. Klein, and M. Hollick,  “Fml: Fast machine learning for 5G mmwave vehicular communications,”  in IEEE INFOCOM 2018-IEEE Conference on Computer Communications. IEEE, 2018, pp. 1961–1969.
  * [17] R. Zhang, H. Zhang, W. Xu, and C. Zhao,  “A codebook based simultaneous beam training for mmwave multi-user mimo systems with split structures,”  in 2018 IEEE Global Communications Conference (GLOBECOM), 2018, pp. 1–6.
  * [18] Y. Xing and T. Rappaport,  “Propagation measurement system and approach at 140 GHz-moving to 6G and above 100 GHz,”  in 2018 IEEE Global Communications Conference (GLOBECOM). IEEE, 2018, pp. 1–6.
  * [19] A. Ekti, A. Boyaci, A. Alparslan, İ. Ünal, S. Yarkan, A. Görçin, H. Arslan, and M. Uysal,  “Statistical modeling of propagation channels for terahertz band,”  in 2017 IEEE Conference on Standards for Communications and Networking (CSCN). IEEE, 2017, pp. 275–280.


# Heegaard Floer Surgery Formula and Cosmetic Surgeries

Alan Du

###### Abstract.

Two Dehn surgeries on a knot are called cosmetic if they yield homeomorphic
three-manifolds. We show for a certain family of null-homologous knots in any
closed orientable three-manifold, if the knot admits cosmetic surgeries with a
pair of positive surgery coefficients, then the coefficients are both greater
than $1$. In addition, for this family of knots, we show that $1/q$ Dehn
surgery for $q$ at least $2$ is not homeomorphic to the original three-
manifold. The proofs of these results use the mapping cone formula for the
Heegaard Floer homology of Dehn surgery in terms of the knot Floer homology of
the knot; we provide a new proof of this formula for integer surgeries in
$\operatorname{Spin}^{c}$ structures with nontorsion first Chern class.

##  1\. Introduction

Heegaard Floer homology is a powerful invariant of closed oriented three-
manifolds introduced by Ozsváth and Szabó [OS04b]. It has been used to prove
many results about three-manifold topology, especially Dehn surgery. In fact,
the Heegaard Floer homology of any surgery on a knot can be computed from the
knot Floer complex of the knot [OS08, OS10, Zem23].

In this paper, we will use Heegaard Floer homology to study cosmetic surgeries
on a certain class of knots in arbitrary closed oriented three-manifolds. We
recall the definition of cosmetic surgeries.

######  Definition 1.1.

If two Dehn surgeries on a knot yield homeomorphic manifolds, then these two
surgeries are cosmetic. If in addition, the two surgeries are orientation-
preserving homeomorphic, then these two surgeries are purely cosmetic.

Cosmetic surgeries are very rare; in fact, the following cosmetic surgery
conjecture has been proposed:

######  Conjecture 1.2 ([Kir95] Problem 1.81).

Suppose $K$ is a knot in a closed oriented three-manifold $Y$ such that $Y-K$
is irreducible and not homeomorphic to the solid torus. If two different Dehn
surgeries on $K$ are purely cosmetic, then there is a homeomorphism of $Y-K$
which takes one slope to the other.

The existence of cosmetic surgeries has been ruled out in many cases for knots
in $S^{3}$. For instance, Ozsváth and Szabó proved that if $K\subset S^{3}$
admits cosmetic rational surgeries, then either the resulting manifold is an
$L$-space or the surgery slopes have opposite signs [OS10]. Much more is known
when considering purely cosmetic surgeries. Hanselman proved that any pair of
purely cosmetic Dehn surgeries on knots in $S^{3}$ must have slopes that are
$\pm 2$ or $\pm 1/q$ for some integer $q$ [Han19].

Less is known for knots in arbitrary three-manifolds. Ni found an obstruction
to cosmetic surgeries on knots in three-manifolds satisfying a certain
condition on their Thurston norms [Ni10].

The main theorem in this paper is an analogue of Ozsváth and Szabó’s result.
We consider a class of knots which were shown by Ni to have special properties
on their knot Floer homology.

######  Definition 1.3.

Let $Y$ be a closed, oriented three-manifold, and $K\subset Y$ a null-
homologous knot. We say that $K$ is blow-up unknotted if there exists a taut
Seifert surface $F$ for $K$, some natural number $N$, and a properly-embedded
smooth disk $G$ inside $X=(Y\times[0,1])\\#N\overline{{\mathbb{C}}P^{2}}$ with
$\partial G=K\times\\{0\\}$ and $[G]=\iota_{*}[F]\in H_{2}(X,K\times\\{0\\})$
for $\iota:(Y,K)\to(X,K\times\\{0\\})$ the inclusion.

######  Remark.

The property of $K$ being blow-up unknotted implies that $K$ is null
homotopic.

For example, by Kirby calculus, $K$ is a blow-up unknotted knot if $K$ is
obtained from the unknot in $Y$ by a sequence of crossing changes that change
a positive crossing to a negative one.

######  Theorem 1.4.

Let $K\subset Y$ be a blow-up unknotted knot, and suppose $0<r<s$ are distinct
positive rational numbers with the property that $Y_{r}(K)\cong\pm Y_{s}(K)$.
Then $r,s>1$ or $K$ is the unknot.

We also provide progress on the knot complement problem for blow-up unknotted
knots.

######  Theorem 1.5.

Let $K\subset Y$ be a nontrivial blow-up unknotted knot and $q\geq 2$, then
$Y_{1/q}(K)\ncong\pm Y$. Furthermore, if $g(K)\geq 2$, then $Y_{1}(K)\ncong\pm
Y$.

The proofs of these theorems rely on a mapping cone formula that relates the
Heegaard Floer homology of Dehn surgery on null-homologous knots in arbitrary
closed, oriented three-manifolds with the knot Floer chain complex. This
result was proven for Heegaard Floer homology in $\operatorname{Spin}^{c}$
structures whose first Chern classes are torsion by Ozsváth and Szabó, and the
formula in nontorsion $\operatorname{Spin}^{c}$ structures for zero surgeries
was proven by Ni [OS08, OS10, Ni20]. A recent result by Zemke provides a very
general surgery formula via an equivalence of certain Lagrangians with local
systems in the Fukaya category [Zem23]. Perhaps of independent interest, we
provide a conceptually-simpler proof of the integer surgery formula for
nontorsion $\operatorname{Spin}^{c}$ structures, which generalizes Ozsváth and
Szabó’s and Ni’s results.

Acknowledgements The author would like to thank Yi Ni for advising him in this
project.

##  2\. Preliminaries on Heegaard Floer Homology

In this section, we establish some notation and collect some results that will
be needed in later sections.

Let $W:Y_{1}\to Y_{2}$ be an oriented cobordism, with $Y_{1},Y_{2}$ connected.
Given $\mathfrak{s}\in\operatorname{Spin}^{c}(W)$, there is a homomorphism

$F_{W,\mathfrak{s}}^{\circ}:HF^{\circ}(Y_{1},\mathfrak{s}|Y_{1})\to
HF^{\circ}(Y_{2},\mathfrak{s}|Y_{2}),$

induced by a chain map

$f_{W,\mathfrak{s}}^{\circ}:CF^{\circ}(Y_{1},\mathfrak{s}|Y_{1})\to
CF^{\circ}(Y_{2},\mathfrak{s}|Y_{2}).$

Among other properties, these maps satisfy a conjugation invariance. For
$\mathfrak{t}$ a $\operatorname{Spin}^{c}$ structure (in either three or four
dimensions), let $\overline{\mathfrak{t}}$ denote its conjugate. There is an
isomorphism $\mathfrak{I}_{Y}:HF^{\circ}(Y,\mathfrak{t})\to
HF^{\circ}(Y,\overline{\mathfrak{t}})$.

######  Theorem 2.1 ([OS06] Theorem 3.6).

Let $W$ be a cobordism from $Y_{1}$ to $Y_{2}$ as above. Then

$F_{W,\mathfrak{s}}^{\circ}=\mathfrak{I}_{Y_{2}}\circ
F_{W,\overline{\mathfrak{s}}}^{\circ}\circ\mathfrak{I}_{Y_{1}}.$

Given a chain map $f:A\to B$, let $MC(f)$ be the mapping cone of $f$, namely,
$MC(f)=A\oplus B$ with the differential

$\partial=\begin{pmatrix}\partial_{A}&0\\\ f&\partial_{B}\end{pmatrix}.$

######  Theorem 2.2 ([OS08] Theorem 3.1).

Let $Y$ be a closed, oriented three-manifold, $K\subset Y$ a null-homologous
knot. Fix an integer $n$ and a positive integer $m$. Then, there is a long
exact sequence

$\cdots\to HF^{+}(Y_{n}(K))\to
HF^{+}(Y_{m+n}(K))\to\bigoplus^{m}HF^{+}(Y)\to\cdots$

and also a corresponding exact sequence using $\widehat{HF}$ in place of
$HF^{+}$. There are ${\mathbb{Z}}[U]$-equivariant chain maps

$f_{1}^{+}:CF^{+}(Y_{n}(K))\to CF^{+}(Y_{m+n}(K))$
$f_{2}^{+}:CF^{+}(Y_{m+n}(K))\to\bigoplus^{m}CF^{+}(Y)$
$f_{3}^{+}:\bigoplus^{m}CF^{+}(Y)\to CF^{+}(Y_{n}(K))$

inducing the maps in the long exact sequence, and
${\mathbb{Z}}[U]$-equivariant quasi isomorphisms

$\phi^{+}:CF^{+}(Y_{n}(K))\to MC(f_{2}^{+})$ $\psi^{+}:MC(f_{2}^{+})\to
CF^{+}(Y_{n}(K)).$

Knot Floer homology associates to a knot $K$ in a closed, oriented three-
manifold $Y$ a ${\mathbb{Z}}\oplus{\mathbb{Z}}$-filtered
${\mathbb{Z}}[U]$-complex $C=CFK^{\infty}(Y,K)$. Write the two factors of the
${\mathbb{Z}}\oplus{\mathbb{Z}}$ filtration as $(i,j)$. Let $S$ be a region in
the plane such that for all $(i,j)\in S$ and all
$(i^{\prime},j^{\prime})\geq(i,j)$, we have that $(i^{\prime},j^{\prime})\in
S$. Define $A_{s}^{+}=C\\{\max(i,j-s)\geq 0\\}$ and $B^{+}=C\\{i\geq 0\\}$ as
in Ozsváth-Szabó. There are two canonical chain maps $v_{s}^{+}:A_{s}^{+}\to
B^{+}$ and $h_{s}^{+}:A_{s}^{+}\to B^{+}$, where $v_{s}^{+}$ is the vertical
projection, and $h_{s}^{+}$ first projects $A_{s}^{+}$ to $C\\{j\geq s\\}$,
then maps to $C\\{j\geq 0\\}$ via $U^{s}$, and finally maps to $B^{+}$ via a
chain homotopy equivalence $C\\{j\geq 0\\}\to C\\{i\geq 0\\}$ [OS08].

Let $K\subset Y$ be null-homologous. Fix a $\operatorname{Spin}^{c}$ structure
$\mathfrak{t}$ over $Y$, and let $n\in{\mathbb{Z}}$. The space of
$\operatorname{Spin}^{c}$ structures over $Y_{n}(K)$ which are
$\operatorname{Spin}^{c}$-cobordant to $\mathfrak{t}$ over the cobordism
$W_{n}^{\prime}(K)$ is identified with ${\mathbb{Z}}/n{\mathbb{Z}}$ by
$\mathfrak{u}\mapsto i$, where $\mathfrak{s}$ is an extension of
$\mathfrak{u}$ over $W_{n}^{\prime}(K)$ with the property that

$\langle c_{1}(\mathfrak{s}),[\hat{F}]\rangle-n\equiv 2i\mod 2n.$

For $i\in{\mathbb{Z}}/n{\mathbb{Z}}$, let $CF^{+}(Y_{n}(K),i,\mathfrak{t})$
denote the corresponding summand of
$CF^{+}(Y_{n}(K))$. Let $A_{s,\mathfrak{t}}^{+},B_{\mathfrak{t}}^{+}$ denote
the corresponding chain complexes associated to the knot filtration on
$CF^{+}(Y,\mathfrak{t})$ [OS08].

Let

${\mathbb{A}}_{i,\mathfrak{t}}^{+}=\prod_{s\in{\mathbb{Z}}|s\equiv i\mod
n}A_{s,\mathfrak{t}}^{+}\hskip 14.22636pt\text{and}\hskip
14.22636pt{\mathbb{B}}_{\mathfrak{t}}^{+}=\prod_{s\in{\mathbb{Z}}|s\equiv
i\mod n}B_{\mathfrak{t}}^{+}$

and let
$D_{n,i,\mathfrak{t}}^{+}:{\mathbb{A}}_{i,\mathfrak{t}}^{+}\to{\mathbb{B}}_{i,\mathfrak{t}}^{+}$
be the map

$D_{n,i,\mathfrak{t}}^{+}(\\{a_{s}\\})=\\{b_{s}\\},$

where here

$b_{s}=h_{s-n}^{+}(a_{s-n})+v_{s}^{+}(a_{s}).$

Let ${\mathbb{X}}_{i,\mathfrak{t}}^{+}(n)$ denote the mapping cone of
$D_{n,i,\mathfrak{t}}^{+}$ [OS08].

Taking the direct sum of the above chain complexes over all
$\mathfrak{t}\in\operatorname{Spin}^{c}(Y)$ and
$i\in{\mathbb{Z}}/n{\mathbb{Z}}$ gives ${\mathbb{A}}^{+}$, ${\mathbb{B}}^{+}$,
and the map $D_{n}^{+}:{\mathbb{A}}^{+}\to{\mathbb{B}}^{+}$. Let
${\mathbb{X}}^{+}(n)$ denote the mapping cone of $D_{n}^{+}$.

Let $K\subset Y$ be a null-homologous knot and let $F$ be a Seifert surface
for $K$ in the homology class $\phi$. Given $n>0$ and
$t\in{\mathbb{Z}}/n{\mathbb{Z}}$, let $W_{n}^{\prime}(K):Y_{n}(K)\to Y$ be the
natural two-handle cobordism, and let $\widehat{F}_{n}\subset
W_{n}^{\prime}(K)$ be the closed surface obtained by capping of $\partial F$
with a disk. Let $\widehat{\phi}_{n}=[\widehat{F}_{n}]\in
H_{2}(W_{n}^{\prime}(K))$. Let $k$ be an integer satisfying $k\equiv t\mod n$
and $|k|\leq n/2$. Let
$\mathfrak{x}_{k},\mathfrak{y}_{k}\in\operatorname{Spin}^{c}(W_{n}^{\prime}(K))$
satisfy $\mathfrak{t}_{k}|Y=\mathfrak{n}_{k}|Y=\mathfrak{s}$ and

$\langle c_{1}(\mathfrak{x}_{k}),\hat{\phi}_{n}\rangle=2k-n\hskip
14.22636pt\langle c_{1}(\mathfrak{y}_{k}),\hat{\phi}_{n}\rangle=2k+n,$

and let
$\mathfrak{s}_{t}=\mathfrak{t}_{k}|Y_{n}(K)=\mathfrak{n}_{k}|Y_{n}(K)$.

######  Theorem 2.3 ([OS08] Theorem 2.3).

When $n\geq 2g(F)$, the chain complex
$CF^{+}(Y_{n}(K),\mathfrak{s}_{t})$ is represented by the chain complex
$A_{k}^{+}$, in the sense that there is an isomorphism

$\Psi_{n}^{+}:CF^{+}(Y_{n}(K),\mathfrak{s}_{t})\to A_{k}^{+}.$

Moreover, the following squares commute:

${{CF^{+}(Y_{n}(K),\mathfrak{s}_{t})}}$${{CF^{+}(Y,\mathfrak{s})}}$${A_{k}^{+}}$${B^{+}}$$\scriptstyle{\Psi_{n}^{+}}$$\scriptstyle{f_{W_{n}^{\prime}(K),\mathfrak{x}_{k}}^{+}}$$\scriptstyle{=}$$\scriptstyle{v_{k}^{+}}$

${{CF^{+}(Y_{n}(K),\mathfrak{s}_{t})}}$${{CF^{+}(Y,\mathfrak{s})}}$${A_{k}^{+}}$${B^{+}.}$$\scriptstyle{\Psi_{n}^{+}}$$\scriptstyle{f_{W_{n}^{\prime}(K),\mathfrak{y}_{k}}^{+}}$$\scriptstyle{=}$$\scriptstyle{h_{k}^{+}}$

In the case of rational surgeries, let $p,q$ be a pair of relatively prime
integers. For $\mathfrak{t}\in\operatorname{Spin}^{c}(Y)$ and
$i\in{\mathbb{Z}}/p{\mathbb{Z}}$, denote

${\mathbb{A}}_{p/q,i,\mathfrak{t}}^{+}=\prod_{r\in{\mathbb{Z}}}(r,A_{\lfloor(i+pr)/q\rfloor,\mathfrak{t}}^{+}(K))\hskip
14.22636pt\text{and}\hskip
14.22636pt{\mathbb{B}}_{p/q,i,\mathfrak{t}}^{+}=\prod_{r\in{\mathbb{Z}}}(r,B_{\mathfrak{t}}^{+}).$

Let
$D_{p/q,i,\mathfrak{t}}^{+}:{\mathbb{A}}_{p/q,i,\mathfrak{t}}^{+}\to{\mathbb{B}}_{p/q,i,\mathfrak{t}}^{+}$
be the map

$D_{p/q,i,\mathfrak{t}}^{+}\\{(r,a_{r})\\}=\\{(r,b_{r})\\},$

where

$b_{r}=v_{\lfloor(i+pr)/q\rfloor}^{+}(a_{r})+h_{\lfloor(i+p(r-1))/q\rfloor}^{+}(a_{r-1}).$

Let ${\mathbb{X}}_{p/q,i,\mathfrak{t}}^{+}$ denote the mapping cone of
$D_{p/q,i,\mathfrak{t}}^{+}$ [OS10].

As before, direct summing over all $\mathfrak{t}\in\operatorname{Spin}^{c}(Y)$
and $i\in{\mathbb{Z}}/p{\mathbb{Z}}$ gives ${\mathbb{A}}_{p/q}^{+}$,
${\mathbb{B}}_{p/q}^{+}$, and
$D_{p/q}^{+}:{\mathbb{A}}_{p/q}^{+}\to{\mathbb{B}}_{p/q}^{+}$. Let
${\mathbb{X}}_{p/q}^{+}$ denote the mapping cone of $D_{p/q}^{+}$. In the case
where $p/q$ is clear from the context, we will drop the subscript $p/q$ from
the notation.

##  3\. Integer Surgeries

In this section, we provide a proof of the mapping cone formula for integer
surgeries in the case where the $\operatorname{Spin}^{c}$ structure has
nontorsion first Chern class. The formula for torsion
$\operatorname{Spin}^{c}$ structures was originally proven by Ozsváth and
Szabó [OS10]. Our proof combines ingredients from their argument as well as
Ni’s proof of the zero surgery formula.

Let $K\subset Y$ be a null-homologous knot. Fix a $\operatorname{Spin}^{c}$
structure $\mathfrak{t}$ over $Y$ with nontorsion first Chern class.

Fix $n>0$. The long exact sequence in Theorem 2.2 decomposes over
$\operatorname{Spin}^{c}$ structures on $Y$. Setting $m=nk$ for large $k$, it
also decomposes over $\operatorname{Spin}^{c}$ structures on $Y_{n}(K)$, i.e.
given $i\in{\mathbb{Z}}/n{\mathbb{Z}}$, $CF^{+}(Y_{n}(K),i,\mathfrak{t})$ is
quasi-isomorphic to $MC(f_{2,i,\mathfrak{t}}^{+})$ where

(3.1) $\displaystyle
f_{2,i,\mathfrak{t}}^{+}=\sum_{\mathfrak{s}_{s}\in\mathcal{X}_{i}}f_{W_{n(k+1)}^{\prime}(K),\mathfrak{s}_{s}}^{+}:CF^{+}(Y_{n(k+1)}(K),[i],\mathfrak{t})\to
CF^{+}(Y,\mathfrak{t}),$
$\displaystyle\mathcal{X}_{i}=\\{\mathfrak{s}_{s}\in\operatorname{Spin}^{c}(W^{\prime}_{n(k+1)}(K))\
$ $\displaystyle|\ \mathfrak{s}_{s}|_{Y}=\mathfrak{t},$ $\displaystyle\langle
c_{1}(\mathfrak{s}_{s}),[\widehat{F}]\rangle-n(k+1)\equiv 2s\mod 2n(k+1),$
$\displaystyle\text{and }s\equiv i\mod n\\},$
$CF^{+}(Y_{n(k+1)}(K),[i],\mathfrak{t})=\bigoplus_{s\in{\mathbb{Z}}/n(k+1){\mathbb{Z}}\
|\ s\equiv i\mod n}CF^{+}(Y_{n(k+1)}(K),s,\mathfrak{t}).$

A strengthening of the proof of Proposition 3.5 in [Ni20] shows the following:

######  Proposition 3.2 ([Ni20] Proposition 3.5).

Let $p=\max\\{k+g(F),-2k+g(F)\\}$. Suppose that $n\geq\max\\{p,2g(F)\\}$,
$\mathfrak{s}\in\mathcal{X}_{k}\setminus\\{\mathfrak{x}_{k},\mathfrak{y}_{k}\\},$
then the map $F_{W_{n}^{\prime}(K),\mathfrak{s}}^{-}$ factorizes through the
map

$U^{n-p}:\operatorname{HF}^{-}(Y_{n}(K),k,\mathfrak{t})\circlearrowright.$

Moreover, if $c_{1}(\mathfrak{t})$ is nontorsion, then
$F_{W_{n}^{\prime}(K),\mathfrak{s}}^{+}$ factorizes through

$U^{n-p}:\operatorname{HF}^{+}(Y_{n}(K),k,\mathfrak{t})\circlearrowright.$

###### Proof.

The proof is exactly the same as in [Ni20], except we note that

$\frac{l(l+1)n}{2}-lk-g\geq n-p$

whenever $l\neq 0,-1$. From here, it follows that
$F_{W_{n}^{\prime}(K),\mathfrak{s}}^{-}$ factorizes through $U^{n-p}$. ∎

######  Theorem 3.3.

For each $i\in{\mathbb{Z}}/n{\mathbb{Z}}$, the mapping cone
${\mathbb{X}}_{i,\mathfrak{t}}^{+}(n)$ of

$D_{n,i,\mathfrak{t}}^{+}:{\mathbb{A}}_{i,\mathfrak{t}}^{+}\to{\mathbb{B}}_{i,\mathfrak{t}}^{+}$

is identified with $CF^{+}(Y_{n}(K),i,\mathfrak{t})$ as
${\mathbb{Z}}[U]$-modules.

###### Proof when $n>0$.

Recall that $CF^{+}(Y_{n}(K),i,\mathfrak{t})$ is quasi-isomorphic to
$MC(f_{2,i,\mathfrak{t}}^{+})$ as in Equation 3.1. Since $c_{1}(\mathfrak{t})$
is nontorsion, $H_{*}(A_{i}^{+})$ is a finitely generated abelian group. So
$U^{p}|H_{*}(A_{i}^{+})=0$ when $p$ is greater than a constant $C_{1}$
independent of $n,k$. By Theorem 2.3, this implies that
$U^{p}|HF^{+}(Y_{n(k+1)}(K),i,\mathfrak{t})=0$ when $p>C_{1}$, $n(k+1)>2g$.

Recall that we choose representatives

$-\frac{n(k+1)}{2}\leq s<\frac{n(k+1)}{2}$

for classes in ${\mathbb{Z}}/n(k+1){\mathbb{Z}}$. Suppose $s\geq 0$. Then for
$l=\max\\{s+g,-2s+g\\}$, we have

$l\leq\frac{n(k+1)}{2}+g\leq n(k+1)$

for large $k$. Proposition 3.2 implies that
$F_{W_{n(k+1)}^{\prime}(K),\mathfrak{s}_{s}}^{+}=0$ when
$\mathfrak{s}_{s}\in\mathcal{X}_{i}\setminus\\{\mathfrak{x}_{s},\mathfrak{y}_{s}\\}$.

Now for $s<0$, let
$\mathfrak{s}_{s}\in\mathcal{X}_{i}\setminus\\{\mathfrak{x}_{s},\mathfrak{y}_{s}\\}$,
then by conjugation invariance, there is a chain equivalence

$F_{W_{n(k+1)}^{\prime}(K),\mathfrak{s}_{s}}^{+}\cong
F_{W_{n(k+1)}^{\prime}(K),\overline{\mathfrak{s}_{s}}}^{+}.$

If $\mathfrak{s}_{s}=\mathfrak{x}_{s}+r\operatorname{PD}[\widehat{F}]$, then

$\overline{\mathfrak{s}_{s}}=\overline{\mathfrak{x}}_{-s}-(r+1)\operatorname{PD}[\widehat{F}],$

in particular, $\overline{\mathfrak{s}_{s}}$ is not one of the two special
$\operatorname{Spin}^{c}$ structures and $-s>0$. By the above argument,
$F_{W_{n(k+1)}^{\prime}(K),\overline{\mathfrak{s}_{s}}}^{+}=0$, hence also
$F_{W_{n(k+1)}^{\prime}(K),\mathfrak{s}_{s}}^{+}=0$.

The conclusion follows by Theorem 2.3. ∎

###  3.1. Proof of Theorem for negative surgery coefficients

Continue to assume $n>0$, only we consider $Y_{-n}(K)$. In this case, let
$W^{\prime}_{n(k-1)}(K)$ be the cobordism from $Y_{n(k-1)}(K)$ to $Y$. For
each $\mathfrak{t}\in\operatorname{Spin}^{c}(Y)$,
$i\in{\mathbb{Z}}/n{\mathbb{Z}}$,

$HF^{+}(Y_{-n}(K),i,\mathfrak{t})\cong
H_{*}\left(MC\left(f_{2,i,\mathfrak{t}}^{+}:CF^{+}(Y_{n(k-1)}(K),[i],\mathfrak{t})\to\bigoplus^{nk}CF^{+}(Y,\mathfrak{t})\right)\right),$

where

$f_{2,i,\mathfrak{t}}^{+}=\sum_{\mathfrak{s}_{s}\in\mathcal{X}_{i}}f_{W_{n(k-1)}^{\prime}(K),\mathfrak{s}_{s}}^{+}:CF^{+}(Y_{n(k-1)}(K),[i],\mathfrak{t})\to
CF^{+}(Y,\mathfrak{t}),$
$\displaystyle\mathcal{X}_{i}=\\{\mathfrak{s}_{s}\in\operatorname{Spin}^{c}(W^{\prime}_{n(k-1)}(K))\
$ $\displaystyle|\ \mathfrak{s}_{s}|_{Y}=\mathfrak{t},$ $\displaystyle\langle
c_{1}(\mathfrak{s}_{s}),[\widehat{F}]\rangle-n(k-1)\equiv 2s\mod 2n(k-1),$
$\displaystyle\text{and }s\equiv i\mod n\\},$
$CF^{+}(Y_{n(k-1)}(K),[i],\mathfrak{t})=\bigoplus_{s\in{\mathbb{Z}}/n(k-1){\mathbb{Z}}\
|\ s\equiv i\mod n}CF^{+}(Y_{n(k-1)}(K),s,\mathfrak{t}).$

Consider the summand $CF^{+}(Y_{n(k-1)}(K),s,\mathfrak{t})$ for

$-\frac{n(k-1)}{2}\leq s<\frac{n(k-1)}{2},\hskip 14.22636pts\equiv i\mod n.$

Then, as above, if
$\mathfrak{s}_{s}\in\mathcal{X}_{i}\setminus\\{\mathfrak{x}_{s},\mathfrak{y}_{s}\\}$,
then $F_{W^{\prime}_{n(k-1)}(K),\mathfrak{s}_{s}}^{+}=0$. The conclusion again
follows by Theorem 2.3.

##  4\. Rational Surgeries

In this section, we sketch the proof of the rational surgery formula:

######  Theorem 4.1.

Let $K\subset Y$ be a null-homologous knot, and let $p,q$ be a pair of
relatively prime integers. Then, for each
$\mathfrak{t}\in\operatorname{Spin}^{c}(Y)$, $i\in{\mathbb{Z}}/p{\mathbb{Z}}$,
there is an isomorphism of ${\mathbb{Z}}[U]$-modules

$H_{*}(\mathbb{X}_{i,\mathfrak{t},p/q}^{+})\cong
HF^{+}(Y_{p/q}(K),i,\mathfrak{t}).$

Ozsváth and Szabó stated and proved this theorem in the case where
$\mathfrak{t}$ has torsion first Chern class [OS10]. In this case, they showed
this isomorphism also preserves ${\mathbb{Z}}$-grading. Their argument still
holds when the $\operatorname{Spin}^{c}$ structure is nontorsion, as long as
grading information is ignored.

By Kirby calculus, $Y_{p/q}(K)$ can be realized by surgery with coefficient
$a$ inside the knot $K\\#O_{q/r}\subset Y\\#(-L(q,r))$, where $a$ is the
greatest integer smaller than or equal to $p/q$, $a=\lfloor p/q\rfloor$, and
$p/q=a+r/q$, and $O_{q/r}\subset-L(q,r)$ is the knot which is obtained by
viewing one component of the the Hopf link as a knot inside the lens space
$-L(q,r)$ thought of as $-q/r$ surgery on the other component of the Hopf link
[OS10]. Thus, one obtains the rational surgery formula by proving a formula
for Morse surgeries on rationally null-homologous knots and then applying a
Kunneth principle for connected sums.

Analogously to the case of integer surgeries on null-homologous knots, for
$K\subset Y$ a rationally null-homologous knot, $\lambda$ a framing on $K$,
and $\mathfrak{s}\in\operatorname{Spin}^{c}(Y_{\lambda}(K))$, Ozsváth and
Szabó define the chain map
$D_{\mathfrak{s}}^{+}:{\mathbb{A}}_{\mathfrak{s}}^{+}(Y,\overline{K})\to{\mathbb{B}}_{\mathfrak{s}}^{+}(Y,\overline{K})$.
Let ${\mathbb{X}}_{\mathfrak{s}}^{+}(Y,\overline{K},\lambda)$ be the mapping
cone of $D_{\mathfrak{s}}^{+}$ [OS10].

######  Theorem 4.2 (Morse surgery formula [OS10]).

Let $K\subset Y$ be a rationally null-homologous knot, and let $\lambda$ be a
framing on $K$. Let $\mathfrak{s}\in\operatorname{Spin}^{c}(Y_{\lambda}(K))$,
then

$HF^{+}(Y_{\lambda}(K),\mathfrak{s})\cong
H_{*}({\mathbb{X}}_{\mathfrak{s}}^{+}(Y,\overline{K},\lambda)$

as ${\mathbb{Z}}[U]$-modules.

We recall the definition of $U$-knots from [OS10].

######  Definition 4.3.

A $U$-knot is a knot in a rational homology three-sphere $Y$ with the property
that the induced filtration of $CF^{\infty}(Y,\mathfrak{s})$ is trivial. More
precisely, given any $\xi\in\underline{\operatorname{Spin}^{c}}(Y,K)$,
$CFK^{\infty}(Y,K,\xi)$ is chain homotopy equivalent via a relatively
${\mathbb{Z}}\oplus{\mathbb{Z}}$-filtered homotopy to the chain complex $R$
which is a free, rank one ${\mathbb{Z}}[U,U^{-1}]$ module with the trivial
differential.

######  Proposition 4.4.

If $K_{2}\subset Y_{2}$ is a $U$-knot, then for each
$\xi_{1}\in\underline{\operatorname{Spin}}^{c}(Y_{1},\overline{K}_{1})$ and
$\mathfrak{s}_{2}\in\operatorname{Spin}^{c}(Y_{2})$, there is some
$\xi_{2}\in\underline{\operatorname{Spin}^{c}}(Y_{2})$ representing
$\mathfrak{s}_{2}$, with the property that

$CFK^{\infty}(Y_{1},\overline{K_{1}},\xi_{1})\cong
CFK^{\infty}(Y_{1}\\#Y_{2},\overline{K_{1}}\\#\overline{K_{2}},\xi_{1}\\#\xi_{2})$

as ${\mathbb{Z}}\oplus{\mathbb{Z}}$-filtered chain complexes [OS10].

######  Lemma 4.5.

$O_{q/r}\subset-L(q,r)$ is a $U$-knot. There is an affine identification
$\phi$ of the $\operatorname{Spin}^{c}$ structures on
$\widehat{HFK}(L(q,r),O_{q,r})$ with ${\mathbb{Z}}$ such that

$\widehat{HFK}(L(q,r),O_{q,r},\phi(i))\cong\begin{cases}{\mathbb{Z}}&\text{if
$0\leq i\leq q-1$,}\\\ 0&\text{otherwise.}\end{cases}$

[OS10]

Let $K\subset Y$ be a null-homologous knot. Fix a $\operatorname{Spin}^{c}$
structure $\mathfrak{t}$ over $Y$, let
$H^{2}(Y,K,\mathfrak{t})\cong{\mathbb{Z}}$ denote the subgroup of $H^{2}(Y,K)$
corresponding to $\mathfrak{t}$, and also
$H^{2}(Y\\#-L(q,r),K\\#O_{q,r},\mathfrak{t})\cong{\mathbb{Z}}$.

######  Lemma 4.6.

Let $K\subset Y$ be a null-homologous knot. Under the connected sum

$(Y,K)\\#(-L(q,r),O_{q,r})\to(Y\\#-L(q,r),K\\#O_{q,r}),$

the following diagram commutes:

${\mathbb{Z}\oplus\mathbb{Z}}$${\mathbb{Z}}$${{H^{2}(Y,K,\mathfrak{t})\oplus
H^{2}(-L(q,r),O_{q,r})}}$${{H^{2}(Y\\#-L(q,r),K\\#O_{q,r},\mathfrak{t})}}$$\scriptstyle{f}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$

where $f(x,y)=qx+y$. Moreover, under this correspondence, if $K_{\lambda}$ is
the pushoff of $K$ with respect to the integral framing $a$, then
$\operatorname{PD}[K_{\lambda}]$ represents the element $p\in{\mathbb{Z}}\cong
H^{2}(Y\\#-L(q,r),K\\#O_{q,r})$, where $p/q=a+r/q$.

###### Proof.

Follows as in [OS10]. ∎

With these ingredients, the proof of Theorem 4.1 follows as in [OS10].

##  5\. Cosmetic Surgeries

In this section, we prove our main results. For simplicity, we consider
coefficients in the field ${\mathbb{F}}={\mathbb{F}}_{2}$. Throughout, we let
$K\subset Y$ be a null-homologous knot in a closed, oriented three-manifold.
We will obstruct the result of Dehn surgeries on $K$ from being homeomorphic
to each other using the rank of $\widehat{HF}(Y_{r}(K))$ for various positive
rational numbers $r$.

We state some lemmas that will be useful in calculating the homology of the
mapping cones for Dehn surgery.

The following lemma was proven in [Ni20].

######  Lemma 5.1.

If $s\leq t$, then
$\operatorname{Im}(\widehat{v}_{s})_{*}\subset\operatorname{Im}(\widehat{v}_{t})_{*}$
and
$\operatorname{Im}(\widehat{h}_{s})_{*}\supset\operatorname{Im}(\widehat{h}_{t})_{*}$.

######  Lemma 5.2.

There is an equivalence between $\widehat{v}_{s}$ and $\widehat{h}_{-s}$ for
all $s$.

This equivalence is induced by changing the roles of the basepoints $w$ and
$z$ in the doubly-pointed Heegaard diagram.

######  Lemma 5.3.

Let $g(K)$ be the genus of $K$, then for all $s\geq g(K)$, $\widehat{v}_{s}$,
$\widehat{h}_{-s}$ induce isomorphisms on homology. For all $s\geq g(K)+1$,
$\widehat{v}_{-s}$ and $\widehat{h}_{s}$ are nullhomotopic.

######  Lemma 5.4 ([Ni20] Lemma 4.1).

Let $K\subset Y$ be a null-homologous knot. Suppose that $F$ is a taut Seifert
surface for $K$. Let $X=(Y\times[0,1])\\#N\overline{{\mathbb{C}}P^{2}}$ for
some $N$, and let $\iota:(Y,K)\to(X,K\times\\{0\\})$ be the inclusion map. Let
$G\subset X$ be a properly-embedded smooth connected surface with $\partial
G=K\times\\{0\\}$, $[G]=\iota_{*}[F]\in H^{2}(X,K\times\\{0\\})$, and
$g(G)<g(F)$. Then

$\operatorname{Im}(v_{k}^{+})_{*}\supset\operatorname{Im}(h_{k}^{+})_{*},$

when $k\geq g(G)$. Moreover, if $G$ is a disk, then

$(v_{0}^{+})_{*}=(h_{0}^{+})_{*}$

Again, call a knot where such $(X,G)$ for $G$ a disk exists a blow-up
unknotted knot. We also consider the unknot itself to be a blow-up unknotted
knot.

The following result and its proof are analogous to [OS10] Proposition 9.7.

######  Proposition 5.5.

Let $K\subset Y$ be a null-homologous knot. Then

$g(K)=\max(0,\\{s\in{\mathbb{Z}}\ |\ (\widehat{v}_{s-1})_{*}\text{ is not an
isomorphism}\\}).$

In particular, $K$ is the unknot if and only if $(\widehat{v}_{s})_{*}$ is an
isomorphism for all $s\geq 0$.

The proof makes use of the $CF^{\delta}$ variant of Heegaard Floer homology
defined in [OS08]. For any nonnegative integer $\delta$, Denote by
$CF^{\delta}(Y)$ the subcomplex of $CF^{+}(Y)$ which is the kernel of
multiplication by $U^{\delta+1}$. Similarly, let $A_{s}^{\delta}$,
$B^{\delta}$ denote the subcomplexes of $A_{s}^{+}$, $B^{+}$, respectively,
which are the kernel of multiplication by $U^{\delta+1}$.

###### Proof.

Let $g=g(K)$, by Lemma 5.3,

$g(K)\geq\max(0,\\{s\in{\mathbb{Z}}\ |\ (\widehat{v}_{s-1})_{*}\text{ is not
an isomorphism}\\}).$

If $g(K)=0$, i.e. $K$ is the unknot, then $(\widehat{v}_{s})_{*}$ is an
isomorphism for all $s\geq 0$. Otherwise, if $g(K)>0$, it suffices to show
$(\widehat{v}_{g-1})_{*}$ is not an isomorphism. Consider the short exact
sequence

(5.6) $\displaystyle 0\to C\\{i<0,j\geq g-1\\}\to C\\{i\geq 0\text{ or }j\geq
g-1\\}\xrightarrow{v_{g-1}^{+}}C\\{i\geq 0\\}\to 0.$

The middle term is $A_{g-1}^{+}$, and $C\\{i\geq 0\\}=B^{+}$. Also,

$H_{*}(C\\{i<0,j\geq g-1\\})\cong H_{*}(C\\{(-1,g-1)\\}\cong
H_{*}(C\\{0,g\\})=\widehat{HFK}(K,g)\neq 0.$

Therefore, $(v_{g-1}^{+})_{*}$ is not an isomorphism.

Suppose for the sake of contradiction that $(\widehat{v}_{g-1})_{*}$ is an
isomorphism. Suppose also that $(v_{g-1}^{\delta})_{*}$ is an isomorphism for
some $\delta\geq 0$. By [OS04a] Theorem 4.4, the diagram in Figure 2 commutes
and the rows are exact, so by the five lemma, $(v_{g-1}^{\delta+1})_{*}$ is
also an isomorphism. By induction, $(v_{g-1}^{\delta})_{*}$ is an isomorphism
for all $\delta$. Thus, by Lemma 2.7 in [OS08], $(v_{g-1}^{+})_{*}$ is an
isomorphism, which is a contradiction.

${H_{*}(A_{g-1}^{\delta})}$${H_{*}(\widehat{A}_{g-1})}$${H_{*}(A_{g-1}^{\delta+1})}$${H_{*}(A_{g-1}^{\delta})}$${H_{*}(\widehat{A}_{g-1})}$${H_{*}(B^{\delta})}$${H_{*}(\widehat{B})}$${H_{*}(B^{\delta+1})}$${H_{*}(B^{\delta})}$${H_{*}(\widehat{B})}$$\scriptstyle{\partial}$$\scriptstyle{(v_{g-1}^{\delta})_{*}}$$\scriptstyle{i_{*}}$$\scriptstyle{(\widehat{v}_{g-1})_{*}}$$\scriptstyle{U_{*}}$$\scriptstyle{(v_{g-1}^{\delta+1})_{*}}$$\scriptstyle{\partial}$$\scriptstyle{(v_{g-1}^{\delta})_{*}}$$\scriptstyle{(\widehat{v}_{g-1})_{*}}$$\scriptstyle{\partial}$$\scriptstyle{i_{*}}$$\scriptstyle{U_{*}}$$\scriptstyle{\partial}$
Figure 2.

∎

######  Lemma 5.7.

Let $K\subset Y$ be a blow-up unknotted knot, $p,q>0$ relatively prime, and
let $x\in H_{*}(\widehat{A}_{j/q})$ for some $j$. If $\lfloor j/q\rfloor\geq
0$, there exists $\alpha\in H_{*}(\oplus_{i\geq j+p}\widehat{A}_{i/q})$ such
that $(\widehat{D}_{p/q})_{*}(x+\alpha)=(\widehat{v}_{j/q})_{*}(x)$ (where we
think of $H_{*}(\widehat{B}_{j/q})$ as including into
$H_{*}(\widehat{\mathbb{B}})$). Similarly, if $\lfloor j/q\rfloor\leq 0$,
there exists $\alpha\in H_{*}(\oplus_{i\leq j-p}\widehat{A}_{i/q})$ such that
$(\widehat{D}_{p/q})_{*}(x+\alpha)=(\widehat{h}_{j/q})_{*}(x)$.

###### Proof.

For $s\geq 0$, we have
$\operatorname{Im}(\widehat{h}_{s})_{*}\subset\operatorname{Im}(\widehat{v}_{s})_{*}\subset\operatorname{Im}(\widehat{v}_{s+1})_{*}\subset\dots$,
and for $s\leq 0$,
$\operatorname{Im}(\widehat{v}_{s})_{*}\subset\operatorname{Im}(\widehat{h}_{s})_{*}\subset\operatorname{Im}(\widehat{h}_{s-1})_{*}\subset\dots$.
Thus, in the first case, there exists $y_{1}\in H_{*}(\widehat{A}_{(j+p)/q})$
such that $(\widehat{h}_{j/q})_{*}(x)=(\widehat{v}_{(j+p)/q})_{*}(y_{1})$. If
$(\widehat{h}_{(j+p)/q})_{*}(y_{1})=0$, then stop and set $\alpha=y_{1}$,
otherwise keep going. Inductively, given $y_{n}$ in
$H_{*}(\widehat{A}_{(j+np)/q})$, there exists $y_{n+1}\in
H_{*}(\widehat{A}_{(j+(n+1)p)/q})$ such that
$(\widehat{h}_{(j+np)/q})_{*}(y_{n})=(\widehat{v}_{(j+(n+1)p)/q})_{*}$. By
Lemma 5.3, this process eventually terminates and gives an element
$\alpha=\sum_{n\geq 1}y_{n}\in H_{*}(\oplus_{i\geq j+p}\widehat{A}_{i/q})$
with the desired condition.

Similarly, in the case where $\lfloor j/q\rfloor\leq 0$, since
$\operatorname{Im}(\widehat{v}_{j/q})_{*}\subset\operatorname{Im}(\widehat{h}_{(j-p)/q})_{*}$,
we can inductively add cancellation terms to the left. ∎

Using the mapping cone formula and the above lemmas, in the case where $K$ is
blow-up unknotted, we may now calculate the rank of $\widehat{HF}(Y_{r}(K))$
for any positive rational number $r$.

######  Lemma 5.8.

Let $K\subset Y$ be a blow-up unknotted knot, let $p,q>0$ coprime. Let
$b=\operatorname{rank}(\widehat{B})=\operatorname{rank}\widehat{HF}(Y)$. Then

(5.9)
$\displaystyle\begin{split}\operatorname{rank}\widehat{HF}(Y_{p/q}(K))&=q(\operatorname{rank}\ker(\widehat{v}_{0})_{*}+b-\operatorname{rank}(\widehat{v}_{0})_{*})\\\
&+2q\sum_{s=1}^{g-1}(\operatorname{rank}\ker(\widehat{v}_{s})_{*}+b-\operatorname{rank}(\widehat{v}_{s})_{*})+2t_{K}^{p/q}-pb,\end{split}$

where

(5.10) $\displaystyle t_{K}^{p/q}$
$\displaystyle=\sum_{j=0}^{p-1}\operatorname{rank}\left(\operatorname{Im}(\widehat{v}_{j/q})_{*}\cap\operatorname{Im}(\widehat{h}_{(j-p)/q})_{*}\right).$

Note that if $Y=S^{3}$, Equation 5.9 agrees with Equation 40 in [OS10].
Indeed, define $\nu(K)$ as in [OS10]. If $\nu(K)>0$, then

$\operatorname{rank}\left(\operatorname{Im}(\widehat{v}_{j/q})_{*}\cap\operatorname{Im}(\widehat{h}_{(j-p)/q})_{*}\right)=\begin{cases}1&j\geq
q\nu(K)\text{ and }j<p-q(\nu(K)-1)\\\ 0&\text{otherwise.}\end{cases}$

Thus,

$t_{K}^{p/q}=\max(0,p-(2\nu(K)-1)q).$

Also, for all $s\geq 0$,

$\operatorname{rank}\ker(\widehat{v}_{s})_{*}=\begin{cases}\operatorname{rank}H_{*}(\widehat{A}_{s})_{*}&s<\nu(K)\\\
\operatorname{rank}H_{*}(\widehat{A}_{s})_{*}-1&s\geq\nu(K),\end{cases}$

so the two equations agree.

If $\nu(K)=0$, then $t_{K}^{p/q}=p$ and
$\operatorname{rank}\ker(\widehat{v}_{s})_{*}=\operatorname{rank}H_{*}(\widehat{A}_{s})_{*}-1$
for all $s\geq 0$, so again the two equations agree.

###### Proof.

By Theorem 4.1, $\widehat{HF}(Y_{p/q}(K))\cong
H_{*}(\widehat{\mathbb{X}}_{p/q})$, the homology of the mapping cone of
$\widehat{D}_{p/q}$ from
$\widehat{{\mathbb{A}}}=\oplus_{i\in{\mathbb{Z}}/p{\mathbb{Z}}}\widehat{\mathbb{A}}_{i}$
to
$\widehat{{\mathbb{B}}}=\oplus_{i\in{\mathbb{Z}}/p{\mathbb{Z}}}\widehat{\mathbb{B}}_{i}$.
We abuse notation to let $\widehat{A}_{j/q}$ denote $\widehat{A}_{\lfloor
j/q\rfloor}$, and also $\widehat{B}_{j/q}$, $\widehat{v}_{j/q}$,
$\widehat{h}_{j/q}$ similarly. So

$\widehat{\mathbb{A}}=\bigoplus_{i\in{\mathbb{Z}}/p{\mathbb{Z}}}\prod_{r\in{\mathbb{Z}}}\widehat{A}_{(i+pr)/q}=\prod_{j\in{\mathbb{Z}}}\widehat{A}_{j/q}$

has $q$ copies of $\widehat{A}_{s}$ for each $s\in{\mathbb{Z}}$, and under the
map $\widehat{v}$, each $\widehat{A}_{j/q}$ corresponds to a copy of
$\widehat{B}$ in $\widehat{\mathbb{B}}$, which we shall denote
$\widehat{B}_{j/q}$. Then $\widehat{h}_{j/q}$ maps $\widehat{A}_{j/q}$ to
$\widehat{B}_{(j+p)/q}$. Thus, for each $s\in{\mathbb{Z}}$, there are $p$
copies of $\widehat{B}_{s}$ in the codomain of $\widehat{h}_{s-1}$ and $q-p$
copies of $\widehat{B}_{s}$ in the codomain of $\widehat{h}_{s}$.

The finiteness of $H_{*}(\widehat{\mathbb{X}}_{p/q})$ is guaranteed by Lemma
5.3; in fact, it implies that $\widehat{\mathbb{X}}_{p/q}$ is isomorphic to a
truncated version of the mapping cone

$\widehat{\mathbb{X}}_{p/q}^{c}=MC(\widehat{D}_{p/q}^{c}:\widehat{\mathbb{A}}_{p/q}^{c}\to\widehat{\mathbb{B}}_{p/q}^{c}),$

for some large $c$, where

$\widehat{\mathbb{A}}_{p/q}^{c}=\bigoplus_{j=-qc+1}^{qc-1}\widehat{A}_{j/q}=\bigoplus_{s=-c+1}^{c-1}\bigoplus_{i=0}^{q-1}\widehat{A}_{s},\hskip
14.22636pt\widehat{\mathbb{B}}_{p/q}^{c}=\bigoplus_{j=-qc+p+1}^{qc-1}\widehat{B}_{j/q},$

and $\widehat{D}_{p/q}^{c}$ is the restriction of $\widehat{D}_{p/q}$ to
$\widehat{A}_{p/q}^{c}$, with the exception that for any
$x\in\widehat{A}_{j/q}$ with $-qc+1\leq j\leq-qc+p$,
$\widehat{D}_{p/q}^{c}(x)=\widehat{h}_{j/q}(x)$, and for any
$x\in\widehat{A}_{j/q}$ with $qc-p\leq j\leq qc-1$,
$\widehat{D}_{p/q}^{c}(x)=\widehat{v}_{j/q}(x)$. Note that it suffices to take
$c\geq g+p/q+1$, and

$\operatorname{rank}H_{*}(\widehat{\mathbb{A}}_{p/q}^{c})=q\sum_{s=-c+1}^{c-1}\operatorname{rank}H_{*}(\widehat{A}_{s}),\hskip
14.22636pt\operatorname{rank}H_{*}(\widehat{\mathbb{B}}_{p/q}^{c})=qb(2c-1)-pb.$

We construct a basis for $\operatorname{rank}\ker(\widehat{D}_{p/q})_{*}$. For
$j\geq 0$, let

$\\{x_{j/q}^{i}\\}_{i=1}^{\operatorname{rank}\ker(\widehat{v}_{j/q})_{*}}$

be a basis of $\ker(\widehat{v}_{j/q})_{*}$. For each such $x_{j/q}^{i}$,
choose an $\alpha_{j/q}^{i}\in H_{*}(\oplus_{k\geq j+p}\widehat{A}_{k/q})$
according to Lemma 5.7 such that
$\widetilde{x}_{j/q}^{i}=x_{j/q}^{i}+\alpha_{j/q}^{i}$ is in
$\ker(\widehat{D}_{p/q})_{*}$. For $j<0$, let

$\\{x_{j/q}^{i}\\}_{i=1}^{\operatorname{rank}\ker(\widehat{h}_{j/q})_{*}}$

be a basis of $\ker(\widehat{h}_{j/q})_{*}$. For each such $x_{j/q}^{i}$,
choose a $\beta_{j/q}^{i}\in H_{*}(\oplus_{k\leq j-p}\widehat{A}_{k/q})$
according to Lemma 5.7 such that
$\widetilde{x}_{j/q}^{i}=x_{j/q}^{i}+\beta_{j/q}^{i}$ is in
$\ker(\widehat{D}_{p/q})_{*}$. Lastly, for $0\leq j\leq p-1$, let

$\\{y_{j/q}^{i}\\}_{i=1}^{\operatorname{rank}\left(\operatorname{Im}(\widehat{v}_{j/q})_{*}\cap\operatorname{Im}(\widehat{h}_{(j-p)/q})_{*}\right)}$

be a basis of the subspace of $H_{*}(\widehat{A}_{j/q})$ that is mapped
isomorphically to
$\operatorname{Im}(\widehat{v}_{j/q})_{*}\cap\operatorname{Im}(\widehat{h}_{(j-p)/q})_{*}$
under $(\widehat{v}_{j/q})_{*}$. Choose $z_{j/q}^{i}\in
H_{*}(\widehat{A}_{(j-p)/q})$ such that
$(\widehat{v}_{j/q})_{*}(y_{j/q}^{i})=(\widehat{h}_{(j-p)/q})_{*}(z_{j/q}^{i})$.
Choose $\gamma_{j/q}^{i}\in H_{*}(\oplus_{k\geq j+p}\widehat{A}_{k/q})$ and
$\delta_{j/q}^{i}\in H_{*}(\oplus_{k\leq j-p}\widehat{A}_{k/q})$ according to
Lemma 5.7 such that
$\widetilde{y}_{j/q}^{i}=y_{j/q}^{i}+z_{j/q}^{i}+\gamma_{j/q}^{i}+\delta_{j/q}^{i}$
is in $\ker(\widehat{D}_{p/q})_{*}$.

The set

$S=\\{\widetilde{x}_{j/q}^{i}\\}\cup\\{\widetilde{y}_{j/q}^{i}\\}$

is linearly independent, and it is finite by Lemma 5.3. Moreover, it spans
$\ker(\widehat{D}_{p/q})_{*}$: let $z\in\ker(\widehat{D}_{p/q})_{*}$ be
nonzero, and suppose first that $z\in H_{*}(\oplus_{k\geq
0}\widehat{A}_{k/q})$. Let $z_{j/q}$ denote the term of $z$ in
$H_{*}(\widehat{A}_{j/q})$, and let

$j=\min\\{k\geq 0\ |\ z_{k/q}\neq 0\\}.$

Then $z\in\ker(\widehat{D}_{p/q})_{*}\cap H_{*}(\oplus_{k\geq
j}\widehat{A}_{k/q})$, so
$z_{j/q}\in\operatorname{Span}\\{x_{j/q}^{i}\\}_{i=1}^{\operatorname{rank}\ker(\widehat{v}_{j/q})_{*}}$:
otherwise if $z_{j/q}\notin\ker(\widehat{v}_{j/q})_{*}$, then
$z\notin\ker(\widehat{D}_{p/q})_{*}$ since $z\in H_{*}(\oplus_{k\geq
j}\widehat{A}_{k/q})$. Thus,

$z-\sum_{i\in J}\widetilde{x}_{j/q}^{i}\in\ker(\widehat{D}_{p/q})_{*}\cap
H_{*}(\oplus_{k>j}\widehat{A}_{k/q})$

for some indexing set
$J\subset\\{1,\dots,\operatorname{rank}\ker(\widehat{v}_{j/q})_{*}\\}$.
Inductively, we can then write

$z=\sum_{k\in I}\sum_{i_{k}\in J_{k}}\widetilde{x}_{k/q}^{i_{k}}$

for some finite indexing sets $I\subset\\{k\geq j\\}$,
$J_{k}\subset\\{1,\dots,\operatorname{rank}\ker(\widehat{v}_{k/q})_{*}\\}$.
Similarly, if $z\in H_{*}(\oplus_{k<0}\widehat{A}_{k/q})$, then we can write

$z=\sum_{k\in I}\sum_{i_{k}\in J_{k}}\widetilde{x}_{k/q}^{i_{k}}$

for some finite indexing set $I\subset\\{k\leq j\\}$,
$J_{k}\subset\\{1,\dots,\operatorname{rank}\ker(\widehat{h}_{k/q})_{*}\\}$.

Now for the general case $z\in\ker(\widehat{D}_{p/q})_{*}$, we claim that for
each $0\leq j\leq p-1$,

$z_{j/q}\in\operatorname{Span}\left(\\{y_{j/q}^{i}\\}\cup\\{x_{j/q}^{i}\\}\right).$

Indeed, suppose this is not true for some $0\leq j\leq p-1$, then
$(\widehat{v}_{j/q})_{*}(z_{j/q})\notin\operatorname{Im}(\widehat{h}_{(j-p)/q})_{*}$,
so $(\widehat{D}_{p/q})_{*}(z)\neq 0$, a contradiction. Thus,

$\displaystyle z-\sum_{j=0}^{p-1}\left(\sum_{i\in
J_{j}}\widetilde{y}_{j/q}^{i}+\sum_{i\in
J_{j}^{\prime}}\widetilde{x}_{j/q}^{i}\right)$
$\displaystyle\in\left(\ker(\widehat{D}_{p/q})_{*}\cap H_{*}(\oplus_{k\geq
p}\widehat{A}_{k/q})\right)$ $\displaystyle\hskip
14.22636pt+\left(\ker(\widehat{D}_{p/q})_{*}\cap
H_{*}(\oplus_{k<0}\widehat{A}_{k/q})\right).$

By the above argument, $z$ is in the span of $S$.

Therefore, $S$ is a basis for $\ker(\widehat{D}_{p/q})_{*}$, so since
$\operatorname{rank}\ker(\widehat{v}_{s})_{*}=\operatorname{rank}\ker(\widehat{h}_{-s})_{*}$,

$\operatorname{rank}\ker(\widehat{D}_{p/q})_{*}=q\operatorname{rank}\ker(\widehat{v}_{0})_{*}+2q\sum_{s=1}^{g-1}\operatorname{rank}\ker(\widehat{v}_{s})_{*}+t_{K}^{p/q}.$

Since
$\operatorname{rank}(\widehat{v}_{s})_{*}=\operatorname{rank}H_{*}(\widehat{A}_{s})-\operatorname{rank}\ker(\widehat{v}_{s})_{*}$,
$\operatorname{rank}H_{*}(\widehat{A}_{s})=b$ for all $|s|\geq g$, and
$\operatorname{rank}\operatorname{coker}(\widehat{D}_{p/q}^{c})_{*}=\operatorname{rank}H_{*}(\widehat{\mathbb{B}}_{p/q}^{c})-\operatorname{rank}H_{*}(\widehat{\mathbb{A}}_{p/q}^{c})+\operatorname{rank}\ker(\widehat{D}_{p/q}^{c})_{*}$
for some large $c$, it follows that

$\displaystyle\operatorname{rank}\operatorname{coker}(\widehat{D}_{p/q}^{c})_{*}$
$\displaystyle=b(2c-1)q-pb-q\sum_{s=-c+1}^{c-1}\operatorname{rank}H_{*}(\widehat{A}_{s})$
$\displaystyle\hskip
14.22636pt+q\operatorname{rank}\ker(\widehat{v}_{0})_{*}+2q\sum_{s=1}^{g-1}\operatorname{rank}\ker(\widehat{v}_{s})_{*}+t_{K}^{p/q}$
$\displaystyle=q(b-\operatorname{rank}H_{*}(\widehat{A}_{0})+\operatorname{rank}\ker(\widehat{v}_{0})_{*})$
$\displaystyle\hskip
14.22636pt+2q\sum_{s=1}^{g-1}(b-\operatorname{rank}H_{*}(\widehat{A}_{s})+\operatorname{rank}\ker(\widehat{v}_{s})_{*})+t_{K}^{p/q}-pb$
$\displaystyle=q(b-\operatorname{rank}(\widehat{v}_{0})_{*})+2q\sum_{s=1}^{g-1}(b-\operatorname{rank}(\widehat{v}_{s})_{*})+t_{K}^{p/q}-pb,$

and therefore Equation 5.9 holds.

∎

Now we are ready to prove our main theorems. The main ingredient for each
result is Equation 5.9.

######  Theorem 5.11.

Let $K\subset Y$ be a blow-up unknotted knot, and suppose $0<r<s$ are distinct
positive rational numbers with the property that $Y_{r}(K)\cong\pm Y_{s}(K)$.
Then $r,s>1$ or $K$ is the unknot.

###### Proof.

Since $H_{1}(Y_{\pm p/q}(K))\cong H_{1}(Y)\oplus{\mathbb{Z}}/p{\mathbb{Z}},$
we can fix $p>0$ throughout. First, let $q$ be relatively prime to $p$ and
$q\geq p$. Then
$\operatorname{Im}(\widehat{v}_{j/q})_{*}\cap\operatorname{Im}(\widehat{h}_{(j-p)/q})_{*}=\operatorname{Im}(\widehat{v}_{0})_{*}$
for all $j=0,\dots,p-1$. Thus
$t_{K}^{p/q}=p\operatorname{rank}(\widehat{v}_{0})_{*}$. Therefore, according
to equation 5.9, for positive integral $q$ with $0<p/q\leq 1$,
$\operatorname{rank}\widehat{HF}(Y_{p/q}(K))$ is a nondecreasing function of
$q$. The function is strictly increasing unless $(\widehat{v}_{s})_{*}$ is an
isomorphism for all $s\geq 0$. Hence, by Proposition 5.5, $K$ is the unknot.

Since the total rank of $\widehat{HF}$ is an invariant of the underlying
(unoriented) three-manifold, the result is true whenever $0<r<s\leq 1$.

Now suppose $q,q^{\prime}$ are relatively prime to $p$ and $0<q<p<q^{\prime}$.
Then

$\displaystyle t_{K}^{p/q}-t_{K}^{p/q^{\prime}}$
$\displaystyle\leq\sum_{j=0}^{p-1}\operatorname{rank}(\widehat{v}_{j/q})_{*}-p\operatorname{rank}(\widehat{v}_{0})_{*}$
$\displaystyle=\sum_{j=q}^{p-1}\operatorname{rank}(\widehat{v}_{j/q})_{*}-(p-q)\operatorname{rank}(\widehat{v}_{0})_{*}$
$\displaystyle\leq(p-q)(b-\operatorname{rank}(\widehat{v}_{0})_{*}).$

Therefore,

$\displaystyle\operatorname{rank}\widehat{HF}(Y_{p/q^{\prime}}(K))$
$\displaystyle-\operatorname{rank}\widehat{HF}(Y_{p/q}(K))=(q^{\prime}-q)(\operatorname{rank}\ker(\widehat{v}_{0})_{*}+b-\operatorname{rank}(\widehat{v}_{0})_{*})$
$\displaystyle+(q^{\prime}-q)\left(2\sum_{s=1}^{g-1}(\operatorname{rank}\ker(\widehat{v}_{s})_{*}+b-\operatorname{rank}(\widehat{v}_{s})_{*})\right)$
$\displaystyle+2t_{K}^{p/q^{\prime}}-2t_{K}^{p/q}$ $\displaystyle\geq
2(q^{\prime}-q)(b-\operatorname{rank}(\widehat{v}_{0})_{*})-2(p-q)(b-\operatorname{rank}(\widehat{v}_{0})_{*})$
$\displaystyle\geq 0,$

with equality only if $(\widehat{v}_{s})_{*}$ is an isomorphism for all $s\geq
0$, hence $K$ is the unknot by Lemma 5.5.

Since the total rank of $\widehat{HF}$ is an invariant of the underlying
(unoriented) three-manifold, the result holds. ∎

In the case where $p$ is $1$ or $2$, the following corollary is immediate,
since there do not exist distinct positive integers $q,q^{\prime}$ coprime to
$p$ with $p/q$ and $p/q^{\prime}$ both greater than $1$.

######  Corollary 5.12.

Let $K\subset Y$ be a blow-up unknotted knot, and suppose $0<q^{\prime}<q$ are
distinct positive numbers with the property that $Y_{p/q}(K)\cong\pm
Y_{p/q^{\prime}}(K)$ for $p\in\\{1,2\\}$, then $K$ is the unknot.

######  Theorem 5.13.

Let $K\subset Y$ be a nontrivial blow-up unknotted knot and $q\geq 2$, then
$Y_{1/q}(K)\ncong\pm Y$. Furthermore, if $g(K)\geq 2$ or
$\operatorname{rank}H_{*}(\widehat{A}_{0})>b$, then $Y_{1}(K)\ncong\pm Y$.

###### Proof.

Let $q\geq 2$, then by Equation 5.9,

$\operatorname{rank}\widehat{HF}(Y_{1/q}(K))\geq(b-\operatorname{rank}(\widehat{v}_{0})_{*})(q-1)+\operatorname{rank}(\widehat{v}_{0})_{*}+q\operatorname{rank}\ker(\widehat{v}_{0})_{*}\geq
b,$

and it is strictly greater unless $(\widehat{v}_{s})_{*}$ is an isomorphism
for all $s\geq 0$. Hence, by Proposition 5.5, $K$ is the unknot.

If $q=1$, then

$\operatorname{rank}\widehat{HF}(Y_{1}(K))\geq\operatorname{rank}(\widehat{v}_{0})_{*}+\operatorname{rank}\ker(\widehat{v}_{0})_{*}=\operatorname{rank}H_{*}(\widehat{A}_{0})\geq
b,$

and it is strictly greater unless
$\operatorname{rank}H_{*}(\widehat{A}_{0})=b$ and $(\widehat{v}_{s})_{*}$ is
an isomorphism for all $s\geq 1$, which implies $g(K)\leq 1$.

Since the total rank of $\widehat{HF}$ is an invariant of the underlying
(unoriented) three-manifold, the result holds. ∎

## References

  * [Han19] Jonathan Hanselman.  Heegaard floer homology and cosmetic surgeries in $S^{3}$.  Journal of the European Mathematical Society, 2019.
  * [Kir95] Rob Kirby.  Problems in low-dimensional topology.  1995\.
  * [Ni10] Yi Ni.  Thurston norm and cosmetic surgeries.  arXiv: Geometric Topology, 2010.
  * [Ni20] Yi Ni.  Property g and the $4$–genus.  arXiv: Geometric Topology, 2020.
  * [OS04a] Peter Ozsváth and Zoltán Szabó.  Holomorphic disks and knot invariants.  Advances in Mathematics, 186(1):58–116, 2004.
  * [OS04b] Peter Ozsváth and Zoltán Szabó.  Holomorphic disks and topological invariants for closed three-manifolds.  Annals of Mathematics, 159(3):1027–1158, 2004.
  * [OS06] Peter Ozsváth and Zoltán Szabó.  Holomorphic triangles and invariants for smooth four-manifolds.  Advances in Mathematics, 202(2):326–400, 2006.
  * [OS08] Peter Ozsváth and Zoltán Szabó.  Knot Floer homology and integer surgeries.  Algebraic & Geometric Topology, 8(1):101 – 153, 2008.
  * [OS10] Peter S Ozsváth and Zoltán Szabó.  Knot floer homology and rational surgeries.  Algebraic & Geometric Topology, 11(1):1–68, January 2010.
  * [Zem23] Ian Zemke.  A general heegaard floer surgery formula, 2023.