# Toward Wheeled Mobility on Vertically Challenging Terrain:
Platforms, Datasets, and Algorithms

Aniket Datar∗, Chenhui Pan∗, Mohammad Nazeri, and Xuesu Xiao ∗Equally
contributing authors. All authors are with the Department of Computer Science,
George Mason University {adatar, cpan7, mnazerir<EMAIL_ADDRESS>
###### Abstract

Most conventional wheeled robots can only move in flat environments and simply
divide their planar workspaces into free spaces and obstacles. Deeming
obstacles as non-traversable significantly limits wheeled robots’ mobility in
real-world, extremely rugged, off-road environments, where part of the terrain
(e.g., irregular boulders and fallen trees) will be treated as non-traversable
obstacles. To improve wheeled mobility in those environments with _vertically
challenging terrain_ , we present two wheeled platforms with little hardware
modification compared to conventional wheeled robots; we collect datasets of
our wheeled robots crawling over previously non-traversable, vertically
challenging terrain to facilitate data-driven mobility; we also present
algorithms and their experimental results to show that conventional wheeled
robots have previously unrealized potential of moving through vertically
challenging terrain. We make our platforms, datasets, and algorithms publicly
available to facilitate future research on wheeled mobility.111Website:
https://cs.gmu.edu/~xiao/Research/Verti-Wheelers/

##  I INTRODUCTION

Building mobile robots that are capable of reaching as many places as possible
has long been a dream for many robotics researchers. Indeed, autonomous mobile
robots have ventured into remote deserts for scientific exploration [1],
explored extraterrestrial planets to look for signs of life [2, 3], and
assisted with search and rescue missions in hazardous or difficult-to-reach
environments [4, 5, 6, 7].

One particular thrust in this area of research is the development of ground
robots capable of navigating vertically challenging terrain (e.g., steep
slopes, rocky outcroppings, and uneven surfaces) [8, 9, 10]. Achieving
reliable and robust mobility in these environments is challenging due to the
intricate nature of the terrain, the complex vehicle-terrain interactions, the
adverse impact caused by gravity, the potential deformation of the vehicle
chassis, and the varing traction between the wheels and the terrain. Despite
these difficulties, such mobility has been made possible mainly through
advancement in hardware, including the development of specialized robot
platforms [4, 10, 11, 3, 12] and the use of new materials [13]. These robots
are capable of climbing walls [13], scaling cliffs [10], and traversing rough
terrain with ease [11, 12], making them suitable for a wide range of real-
world applications.

However, despite these hardware advances, the vast majority of currently
available ground robots are still conventional wheeled platforms, whose
mobility is mostly limited to flat surfaces. Most existing autonomous
navigation systems for wheeled robots merely divide their presumably planar
workspaces into free spaces (traversable) or obstacles (non-traversable),
significantly reducing these robots’ reachability in the real world,
especially outdoor off-road environments where vertical protrusions from the
ground are not uncommon. Our work is motivated by such limitations and aims at
expanding the mobility of these widely available wheeled robot platforms so
that they can venture into vertically challenging environments, which would
otherwise be deemed as obstacles (non-traversable) or require specialized
hardware. Note that the _vertically challenging_ terrain we are interested in
conquering requires driving wheeled robots over irregular and complex
obstacles and is therefore much more difficult compared to simply driving on
_non-flat_ environments.

Figure 1: The Verti-Wheelers: Conventional Wheeled Vehicles Moving through
Vertically Challenging Terrain.

To this end, we first present an open-source design of two wheeled robot
platforms, the Verti-Wheelers, which are representative of the majority of
existing conventional ground mobile robot platforms, and hypothesize that
conventional wheeled robots can also navigate many vertically challenging
terrain, which are normally considered as non-traversable obstacles (Fig. 1).
Second, considering the difficulty in creating accurate analytical models to
solve this problem, we collect and make publicly available datasets to
facilitate data-driven approaches. Third, we present three algorithms to
autonomously drive wheeled robots over vertically challenging terrain: an
open-loop, a classical rule-based, and an end-to-end learning-based approach,
which can be used as baselines to benchmark future research. We provide
extensive experimental results of all three algorithms in both an indoor
controlled testbed and outdoor natural environments, confirm our hypothesis,
and point out future research directions for wheeled mobility on vertically
challenging terrain. To the best of our knowledge, no existing work has
tackled vertically challenging terrain with conventional wheeled robots in the
real world.

##  II RELATED WORK

This section discusses conventional wheeled robot mobility, reviews novel
hardware to expand ground mobility beyond the capability of conventional
wheeled platforms, and surveys data-driven approaches to improve robot
mobility.

###  II-A Conventional Wheeled Mobility

Due to their simplicity and efficiency, wheeled robots have been widely used
in various applications such as scientific exploration [1], autonomous
delivery [14], and search and rescue [8]. Equipped with differential-drive
mechanism [15], Ackermann steering [16], or omnidirectional wheels [17], these
robots can move through their planar workspaces and reach their goals.

One basic capability for wheeled robots is obstacle avoidance. Roboticists
have developed autonomous navigation systems which first divide their planar
workspaces into free spaces and obstacles and then move robots to their goals
without collisions [18, 19, 20, 21, 22]. Researchers have also investigated
off-road navigation [23, 24, 25, 26], in which the planar workspace includes a
variety of terrain types, such as gravel, grass, mud, sand, and snow. Instead
of the binary free/obstacle depiction of the planner workspace, off-road
navigation systems usually build semantic or traversability maps [27, 28],
e.g., gravel is better to drive on than grass, which is better than mud.
However, for vertical protrusions from the ground, e.g., large boulders or
fallen tree trunks, these systems still treat them as non-traversable
obstacles due to mobility limitations caused by the wheels.

However, in real-world unstructured environments, vertical protrusions from
the ground are not necessarily non-traversable, even for wheeled robots.
Consider the motor-sport of rock crawling, in which human drivers are able to
drive off-road vehicles over obstacles of similar size to the vehicles
themselves. Deeming all these vertically challenging terrain as completely
non-traversable largely limits the mobility of wheeled robots in the real
world. In this work, we aim at expanding wheeled mobility to these previously
impossible, vertically challenging terrain.

###  II-B Novel Hardware for Vertically Challenging Terrain

One way to improve ground mobility is through novel hardware design. To
overcome the limitations of wheels, researchers have developed robots with
alternative actuation: wheels can be replaced by legs when facing extremely
rugged terrain [29, 30]; active suspensions are widely used on planetary
rovers to achieve better maneuverability, reconfigurability, and therefore
mobility by allowing the chassis to actively conform to different underlying
terrain [31, 32]; tracked robots are not particularly novel, but the increased
track surface contact compared to wheels makes them more appropriate on
vertically challenging terrain, while compromising efficiency on flat
surfaces. Another thrust is to develop new adhesive materials so that robots
can adhere to steep or even vertical surfaces [13].

Despite their superior mobility on vertically challenging environments, these
costly, specialized hardware require extra engineering effort and may
adversely affect vehicles’ mobility and efficiency on flat environments.
Considering that most ground robots are wheeled with no or passive suspension
systems, our research aims at equipping those conventional wheeled platforms
with enhanced mobility to autonomously move through vertically challenging
terrain.

###  II-C Data-Driven Robot Mobility

Thanks to the recent advancement in machine learning, data-driven approaches
have been leveraged to improve robot mobility [33]. Researchers have
investigated using learning to achieve adaptive navigation in a variety of
environments [34, 35, 36, 37, 38, 39, 40], agile navigation in highly
constrained spaces [41, 42, 43], high-speed off-road navigation [23, 24, 25,
26], visual-only navigation [44, 45, 46, 47], and socially compliant
navigation [48, 49, 50, 51, 34, 52, 53, 54]. Learning from data using either
imitation learning [55, 36, 44] or reinforcement learning [56, 57, 58] removes
the necessity of building analytical models of the environments, such as
vehicle-terrain or human-robot interactions, and alleviates the burden of
crafting delicate cost functions [34, 59] or tuning unintuitive parameters
[35, 36, 37, 38, 39].

The problem of autonomously driving wheeled robots through vertically
challenging terrain shares many aforementioned difficulties. Therefore, we
hypothesize that data-driven approaches are one avenue toward enabling
enhanced wheeled mobility on previously impossible, vertically challenging
terrain.

##  III PLATFORMS AND DATASETS

Figure 2: Components of the Verti-Wheelers.

Considering that most existing mobile robots are wheeled, our goal is to equip
these easily available wheeled platforms with no or little specialized
hardware modification. To assure it is physically feasible for wheeled
platforms to move through vertically challenging terrain, we identify the
following seven desiderata for their hardware:

All-Wheel Drive (D1): Due to the high possibility of losing wheel contact with
the irregular terrain, our wheeled platforms need to be all-wheel drive to
maximize traction.

Independent Suspensions (D2): Considering the non-flatness of vertically
challenging terrain, independent suspensions allow the chassis to conform to
and most wheels to contact with the ground to maintain stability and traction.

Differential Lock (D3): While traditional differentials aim at increasing
efficiency during vehicle turning on flat surfaces, loosing left wheel contact
will cause the right wheel to lose traction, or vice versa. Therefore, our
wheeled platforms need to be able to lock the differential(s) when necessary.

Low/High Gear (D4): Although high gear improves energy efficiency on flat
surfaces, to mitigate downward force caused by gravity, our platforms require
high torque and low gear on vertically challenging terrain.

Wheel Speed / RPM Sensing (D5): Precise end-point wheel speed measurement
(instead of motor encoder readings that will be skewed by the drivetrain and
differential) allows the vehicle to sense wheel slippage before losing wheel
contact.

Ground Speed Sensing (D6): Our platforms need an easy way to identify when the
vehicle is getting stuck, which is very likely to happen on vertically
challenging terrain.

Actuated Perception (D7): Due to the increased vehicle pose variability (i.e.,
pitch and roll in addition to yaw) when crawling over vertically challenging
terrain, traditional fixed onboard sensors may lose sight of the terrain.
Therefore, our platforms need to be able to point their sensors toward the
terrain, regardless of how the vehicle pose varies.

###  III-A Verti-6-Wheeler (V6W)

For the mechanical components in D1 to D4, we base our platform on an off-the-
shelf, three-axle, six-wheel, all-wheel-drive, off-road vehicle chassis from
Traxxas. The length of the chassis is 0.863m, with 0.471m front-to-middle and
0.603m front-to-rear axle wheelbase. The total height and width of V6W after
outfitting all mechanical and electrical components is 0.200m and 0.249m. D1
and D2 are therefore achieved. We use an Arduino Mega micro-controller to
lock/unlock the front and rear differential (D3) and switch between low and
high gear (D4) through three servos. For D5, we install four magnetic sensors
on the front and middle axles, and eight magnets per wheel to sense the wheel
rotation. We choose not to install magnetic sensors on the rear axle
considering the rear differential lock is shared by both middle and rear
axles. For D6, we install a Crazyflie Flow deck v2 sensor on the chassis
facing downward, providing not only 2D ground speed ($x$ and $y$) but also
distance between the sensor and the ground ($z$). We choose an Azure Kinect
RGB-D camera due to its high-resolution depth perception at close range. For
D7, we add a tilt joint for the camera actuated by a servo. We use a
complementary filter and camera Inertia Measurement Unit (IMU) readings to
estimate the camera orientation and a PID controller to regulate the camera
pitch angle. For the core computation unit, we use an NVIDIA Jetson AGX Orin
to provide both onboard CPU and GPU computation. To interface all low-level
sensors and actuators, we use an Arduino Mega micro-controller.

Figure 3: System Architecture of the Verti-Wheelers.

###  III-B Verti-4-Wheeler (V4W)

Considering that most wheeled robots have four wheels, we also build a four-
wheeled platform based on an off-the-shelf, two axle, four-wheel-drive, off-
road vehicle from Traxxas. The length of the V4W chassis is 0.523m, with a
0.312m wheelbase. The total height and width of V4W is 0.200m and 0.249m. Most
components remain the same as the six-wheeler, but to accommodate the small
footprint and payload capacity of the four-wheeler, we replace the NVIDIA
Jetson AGX Orin with a Xavier NX. The mechanical and electrical components and
system architecture for both V4W and V6W are shown in Fig. 2 and 3
respectively.222https://github.com/RobotiXX/Verti-Wheelers

###  III-C Testbed

Figure 4: Custom-Built Testbed with V6W and V4W and Example Traversals by the
Three Algorithms (OL, RB, and BC).

In addition to outdoor field experiments, we construct a custom-built indoor
testbed for vertically challenging terrain for controlled and repeatable
experiments: hundreds of rocks and boulders of an average size of 30cm (at the
same scale of the V6W and V4W) are randomly laid out and stacked up on a
3.1$\times$1.3m test course. The highest elevation of the test course can
reach up to 0.5m, more than twice the height of both vehicles, as shown in
Fig. 4. The testbed is highly reconfigurable by shuffling the pose of each
rock/boulder.

###  III-D Datasets

Considering the difficulty in representing surface topography and modeling
complex vehicle dynamics and the recent success in data-driven mobility [33],
we collect two datasets with the two wheeled robots on our custom-built
testbed.333https://dataverse.orc.gmu.edu/dataset.xhtml?persistentId=doi:10.13021/orc2020/QSN50Q
We reconfigure our testbed multiple times and both robots are manually driven
through different vertically challenging terrain. We collect the following
data streams from the onboard sensors and human teleoperation commands: RGB
($1280\times 720\times 3$) and depth ($512\times 512$) images $i$, wheel speed
$w$ (4D float vector for four wheels), ground speed $g$ (relative movement
indicators along $\Delta x$ and $\Delta y$ and displacement along $z$, along
with two binary reliability indicators for speeds and displacement),
differential release/lock $d$ (2D binary vector for both front and rear
differentials), low/high gear switch $s$ (1D binary vector), linear velocity
$v$ (scalar float number), and steering angle $\omega$ (scalar float number).
Each dataset $\mathcal{D}$ is therefore
$\mathcal{D}=\\{i_{t},w_{t},g_{t},d_{t},s_{t},v_{t},\omega_{t}\\}_{t=1}^{N}$,
where $N$ indicates the total number of data frames.

The initial release of the V6W and V4W datasets include 50 and 64 teleoperated
trials of 46667 and 70143 data frames of the 6-wheeler and 4-wheeler crawling
over different vertically challenging rock/boulder courses respectively. To
assure the human demonstrator has only access to the same perception as the
robot, teleoperation is conducted in a first-person-view from the onboard
camera, rather than a third-person-view. Fig. 5 shows an example data frame of
the RGB and depth images $i$ of the V4W dataset.

With a more mechanically capable chassis and two more wheels, the demonstrator
can demonstrate crawling behaviors on V6W at ease, i.e., mostly driving
forward, slowing down when approaching an elevated terrain patch, and only
using the steering to circumvent very difficult obstacles or ditches in front
of V6W. On the other hand, the demonstration of V4W takes much more effort,
and the demonstrator needs to carefully control both the linear velocity and
steering angle at fine resolution to negotiate through a variety of vertically
challenging terrain.

Figure 5: Example RGB (Left) and Depth (Right) Images in the Verti-Wheelers
Dataset.

##  IV ALGORITHMS

Requiring no or little specialized hardware in addition to conventional
wheeled platforms, we develop three algorithms to drive our robots through
vertically challenging terrain: an open-loop, a classical rule-based, and an
end-to-end learning-based controller.

###  IV-A Open-Loop Controller

As a baseline, we implement an open-loop controller that drives the robots
toward vertically challenging terrain previously deemed as non-traversal
obstacles, simply treating them as free spaces. Our open-loop controller locks
the differentials and uses the low gear all the time. We set a constant linear
velocity to drive the robot forward. No onboard perception is used for the
open-loop controller.

###  IV-B Classical Rule-Based Controller

We design a classical rule-based controller based on our heuristics on off-
road driving: we lock the corresponding differential when we sense wheel
slippage; we use the low gear when ascending steep slopes; when getting stuck
on rugged terrain, we first increase the wheel speed and attempt to move the
robot forward beyond the stuck point; if unsuccessful, we then back up the
robot to get unstuck, and subsequently try a slightly different route. With
the hardware described in Sec. III, our robots are able to perceive all
aforementioned information with the onboard sensors and initiate corresponding
actions. The finte state machine of the rule-based controller is shown in Fig.
6.

Figure 6: Finite State Machine of the Rule-Based Controller

###  IV-C End-to-End Learning-Based Controller

We also develop an end-to-end learning-based controller to enable data-driven
mobility. We aim at learning a motion policy that maps from the robots’
onboard perception to raw motor commands to drive the robots over vertically
challenging terrain. Utilizing the datasets we collect, we adopt an imitation
learning approach, i.e., Behavior Cloning [60], to regress from perceived
vehicle state information to demonstrated actions.

TABLE I: Number of Successful Trials (Out of 10) and Mean Traversal Time (of
Successful Trials in Seconds) with Variance

V6W V4W OL RB BC6 BC4 OL RB BC6 BC4 Easy 5 ($20.7\pm 1.7$) 8 ($19.2\pm 3.9$) 9
($13.8\pm 8.2$) 10 ($11.6\pm 1.9$) 6 ($17.7\pm 3.8$) 6 ($13.4\pm 2.5$) 7
($17.2\pm 6.7$) 9 ($14.1\pm 7.7$) Medium 6 ($15.4\pm 0.9$) 9 ($14.8\pm 2.2$) 9
($14.6\pm 11.2$) 10 ($13.6\pm 2.3$) 4 ($15.6\pm 14.2$) 6 ($12.9\pm 1.8$) 3
($19.2\pm 10.6$) 8 ($13.7\pm 1.6$) Difficult 3 ($24.1\pm 2.6$) 6 ($14.3\pm
1.9$) 6 ($15.7\pm 18.5$) 9 ($14.9\pm 2.9$) 3 ($19.7\pm 29.4$) 5 ($16.8\pm
20.5$) 3 ($23.3\pm 43.4$) 7 ($14.9\pm 8.2$)

Our motion policy $\pi_{\theta}:\mathcal{X}\rightarrow\mathcal{A}$, is a
mapping from vehicle state space $\mathcal{X}$ to vehicle action space
$\mathcal{A}$, parameterized by a set of parameters $\theta$. To instantiate a
vehicle state $x\in\mathcal{X}$, we use all (or a subset of) onboard
perception, including RGB and depth image $i$, wheel speed $w$, and ground
speed $g$, while the vehicle action $a\in\mathcal{A}$ includes differential
release/lock $d$, low/high gear switch $s$, and most importantly, linear
velocity $v$ and steering angle $\omega$:

$x=(i,w,g),~{}a=(d,s,v,\omega).$ (1)

To train $\pi_{\theta}$, we minimize an end-to-end behavior cloning loss to
seek for the optimal parameter set $\theta^{*}$:

$\begin{split}\theta^{*}&=\operatorname*{argmin}_{\theta}\sum_{(x,a)\in\mathcal{D}}||a-\pi_{\theta}(x)||_{H},\end{split}$
(2)

where $||v||_{H}=v^{T}Hv$ is the induced norm by a diagonal matrix $H$ with
positive real entries to weigh each dimension of the action. We instantiate
$\pi_{\theta}$ as a neural network and use backpropagation to find the
appropriate set of parameters $\theta^{*}$.

##  V EXPERIMENTS

All three algorithms in Sec. IV are implemented on both V6W and V4W in Sec.
III, with the end-to-end learning-based controller learned from the collected
datasets.

###  V-A Implementations

Considering the extremely rugged testbed (Fig. 4), all three methods lock both
differentials and use low gear. The open-loop controller (OL) uses a constant
linear velocity of 0.5m/s. When the rule-based (RB) controller gets stuck for
2 seconds, the wheel speed is gradually increased from 0.5m/s to 0.75m/s. If
unsuccessful for 3 seconds, the robots back up for 2 seconds at 0.5m/s, and
then continue forward at 0.75m/s steering 18° left for 2 seconds and right for
2 seconds.

For the end-to-end Behavior Cloning (BC) controller, we use both datasets on
both vehicles. Specifically, in addition to using their own corresponding
dataset, we also cross-deploy the learned model on the other platform, i.e.,
learned model from V6W data deployed on V4W and vice versa. Such a cross-
deployment aims at revealing the relationship between different mechanical
capabilities and qualities of the training data. Similar to OL and RB, BC
maintains locked differentials and low gear. Our BC implementation takes
vehicle state $x=(i)$ (depth image only) and action $a=(v,\omega)$ (Eqn. 1)
and seeks optimal parameter $\theta^{*}$ (Eqn. 2) in a neural network. BC uses
a RESNET18 to process the $224\times 224$ depth images (down sampled from
$512\times 512$) before feeding the 512-dimensional embedding into four fully
connected layers with 256, 128, 64, and 2 neurons with ReLU activation except
the last layer.

###  V-B Results

Three different test courses are built by reconfiguring the rocks/boulders on
the testbed (Fig. 4), whose difficulty levels range from easy, medium, to
difficult. For the difficult level, we add wooden blocks to the rock/boulder
course, which do not exist in the datasets, to test BC’s generalizability. For
each test course, we run four different approaches, i.e., OL, RB, BC, and BC
with cross-deployment (we denote the model trained with the V6W and V4W
dataset as BC6 and BC4 respectively), each ten trials, running from both
directions of the course. We report both number of successful trials (out of
10 attempts) and mean traversal time (for the successful trials in seconds)
with variance of all 240 experiment trials in Tab. I. A failure trial can
either be the vehicle getting stuck or tipping over on the test course. For
all four approaches, we start the vehicles at the same starting location and
orientation facing the test course.

For a certain vehicle on a certain difficulty level, in general BC achieves
higher success rate than both OL and RB, with OL most frequently getting stuck
or tipping over. Among all successful trials, BC mostly achieves the shortest
traversal time, but not always, because BC learns to slow down to smoothly go
through rugged terrain while OL and RB may drive aggressively. With increasing
difficulty level, success rate decreases and traversal time increases,
although V6W’s performance is not very sensitive to the Easy and Medium
levels, considering its superior mechanical capability.

One very interesting finding is the cross-deployment results. While V4W
trained with its own dataset (BC4) outperforms V4W cross-trained with V6W’s
dataset (BC6), V6W’s performance is opposite: V6W performs better when cross-
trained with V4W’s dataset (BC4) compared to when being trained with its own
dataset (BC6). Such a discrepancy is caused by the different demonstrations in
the V6W and V4W datasets, which are further caused by the difference in the
two vehicles’ mechanical capabilities: as mentioned above, the demonstrator
can drive the more mechanically capable V6W through different test courses at
ease without the need to constantly adjust the speed and steering, while the
limited mechanical capability of V4W requires fine-grained control, causing
the quality of the BC4’s dataset to exceed that of BC6’s. Therefore, even V6W,
a different vehicle, can achieve improved mobility with BC4, compared to BC6
with its own dataset. On the other hand, learning from the less careful
driving demonstrations with BC6 jeopardizes the learned mobility of V4W.

###  V-C Outdoor Mobility Demonstration

Figure 7: Outdoor Demonstration: V4W’s right front wheel suspends in the air
(left) and V6W climbs a steeper rock pile than the indoor testbed.

We also deploy both Verti-Wheelers in outdoor vertically challenging
environments beyond the controlled indoor testbed (Fig. 7). The outdoor
environments contain different rock sizes, including both smaller pebbles and
larger boulders compared to the indoor testbed. Furthermore, the outdoor
environments are more jagged, including larger ditches between rocks that may
easily get the vehicles stuck (Fig. 7 left) and steeper slopes that require
stable traction between all wheels and the terrain (Fig. 7 right). While OL
and RB can drive smoothly on small pebbles, they often get stuck on larger and
more jagged boulders. BC is able to generalize well to rocks of similar sizes
as in the indoor testbed, but may occasionally get stuck on large boulders,
wide ditches, and steep slopes as well, showing future research is still
required to improve wheeled mobility on vertically challenging terrain in the
real world.

##  VI CONCLUSIONS AND FUTURE WORK

This paper presents two vehicle platforms, demonstration datasets, and three
algorithms to unlock the previously unrealized potential of wheeled mobility
on vertically challenging terrain. The two platforms, with minimal hardware
modification to conventional wheeled robots, achieve seven desiderata to
tackle such challenging environments. We open-source our hardware design,
software implementation, and demonstration datasets to facilitate future
research. The experiment results of the three algorithms confirm our
hypothesis that conventional wheeled robots have the mechanical capability of
navigating many vertically challenging terrain, which are normally considered
as non-traversable obstacles, especially with the help of data-driven
approaches.

This paper opens up a new direction of research to achieve superior robot
mobility beyond their original design with only limited mechanical capability.
Despite the fact that the simple Behavior Cloning algorithm works well on many
vertically challenging terrain, it still fails in many more challenging
scenarios, such as in more difficult test courses than the Difficult level in
Tab. I or in the natural outdoor environments (Fig. 7). However, these
environments can be conquered by human teleoperation and are therefore within
the vehicles’ mechanical limit. One promising future research direction is to
explicitly model terrain traversability instead of implicitly learning it in a
motion policy. It is also possible to design or learn a vehicle dynamics model
on vertically challenging terrain. Another interesting direction is to combine
autonomous crawling and goal-oriented navigation with obstacle avoidance,
i.e., real obstacles that are absolutely beyond the robots’ mechanical
capability. Last but not least, considering the interesting findings in the BC
cross-deployment experiments, Verti-Wheelers provide ideal physical robot
platforms for research in transfer learning for cross-platform mobility,
learning with embodiment mismatch, and meta learning for general vehicle
mobility and maneuverability.

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# Chainmail links and $L$-spaces

Ian Agol  University of California, Berkeley
970 Evans Hall #3840
Berkeley, CA 94720-3840<EMAIL_ADDRESS>
(Date: June 2023)

###### Abstract.

In this note we prove that alternating chainmail links are $L$-space links.
The proof is inspired by corresponding proofs for double branched covers of
alternating links. We also more generally show that flat augmented chainmail
links are generalized $L$-space links. Some other properties of these links
are also considered.

###### 2010 Mathematics Subject Classification:

57M

Ian Agol is supported by a Simons Investigator grant #376200

##  1\. Introduction

In the process of exploring lens space Dehn fillings on knots, Ozsvath and
Szabo introduced the notion of $L$-spaces [OS3, Definition 1.1] and proved
that double branched covers of non-split alternating links are $L$-spaces
[OS4, Lemma 3.2, Proposition 3.3]. $L$-spaces are closed rational homology
3-spheres $Y$ whose Heegaard Floer homology is minimal (like a lens space), in
the sense that $rank(\widehat{HF}(Y))=|H_{1}(Y;\mathbb{Z})|$. $L$-spaces are
interesting manifolds since they do not admit taut orientable foliations [OS1,
Theorem 1.4], [KR, Corollary 1.6]; the converse has been conjectured for
irreducible closed 3-manifolds by Juhasz [Juh, Conjecture 5]. Related concepts
have been defined for other versions of Floer homology, for example an
instanton $L$-space $Y$ satisfies $dimI^{\\#}(Y)=|H_{1}(Y;\mathbb{Z})|$ [BS,
Definition 1.13]. It is conjectured that
$dimI^{\\#}(Y)=dim\widehat{HF}(Y;\mathbb{C})$ [KM, Conjecture 7.24].

Quasi-alternating links [OS4, Definition 3.1] are a class of links whose
double branched covers are $L$-spaces, leading to the question of whether
there are $L$-spaces which are not double branched covers of links? Examples
of hyperbolic $L$-spaces were given which do not admit any symmetries, and
hence cannot be double branched covers of links [DHL, Theorem 1.2].
Subsequently asymmetric hyperbolic $L$-space knots were discovered leading to
further examples [BL], [ABGK+]. Meanwhile, Boyer-Gordon-Watson conjectured
that $L$-spaces do not have orderable fundamental group [BGW, Conjecture 1],
and they proved this for branched double covers of alternating links [BGW,
Theorem 4], with a subsequent simpler proof given by Greene [Gre2, Theorem
2.1]. Greene’s proof relies on a presentation of the fundamental group
associated to a certain Dehn filling on a link described by Ozsvath-Szabo
which gives a Kirby diagram for a double branched cover of a link (described
in [Gre1, Section 3.1]).

Polyak introduced the notion of chainmail graphs and corresponding chainmail
links, such that surgery on this class of links gives all closed 3-manifolds
[Pol]. He indicated that there exists a finite set of moves on chainmail
graphs (analogous to Reidemeister moves) generating the equivalence relation
of chainmail graphs giving the same 3-manifold. This is just a restricted
class of Kirby diagrams, motivated by an alternative proof of Kirby’s theorem
given by Matveev-Polyak using a tangle presentation for the mapping class
group [MP]. The surgery diagrams of Ozsvath-Szabo for double branched covers
of links (mentioned in the last paragraph) are certain surgeries on chainmail
links. We learned about these topics from a Mathoverflow post by Lucas Culler
[hc].

Gorsky and Nemethi introduced the terminology of $L$-space link (links such
that sufficiently large positive surgeries are $L$-spaces) and proved that
algebraic links are $L$-space links [GN2, Theorem 2]. Subsequently Yajing Liu
explored the properties of $L$-space links and introduced the terminology of
generalized $L$-space links which also admit infinitely many $L$-space
fillings [Liu, Definition 2.9]. He asked whether every 3-manifold is a surgery
on a generalized $L$-space link [Liu, Question 1.19]? The main result of this
paper is that alternating chainmail links are $L$-space links. Thus each
double branched cover associated to an alternating diagram of a link fits into
infinite families of $L$-spaces. More generally, we prove that fully augmented
chainmail links are generalized $L$-space links, hence answering Liu’s
question positively by Polyak’s observation that every 3-manifold is surgery
on a chainmail link (and hence on a fully augmented chainmail link). Since the
symmetries of flat augmented links are understood, we may also therefore
conclude the existence of asymmetric $L$-space links with arbitrarily many
components.

We show that certain positive surgeries on negative alternating chainmail
links do not have orderable fundamental group. The proof follows closely the
strategy of Greene, using a related presentation of the fundamental group.
Finally, we note that all known examples in the literature of $L$-space links
have complements that fiber over the circle (this is true for $L$-space knots
by [Ni1]). We observe that alternating chainmail links have fibered
complements, continuing this trend. In the final section we ask some natural
questions stemming from these examples.

Background: Different sections of this paper assume different background
knowledge. Because some of the proofs are fairly short modifications of
results in the literature, we will use standard terminology as defined in the
primary sources for each section to avoid a plethora of definitions which are
only used once. Hopefully the reader will be able to black-box as much as
possible and refer to the sources for more details when necessary.

Acknowledgements: We thank Qiuyu Ren, Joshua Greene, Eugene Gorsky, Zhenkun Li
and Francesco Lin for helpful comments.

##  2\. Chainmail links

###  2.1. Chainmail graphs

Polyak introduced the terminology of chainmail link [Pol] and pointed out that
a result of Matveev-Polyak implies that every closed connected orientable
3-manifold is realized by surgery on a chainmail link [MP]. Since Polyak’s
results have not been published, we review some of his notation and
terminology which, although not strictly necessary, will be a useful way to
formalize the proofs in this paper.

######  Definition 2.1.

A chainmail graph is a finite planar graph $G\subset S^{2}$ whose edges and
vertices are decorated with weights in $\mathbb{Z}$. Note that $G$ may have
loops and multiedges. Each vertex $v\in V(G)$ is decorated with a weight
$\nu(v)\in\mathbb{Z}$, and each edge $e\in E(G)$ is given a weight
$\epsilon(e)\in\mathbb{Z}$, where $V(G)$ and $E(G)$ are the set of vertices
and edges of $G$ respectively. Thus a chainmail graph consists of a triple
$(G,\nu,\epsilon)$, but much of the time we will abbreviate this just as $G$
with the weights implicit (see also [AWvW] for this notation).

Given a chainmail graph $G$, one may define a chainmail link $L_{G}$ in two
steps in the following way. For each $v_{i}\in V(G)$, place a planar unknot
$L_{i}$. Orient $L_{i}$ counterclockwise. For each edge $e$ connecting
vertices $v_{i},v_{j}$, introduce a $\epsilon(e)$-clasped ribbon linking
$L_{i}$ and $L_{j}$. Let $w_{ij}$ be the sum of edge weights of edges
connecting $v_{i}$ and $v_{j}$, with the convention that if distinct vertices
$v_{i}$ and $v_{j}$ are not connected by an edge, then define $w_{ij}=0$.

Now we assign framings to the components $L_{i}$ with framing
$w_{ii}=\nu(v_{i})-\sum_{k\neq i}w_{ik}$. This framing is chosen in such a way
that the linking number of the framing slope of $L_{i}$ with the chainmail
link is $w_{ii}$. In turn this gives a framing matrix $\Lambda(G)=(w_{ij})$.
Surgery on $L_{G}$ with these framings gives a 3-manifold $M_{G}$. Moreover,
$det(\Lambda(G))\neq 0$ iff $M_{G}$ is a rational homology 3-sphere, in which
case $|H_{1}(M_{G};\mathbb{Z})|=|det(\Lambda(G))|$. We may also view
$\Lambda(G)$ as the intersection form of $H_{2}(W_{G})$ of the 4-manifold
$W_{G}$ obtained by adding two handles to the 4-ball along the link $L_{G}$
with given framings.

One special property of chainmail links associated to simplicial graphs is
that the crossing number is the minimal possible given the linking numbers
between components. Chainmail graphs generalize plumbing diagrams for graph
manifolds studied by Scharf [Sch] and Fintushel-Stern [FS] who considered the
case of trees with edges of weight $\pm 1$ and allowing rational weights on
vertices (note however that the vertex labeling convention here may be
different).

We briefly review some of the notation from [MP]. Consider framed tangles
$T_{2n}$ in $\mathbb{R}^{2}\times[0,1]$ with $2n$ framed endpoints on
$\mathbb{R}^{2}\times\\{1\\}$ and on $\mathbb{R}^{2}\times\\{0\\}$. We may
draw these tangles with blackboard framing, and compose the tangles by
stacking.

Figure 1. Generators for the genus 2 mapping class group Figure 2. Closure of
a tangle

One may close up a tangle in $T_{2n}$ to get a framed link as in Figure 2 by
adding strands on top of $\mathbb{R}^{2}\times\\{1\\}$ and removing the
strands meeting the bottom $\mathbb{R}^{2}\times\\{0\\}$, and hence a Kirby
diagram for a 3-manifold. Matveev and Polyak prove that one may obtain all
genus 2 manifolds by stacking the tangles and their mirror images in Figure 1,
then closing. In fact they show that these tangles generate the genus two
mapping class group when one mods out by Kirby moves, such that the generators
in Figure 1 correspond to the Dehn twists about curves in Figure 11. For
example, the manifold associated to
$\beta_{2}^{-1}\alpha_{1}\alpha_{2}\beta_{1}^{-1}\delta_{2}\beta_{2}^{-1}\alpha_{1}\alpha_{2}\beta_{1}^{-1}\beta_{2}^{-1}$
is shown in Figure 3.

Figure 3. A plat closure associated to the tangle
$\beta_{2}^{-1}\alpha_{1}\alpha_{2}\beta_{1}^{-1}\delta_{2}\beta_{2}^{-1}\alpha_{1}\alpha_{2}\beta_{1}^{-1}\beta_{2}^{-1}$

######  Theorem 2.2 ([Pol]).

Any closed oriented 3-manifold $Y\cong M_{G}$ for some chainmail graph $G$.

###### Proof.

The plat representation of a Kirby diagram for a 3-manifold $Y$ given in [MP,
Theorem 7.1] is easily seen to be an integral surgery on a chainmail link
[Pol] as in Figure 3. ∎

We remark on a few moves on chainmail graphs that give rise to the same
manifold. These may be verified by using flypes of the corresponding diagram
keeping track of framings or Kirby calculus. A zero-weighted edge may be
erased. Two edges $e_{1},e_{2}$ connecting the same pair of vertices may be
redrawn as one edge of weight $\epsilon(e_{1})+\epsilon(e_{2})$. A loop may be
erased. A vertex $v$ with weight $\nu(v)=0$ of degree 1 adjacent to the edge
$e$ with $\epsilon(e)=\pm 1$ may be removed along with the edge $e$ (see
Figure 4; see also [EN, Theorem 18.3]). Even though we may get rid of loops
and multiedges, it is convenient to allow them in chainmail graphs since they
can get created under edge contraction.

Figure 4. Moves on chainmail graphs $G$ preserving the 3-manifold $M_{G}$

###  2.2. Alternating chainmail links

When all of the edge weights of a chainmail graph $G$ are of the same sign,
then the associated chainmail link is alternating, see Figure 5. If the edge
weights are all negative, then we call the associated link a negative
alternating chainmail link. If the graph is balanced and connected ($\nu(v)=0$
for all $v\in V(G)$), then it was shown by Ozsvath-Szabo that $M_{G}$ is
homeomorphic to the connected sum of the 2-fold branched cover of the link
$K_{G}$ and $S^{2}\times S^{1}$, where $K_{G}$ is obtained from $G$ as in
Figure 6 [Gre1, §3.1].

Figure 5. From a graph with negative edge weights to an alternating chainmail
link Figure 6. From an edge-weighted graph $G$ to a link $K_{G}$

###  2.3. Augmented chainmail links

Assume the graph $G$ is simplicial (so that there are no loops or edges with
the same endpoints). We may insert crossing circles into $L_{G}$ then twist to
get an augmented link (Figure 7), possibly at only some of the edges. If we
insert crossing circles associated to every edge, then it is called a flat
fully augmented (chainmail) link, otherwise it is a partially augmented link.
Thus one obtains a flat augmented link for every planar graph (Figure 8). The
labels on a chainmail graph then correspond to framing coefficients for the
loops corresponding to vertices and the reciprocals edge weights of the flat
fully augmented chainmail link. Thus every 3-manifold is surgery on a fully
augmented (flat) link by Theorem 2.2, since every chainmail link is surgery on
such a link.

Figure 7. Augmenting a chainmail link Figure 8. Going from a planar graph to a
flat augmented link

The following theorem may be deduced from the main result of [MT]:

######  Theorem 2.3.

A symmetry of a flat fully augmented link complement which sends crossing
loops to crossing loops is induced by a symmetry of the link in $S^{3}$.

This will be used in the proof of Theorem 6.1.

##  3\. Alternating chainmail links are $L$-space links

In this section, we prove that alternating chainmail links are $L$-space
links. Although Theorem 3.2 is a corollary of Theorem 4.2 by taking the
coefficients on each crossing loop to be $-1$, we prove it separately as it is
serves as a base case of the induction for the more general theorem.

######  Definition 3.1.

[GN1, Definition 2.2] An $l$-component link $L\subset S^{3}$ is called an
(instanton) $L$-space link if there exist integers $p_{1},\ldots,p_{l}$ such
that $S^{3}_{n_{1},\ldots,n_{l}}(L)$ is an (instanton) $L$-space for all
$n_{1},\ldots,n_{l}$ with $n_{i}>p_{i}$ for all $1\leq i\leq l$.

Given a chainmail graph $(G,\nu,\epsilon)$, we may form minors. For an edge
$e\in E(G)$ with $\epsilon(e)=-1$, let $G-e$ be the graph obtained by deleting
the edge $e$ and keeping the weight functions the same on vertices and edges.
If $e$ is not a loop, and $e$ has endpoints $v_{1},v_{2}$, let $G/e$ be the
graph obtained by quotienting $e$ to a point (which preserves planarity). The
weight function on vertices $V(G)-\\{v_{1},v_{2}\\}$ is the restriction of
$\nu$, and the weight function $\epsilon$ on edges are the same. There is a
new vertex $v^{\prime}$ of $G/e$ obtained by identifying $v_{1}$ and $v_{2}$.
We define $\nu(v^{\prime})=\nu(v_{1})+\nu(v_{2})$ (Figure 9).

Figure 9. Minors of $G$ obtained by deleting and contracting the edge $e$

Let us see how the framed links / 3-manifolds associated to the graphs
$G,G-e,G/e$ differ, with $\epsilon(e)=-1$. Consider the augmented link
obtained by inserting a crossing loop $c$ about $L_{1}$ and $L_{2}$, the
components corresponding to $v_{1}$ and $v_{2}$, then do a twist to remove the
double crossing. Then $L_{G}$ corresponds to doing $-1$ surgery on $c$,
$L_{G-e}$ corresponds to $\infty$ surgery on $c$ (which corresponds to erasing
$c$), and $L_{G/e}$ corresponds to $0$ surgery on $c$ (see Figure 10). This
last claim follows from a Kirby calculus operation $K_{3}$ described in [MP,
Fig. 3 p. 539].

Figure 10. Links and surgeries on augmented link corresponding to minors of
$G$

Thus we see that $M_{G},M_{G-e},$ and $M_{G/e}$ are related by surgeries on
the knot $c$ with slopes which have pairwise intersection number $1$. Hence we
have an exact triangle

$\widehat{HF}(M_{G})\to\widehat{HF}(M_{G-e})\to\widehat{HF}(M_{G/e})\to\widehat{HF}(M_{G})$

[OS2, Theorem 9.16].

The following theorem generalizes [Liu, Example 3.15]

######  Theorem 3.2.

Negative alternating chainmail links are $L$-space links. More specifically,
for a chainmail graph $G$ with all $\nu(v)\geq 0,v\in V(G)$, and at least one
$\nu(v)>0$ in each component of $G$, and $\epsilon(e)<0$ for all $e\in E(G)$,
$M_{G}$ is an $L$-space and an instanton $L$-space.

###### Proof.

The proof is similar to the proof that double branched covers of
(quasi-)alternating links are $L$-spaces in [OS3, Proposition 3.3]. By
replacing edges $e$ with weight $\epsilon(e)<-1$ with parallel copies, we may
assume that $\epsilon(e)=-1$ for every edge $e\in E(G)$, and we may assume
that there are no loops by removing them without changing the manifold.

If $G$ is a chainmail graph and $\nu(v)=0$ for all vertices and
$\epsilon(e)=-1$ for all edges, then the linking matrix $\Lambda(G)$ is the
matrix of the graph Laplacian of the underlying unweighted graph, hence all
eigenvalues are $\geq 0$, and the $0$-eigenspace has dimension the number of
components of $G$ (essentially the combinatorial Hodge theorem for
$0$-cocycles). If $G$ has $\nu(v)\geq 0$, and at least one $\nu(v)>0$ in each
component of $G$ and $\epsilon(e)=-1$, then when we add the diagonal matrix
with diagonal entries $\nu(v_{i})$ to the Laplacian matrix to get
$\Lambda(G)$, we get a symmetric positive definite matrix. Thus
$det(\Lambda(G))>0$ for such graphs.

The proof is by induction on $|E(G)|$, the number of edges of $G$. The base
case is an edgeless graph $G$, in which case all vertices have positive weight
by hypothesis and $M_{G}$ is a connected sum of lens spaces.

Now assume that $E(G)=n$, and all such chainmail graphs with fewer edges
represent $L$-spaces. Choose an edge $e$ of $G$, and delete the edge to get
$G-e$ and contract the edge to get $G/e$, both of which have fewer edges. If
there are parallel edges to $e$ (with the same vertices), then $G/e$ may have
loops, but we may delete them in order to satisfy the induction hypothesis
while still representing the same manifold (see Figure 4).

We may assume that $e$ is not an isthmus, unless $G$ is a tree (equivalently
every edge is an isthmus). If $G$ has at least two vertices with $\nu(v)>0$,
then we may choose $e$ so that both components of $G-e$ satisfy the hypothesis
that there is at least one vertex $v$ with $\nu(v)>0$. Otherwise, there is a
single vertex $v$ with $\nu(v)>0$. Choose any leaf vertex $v^{\prime}$ with
$\nu(v^{\prime})=0$ (a tree with non-empty edge set must have at least two
leaf vertices, so such a vertex exists). Then we may delete $v^{\prime}$
keeping the same manifold $M_{G-v^{\prime}}$ by the last move in Figure 4.
Thus in the latter case $M_{G}\cong M_{G-v^{\prime}}$ will be an $L$-space by
induction, and thus we may assume we are in the former case in which case
$G-e$ is a chainmail graph satisfying the hypotheses. We will assume that the
endpoints of $e$ are labeled $v_{1}$ and $v_{2}$.

Claim: $det\Lambda(G)=det\Lambda(G-e)+det\Lambda(G/e)$.

Proof: Consider $\Lambda(G)$ and $\Lambda(G-e)$ as linear operators on
functions on $V(G)=V(G-e)$. We may write the symmetric matrices for these as

$\Lambda(G)=\left(\begin{array}[]{ccc}w_{11}&w_{12}&x_{1}^{T}\\\
w_{12}&w_{22}&x_{2}^{T}\\\
x_{1}&x_{2}&A\end{array}\right),\Lambda(G-e)=\left(\begin{array}[]{ccc}w_{11}-1&w_{12}+1&x_{1}^{T}\\\
w_{12}+1&w_{22}-1&x_{2}^{T}\\\ x_{1}&x_{2}&A\end{array}\right),$

differing only in the upper left block corresponding to vertices $v_{1}$ and
$v_{2}$. Since we are only interested in the determinant, we may do elementary
row operations keeping the determinants the same. Add the first row to the
second row and first column to the second column (in either order) of
$\Lambda(G),\Lambda(G-e)$ to get matrices with the same determinant

$\Lambda(G)^{\prime}=\left(\begin{array}[]{ccc}w_{11}&w_{11}+w_{12}&x_{1}^{T}\\\
w_{11}+w_{12}&w_{11}+w_{22}+2w_{12}&x_{1}^{T}+x_{2}^{T}\\\
x_{1}&x_{1}+x_{2}&A\end{array}\right),$
$\Lambda(G-e)^{\prime}=\left(\begin{array}[]{ccc}w_{11}-1&w_{11}+w_{12}&x_{1}^{T}\\\
w_{11}+w_{12}&w_{11}+w_{22}+2w_{12}&x_{1}^{T}+x_{2}^{T}\\\
x_{1}&x_{1}+x_{2}&A\end{array}\right).$

Then we see that

$\Lambda(G)^{\prime}-\Lambda(G-e)^{\prime}=\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right).$

By linearity of determinants (say with respect to the first row), we see that

$det(\Lambda(G))-det(\Lambda(G-e))=\left|\begin{array}[]{ccc}1&0&0\\\
w_{11}+w_{12}&w_{11}+w_{22}+2w_{12}&x_{1}^{T}+x_{2}^{T}\\\
x_{1}&x_{1}+x_{2}&A\end{array}\right|=det(\Lambda(G/e)),$

since the lower block matrix is $\Lambda(G/e)$.

This establishes the determinant identity. One may also prove this using
[AWvW, Theorem 4.4].

Hence $|H_{1}(M_{G})|=|H_{1}(M_{G-e})|+|H_{1}(M_{G/e})|$. By our induction
hypothesis, $M_{G-e}$ and $M_{G/e}$ are $L$-spaces, and they are related by an
exact triangle. Then by [OS3, Proposition 2.1], $M_{G}$ is also an $L$-space,
where the knot is the crossing loop $c$. The proof is very similar to the
proof of [OS4, Lemma 3.2, Proposition 3.3].

For the framed instanton homology case, the base case is the same (lens spaces
are instanton $L$-spaces). We have $dimI^{\\#}(Y)\geq|H_{1}(Y)|$ by [Sca,
Corollary 1.4]. There is an exact triangle (with $\lambda=0$) given in [Sca,
Section 7.5]. Applied to the case at hand, we get

$I^{\\#}(M_{G})\to I^{\\#}(M_{G-e};\mu)\to I^{\\#}(M_{G/e})\to I^{\\#}(M_{G})$

where $\mu$ is a loop representing the core of the Dehn filling of $M_{G-e}$
on the crossing loop $c$ (Figure 10), and similar exact sequences by permuting
the role of $\mu$ for $M_{G}$ and $M_{G/e}$. But in $M_{G}-c$, two out of the
three slopes $\\{0,-1,\infty\\}$ will be non-trivial in
$H_{1}(M_{G}-c;\mathbb{F}_{2})$ and the third will be trivial by half-lives,
half-dies. Filling along a homologically non-trivial slope, the homologically
trivial slope will be isotopic to the core, and hence $[\mu]=0\in
H_{1}(M_{G^{\prime}};\mathbb{F}_{2})$, where $G^{\prime}=G,G-e,$ or $G/e$
depending on the filling slope. So in this case we get an exact triangle

$I^{\\#}(M_{G})\to I^{\\#}(M_{G-e})\to I^{\\#}(M_{G/e})\to I^{\\#}(M_{G}).$

Now we apply the same argument as for Heegaard Floer homology to conclude that

$|H_{1}(M_{G})|\leq dimI^{\\#}(M_{G})\leq
dimI^{\\#}(M_{G-e})+dimI^{\\#}(M_{G/e})$
$=|H_{1}(M_{G-e})|+|H_{1}(M_{G/e})|=|H_{1}(M_{G})|,$

and hence $M_{G}$ is an instanton $L$-space. ∎

Remark: Zhenkun Li pointed out that one can take a connected sum of any
component of a negative alternating chainmail link with an $L$-space knot to
get another $L$-space link. The point is that the proof goes through as
before, with the vertex label adjusted larger, with the base case of the
induction including a split link with one component an $L$-space knot so that
surgery still gives an $L$-space for large enough coefficients.

One may give a slightly different characterization of some of the $L$-spaces
described in Theorem 3.2 in terms of Heegaard splittings. Consider a set of
generators $\alpha_{i},\beta_{i},\delta_{i}$ for the mapping class group
$Mod(\Sigma)$ of the Heegaard surface $\Sigma$ by right-hand Dehn twists about
curves $a_{i},b_{i},d_{i}$ respectively shown in Figure 11, which represents a
standard Heegaard splitting $H_{1}\cup_{\Sigma}H_{2}$ of $S^{3}$.

Figure 11. Generators for the mapping class group of a Heegaard surface
$\Sigma$ for $S^{3}$

######  Corollary 3.3.

Choose $\sigma\in\\{\alpha_{i},\beta_{i}^{-1},\delta_{i}\\}^{\ast}$, the
submonoid of products of these elements. Then the resulting manifold
$H_{1}\cup_{\sigma}H_{2}$ is a connected sum of an $L$-space and
$\\#^{k}S^{2}\times S^{1}$ for some $k$.

###### Proof.

The Kirby diagram produced from the products of these generators by [MP,
Theorem 7.1] is a negative alternating chainmail link corresponding to a
chainmail graph with non-negative vertex weights (see Figure 3 for an
example). In each component of the chainmail graph with vertex weights $0$,
there will be an $S^{2}\times S^{1}$ summand connected sum with a double-
branched cover of an alternating link in the corresponding component of the
Kirby diagram. Hence the manifold will be of the claimed type in each connect
summand, and therefore in total. ∎

##  4\. Augmented negative alternating chainmail links are generalized
$L$-space links

Next we consider augmented chainmail links.

######  Definition 4.1.

[Liu, Definition 2.9] Let $L=\cup_{i=1}^{l}L_{i}\subset S^{3}$ be a link, and
for each $i$ choose a sign $\epsilon_{i}\in\\{\pm\\}$. Then $L$ is called a
generalized (instanton) $(\epsilon_{1}\cdots\epsilon_{l})$ $L$-space link if
there exist integers $p_{1},\ldots,p_{l}$ such that
$S^{3}_{\epsilon_{1}n_{1},\ldots,\epsilon_{l}n_{l}}(L)$ is a (instanton)
$L$-space for all $n_{1},\ldots,n_{l}$ with $n_{i}>p_{i}$ for all $1\leq i\leq
l$. If we do not specify the sign, then $L$ is denoted a generalized $L$-space
link.

######  Theorem 4.2.

Partially augmented negative alternating chainmail links are generalized
$L$-space links with a $+$ sign associated to each vertex loop and $-$ to each
edge (crossing) loop.

As a corollary of Theorems 2.2 and 4.2, we answer a question of Yajing Liu
[Liu, Question 1.9].

######  Corollary 4.3.

Any closed orientable connected 3-manifold is obtained by surgery on a
generalized $L$-space link.

###### Proof.

4.2 First, we need a lemma.

######  Lemma 4.4.

Consider a partially augmented chainmail link simplicial graph $G$, with
negative weights on the vertices $V_{c}(G)\subset V(G)$ corresponding to
crossing loops and non-negative weights on the other vertices $V(G)-V_{c}(G)$.
Moreover, the edges between vertices of $V(G)-V_{c}(G)$ have weight $-1$. Then
the sign of the determinant is $(-1)^{|V_{c}(G)|}$ if non-zero.

###### Proof.

The proof is by induction on the sum of the negative crossing loop weights. If
there are no crossing loops, then $\Lambda(G)$ is is the linking matrix of a
negative alternating chainmail graph, and hence the sign of the determinant is
non-negative by the proof of Theorem 3.2, establishing the base case.

Now consider a single crossing loop of the flat augmented link with weight
$-c$, $c>0$, and its two adjacent loops having framings $a^{\prime}$ and
$b^{\prime}$. The matrix $\Lambda(G)$ looks like

$\left(\begin{array}[]{cccc}-c&1&-1&0\\\ 1&a^{\prime}&0&\ast\\\
-1&0&b^{\prime}&\ast\\\ 0&\ast&\ast&A\end{array}\right).$

When $c=1$, we do column operations adding the first column to the second and
subtracting the first column from the third to get

$\left|\begin{array}[]{cccc}-1&0&0&0\\\ 1&a^{\prime}+1&-1&\ast\\\
-1&-1&b^{\prime}+1&\ast\\\
0&\ast&\ast&A\end{array}\right|=(-1)\left|\begin{array}[]{ccc}a^{\prime}+1&-1&\ast\\\
-1&b^{\prime}+1&\ast\\\ \ast&\ast&A\end{array}\right|.$

The last determinant is that of a partially augmented chainmail link by doing
a $-1$ twist along the crossing loop (see the left diagrams of Figure 10). So
the sign of this determinant is $(-1)^{(|V_{c}(G)|-1)}$ by induction. Thus it
is true by induction for $c=1$.

When $c>1$, we have

$\left|\begin{array}[]{cccc}-c-1&1&-1&0\\\ 1&a^{\prime}&0&\ast\\\
-1&0&b^{\prime}&\ast\\\
0&\ast&\ast&A\end{array}\right|=\left|\begin{array}[]{cccc}-c&1&-1&0\\\
1&a^{\prime}&0&\ast\\\ -1&0&b^{\prime}&\ast\\\
0&\ast&\ast&A\end{array}\right|-\left|\begin{array}[]{cccc}1&0&0&0\\\
1&a^{\prime}&0&\ast\\\ -1&0&b^{\prime}&\ast\\\
0&\ast&\ast&A\end{array}\right|.$

The matrix on the right has determinant

$\left|\begin{array}[]{ccc}a^{\prime}&0&\ast\\\ 0&b^{\prime}&\ast\\\
\ast&\ast&A\end{array}\right|$

which is $det\Lambda(G-v)$. By induction, this matrix has the opposite sign,
so we get by induction on $c$ that the sign with weight $-c-1$ is
$(-1)^{|V_{c}(G)|}$. ∎

Now we apply the exact triangle to see if $M_{G}$ with $\nu(v)=-c$ is an
$L$-space, then $M_{G}$ with $\nu(v)=-c-1$ is an $L$-space. These are in an
exact triangle with the loop erased corresponding to $M_{G-v}$, so we get this
by induction again applying [OS3, Proposition 2.1]. The framed instanton case
is similar to the argument at the end of the proof of Theorem 3.2. ∎

##  5\. Non-orderable positive fillings on negative alternating chainmail
links

We are not able to show that the $L$-spaces described in Theorem 4.2 have non-
orderable fundamental group, addressing [BGW, Conjecture 1], but we can show
non-orderability in special cases.

######  Theorem 5.1.

For a chainmail graph $G$ with edge weights $-1$, and non-negative vertex
weights with at most one vertex weight non-zero, $\pi_{1}(M_{G})$ is non-
orderable.

###### Proof.

The proof is modeled on the proof of Greene that double-branched covers of
alternating links have non-orderable fundamental group [Gre2, Theorem 2.1]
(proved originally by Boyer, Gordon and Watson [BGW, Theorem 4]).

Figure 12. (a) Planar graph (b) loops associated to oriented edges (c) acyclic
orientation from group ordering

Each edge of $G$ with a chosen orientation corresponds to an element of
$\pi_{1}(S^{3}-L_{G})$ as shown in Figure 12(b). If a vertex $v_{i}$ has
weight $\nu(v_{i})=0$, then the relation associated to Dehn filling will be
the product of the incoming edges (see Figure 13).

Figure 13. The zero-framing loop is a product of incoming edge loops

Suppose that $\pi_{1}(M_{G})$ admits a left ordering. Then we may orient each
edge so that the associated element is $>1$ (assuming all of these elements
are non-trivial first). This gives an acyclic orientation of the graph 12(c);
any oriented loop corresponds to the trivial element but is also a product of
elements $>1$, a contradiciton. Then there must be a sink vertex and a source
vertex. Since only one vertex $v$ has weight $\nu(v)>0$ by assumption, we may
choose a sink or source vertex $v^{\prime}$ with $\nu(v^{\prime})=0$. In this
case, the product of the edges oriented into this will be trivial but also
$>1$ ( or $<1$), a contradiction.

In the case that some of the edges are trivial, we may still locate a sink or
source vertex, allowing that some of the adjacent edges are trivial. If all
edges are trivial, then either all vertex weights are zero, and we are in the
case considered by Greene [Gre2], or one vertex has non-zero weight. In this
case the relator from Dehn filling will give a power of the meridian of that
loop is trivial. If the meridian is torsion, then the group cannot be
orderable. If the meridian is trivial, the one can see that all meridians are
trivial (since the edge loops are products of meridians), and hence the
fundamental group is trivial. But by definition the trivial group is not left-
orderable. ∎

A theorem of Tao Li implies that a 3-manifold $Y$ with Heegaard genus 2 that
is an $L$-space does not have orderable fundamental group [Li, Theorem 1.2].
One may apply this to conclude that the manifolds in Corollary 3.3 whose
gluing map lies in the monoid
$\\{\alpha_{1},\alpha_{2},\beta_{1}^{-1},\beta_{2}^{-1},\delta_{2}\\}^{*}$ do
not have orderable fundamental group unless it is a connected sum of copies of
$S^{1}\times S^{2}$. Such manifolds are realized by surgery on the framed
links made by composing the tangles in Figure 1, such as in Figure 3.

##  6\. Asymmetric $L$-space links

It was shown that there exist asymmetric $L$-spaces and asymmetric $L$-space
knots [DHL, Theorem 1.2], [BL]. Here we show the existence of asymmetric
$L$-space links.

######  Theorem 6.1.

There are alternating hyperbolic chainmail links of arbitrarily many
components which have trivial symmetry group.

###### Proof.

Start with a flat augmented chainmail link associated to a graph $G$ such as
in Figure 8 with hyperbolic complement (so no parallel crossing loops).
Perform $-1/n_{i}$ surgeries with $n_{i}$ large on each of the crossing loops
$c_{i}$ to make a negative alternating chainmail link. By choosing the $n_{i}$
sufficiently large, we may assume that the cores of the Dehn fillings on each
of the crossing loops $c_{i}$ give the $|E(G)|$ shortest loops in the
hyperbolic metric on the manifold, and they have distinct lengths [Thu,
Theorem 5.8.2]. Thus, any isometry of this link complement must preserve the
Margulis tubes about these short loops. Thus it also preserves the complements
of these loops which is the initial flat augmented chainmail link complement.
By [MT], any homeomorphism of this link complement must be induced by a
homeomorphism of the link. Moreover, cusps are taken to cusps, and hence cusps
corresponding to crossing loops are fixed.

Figure 14. A flat augmented chainlink with two crossing disks must have a
separating edge

Choose a simplicial planar graph $G$ which contains no triangles and no
vertices of degree 2 and no edge that separates the graph. By [MT, Proposition
2.7], the only thrice punctured spheres in the fully augmented chainmail link
complement associated to $G$ are crossing disks. Moreover, each crossing
circle bounds a unique twice-punctured disk intersecting two knot loops (one
can see that for a flat augmented chainmail link that a second crossing circle
would give a separating edge, see Figure 14). We may also assume that these
have a unique reflection surface, since our assumptions rule out the examples
in that paper with multiple reflection surfaces [MT, Theorem 1.2]. Thus, any
isometry of $S^{3}-L$ must send the reflection surface to the reflection
surface and the crossing disks to crossing disks. Each loop in the reflection
surface bounds a disk punctured by crossing loops and intersecting the
crossing disks in arcs. One may recover the graph $G$ from this pattern of
intersections, as well as its planar embedding from the reflection surface. So
the only orientation preserving symmetries of the link complement must
preserve the graph fixing each edge and hence are trivial. There is an
orientation reversing symmetry by reflecting in the reflection surface, but
this will not preserve the slopes of the fillings of the crossing loops, and
hence does not extend to a symmetry of the alternating link. Thus we have
shown the existence of negative alternating chainmail links which admit no
non-trivial symmetry. ∎

Remark: Further Dehn fillings on asymmetric negative alternating chain links
give rise to more examples of asymmetric $L$-spaces. Compare to [DHL, Theorem
1.2] and [ABGK+] which require computation to verify the results, and [BL]
which uses an intricate construction to give asymmetric $L$-space knots.

##  7\. Alternating chainmail links are fibered

It was proved that $L$-space knots are fibered [Ni1]. Moreover, all of the
examples in the literature of $L$-space links have fibered complement. Note
that [Liu, Example 3.9] shows that the links do not admit fibered Seifert
surfaces; nevertheless the complements fiber.

######  Theorem 7.1.

Non-split negative alternating chainmail links have fibered complement.

###### Proof.

Take the checkerboard surface for the alternating chainmail link $L_{G}$ which
is a Seifert surface by applying Seifert’s algorithm to the orientation of the
link in which all of the loops are oriented counterclockwise (Figure 15). The
complement of this surface is a sutured handlebody $(H,\gamma)$ which Gabai
proved is disc-decomposable [Gab, Theorem 6.1]. Moreover, one sees that the
restriction maps $H^{1}(S^{3}-L_{G})\to H^{1}(H)$ is surjective. Choose a
cutset of oriented disks $D\subset H$ giving a taut sutured decomposition, and
a cohomology class $\alpha^{\prime}\in H^{1}(H)$ Poincaré dual to $[D,\partial
D]\subset H_{2}(H,\partial H)$. Then there exists a cohomology class
$\alpha\in H^{1}(S^{3}-L_{G})$ so that $\alpha_{|H}=\alpha^{\prime}$. Let
$\mu\in H^{1}(S^{3}-L_{G})$ denote the cohomology class dual to the Seifert
surface. Then $k\mu+\alpha$ will lie in a cone over a face of the Thurston
norm ball adjacent to that containing $\mu$. By [AZ, Theorem 1.4] the guts
$\Gamma(k\mu+\alpha)=\Gamma(\alpha^{\prime})=\emptyset$, and hence
$k\mu+\alpha$ is a fibered class (see [AZ, Definition 2.17] for the definition
of guts). ∎

Figure 15. A Seifert surface for a chainmail link

##  8\. Conclusion

We conclude with some questions and challenges stemming from the examples in
this paper.

  1. (1)

Are any of the new examples of $L$-spaces described in this paper strong
$L$-spaces [GL]?

  2. (2)

Are there non-fibered $L$-space link complements?

  3. (3)

Describe all $L$-space surgeries on alternating chainmail and augmented
chainmail links. Since this includes all 3-manifolds, such a characterization
will include a description of all $L$-spaces and thus is likely to be
intricate [GN3].

  4. (4)

Find the foliar surgeries on these links. If the $L$-space conjecture is true,
then this should consist of the complement of the reducible and $L$-space
fillings [Juh, Conjecture 5].

  5. (5)

Prove that the $L$-space surgeries on alternating and augmented chainmail
links described in these theorems have non-orderable fundamental group
(Question (1) is relevant if there is a positive answer by [LL]). If the
$L$-space conjecture is true, then one would expect the fundamental groups to
be non-orderable.

  6. (6)

Presumably one could prove Theorem 7.1 by computing the multi-variable
Alexander polynomial of negative alternating chainmail links which allows one
to detect if the link complement fibers [Ni1, Corollary 1.2], [Ni2], [OS5,
Theorem 1.2]. It might be interesting to find a formula for these polynomials
and use it to reprove Theorem 7.1. It would also be interesting to know which
augmented alternating links have fibered complement?

  7. (7)

Are there more $L$-space links that may be found using this approach? Which
chainmail links are (generalized) $L$-space links? Complete this analogy:

alternating links: alternating chainmail links :: quasi-alternating links: ?

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