# A Machine Learning Smartphone-based Sensing for Driver Behavior
Classification

Sarra Ben Brahim1, Hakim Ghazzai2, Hichem Besbes1, and Yehia Massoud2
1Higher School of Communications of Tunis, University of Carthage, Tunis,
Tunisia
2CEMSE Division, King Abdullah University of Science and Technology (KAUST),
Thuwal, Saudi Arabia

###### Abstract

Driver behavior profiling is one of the main issues in the insurance
industries and fleet management, thus being able to classify the driver
behavior with low-cost mobile applications remains in the spotlight of
autonomous driving. However, using mobile sensors may face the challenge of
security, privacy, and trust issues. To overcome those challenges, we propose
to collect data sensors using Carla Simulator available in smartphones
(Accelerometer, Gyroscope, GPS) in order to classify the driver behavior using
speed, acceleration, direction, the 3-axis rotation angles (Yaw, Pitch, Roll)
taking into account the speed limit of the current road and weather conditions
to better identify the risky behavior. Secondly, after fusing inter-axial data
from multiple sensors into a single file, we explore different machine
learning algorithms for time series classification to evaluate which algorithm
results in the highest performance.

###### Index Terms:

Artificial intelligence, intelligent transportation systems, driver profiling,
time series classification.

##  I Introduction

Over the last two decades, Road Traffic Accidents (RTAs) are increasingly
being recognised as a growing public health problem. A statistical projection
of traffic fatalities for 2020 from the National Highway Traffic Safety
Administration (NHTSA) shows that an estimated 38,680 people died in motor
vehicle traffic crashes on U.S. roadways, an increase of 7.2 percent compared
to the previous year [1]. Aggressive driving is one of the major causes of
traffic crashes, according to the NHTSA, which listed impaired driving, over-
speeding and sudden lane change as ones of main behaviors causing fatal
accidents [2]. Therefore, urgent solutions to control road traffic and monitor
driving behavior are becoming a necessity for many countries to cope with RTAs
[3].

Road safety systems and strategies have been developed to safeguard road users
and avoid inevitable errors that they might make. They aim to reduce risky
behaviors and identify the factors of accidents if they occur. Another
approach to promote safe driving is adopted by insurance companies. Premium
plans such Usage-Based Insurance (UBI) or Pay-How-You-Drive (PHYD) are
proposed to customers to reduce automobile insurance costs by rewarding those
who adopt safe driving behavior [4]. Nevertheless, tracking the driving
behavior in real-time and providing feedback on how a driving is behaving on
road is not a straightforward task. Traditional techniques such as the ones
using exclusively GPS data to identify the driver behavior are not always
effective [5]. However, Artificial Intelligence (AI) and the Internet-of-
Things have shown promising results in tackling this complex task [6, 7, 8, 9,
10].

Two categories of sensors are usually involved to monitor the driving
behavior: vehicle dynamics and driver dynamics [6]. Vehicle dynamics mostly
rely on the signals of sensors, such as Global Positioning System (GPS),
accelerometers, gyroscopes, and magnetometers. On the other hand, driver
dynamics mostly rely on the signals of active sensors, such as video-cameras
located inside the vehicle that observe the driver’s behavioral biometrics.
Using vehicle dynamics, more precisely smartphone sensors, for driving
behavior analysis is the focus of this study due to the fact it has been shown
that it is more suitable to use than CAN-bus sensors for driver behavior
classification [7]. Furthermore, we are looking to design reliable and low-
cost solution.

Figure 1: Proposed workflow of the AI-based driver behavior classification.

Recently, the authors of [11] have conducted a complete literature review on
existing studies and techniques about various sensors, technologies, data, and
machine learning algorithms used to investigate driver behavior. They also
suggested a conceptual framework that uses smartphones to identify the driver
behavior. Another study utilized UAH-DriveSet which is a public dataset that
provides a wide range of smartphone sensors (e.g., GPS, accelerometers) and
even video recordings to distinguish between three driving behaviors; normal,
aggressive, and drowzy using Long Term Short Memory (LSTM) and LSTM-Fully
Convolutional Network [9, 6]. In order to examine the performance of different
types of Recurrent Neural Networks (RNNs), the authors of [12] used only data
that can be obtained from the smartphone’s sensors, such as acceleration and
GPS. They gathered information from four distinct car routes driven by two
different drivers in identical weather conditions. The RNNs were evaluated
using seven different driving events, and it was shown that the Gated
Recurrent Unit (GRU) outperformed the LSTM and simpleRNN methods for this
dataset.

In this paper, we propose to classify the driving behavior into four different
classes (normal, intermediate, aggressive, and dangerous) in various external
conditions (speed limits, weather conditions, traffic signs) using data
collected from smartphones’ sensors only such as accelerometer and gyroscope.
The data is collected in the form of time series which are analyzed to train
an AI-based classifier. Two category of classifiers are investigated including
three Gradient Boosting Decision Trees (GBDT) machine learners and three LSTM
deep learners. Unlike previous work which used one major dataset, namely the
UAH-DriveSet which considers only three behavior classes (normal, drowzy and
aggressive) on two different routes, we propose to simulate the driving
environment on the Carla simulator and collect the time series data for
training and testing. Selected results compare the different implemented
methods and show an effective ability to detect the driving behavior using
limited sensor sources.

##  II Driver Behavior Classification Method

As showcased in Fig. 1, the proposed method encompasses four phases: i) data
collection, ii) data pre-processing, iii) parameter tuning, and iv) machine
learning model design.

In this study, we propose to extract information about the driver’s risky
behavior from the smartphone sensors only as they are equipped with reliable
and low-cost sensors that are available with most Android and IOS operating
systems such as Global Navigation Satellite System (GNSS), accelerometers, and
gyroscopes. To achieve a more complete and accurate movement tracking of the
vehicle, sensor fusion is therefore required. Collected signals can be
detected at unequal intervals so resampling them is an essential step, which
will be investigated in the pre-processing phase of the framework.

Once data is processed, we obtain time series of data points resampled at the
same frequency. Before feeding them to the classification model, we perform a
parameter tuning to find the best combination of hyperparameters for the
machine learner that results in the most accurate classification performance.
Finally, we implement a machine learning algorithm for the classification
task. In this paper, we propose a comparative study between different machine
learners that are the most suitable for the driving behavior classification.

For a better understanding of the driver behavior, we distinguish between four
types or classes of the driver behavior:
$\bullet$ Normal: the driver is respecting the speed limit given weather
conditions and that he/she is safe for other drivers.
$\bullet$ Intermediate: the driver is showing few aggressive actions on the
road but he/she is not putting other drivers in danger.
$\bullet$ Aggressive: the driver is over-speeding, over-steering, weaving in
and out of traffic.
$\bullet$ Dangerous: the driver is performing many dangerous actions, e.g.,
dangerous steering, dangerous over-speeding, with a high risk of causing a
crash.

This study does not assume that a driver behavior will remain the same over
the whole trip. Indeed, we aim to detect different actions performed by the
drivers during the same trip and classify them accordingly, e.g., every one
minute.

TABLE I: Impact of weather on the speed limit Weather | Speed reduc-
---|---
type | tion factor
Sunny | 0%
Soft rain | 15%
Foggy | 30%
Stormy | 40%

TABLE II: Degree of severity of the speeding Speed | Degree of
---|---
Behavior | aggressiveness
Under speed limit | Normal
Overspeed [10,33]% | Intermediate
Overspeed [33,66]% | Aggressive
Overspeed 66%+ | Dangerous

TABLE III: Degree of severity of the steering Allowed number of waving in and out per minute |  Absolute value of the steering impulse |  Steering behavior
---|---|---
Under 5 |  Less than 0.15 |  Intermediate
Between 5 and 10 |  Between 0.16 and 0.45 |  Aggressive
Over 10 |  More than 0.45 |  Dangerous

##  III Driver Behavior Classification Phases

###  III-A Data collection

In this phase, we define the features considered in the driver behavior
classification. Then, we identify the rules of modeling the driver behavior
classes and how they will be labeled.

####  III-A1 Feature description

Data collection in driver behavior analysis can be performed either in real
world environment or in a virtual environment. In the first case, although the
collected data is more close to the reality, it requires more time and
financial effort to gather a dataset. In the second case, driver behavior can
be simulated on powerful software that allows a faster collection of data,
while providing a full control on the investigated scenarios, e.g., multiple
type of roads, traffic situation, and weather conditions. In this study, we
perform the data collection using the Carla simulation that we will be
discussed further in the next section.

As mentioned earlier, we maintain the gyroscope and the accelerometer to
identify the risky behavior [12]. Hence, those motion sensors are considered
to measure acceleration and rotational forces along three axes. The magnitude
of the acceleration is also an important feature that will be exploited with
lateral and longitudinal accelerations to detect aggressive driving behavior
[13]. The GNSS supplies the longitude and latitude of the vehicle. Steering
angle and throttle are also considered in our model since they can reflect a
certain degree of aggressiveness [14]. Moreover, every trajectory is enriched
with weather information (e.g., rain, fog, humidity) and road data (e.g.,
traffic light location and speed limits) which are provided by environmental
sensors and an offline map available in the smartphone.

All features, described in Fig. 1, will be gathered in one output as time-
series data collected from the vehicle trip.

####  III-A2 Selected Rules about Modeling the Driver Behavior

Over speeding is considered as one of the major factors contributing to severe
car crashes on highways and motorways. Thus, speed limits set by local
authorities represent the most popular countermeasure. We summarize, in Table
II, the impact of the weather on the speed after considering the reduction of
visibility and pavement friction due to bad weather conditions. On the other
hand, based on California speeding tickets fees that vary from the range of
35$ to penalize an overspeed between 1.5 km/h and 25 km/h to 200$ for an
overspeed of 160 km/h or more, we have constructed, in Table II, the degree of
aggressiveness of the driving based on the speed behavior. Due to the fact
that cutting off other motorists in traffic is dangerous and reckless
behavior, in Table III, we present some examples of aggressive steering based
on our approach, assuming, for simplification, that the driver sessions are
recorded on highways only and that there is no turning, and the steering angle
is between 1 and -1 (maximum right and left steering).

###  III-B Data Pre-processing

We propose to pre-process the collected data before initiating the driver
behavior analysis.

####  III-B1 Preparation step

Data captured from different sensors are measured at irregular intervals due
to latency or other external factors. Hence, resampling the time series data
is essential as it involves changing the frequency of our time series
observations so that it can be at the same frequency. In our case, resampling
will occur every 250 milliseconds so all captured signals from sensors will be
available every 250 milliseconds which is sufficient since not major events
from driving sessions can happen during that period, e.g., sudden
acceleration, sudden deceleration, over-steering can be detected.

####  III-B2 Augmentation phase and normalization

In this driving behavior study, we resort to the sliding window principle as a
feature engineering tool to achieve more accurate results in the
classification. Actually, sliding windows based algorithms are widely used for
time-series segmentation, in order to reveal the underlying properties of the
signals. Previous studies using the sliding window principle in driver
behavior classification [6, 7, 9] have shown that it provides more accurate
results. The sliding window principle means dividing the original time series
into multiple equal length subsequences to give more insights about the data
and increase the number of samples.

We choose to employ a 1-minute sliding window with 50% overlap. Each sliding
window is then labeled according to the four potential driver behavior
classes. Then, all sliding windows are assembled into a single dataset. Since
our data contains multiple measurements from different sensors, we proceed
with its normalization and shift and rescale the collected features to the
same range.

###  III-C Parameter Tuning

Parameter tuning allows to customize the classification models so that they
generate the most precise outputs. For the same machine learning model,
different parameters are required to control the convergence speed and avoid
overfitting. For instance, in gradient boosting decision trees algorithms,
some parameters should be tuned to get the best combinations that give better
results in the driver behavior classification such as the number of trees, the
learning rate, and the tree depth. For deep learning algorithms, we choose to
optimize the number of epochs and batch size.

###  III-D Machine Learning Models

In this study, we investigate six classification models to identify the driver
behavior and distinguish between the four potential driving behavior classes.
The first category is the GBDT, which combines multiple weak learners,
typically a decision tree, to form a stronger model that outperforms the base
model. When a boosting decision tree is added, it learns from the error of
previous individual tree. All the trees are connected in series and each tree
tries to minimize the error of the previous tree. Therefore, GBDT uses an
incremental learning process as it learns from the mistake of each individual
weak learner. Due to this sequential connection, boost algorithms are
generally slow to learn, but show great performance on the classification
task. Yet, different algorithms for gradient boosting on decision trees, e.g.,
CatBoost, XGBoost, and LightGBM, have proven their efficiency in the multi-
classification task in terms of accuracy and training time [15].

Similar to GBDT, we propose to classify the driver behavior with three
different architectures of LSTM. The first model is constructed based on a
custom design of LSTM architecture consisting of one LSTM cell layer called
Stacked-LSTM and organized as a many-to-one structure since our dataset
contains multiple different trips with fixed duration. The second model is a
combination of a Convolutional Neural Network (CNN) and LSTM, defined as CNN-
LSTM. It can be seen as two sub-models: the CNN model (on the front) is used
feature extraction and then followed by the LSTM layers to interpret features
across time steps. In the case of this study, the input will be subsequences
equal in length instead of one sequence. Each will be considered for the CNN
layer as a block. CNN will use those blocks to detect common patterns for each
feature. This technique enables the identification and analysis of the
patterns of each of the subsequence and hence, learn in a more efficient way
the temporal features to achieve better classification results using LSTM.

The third model is a combination of a Fully Convolutional Network (FCN) and an
LSTM. It is denoted as FCN-LSTM and utilizes temporal convolutions layers as
feature extractors. The LSTM-FCN model achieves the state-of-the-art in many
sequence classification problems and has received a lot of attention from the
time series classification community. The ability of these models to compute
features on their own presents a significant advantage, since they eliminate
the need for extensive domain expertise and manual feature extraction.
Furthermore, both of these models are easily scalable to huge amounts of time
series data generated daily by automated procedures. The FCN block consists of
three convolution layer followed by a batch normalization, and an activation
function. Following the final convolution block, a global average pooling is
used to reduce the number of parameters in the model before classification.
For the LSTM block, it consists of one layer of LSTM which is followed by a
dropout to prevent overfitting. The outputs of the global pooling layer and
the LSTM block are concatenated and fed into a softmax classification layer
which will predict the corresponding class.

Thus, we propose a comparative study between gradient boosting algorithms
(Catboost, XGBoost, LightGBM) and different architectures of LSTM (LSTM, CNN-
LSTM, FCN-LSTM) with the aim of achieving effective classification
performance, specifically the macro F1-score, which considers false alarms and
imbalanced class distribution.

##  IV Results and Discussions

###  IV-A CARLA Simulator and Dataset Generation

After modeling multiple driving behaviors and generating different scenarios
of trips, we collect more than 500 minutes using the Carla simulator as shown
in Fig. 2 [16, 17, 18]. The driving sessions are recorded on four different
towns with two types of road (motorway and secondary) with four different
weather conditions (sunny, soft rain, cloudy & foggy, and stormy) and four
styles of driving (normal, intermediate, aggressive, and dangerous) that occur
along the trips such that a reasonable balanced dataset between the different
classes is generated.

Figure 2: Smartphone sensors collected using the Carla simulator.

###  IV-B Selected Results

In this section, we compare the performances of the CatBoost, the XGBoost, and
the LightGBM to evaluate their efficiency in driver behavior classification.
In our evaluation, we consider the following metrics:

$\text{Average
Accuracy}=\frac{1}{N}\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{N}\frac{{TP_{i}}+{TN_{i}}}{{TP_{i}}+{TN_{i}}+{FP_{i}}+{FN_{i}}},$
(1) $\text{Macro
Precision}=\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{N}\text{Precision}_{i}=\frac{1}{N}\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{N}\frac{{TP_{i}}}{{TP_{i}}+{FP_{i}}},$
(2) $\text{Macro
Recall}=\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{N}\text{Recall}_{i}=\frac{1}{N}\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{N}\frac{{TP_{i}}}{{TP_{i}}+{FN_{i}}},$
(3) $\text{Macro
F1-score}=\frac{1}{N}\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{N}\frac{\text{Precision}_{i}\;\text{Recall}_{i}}{\text{Precision}_{i}+\text{Recall}_{i}},$
(4)

where $N$ is the number of classes which is equal to four in our case,
$TP_{i}$, $FP_{i}$, $TN_{i}$, and $FN_{i}$ are the true positives, false
positives, true negatives, and false negatives per class. We use macro-
averaging technique when calculating the accuracy, recall, precision and
F1-score to cope with the problem of imbalanced data.

In Table IV, we present the best combinations of hyperparameters as a result
from the random search technique which tests different parameters including
the number of trees, the learning rate, the tree depth, and the
L2-regularization in addition to the used loss function.

TABLE IV: Results of the hyperparameter optimization Training parameters |  CatBoost |  XGBoost |  LightGBM
---|---|---|---
Loss function |  ’MultiClass’ |  ’mlogloss’ |  ’multi-logloss’
Best parameters |  learning rate=0.73,
max depth=8,
n estimators=88,
random state=10 |  learning rate=0.28,
max depth=4,
n estimators=13,
random state=10, |  learning rate=0.04,
max depth=8,
n estimators=53,
random state=10
TABLE V: Comparison between the GBDT algorithms Metrics |  CatBoost |  XGBoost |  LightGBM
---|---|---|---
Average Accuracy |  80% |  82% |  88%
Macro Precision |  83% |  83% |  88%
Macro Recall |  81% |  83% |  89%
Macro F1-score |  81% |  83% |  88%

Based on the comparison between the three different gradient boosting
algorithms described in Table V, and after splitting our dataset into 70% of
training and 30% of testing, we find that LightGBM has achieved an F1-score of
88% with testing sets. Nore that, XGBoost is the fast in training taking only
few seconds to choose the best parameters and to classify the behavior of the
driver.

In Table VI, we test different numbers of epochs and batch sizes before
concluding that training on 150 epochs and a batch size of 20 is the best
configuration for our three models due to the fact that we have 500 samples.
The table evaluates the performance of the different LSTM architectures on the
testing set to identify which architecture achieves the best F1-score and thus
the least false positive and false negative rates. We conclude that the best
architecture among LSTM is FCN-LSTM which achieves an F1-score of 74% which is
higher than LSTM that achieves 58% and CNN-LSTM 68%. Indeed, FCN-LSTM are
effective in sequence classification and hence, more suitable for driver
behavior classification.

TABLE VI: Performance of the different LSTM architectures Training parameters |  LSTM |  CNN-LSTM |  LSTM-FCN
---|---|---|---
Loss function |  ’categorical
crossentropy’ |  ’categorical
crossentropy’ |  ’categorical
crossentropy’
Optimizer |  Adam |  Adam |  Adam
Average accuracy |  70% |  76% |  79%
Macro Precision |  68% |  79% |  75%
Macro recall |  60% |  69% |  74%
Macro F1-score |  58% |  68% |  74%
Figure 3: Example of test data collected from the simulator using the LightGBM
classifier.

Fig. 3 shows how the proposed solution, in this case the LightGBM, can
classify the driver behavior in practice. In this example, we record a ten-
minute trip. The classifier observes this trip as ten sub-trips of one-minute
duration using the sliding window principle and aims to predict the
corresponding class. Fig. 3 illustrates one test example of the simulated data
with $W_{j}$ denotes the $j^{\text{th}}$ sliding window and ”P.C” refers to
the predicted class with 0 for normal behavior, 1 for intermediate, 2 for
aggressive and 3 for dangerous. The figure illustrates the most influential
features: steering, speed (km/h), and the three acceleration components. In
this testing phase, the predicted class of $W_{2}$ is 3, which can be
justified by the detected over speeding noticeable specifically in the AccelX.
This means if one of the features is detected abnormal given the weather
condition and the road status, the whole sub-trip can be classified as
dangerous. The predicted classes of those sub-trips can be further analyzed to
get a complete overview of the driver behavior along its whole journey.

##  V Conclusion

In this paper, we have investigated the driving behavior classification
problem using smartphone-based sensing only.After collection of our time
series data under different road and weather conditions, we fed our simulated
data into our AI-agent to detect different driving behavior classes and to
identify every driver profile. Results have shown the ability of both machine
and deep learning models to achieve an accuracy greater than 88% in detecting
driving profile for a one-minute duration trip.

This framework is a low-cost solution for many fleet management and monitoring
problems and can further be used to prevent cars from accidents. Although the
collected data from simulations mimics real-life situations in terms of
weather conditions, types of aggressive behaviors, in the future extension of
this work, we aim to focus on collecting realistic data and provide more
generalized driver behavior classification framework that considers other road
types other than highways.

## References

  * [1] NHTSA Media, in _2020 Fatality Data Show Increased Traffic Fatalities During Pandemic_ , June 3, 2021 — Washington, DC.
  * [2] U.S. Department of Transportation, National Highway Traffic Safety Administration., in _Facts + Statistics: Aggressive driving_ , 2019.
  * [3] M. C. Lucic, X. Wan, H. Ghazzai, and Y. Massoud, “Leveraging intelligent transportation systems and smart vehicles using crowdsourcing: An overview,” _Smart Cities_ , vol. 3, no. 2, pp. 341–361, May 2020.
  * [4] M. Carfora, F. Martinelli, F. Mercaldo, F. Nardone, V. Orlando, A. Santone, and G. Vaglini, “A “pay-how-you-drive” car insurance approach through cluster analysis,” in _Soft Computing._ , 2019.
  * [5] Grengs, J., Wang, X., and Kostyniuk, L., “Using gps data to understand driving behavio,” in _Journal of Urban Technology_ , 19 Jan 2009, pp. 33–53.
  * [6] K. Saleh, M. Hossny, and S. Nahavandi, “Driving behavior classification based on sensor data fusion using $lstm$ recurrent neural networks,” in _2017 IEEE 20th International Conference on Intelligent Transportation Systems (ITSC)_ , October 2017.
  * [7] V. Vaitkus, P. Lengvenis, and G. Žylius, “Driving style classification using long-term accelerometer information,” in _2014 19th International Conference on Methods and Models in Automation and Robotics (MMAR)_ , September 2014.
  * [8] A. Abdelrahman, , H, S. Hassanein, and N. Abu Ali, “Robust data-driven framework for driver behavior profiling using supervised machine learning,” in _IEEE Transactions on Intelligent Transportation Systems_ , November 2020\.
  * [9] Y. Moukafih, H. Hafidi, and M. Ghogho, “Aggressive driving detection using deep learning-based time series classification,” in _2019 IEEE International Symposium on INnovations in Intelligent SysTems and Applications (INISTA)_ , July 2019.
  * [10] X. Wan, M. C. Lucic, H. Ghazzai, and Y. Massoud, “Empowering real-time traffic reporting systems with nlp-processed social media data,” _IEEE Open Journal of Intelligent Transportation Systems_ , vol. 1, pp. 159–175, 2020.
  * [11] F. Lindow and A. Kashevnik, “Driver behavior monitoring based on smartphone sensor data and machine learning methods,” in _Conference of Open Innovations Association (FRUCT’19)_ , 2019.
  * [12] J. Ferreira Ju´nior, E. Carvalho, B, V. Ferreira, C. de Souzaand Y. Suhara, A. Pentland, and G. Pessin, “Driver behavior profiling: An investigation with different smartphone sensors and machine learning,” April,2017.
  * [13] S. Chigurupati, S. Polavarapu, Y. Kancherla, and A. Kousar Nikhath4, “Integrated computing system for measuring driver safety index,” in _International Journal of Emerging Technology and Advanced Engineering_ , Juin 2012.
  * [14] C. G. Quintero M., , J. Oñate López, and A, C. Cuervo Pinilla, “Driver behavior classification model based on an intelligent driving diagnosis system,” in _International IEEE Conference on Intelligent Transportation Systems_ , 25 October 2012.
  * [15] J. Tanha, Y. Abdi , N. Samadi, N. Razzaghi , and M. Asadpour , “Boosting methods for multi-class imbalanced data classification: an experimental review,” in _Journal of Big Data_ , November 2020.
  * [16] A. Dosovitskiy, G. Ros, F. Codevilla, A. Lopez, and V. Koltun, “CARLA: An open urban driving simulator,” in _Proceedings of the 1st Annual Conference on Robot Learning_ , 2017.
  * [17] H. Friji, H. Ghazzai, H. Besbes, and Y. Massoud, “A dqn-based autonomous car-following framework using rgb-d frames,” in _2020 IEEE Global Conference on Artificial Intelligence and Internet of Things (GCAIoT)_ , 2020, pp. 1–6.
  * [18] M. Masmoudi, H. Friji, H. Ghazzai, and Y. Massoud, “A reinforcement learning framework for video frame-based autonomous car-following,” _IEEE Open Journal of Intelligent Transportation Systems_ , vol. 2, pp. 111–127, 2021.


# On the slice genus of quasipositive knots in indefinite 4-manifolds

David Baraglia  School of Mathematical Sciences, The University of Adelaide,
Adelaide SA 5005, Australia<EMAIL_ADDRESS>
###### Abstract.

Let $X$ be a closed indefinite $4$-manifold with $b_{+}(X)=3\;({\rm mod}\;4)$
and with non-vanishing mod $2$ Seiberg–Witten invariants. We prove a new lower
bound on the genus of a properly embedded surface in $X\setminus B^{4}$
representing a given homology class and with boundary a quasipositive knot
$K\subset S^{3}$. In the null-homologous case our inequality implies that the
minimal genus of such a surface is equal to the slice genus of $K$. If $X$ is
symplectic then our lower bound differs from the minimal genus by at most $1$
for any homology class that can be represented by a symplectic surface. Along
the way, we also prove an extension of the adjunction inequality for closed
$4$-manifolds to classes of negative self-intersection without requiring $X$
to be of simple type.

##  1\. Introduction

An important problem in the study of smooth $4$-manifolds is to determine the
minimal genus of an embedded surface representing a given homology class. The
relative version of this problem for $4$-manifolds with boundary has also
received considerable attention [27, 25, 7, 32, 23, 8, 16, 22, 19, 20, 18].

Consider a closed, connected, oriented smooth $4$-manifold $X$ and let $X_{0}$
be the $4$-manifold with boundary obtained by removing an open ball from $X$.
Given a knot $K\subset S^{3}=\partial X_{0}$ and a relative homology class
$a\in H_{2}(X_{0},\partial X_{0};\mathbb{Z})\cong H_{2}(X;\mathbb{Z})$, we
seek the minimal genus of a properly embedded surface $\Sigma\subset X$ with
$\partial\Sigma=K$ and $[\Sigma]=a$. We define the slice genus of $K$ with
respect to $(X,a)$ to be the minimum genus of such a surface and denote it by
$g_{4}(K,X,a)$ or $g_{4}(K,X_{0},a)$. Of particular interest is the case that
$[\Sigma]$ is null-homologous. In this case we call the minimal genus of such
a surface the $H$-slice genus of $K$ in $X$ and denote it by $g_{H}(K,X)$ or
$g_{H}(K,X_{0})$. In the case that $X=S^{4}$, $X_{0}=B^{4}$ is the $4$-ball
and $g_{H}(K,X)=g_{4}(K)$ is the usual slice genus of $K$.

To motivate our results we first recall the following relative genus bound for
surfaces in definite $4$-manifolds:

######  Theorem 1.1 (Ozsváth–Szabó [27]).

Let $X_{0}$ be a smooth, compact, oriented, negative-definite $4$-manifold
with $b_{1}(X_{0})=0$ and $\partial X_{0}=S^{3}$. For any smooth, properly
embedded surface $\Sigma\subset X_{0}$ bounding a knot $K$, we have

$2g(\Sigma)\geq[\Sigma]^{2}+\left|[\Sigma]\right|+2\tau(\Sigma).$

Here $\left|[\Sigma]\right|$ is defined as
$\sum_{i=1}^{n}|\langle[\Sigma],e_{1}\rangle|$, where $e_{1},\dots,e_{n}$ is
an orthonormal basis for $H^{2}(X_{0};\mathbb{Z})$, (which exists by
Donaldson’s diagonalisation theorem) and $\tau(K)$ is the Ozsváth–Szabó tau-
invariant, defined in [27]. Taking $\Sigma$ to be null-homologous, we obtain a
lower bound on the $H$-slice genus of $K$:

(1.1) $g_{H}(K,X_{0})\geq\tau(K).$

A surprising feature of this bound is that it does not depend on the
$4$-manifold $X_{0}$.

Suppose now that $K$ is quasipositive. This means that $K$ is the braid
closure of a braid that is a product of the standard generators
$\sigma_{1},\dots,\sigma_{n-1}$ and their conjugates [29]. It is shown in [15]
that as a consequence of the inequality given in [28], one has
$\tau(K)=g_{4}(K)$ for any quasipositive knot. For instance, if $T_{p,q}$
denotes the $(p,q)$-torus knot for positive coprime integers $p,q$, then
$\tau(T_{p,q})=g_{4}(T_{p,q})=(p-1)(q-1)/2$. Thus for quasipositive knots the
inequality (1.1) gives $g_{H}(K,X)\geq g_{4}(K)$. But we obviously have
$g_{H}(K,X)\leq g_{4}(K)$ and so we deduce that the $H$-slice genus of any
quasipositive knot in any negative definite $4$-manifold with $b_{1}(X)=0$ is
equal to the slice genus.

In this paper we prove an analogue of Theorem 1.1 for quasipositive knots and
links in a large class of indefinite $4$-manifolds. Let $L\subset
S^{3}=\partial B$ be an oriented link. Define the Murasugi characteristic
$\chi_{4}(L)$ of $L$ to be the maximum of the Euler characteristic of any
smooth, properly embedded, oriented surface in $B$ bounding $L$ and having no
closed components [24, 4]. If $L$ is a knot, then $\chi_{4}(L)=1-2g_{4}(L)$,
where $g_{4}(L)$ is the slice genus.

Given a spinc-structure $\mathfrak{s}$ on a compact, oriented smooth
$4$-manifold $X$ with $b_{+}(X)>1$, we let $SW(X,\mathfrak{s})\in\mathbb{Z}$
denote the Seiberg–Witten invariant of $(X,\mathfrak{s})$. We also let

$d(X,\mathfrak{s})=\frac{c_{1}(\mathfrak{s})^{2}-\sigma(X)}{4}-1+b_{1}(X)-b_{+}(X)$

denote the expected dimension of the Seiberg–Witten moduli space.

######  Theorem 1.2.

Let $X$ be a smooth, closed, oriented $4$-manifold with $b_{1}(X)=0$ and
$b_{+}(X)=3\;({\rm mod}\;4)$. Suppose that there is a spinc-structure
$\mathfrak{s}$ with $SW(X,\mathfrak{s})=1\;({\rm mod}\;2)$. Let $X_{0}$ be the
$4$-manifold with boundary $S^{3}$ obtained by removing an open ball from $X$
and let $L\subset S^{3}$ be a quasipositive link. Then for any connected,
smooth, oriented, properly embedded surface $\Sigma\subset X_{0}$ bounding
$L$, we have

$-\chi(\Sigma)\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|-\chi_{4}(L).$

In particular, if $L=K$ is a quasipositive knot, then

$2g(\Sigma)\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|+2g_{4}(K).$

Taking $\Sigma$ in Theorem 1.2 to be null-homologous and $L$ to be a knot, we
obtain:

######  Corollary 1.3.

Let $X$ and $K$ be as in Theorem 1.2. Then $g_{H}(K,X)=g_{4}(K)$.

In the case that $a\in H_{2}(X;\mathbb{Z})$ can be represented by a closed
surface $S$ for which the adjunction inequality is sharp in the sense that
$2g(S)-2=a^{2}+\langle a,c_{1}(\mathfrak{s})\rangle$, we find that the lower
bound given by Theorem 1.2 differs from the minimal genus by at most $1$:

######  Corollary 1.4.

Let $X,\mathfrak{s}$ and $K$ be as in Theorem 1.2. If there exists a closed
embedded surface $S\subset X$ representing the homology class $a\in
H_{2}(X;\mathbb{Z})$ and satisfying $2g(S)-2=a^{2}+\langle
a,c_{1}(\mathfrak{s})\rangle$, then we have

$0\leq g_{4}(K,X,a)-\frac{a^{2}+\langle
a,c_{1}(\mathfrak{s})\rangle}{2}-g_{4}(K)\leq 1$

######  Example 1.5.

Consider the elliptic surface $X=E(2n)$. We have $b_{+}(X)=4n-1$ [13,
Proposition 3.1.11], so $b_{+}(X)=3\;({\rm mod}\;4)$. Suppose that $a\in
H_{2}(X;\mathbb{Z})$ is orthogonal to the canonical class and $a^{2}\geq-2$.
Then according to [14, Theorem 1.1], $a$ can be represented by a surface of
genus $g$, where $2g-2=a^{2}$. Thus Corollary 1.4 gives

$0\leq g_{4}(K,E(2n),a)-\frac{1}{2}a^{2}-g_{4}(K)\leq 1$

for any quasipositive knot.

When $X$ is symplectic and the class $a\in H_{2}(X;\mathbb{Z})$ can be
represented by a symplectic surface, we can apply the positive solution of the
symplectic Thom conjecture [26] to Corollary 1.4:

######  Theorem 1.6.

Let $X$ be a smooth, closed symplectic $4$-manifold with $b_{1}(X)=0$ and
$b_{+}(X)=3\;({\rm mod}\;4)$. If the homology class $a\in H_{2}(X;\mathbb{Z})$
can be represented by a closed embedded symplectic surface, then for any
quasipositive knot $K$, we have

$0\leq g_{4}(K,X,a)-\frac{a^{2}+\langle a,K_{X}\rangle}{2}-g_{4}(K)\leq 1,$

where $K_{X}$ is the canonical class of $X$.

###### Proof.

Let $\mathfrak{s}$ be the anti-canonical spinc-structure, so
$c_{1}(\mathfrak{s})=K_{X}$. Recall that for a symplectic $4$-manifold with
$b_{+}(X)>1$, one has $SW(X,\mathfrak{s})=\pm 1$. Let $S\subset X$ be a closed
symplectic surface representing $a$. Then by the adjunction formula, we have
$2g(S)-2=a^{2}+\langle a,K_{X}\rangle$. Hence we may apply Corollary 1.4. ∎

######  Remark 1.7.

In addition to the smooth $H$-slice genus $g_{H}(K,X)$, one can also consider
the topological equivalent $g_{H}^{top}(K,X)$ defined as the minimal genus of
a connected, properly embedded, topologically locally flat, null-homologous
surface in $X_{0}=X\setminus B^{4}$ bounding $K$. When $X=S^{4}$,
$g_{H}^{top}(K,S^{4})=g_{4}^{top}(K)$ is the topological $4$-genus of $K$. For
a torus knot $K=T_{p,q}$ where $p,q$ are positive coprime integers, it is
known that $g_{4}^{top}(K)<g_{4}(K)$, except in the cases $K=T_{2,n},T_{3,4}$
and $T_{3,5}$ [30, 1]. In fact, for such knots one has
$g_{4}^{top}(K)\leq\tfrac{6}{7}g_{4}(K)$ [1]. From this and Corollary 1.3 we
deduce the following: let $K$ be a torus knot other than $T_{2,n}$, $T_{3,4}$
or $T_{3,5}$ and let $X$ be a smooth, closed, oriented $4$-manifold with
$b_{1}(X)=0$, $b_{+}(X)=3\;({\rm mod}\;4)$ and having a non-vanishing mod $2$
Seiberg–Witten invariants. Then

$g_{H}^{top}(K,X)\leq
g_{4}^{top}(K)\leq\frac{6}{7}g_{4}(K)=\frac{6}{7}g_{H}(K,X).$

In particular, for such an $X$ the difference $g_{H}(K,X)-g_{H}^{top}(K,X)$
can be arbitrarily large.

Taking $L$ to be the unkot, Theorem 1.2 also implies an extension of the
adjunction inequality to classes of negative self-intersection without
requiring $X$ to be of simple type:

######  Theorem 1.8.

Let $X$ be a smooth, closed, oriented $4$-manifold with $b_{1}(X)=0$ and
$b_{+}(X)=3\;({\rm mod}\;4)$. Let $\mathfrak{s}$ be a spinc-structure on $X$
with $SW(X,\mathfrak{s})=1\;({\rm mod}\;2)$. Suppose $\Sigma$ is a compact,
connected, smoothly embedded surface in $X$ of genus $g(\Sigma)$. Then

$2g(\Sigma)\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|.$

Note that this is slighly weaker than the usual adjunction inequality when
$[\Sigma]^{2}>0$, which takes the form
$2g(\Sigma)-2\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|$.
When $X$ is of simple type the adjunction inequality in the stronger form with
$2g(\Sigma)-2\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|$
was proven in [26].

Lastly, applying the same type of argument used in the proof of Theorem 1.2 to
a negative definite $4$-manifold, we obtain an extension of the Ozsváth–Szabó
inequality to quasipositive links.

######  Theorem 1.9.

Let $X$ be a smooth, compact, oriented negative-definite $4$-manifold with
$b_{1}(X)=0$ and $\partial X=S^{3}$. Let $L\subset S^{3}$ be a quasipositive
link. Then for any connected, smooth, properly embedded surface $\Sigma\subset
X$ bounding $L$, we have

$-\chi(\Sigma)\geq[\Sigma]^{2}+\left|[\Sigma]\right|-\chi_{4}(L).$

We note here that an extension of the Ozsváth–Szabó $\tau$ invariant for links
was defined in [6] and one has that $-\chi(\Sigma)\geq 2\tau(L)-n$ for a
smooth, connected, properly embedded surface $\Sigma$ in $X$ bounding $L$,
where $n$ is the number of components of the link [6, Proposition 1.4]. This
suggests that there should be a generalisation of the Ozsváth–Szabó inequality
to arbitrary links taking the form
$-\chi(\Sigma)\geq[\Sigma]^{2}+\left|[\Sigma]\right|+2\tau(L)-n$.

###  1.1. Structure of the paper

In Section 2 we prove some results concerning surfaces in
$\mathbb{CP}^{2}\setminus B^{4}$ bounding the mirror $\overline{L}$ of a
quasipositive link $L\subset S^{3}$, leading to Lemma 2.2. In Section 3 we
first prove an extension of the Seiberg–Witten adjunction inequality to the
case of non-vanishing Bauer–Furuta invariant (Proposition 3.1). We then use
this and Lemma 2.2 to prove the main results of the paper.

###  1.2. Acknowledgements

We thank Hokuto Konno for comments on a draft of this paper.

##  2\. Quasipositive links

Let $B_{n}$ denote the braid group on $n$ strands and
$\sigma_{1},\dots,\sigma_{n-1}$ the standard generators. A link $L$ is said to
be quasipositive if it can be realised as the braid closure of a braid $\beta$
which is a product of $\sigma_{1},\dots,\sigma_{n-1}$ and their conjugates
[29]. Every quasipositive link can be realised as the transverse intersection
of a plane algebraic curve $\Gamma\subset\mathbb{C}^{2}$ with a $3$-sphere
$S^{3}=\partial B\subset\mathbb{C}^{2}$ bounding a ball
$B\subset\mathbb{C}^{2}$ [29]. Conversely, any link $L\subset S^{3}$
constructed in this manner is quasipositive [5].

Let $L\subset S^{3}$ be a quasipositive link. Hence we can realise $L$ as the
transverse intersection $L=\partial B\cap\Gamma$ of a plane algebraic curve
$\Gamma\subset\mathbb{C}^{2}$ with an open ball $B\subset\mathbb{C}^{2}$. Let
$\overline{\Gamma}\subset\mathbb{CP}^{2}$ be the projective completion of
$\Gamma$. Here we identify $\mathbb{C}^{2}$ as the complement
$\mathbb{C}^{2}=\mathbb{CP}^{2}\setminus L_{\infty}$ of a projective line
$L_{\infty}$ in $\mathbb{CP}^{2}$.

######  Lemma 2.1.

There exists a non-singular sextic curve $\Sigma\subset\mathbb{CP}^{2}$ which
is disjoint from $\overline{B}\cup Sing(\overline{\Gamma})$ and which meets
$\overline{\Gamma}$ transversally.

###### Proof.

Let $P=\mathbb{P}(H^{0}(\mathbb{CP}^{2},\mathcal{O}(1)))$ be the dual
projective space of $\mathbb{CP}^{2}$, the space of lines in
$\mathbb{CP}^{2}$. Let $U\subset P$ be the set of lines disjoint from
$\overline{B}$. This is an open subset of $P$ and is non-empty, since the line
at infinity is disjoint from $\overline{B}$. For each $x\in
Sing(\overline{\Gamma})$, let $\ell_{x}\subset P$ be the set of lines passing
through $x$. Then $\ell_{x}$ is a projective line in $P$. Now since
$Sing(\overline{\Gamma})$ is finite, it follows that
$V=P\setminus\left(\cup_{x\in Sing(\overline{\Gamma})}\ell_{x}\right)$ is a
dense open subset of $P$. Hence $W=U\cap V$ is non-empty. So there exists a
non-empty open subset $W$ of lines in $\mathbb{CP}^{2}$ disjoint from
$\overline{B}$ and $Sing(\overline{\Gamma})$.

Now consider the linear system
$S=\mathbb{P}(H^{0}(\mathbb{CP}^{2},\mathcal{O}(6)))$ of sextic curves in
$\mathbb{CP}^{2}$. Let $U_{1}\subset S$ denote the open subset of sextic
curves disjoint from $\overline{B}\cup Sing(\overline{\Gamma})$. Then $U_{1}$
is non-empty. Indeed, let $N$ be a projective line in $\mathbb{CP}^{2}$
disjoint from $\overline{B}$ and $Sing(\overline{\Gamma})$. Then the sextic
with divisor $6N$ belongs to $U_{1}$.

Choose a point $p\in\mathbb{CP}^{2}$ not lying $\overline{\Gamma}$. The space
of lines in $\mathbb{CP}^{2}$ through $p$ can be identified with
$\mathbb{CP}^{1}$ and this determines a regular map
$\rho:\overline{\Gamma}\to\mathbb{CP}^{1}$ which sends a point
$q\in\overline{\Gamma}$ to the line joining $p$ and $q$. Let
$T\subset\mathbb{CP}^{1}$ be the finite set $\rho(Sing(\overline{\Gamma}))$.
Then $\rho:\overline{\Gamma}\setminus\rho^{-1}(T)\to\mathbb{CP}^{1}\setminus
T$ is a branched covering. Any point in $\mathbb{CP}^{1}\setminus T$ which is
not a branch point of $\rho$ corresponds to a line in $\mathbb{CP}^{2}$ such
that each point of intersection with $\overline{\Gamma}$ has multiplicity $1$.
Let $U_{2}\subset S$ be the set of sextics whose intersection multiplicities
with $\overline{\Gamma}$ all equal $1$. Then $U_{2}$ is a non-empty open
subset of $S$ because we can take a sextic which is the union of six lines
corresponding to six distinct non-branch points of $\rho$ in
$\mathbb{CP}^{1}\setminus T$. So $U_{1}\cap U_{2}$ is a non-empty open subset
of $S$. Bertini’s theorem implies that there exists a non-singular sextic
which belongs to $U_{1}\cap U_{2}$. ∎

Let $S\subset\mathbb{CP}^{2}$ be a non-singular sextic curve which is disjoint
from $\overline{B}\cup Sing(\overline{\Gamma})$ and which meets
$\overline{\Gamma}$ transversally. Note that $S$ is connected, since any two
distinct components of $S$ would intersect in a singular point. Suppose
$\overline{\Gamma}$ has degree $d$. Then $\overline{\Gamma}$ and $S$ meet in
exactly $6d$ points, by Bézout’s theorem. Let $W=\mathbb{CP}^{2}\setminus B$.
Then $W$ is a $4$-manifold with boundary $\partial W=S^{3}$. Consider
$\Gamma_{0}=\overline{\Gamma}\setminus B$. Then $\Gamma_{0}$ is a properly
embedded surface in $X$ meeting $\partial X$ in the link
$\overline{L}=\partial W\cap\Gamma_{0}$ (we obtain the mirror $\overline{L}$
of $L$, because the orientation on $S^{3}=\partial W$ is opposite to the
orientation obtained by viewing $S^{3}$ as the boundary of $B$).

######  Lemma 2.2.

There exists a smoothly embedded, connected, oriented surface $C_{0}\subset W$
having the following properties:

  * (1)

$C_{0}$ meets $\partial W$ transversally in $\overline{L}$,

  * (2)

$C_{0}$ meets $S$ transversally in $6d$ points,

  * (3)

$\chi(C_{0})=3d-d^{2}-\chi_{4}(L)$,

  * (4)

The homology class of $C_{0}$ in $H_{2}(W,\partial
W;\mathbb{Z})\cong\mathbb{Z}$ is $d[\ell]$, where $[\ell]$ is the class of a
projective line,

  * (5)

Each connected component of $C_{0}$ meets $\overline{L}$.

###### Proof.

First note that it is enough to find a smoothly embedded, connected oriented
surface $C_{0}\subset W$ satisfying (2)-(5) and such that $C_{0}$ meets
$\partial W$ transversally in a link which is isotopic to $\overline{L}$, for
then we can perform an isotopy on $C_{0}$ supported in a neighbourhood of
$\partial W$ so as to satisfy (1) while still maintaining conditions (2)-(5).

Let $C\subset\mathbb{CP}^{2}$ be a non-singular algebraic curve of degree $d$,
obtained by deforming $\overline{\Gamma}$ within the space of degree $d$
algebraic curves. In particular, $C$ is connected. By choosing $C$ to be
sufficiently close to $\overline{\Gamma}$, we can assume that the intersection
of $C$ with $S^{3}=\partial B$ remains transverse and that $C\cap\partial B$
is isotopic to $L$. Similarly, we can assume that $C$ meets $S$ transversally.
Now take $C_{0}=C\setminus B$. Then $C_{0}$ meets $\partial W$ transversally
in a link which is isotopic to $\overline{L}$.

We claim that $C_{0}$ satisfies (5). For if not, $C_{0}$ contains a connected
component $U$ which has no boundary. Then $U$ is a complex submanifold of
$\mathbb{CP}^{2}$. Chow’s theorem implies that $U$ is algebraic, so $U$ is an
irreducible component of $C$. But $C$ is irreducible, which implies $C=U$. But
this is impossible as $C$ has non-empty intersection with $\overline{B}$.

$C$ is an algebraic curve of degree $d$, so $C$ and $S$ meet in $6d$ points.
Since $S$ is disjoint from $\overline{B}$, it follows that $C_{0}$ satisfies
(2). Moreover, this implies (4), since $S$ represents $6[\ell]$ and all the
intersections of $S$ and $C_{0}$ are positive.

As $C$ is an algebraic curve of degree $d$, we have $\chi(C)=3d-d^{2}$ by the
degree-genus formula and hence $\chi(C_{0})=3d-d^{2}-\chi(C\cap\overline{B})$.
So to prove (3) it remains to show that $\chi(C\cap\overline{B})=\chi_{4}(L)$.
In fact, this proven in [31]. ∎

##  3\. Adjunction inequalities

We will need an extension of the adjunction inequality to the case of non-
vanishing Bauer–Furuta invariants. Recall that to a compact oriented smooth
manifold $M$ with $b_{1}(M)=0$ and a spinc-structure $\mathfrak{s}$, one may
define an invariant called the Bauer–Furuta invariant, which takes values in a
certain equivariant stable cohomotopy group [2]. In what follows, we will
concern ourselves only with the corresponding non-equivariant Bauer–Furuta
invariant

$BF(M,\mathfrak{s})\in\pi^{st}_{d(M,\mathfrak{s})+1},$

where $\pi^{st}_{k}$ denotes the $k$-th stable homotopy group of spheres and

$d(M,\mathfrak{s})=\frac{c_{1}(\mathfrak{s})^{2}-\sigma(M)}{4}-1+b_{+}(X)$

is the expected dimension of the Seiberg–Witten moduli space.

The following result is the adjunction inequality for Bauer–Furuta invariants.
Special cases of this result have appeared in [10, 11, 12, 22].

######  Proposition 3.1.

Let $M$ be a smooth, compact oriented $4$-manifold with $b_{1}(M)=0$ and
$b_{+}(M)>1$. Let $\mathfrak{s}$ be a spinc-structure on $M$ for
$BF(M,\mathfrak{s})\neq 0$. Suppose $\Sigma$ is a compact, connected, smoothly
embedded surface in $M$ and that $[\Sigma]^{2}\geq 0$. Then:

  * (1)

If $g\geq 1$ then

$2g(\Sigma)-2\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|$

  * (2)

If $g=0$, then $[\Sigma]$ is a torsion class.

###### Proof.

The proof is similar to the case where the Seiberg–Witten invariant of
$(M,\mathfrak{s})$ is non-zero. The main point is that since
$BF(M,\mathfrak{s})\neq 0$, it follows that the Seiberg–Witten equations for
$(M,\mathfrak{s})$ admits a solution for any metric and any perturbation.

Consider first the case that $g\geq 1$ and $[\Sigma]^{2}=0$. The standard neck
stretching argument of Kronheimer–Mrowka [21] implies that there exists a
translation invariant solutions to the Seiberg–Witten equations on
$\Sigma\times\mathbb{R}$, where $\Sigma$ is given a constant scalar curvature
metric and $\mathbb{R}$ the standard Euclidean metric. As in [21], this
implies that $2g(\Sigma)-2\geq\langle[\Sigma],c_{1}(\mathfrak{s})\rangle$.
Reversing orientation on $\Sigma$ if necessary, we obtain
$2g(\Sigma)-2\geq|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|$, which proves
the result in this case.

The case $g=0$ was proven in [22, Theorem 4.5] (note however that the
condition $[\Sigma]\neq 0$ in the statement of [22, Theorem 4.5] should be
replaced with the stronger condition that $[\Sigma]$ is non-torsion. This can
be seen by examination of the proof of [9, Lemma 5.1], on which the proof of
[22, Theorem 4.5] is based).

Lastly, suppose that $g\geq 1$ and $[\Sigma]^{2}=n>0$. By possibly reversing
orientation on $\Sigma$, we can assume that
$\langle[\Sigma],c_{1}(\mathfrak{s})\rangle\geq 0$. Let $M_{n}$ be maniold
obtained by blowing up $M$ at $n$ points disjoint from $\Sigma$. So $M_{n}$ is
diffeomorphic to $M\\#^{n}\overline{\mathbb{CP}^{2}}$. Let $\mathfrak{s}_{i}$
be a spinc-structure on the $i$-th copy of $\overline{\mathbb{CP}^{2}}$ such
that $S_{i}=c_{1}(\mathfrak{s}_{i})$ represents the $i$-th exceptional divisor
and let $\widetilde{\Sigma}$ denote the connected sum of $\Sigma$ with the
$2$-spheres representing $-S_{1},\dots,-S_{n}$. Then $\widetilde{\Sigma}$ has
the same genus as $\Sigma$ and $[\widetilde{\Sigma}]^{2}=0$. Define a spinc-
structure $\widetilde{\mathfrak{s}}$ on $M_{n}$ by gluing together the spinc-
structures $\mathfrak{s},\mathfrak{s}_{1},\dots,\mathfrak{s}_{n}$. Then
$c_{1}(\widetilde{\mathfrak{s}})=c+S_{1}+\cdots+S_{n}$. Since
$\overline{\mathbb{CP}^{2}}$ is negative definite, has vanishing first Betti
number and $d(\overline{\mathbb{CP}^{2}},\mathfrak{s}_{i})=-1$, it follows
that $BF(\overline{\mathbb{CP}^{2}},\mathfrak{s}_{i})\in\pi^{st}_{0}$ is the
identity map. The connected sum formula for Bauer–Furuta invariants [3,
Theorem 1.1], implies that $BF(M_{n},\widetilde{\mathfrak{s}})\neq 0$.
Therefore, we are in the self-intersection zero case, so as shown above we
have

$2g(\Sigma)-2\geq|\langle[\widetilde{\Sigma}],c_{1}(\widetilde{\mathfrak{s}})\rangle|=|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle+n|=|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|+[\Sigma]^{2},$

where we used $\langle[\Sigma],c_{1}(\mathfrak{s})\rangle\geq 0$ and
$[\Sigma]^{2}=n$. This is the adjunction inequality. ∎

######  Theorem 3.2.

Let $X$ be a smooth, compact, oriented $4$-manifold with $b_{1}(X)=0$ and
$b_{+}(X)=3\;({\rm mod}\;4)$. Suppose that there is a spinc-structure
$\mathfrak{s}$ with $SW(X,\mathfrak{s})=1\;({\rm mod}\;2)$. Let $X_{0}$ be the
$4$-manifold with boundary $S^{3}$ obtained by removing an open ball from $X$.
Let $L\subset S^{3}$ be a quasipositive link. Then for any connected,
oriented, smooth, properly embedded surface $\Sigma\subset X_{0}$ bounding
$L$, we have

$-\chi(\Sigma)\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|-\chi_{4}(L).$

###### Proof.

First note that by blowing up $X$ at points disjoint from $\Sigma$, we can
assume $d(X,\mathfrak{s})=0$. Doing this does not alter the value of
$[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|$.

Let $Y=X\\#\mathbb{CP}^{2}$ be the connected sum of $\mathbb{CP}^{2}$ and $X$.
More precisely, we obtain $Y$ by identifying $W=\mathbb{CP}^{2}\setminus B$
and $X_{0}$ along their boundary $S^{3}$. Here we orient $S^{3}$ so that it is
an ingoing boundary of $\mathbb{CP}^{2}$ and an outgoing boundary of $X_{0}$.
Since $S$ is disjoint from $\overline{B}$, we may regard it an an embedded
surface in $Y$. Let $\pi:\widetilde{Y}\to Y$ be the double cover of $Y$
branched over $S$. To see that the branched cover exists, first consider the
double cover $\pi_{Z}:Z\to\mathbb{CP}^{2}$ branched along $\Sigma$. In fact
$Z$ is a $K3$ surface [13, Corollary 7.3.25] ($Z$ is a compact complex surface
with $b_{1}(Z)=0$ and using the adjunction formula, one finds that the
canonical bundle of $Z$ is trivial). Then $\pi^{-1}(B)$ consits of two balls
in $Z$. By removing these balls and gluing in two copies of $X_{0}$, we obtain
$\widetilde{Y}$. Further, this shows that $\widetilde{Y}$ is diffeomorphic to
the connected sum $Z\\#X\\#X$ of a $K3$ surface and two copies of $X$.

Recall that $\pi^{st}_{1}\cong\mathbb{Z}_{2}$ and is generated by the Hopf map
$\eta:S^{3}\to S^{2}$. According to [3, Proposition 4.4], if
$b_{+}(M)=3\;({\rm mod}\;4)$ and $SW(M,\mathfrak{s})=1\;({\rm mod}\;2)$, then
$BF(M,\mathfrak{s})=\eta\in\pi^{st}_{1}$. In particular, this is the case for
$(X,\mathfrak{s})$ and also for the $K3$ surface $Z$, equipped with the unique
spinc structure $\mathfrak{s}_{Z}$ which comes from a spin-structure. Let
$\mathfrak{s}_{\widetilde{Y}}$ denote the spinc-structure on
$\widetilde{Y}=Z\\#X\\#X$ obtained by gluing together the spinc-structures
$\mathfrak{s}_{Z},\mathfrak{s},\mathfrak{s}$. The connected sum formula for
Bauer–Furuta invariants implies that
$BF(\widetilde{Y},\mathfrak{s}_{\widetilde{Y}})=\eta^{3}\in\pi^{st}_{3}$. Now
since $\eta^{3}\neq 0$, we have that the Bauer–Furuta invariant of
$(\widetilde{Y},\mathfrak{s}_{\widetilde{Y}})$ is non-zero.

Let $C_{0}\subset W$ be as in Lemma 2.2 and let
$\Sigma^{\prime}=\Sigma\cup_{L}C_{0}$ be the surface in $Y$ obtained by
attaching $\Sigma$ and $C_{0}$ along their boundaries. We have that
$\Sigma^{\prime}$ is connected, since $\Sigma$ is connected and every
component of $C_{0}$ meets $L$. Let
$\widetilde{\Sigma}=\pi^{-1}(\Sigma^{\prime})$. Since $C_{0}$ meets $S$
transversally, it follows that $\widetilde{\Sigma}$ is a smooth, compact,
embedded surface in $\widetilde{Y}$ and that the restriction of
$\pi:\widetilde{\Sigma}\to\Sigma$ is a branched double cover with $6d$ branch
points. We have that

$[\widetilde{\Sigma}]^{2}=2[\Sigma^{\prime}]^{2}=2[C_{0}]^{2}+2[\Sigma]^{2}=2d^{2}+2[\Sigma]^{2}.$

Now we observe that for any given quasipositive link $L$, we can choose the
algebraic curve $\Gamma$ to have arbitrarily large degree. Indeed, we can
replace $\Gamma$ by the union of $\Gamma$ with any number of lines which are
disjoint from $\overline{B}$. Thus we can take $d$ large enough that
$[\widetilde{\Sigma}]^{2}=2d^{2}+2[\Sigma]^{2}>0$. Having chosen such a
$\Gamma$, we may apply Proposition 3.1 to obtain

$\displaystyle-\chi(\widetilde{\Sigma})$
$\displaystyle\geq[\widetilde{\Sigma}]^{2}+|\langle[\widetilde{\Sigma}],c_{1}(\mathfrak{s}_{\widetilde{Y}})\rangle|$
$\displaystyle\geq
2[\Sigma]^{2}+2d^{2}+2|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|.$

Furthermore, by Riemann–Hurwitz, we have

$\chi(\widetilde{\Sigma})=2\chi(\Sigma^{\prime})-6d=2\chi(\Sigma)+2\chi(C_{0})-6d=2\chi(\Sigma)-2d^{2}-2\chi_{4}(L),$

where we used that $\chi(C_{0})=3d-d^{2}-\chi_{4}(L)$ from Lemma 2.2. Putting
these together, we get

$-\chi(\Sigma)+d^{2}+\chi_{4}(L)\geq[\Sigma]^{2}+d^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|,$

which gives the result. ∎

######  Corollary 3.3.

Let $X$ be a smooth, compact, oriented $4$-manifold with $b_{1}(X)=0$ and
$b_{+}(X)=3\;({\rm mod}\;4)$. Suppose that there is a spinc-structure
$\mathfrak{s}$ with $d(X,\mathfrak{s})=0$ and $SW(X,\mathfrak{s})=1\;({\rm
mod}\;2)$. Let $X_{0}$ be the $4$-manifold with boundary $S^{3}$ obtained by
removing an open ball from $X$. Let $K\subset S^{3}$ be a quasipositive knot.
Then for any connected, oriented, smooth, properly embedded, homologically
trivial surface $\Sigma\subset X_{0}$ bounding $K$, we have $g(\Sigma)\geq
g_{4}(K)$. Thus $g_{4}(K)$ is the minimal genus of a homologically trivial
surface in $X$ bounding $K$.

######  Remark 3.4.

We make some comments on the proof of Theorem 3.2:

  * •

In the proof of Theorem 3.2 it is essential that we pass to the branched cover
of $Y=X\\#\mathbb{CP}^{2}$. This is because $b_{+}(\mathbb{CP}^{2})=1\;({\rm
mod}\;4)$ and so the adjunction inequality Proposition 3.1 can not be directly
applied to $Y$.

  * •

To obtain a bound on $\chi(\Sigma)$, one may consider a simpler strategy of
capping off $X_{0}$ with a $4$-ball $B^{4}$ and closing up $\Sigma$ with a
surface in $B^{4}$ with Euler characteristic equal to $\chi_{4}(L)$. This will
give

$-\chi(\Sigma)\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|+\chi_{4}(L)$

provided $[\Sigma]^{2}\geq 0$ and $[\Sigma]$ is non-torsion. Notice that this
differs from Theorem 3.2 in that the right hand side has $+\chi_{4}(L)$
whereas in Theorem 3.2 we have $-\chi_{4}(L)$. This inequality is usually a
much weaker bound on $\chi(\Sigma)$ than Theorem 3.2. For example, if $L$ is a
knot then $\chi_{4}(L)=1-2g_{4}(L)$, so $-\chi_{4}(L)\geq\chi_{4}(L)$, except
when $g_{4}(L)=0$.

  * •

Our inequality may be compared with similar adjunction-type inequalities in
[22]. For each spinc-structure for which the Ozsváth–Szabó mixed invariant
$\Phi_{X,\mathfrak{s}}$ is non-zero, we get

$2g(\Sigma)-2\geq[\Sigma]^{2}+\langle[\Sigma],c_{1}(\mathfrak{s})\rangle-2\nu^{+}(\overline{K})$

provided either $[\Sigma]^{2}\geq 2\nu^{+}(\overline{K})$ or $X$ has
Ozsváth–Szabó simple type [22, Theorem 1.1]. Here $\nu^{+}(K)$ is the
concordance invariant constructed by Hom and Wu [17]. Note that $\nu^{+}$ is
always non-negative, so this inequality is weaker than Theorem 3.2, unless
$2-2\nu^{+}(\overline{K})>2g_{4}(K)$ which can only happen if $K$ is slice.

  * •

In the proof of Theorem 3.2, it is crucial that $L$ is quasipositive. This
ensures that the number of intersection points of
$\Sigma^{\prime}=\Sigma\cup_{L}C_{0}$ is exactly
$6d=|\langle[\Sigma^{\prime}],[S]\rangle|$. For non-quasipositive knots, Lemma
2.2 must generally fail since there are knots which do not satisfy Theorem
3.2. For example, the left handed trefoil $K=\overline{T_{2,3}}$ is $H$-slice
in $K3$ [22, Example 2.5], but $g_{4}(K)=1$. This implies that Lemma 2.2 fails
for $K$.

###### Proof of Corollary 1.4:.

Suppose $a$ can be represented by a surface $S\subset X$ satisfying
$2g(S)-2=a^{2}+\langle a,c_{1}(\mathfrak{s})\rangle$. Now let
$X_{0}=X\setminus B$, where the ball $B$ is chosen with $\overline{B}$
disjoint from $S$. Let $K\subset S^{3}$ be a quasipositive knot. Choose a
connected, smooth, properly embedded surface $\Sigma\subset X_{0}$ bounding
$K$, supported in a collar neighbourhood of $\partial X_{0}$ and having genus
equal to $g_{4}(K)$. Let $\Sigma^{\prime}\subset X_{0}$ be the surface
obtained by attaching a handle joining $\Sigma$ to $S$. Then
$g(\Sigma^{\prime})=g(S)+g_{4}(K)$ and $\Sigma^{\prime}$ is a properly
embedded surface representing the class $a$ and bounding $K$. Therefore

$g_{4}(K,X,a)\leq g(\Sigma^{\prime})=g(S)+g_{4}(K)=\frac{a^{2}+\langle
a,c_{1}(\mathfrak{s})\rangle}{2}+g_{4}(K)+1.$

The inequality $g_{4}(K,X,a)\geq\dfrac{a^{2}+\langle
a,c_{1}(\mathfrak{s})\rangle}{2}+g_{4}(K)$ is obtained by applying Theorem
3.2. ∎

###### Proof of Theorem 1.9:.

Let $\mathfrak{s}$ be a spinc-structure on $X$ such that
$d(X,\mathfrak{s})=-1$. Since $X$ is negative definite this is equivalent to
$c_{1}(\mathfrak{s})^{2}=b_{2}(X)$. Consider the $4$-manifold
$X^{\prime}=K3\\#X$ with spinc-structure
$\mathfrak{s}^{\prime}=\mathfrak{s}_{0}\\#\mathfrak{s}$, where
$\mathfrak{s}_{0}$ is the unique spinc-structure on $K3$ coming from the spin-
structure. Then $b_{+}(X^{\prime})=3$ and
$SW(X^{\prime},\mathfrak{s}^{\prime})=SW(K3,\mathfrak{s}_{0})=1$, by the
blowup formula for Seiberg–Witten invariants. So we may apply Theorem 3.2 to
deduce the inequality

$-\chi(\Sigma)\geq[\Sigma]^{2}+|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|-\chi_{4}(L),$

for any spinc-structure $\mathfrak{s}$ on $X$ for which
$c_{1}(\mathfrak{s})^{2}=b_{2}(X)$. It is easily seen that the maximum of
$|\langle[\Sigma],c_{1}(\mathfrak{s})\rangle|$ over all such spinc-structures
on $X$ equals $\left|[\Sigma]\right|$. Thus we find that

$-\chi(\Sigma)\geq[\Sigma]^{2}+\left|[\Sigma]\right|-\chi_{4}(L).$

∎

## References

  * [1] S. Baader, P. Feller, L. Lewark, L. Liechti, On the topological $4$-genus of torus knots. Trans. Amer. Math. Soc. 370 (2018), no. 4, 2639-2656.
  * [2] S. Bauer, M. Furuta, A stable cohomotopy refinement of Seiberg–Witten invariants. I. Invent. Math. 155 (2004), no. 1, 1-19.
  * [3] S. Bauer, A stable cohomotopy refinement of Seiberg–Witten invariants. II. Invent. Math. 155 (2004), no. 1, 21-40.
  * [4] M. Boileau, L. Fourrier, Knot theory and plane algebraic curves. Chaos Solitons Fractals 9 (1998), no. 4-5, 779-792.
  * [5] M. Boileau, S. Orevkov, Quasi-positivité d’une courbe analytique dans une boule pseudo-convexe. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 9, 825-830.
  * [6] A. Cavallo, The concordance invariant tau in link grid homology. Algebr. Geom. Topol. 18 (2018), no. 4, 1917-1951.
  * [7] T. D. Cochran, S. Harvey, P. Horn, Filtering smooth concordance classes of topologically slice knots. Geom. Topol. 17 (2013), no. 4, 2103-2162.
  * [8] A. Conway, M. Nagel, Stably slice disks of links. J. Topol. 13 (2020), no. 3, 1261-1301.
  * [9] R. Fintushel, R. J. Stern, Immersed spheres in $4$-manifolds and the immersed Thom conjecture. Turkish J. Math. 19 (1995), no. 2, 145-157.
  * [10] M. Furuta, Y. Kametani, H. Matsue, H. Spin $4$-manifolds with signature $=-32$. Math. Res. Lett. 8 (2001), no. 3, 293-301.
  * [11] M. Furuta, Y. Kametani, H. Matsue, N. Minami, Homotopy theoretical considerations of the Bauer–Furuta stable homotopy Seiberg–Witten invariants. Proceedings of the Nishida Fest (Kinosaki 2003), 155-166, Geom. Topol. Monogr., 10, Geom. Topol. Publ., Coventry, (2007).
  * [12] M. Furuta, Y. Kametani, N. Minami, Stable-homotopy Seiberg–Witten invariants for rational cohomology $K3\\#K3$’s. J. Math. Sci. Univ. Tokyo 8 (2001), no. 1, 157-176.
  * [13] R. E. Gompf, A. Stipsicz, $4$-manifolds and Kirby calculus. Graduate Studies in Mathematics, 20. American Mathematical Society, Providence, RI, (1999). xvi+558 pp.
  * [14] M. J. D. Hamilton, The minimal genus problem for elliptic surfaces. Israel J. Math. 200 (2014), no. 1, 127-140.
  * [15] M. Hedden, Notions of positivity and the Ozsváth–Szabó concordance invariant. J. Knot Theory Ramifications 19 (2010), no. 5, 617-629.
  * [16] M. Hedden, K. Raoux, Knot Floer homology and relative adjunction inequalities, arXiv:2009.05462 (2020).
  * [17] J. Hom, Z. Wu, Four-ball genus bounds and a refinement of the Ozsváth–Szabó tau invariant. J. Symplectic Geom. 14 (2016), no. 1, 305-323.
  * [18] N. Iida, A. Mukherjee, M. Taniguchi, An adjunction inequality for the Bauer–Furuta type invariants, with applications to sliceness and $4$-manifold topology, arXiv:2102.02076 (2021).
  * [19] A. Kjuchukova, A. N. Miller, A. Ray, S. Sakallı. Slicing knots in definite 4-manifolds, arXiv:2112.14596 (2021).
  * [20] H. Konno, J. Miyazawa, M. Taniguchi, Involutions, knots, and Floer K-theory, arXiv:2110.09258 (2021).
  * [21] P. B. Kronheimer, T. S. Mrowka, The genus of embedded surfaces in the projective plane. Math. Res. Lett. 1 (1994), no. 6, 797-808.
  * [22] C. Manolescu, M. Marengon, L. Piccirillo. Relative genus bounds in indefinite four-manifolds, arXiv:2012.12270 (2020).
  * [23] C. Manolescu, M. Marengon, S. Sarkar, M. Willis. A generalization of Rasmussen’s invariant, with applications to surfaces in some four-manifolds. Duke Math. J. (to appear), arXiv:1910.08195 (2019).
  * [24] K. Murasugi, On a certain numerical invariant of link types. Trans. Amer. Math. Soc. 117 (1965), 387-422.
  * [25] M. A. Nouh, Genera and degrees of torus knots in $\mathbb{CP}^{2}$. J. Knot Theory Ramifications 18 (2009), no. 9, 1299-1312.
  * [26] P. Ozsváth, Z. Szabó, The symplectic Thom conjecture. Ann. of Math. (2) 151 (2000), no. 1, 93-124.
  * [27] P. Ozsváth, Z. Szabó, Knot Floer homology and the four-ball genus. Geom. Topol. 7 (2003), 615-639.
  * [28] O. Plamenevskaya, Bounds for the Thurston–Bennequin number from Floer homology. Algebr. Geom. Topol. 4 (2004), 399-406.
  * [29] L. Rudolph, Algebraic functions and closed braids. Topology 22 (1983), no. 2, 191-202.
  * [30] L. Rudolph, Some topologically locally-flat surfaces in the complex projective plane. Comment. Math. Helv. 59 (1984), no. 4, 592-599.
  * [31] L. Rudolph, Quasipositivity as an obstruction to sliceness. Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 1, 51-59.
  * [32] K. Sato, Topologically slice knots that are not smoothly slice in any definite $4$-manifold. Algebr. Geom. Topol. 18 (2018), no. 2, 827-837.