11institutetext: Institute for Mathematics, Brandenburg Technical University,
Platz der Deutschen Einheit 1, 03046 Cottbus, Germany
11email<EMAIL_ADDRESS>
# Towards Efficient Time Stepping for
Numerical Shape Correspondence

Alexander Köhler 11 Michael Breuß 11

###### Abstract

The computation of correspondences between shapes is a principal task in shape
analysis. To this end, methods based on partial differential equations (PDEs)
have been established, encompassing e.g. the classic heat kernel signature as
well as numerical solution schemes for geometric PDEs. In this work we focus
on the latter approach.

We consider here several time stepping schemes. The goal of this investigation
is to assess, if one may identify a useful property of methods for time
integration for the shape analysis context. Thereby we investigate the
dependence on time step size, since the class of implicit schemes that are
useful candidates in this context should ideally yield an invariant behaviour
with respect to this parameter.

To this end we study integration of heat and wave equation on a manifold. In
order to facilitate this study, we propose an efficient, unified model order
reduction framework for these models. We show that specific $l_{0}$ stable
schemes are favourable for numerical shape analysis. We give an experimental
evaluation of the methods at hand of classical TOSCA data sets.

###### Keywords:

shape analysis second order time integration heat equation wave equation model
order reduction

##  1 Introduction

The computation of shape correspondences is a fundamental task in computer
vision with many potential applications, cf. [8, 4]. In the setting of three-
dimensional shape analysis, the underlying problem amounts to identify an
explicit relation between the surface elements of two or more shapes. The
variety of possible shape correspondence mappings that is of interest in
applications includes non-rigid transformations where shapes are just almost
isometric, allowing e.g. to match different poses of human or animal shapes.

An important solution strategy is to achieve a pointwise shape correspondence
using so called descriptor based methods. For this, a feature descriptor has
to be computed that characterizes each point on a shape by describing the
surrounding shape surface geometry. A mathematically sound approach to compute
such shape signatures is to make use of the spectral decomposition of the
Laplace-Beltrami operator, see e.g. [14, 15, 4]. To this end, the Laplace-
Beltrami operator may be incorporated in a certain variety of partial
differential equations (PDEs) that are potentially useful as models for
feature computation. The arguably most important classic signature is the heat
kernel signature (HKS) [15] which relies on the heat equation, however also
versions of the Schrödinger equation leading to the wave kernel signature
(WKS) [1] as well as the hyperbolic wave equation [6] have already been
proposed.

For the computation of such PDE-based signatures, the basic task amounts to
resolve the underlying PDEs on a manifold representing a shapes’ boundary. The
HKS and WKS both rely on the eigenfunction expansion of the Laplace-Beltrami
operator to tackle this task and for achieving efficient algorithms. An
alternative to the spectral approach is to consider the numerical integration
of the underlying PDEs as proposed in [6, 5, 2]. However, when following this
path it is highly advocated to employ very efficient computational means such
as the model order reduction framework presented in [2, 3] in order to avoid
high computational times.

In previous works based on numerical integration, first order implicit time
integration has been studied in detail [5]. For the wave equation model, it
has been shown that backward differencing in time may yield favorable results
over simple central differences [6]. Especially, as has been illustrated in
[6], the classic wave equation may yield in some experiments results of higher
correspondence quality compared to the other mentioned models. Let us note
that this holds again in particular when assessing the models using numerical
integration with implicit first order time stepping.

Our contributions. In this paper we give an account of ongoing work on
improving numerical PDE-based shape descriptors with respect to computational
efficiency. The underlying question addressed in this paper is, if the class
of implicit schemes has particularly useful key properties that should be
respected in the shape analysis context. We present a study of three implicit
time stepping schemes for heat and wave equation on manifolds, respectively.
We show that standard finite differences may not be appropriate since the
typical initial condition used for the construction of shape signatures, which
is a discrete Dirac delta function, may yield oscillatory artefacts that may
spoil correspondence quality. In order to resolve this issue, we propose to
adopt $l_{0}$-stable methods. For performing the study, we have developed a
unified numerical model order reduction framework based on [2]. We give an
experimental account of our investigations at hand of selected shape data
sets.

##  2 Modelling of the Shape Correspondence Framework

The shape of three-dimensional geometric object can be described by its
bounding surface, given as a compact two-dimensional Riemannian manifold
$\mathcal{M}\subset\bbbr^{3}$. Two such shapes $\mathcal{M}$ and
$\widetilde{\mathcal{M}}$ are considered to be isometric if there is a smooth
homeomorphism between the corresponding object surfaces that preserves the
intrinsic distances between surface points. For many applications, isometry
may only hold approximately, leading to the notion of almost isometric shapes.

As indicated, a widely used descriptor class that can handle almost isometric
transformations relies on PDEs on manifolds. We will consider in this paper
the corresponding heat and wave equation, respectively. These PDEs may be put
together in a more general form as described below.

All the mentioned PDEs rely on the Laplace-Beltrami operator as a fundamental
building block. This operator is the geometric version of the Laplace
operator, meaning that it takes into account the local curvature of a smooth
manifold in 3D. Applied to a scalar-valued function $u$, it can be written as:

$\Delta_{\mathcal{M}}u=\frac{1}{\sqrt{|g|}}\sum_{i,j=1}^{2}\partial_{i}\left(\sqrt{|g|}g^{ij}\partial_{j}u\right)$
(1)

where $|g|$ is the determinant of the metric tensor $g\in\bbbr^{2\times 2}$
that describes locally the geometry, and $g^{ij}$ are the entries of its
inverse, see e.g. [7] for more details on the differential geometric notions.

### Heat Equation

The heat equation on a manifold is a proper shape descriptor [15] and reads as

$\partial_{t}u(x,t)=\Delta_{\mathcal{M}}u(x,t).\quad x\in\mathcal{M},~{}t\in
I$ (2)

This PDE basically describes heat flow along a surface $\mathcal{M}$.

### Wave Equation

Proposed in [6] for computing shape correspondences, this equation can be
written as

$\partial_{tt}u(x,t)=\Delta_{\mathcal{M}}u(x,t).\quad x\in\mathcal{M},~{}t\in
I$ (3)

### Meta PDE

We can combine the PDEs from above into a single equation as

$\phi\partial_{tt}u(x,t)+\psi\partial_{t}u(x,t)=\Delta_{\mathcal{M}}u(x,t).\quad
x\in\mathcal{M},~{}t\in I$ (4)

We see if we choose $\phi=0$ and $\psi=1$ we receive heat equation (2) and for
$\phi=1$ and $\psi=0$ it will lead us to the wave equation (3). The PDE
formulation (4) is useful for us in this paper, since it allows us to describe
in a compact way numerical developments in later sections.

### Initial condition for feature computation

Both equations need initial conditions to be solved accurately. The classic
initial condition used for PDE-based feature generation is the Dirac Delta
function $u(x,0)=u_{0}(x)=u_{x_{i}}$ centred around a point
$x_{i}\in\mathcal{M}$, cf. [15]. We also adopt this initial condition in the
numerical integration framework. Because equation (3) is a PDE of second order
in time we need an additional condition at the initial velocity. A canonical
choice is to consider the zero initial velocity condition
$\partial_{t}u(x,0)=0$.

### Feature Descriptor

For establishing shape correspondence we consider for each point on an
objects’ surface a feature descriptor, which is a computational object that
contains geometric shape information. Within our framework, a simple way to
construct it is to restrict the spatial component of $u(x,t)$ to

$f_{x_{i}}(t):=u(x,t)|_{x=x_{i}}\quad\mbox{with}\quad
u(x,0)|_{x=x_{i}}=u_{x_{i}}$ (5)

and call $f_{x_{i}}$ the feature descriptor at the location
$x_{i}\in\mathcal{M}$.

With this definition we may adopt a physical interpretation of (5). The
feature descriptor $f_{x_{i}}$ describes heat transferred away from $x_{i}$
respectively the motion amplitudes of a wave front emitted and observed again
at $x_{i}$. Thereby the processes run on the surface $\mathcal{M}$, as
indicated via the Laplace-Beltrami operator.

### Shape Correspondence

To compare the feature descriptors for different locations
$x_{i}\in\mathcal{M}$ and $\tilde{x}_{j}\in\widetilde{\mathcal{M}}$ on two
different shapes $\mathcal{M}$ and $\widetilde{\mathcal{M}}$, we define a
distance $d_{f}(x_{i},\tilde{x}_{j})$ via the $L_{1}$ norm

$d_{f}(x_{i},\tilde{x}_{j})=\int\limits_{I}|f_{x_{i}}-f_{\tilde{x}_{j}}|\textmd{d}t$
(6)

The tuple of locations
$(x_{i},\tilde{x}_{j})\in\mathcal{M}\times\widetilde{\mathcal{M}}$ with the
smallest distance belong together.

### Basic Discretisation

Figure 1: Continuous $\&$ discrete shape

The discrete surface representation is given by a set of points
$P:=(x_{1},\ldots,x_{N})$ containing coordinates of each point $x_{i}$ and one
for triangles $T$. $T$ stores triples of points forming a triangle. We denote
this data set tuple as $\mathcal{M}_{d}=(P,T)$. Furthermore we denote by
$\Omega_{i}$ the barycentric cell volume surround the point $x_{i}$. The
discrete shape is given by non-uniform linear triangles. Compare also Figure 1
(figure adopted from [2]).

##  3 Spatial Discretisation

We employ the equation (4) with a finite volume method for discretisation.
First, we consider (4) over a cell $\Omega_{i}$ and a time interval $I_{k}$ so
that the integration in time and space will lead to

$\int\limits_{I_{k}}\int\limits_{\Omega_{i}}\phi\partial_{tt}u(x,t)+\psi\partial_{t}u(x,t)\,\mathrm{d}x\,\mathrm{d}t=\int\limits_{I_{k}}\int\limits_{\Omega_{i}}\Delta_{\mathcal{M}}\,u(x,t)\,\mathrm{d}x\,\mathrm{d}t$
(7)

The definition of the cell average

$u_{i}(t)=u(\bar{x}_{i},t)=\frac{1}{|\Omega_{i}|}\int\limits_{\Omega_{i}}u(x,t)\,\mathrm{d}x$
(8)

where $|\Omega_{i}|$ denotes the area of $\Omega_{i}$, is used to define the
averaged Laplacian as

$Lu_{i}(t)=\frac{1}{|\Omega_{i}|}\int\limits_{\Omega_{i}}\Delta_{\mathcal{M}}\,u(x,t)\,\mathrm{d}x$
(9)

Evaluation of this integral making use of the divergence theorem will lead to
a line integral over the boundary of each cell. Application of the cotangent
weight scheme [10] turns the resulting integrals over a full shape into the
ODE system

$\phi\mathbf{\ddot{u}}(t)+\psi\mathbf{\dot{u}}(t)=L\mathbf{u}(t)$ (10)

The cotangent weight scheme will give us the discrete Laplace-Beltrami
operator $L\in\bbbr^{N\times N}$ via the sparse matrix representation
$L=D^{-1}W$. The matrix $W$ represents spatial weights computed from the
cotangent formulae, encoding spatial connectivity and local geometry
information. The matrix
$D=\text{diag}(|\Omega_{1}|,\ldots,|\Omega_{i}|,\ldots,|\Omega_{N}|)$ contains
the local cell areas and all functions $u_{i}(t)$ are stored in the
$N$-dimensional vector $\mathbf{u}(t)=(u_{1}(t),\ldots,u_{N}(t))^{\top}$.

Revisiting equation (10), we may notice that for $\phi=0$ we have an ODE
system of first order. Additionally we want to transform the equation for
$\phi\neq 0$ into an ODE system of first order as well, since this is
advantageous for the proceeding. We find

$\mathbf{\dot{q}}(t)=H\mathbf{q}(t)$ (11)

with
$\mathbf{q}(t)=\big{(}\mathbf{u}(t),\mathbf{\dot{u}}(t)\big{)}^{\top}\in\bbbr^{2N}$,

$H=\begin{pmatrix}0&I\\\
\frac{1}{\phi}L&-\frac{\psi}{\phi}I\end{pmatrix}\stackrel{{\scriptstyle\phi=1}}{{=}}\begin{pmatrix}0&I\\\
L&-\psi I\end{pmatrix}\in\bbbr^{2N\times 2N}$ (12)

and $I\in\bbbr^{N\times N}$ the identity matrix.

Let us turn to the discrete initial conditions. Choosing
$\mathbf{\dot{u}}(0)=\mathbf{0}$ let us only need to consider the initial
spatial conditions $\mathbf{u}(0)$. Again using cell averages, the normalised
initial condition at the location $x_{i}$ can be written as

$\mathbf{u_{x_{i}}}=\mathbf{u}(x_{i},0)=(0,\dots,0,|\Omega_{i}|^{-1},0,\dots,0)^{\top}$
(13)

### Eigenproblem and Modal Coordinate Reduction

Systems like (10) have to deal with large sparse matrices and consequently
high cost of computation time. With the Model Order Reduction (MOR) we can
approximate our high dimensional system with lower dimensional one. In [12,
13] one may find an overview about the MOR topic. For this work we are
interested in the specific Modal Coordinate Reduction (MCR) technique as
presented in [3], which we briefly recall now.

Solving equation (10) via MCR, it is essential to calculate the eigenvalues
from the Laplace-Beltrami Operator $L\mathbf{v}=\lambda\mathbf{v}$. With
$L=D^{-1}W$ we can transform the eigenvalue problem to the generalised
eigenvalue Problem

$W\mathbf{v}=\lambda D\mathbf{v}$ (14)

By the properties of $W$ and $D$ all eigenvalues are real and the eigenvectors
are linear independent. Additionally they are $D$-orthogonal with
$\mathbf{v}_{i}^{\top}D\mathbf{v}_{j}=\delta_{ij}$. Thus we obtain the
equalities

$I=V^{\top}DV,\qquad L=V\Lambda V^{\top}D,\qquad\Lambda=V^{\top}WV$ (15)

with $V$ being the right eigenvector matrix of $L$ and $\Lambda$ the diagonal
matrix of eigenvalues.

To execute the MCR transformation we take advantage of these considerations
and substitute equation (10) with $\mathbf{u}=V\mathbf{w}$ to obtain:

$\phi V\ddot{\mathbf{w}}+\psi V\dot{\mathbf{w}}=LV\mathbf{w}$ (16)

Multiplication with $V^{\top}D$ from the left results in

$\phi\ddot{\mathbf{w}}+\psi\dot{\mathbf{w}}=\Lambda\mathbf{w}$ (17)

Equation (17) is still the full system. To reduce it we are only interested in
the first $r\ll N$ ordered eigenvalues
$0=\lambda_{1}<\lambda_{2}\leq\ldots\leq\lambda_{r}$. This gives us the
reduced matrix $\Lambda_{r}\in\bbbr^{r\times r}$ extracted from $\Lambda$ and
$V_{r}\in\bbbr^{N\times r}$. Consequently we obtain the reduced model

$\phi\ddot{\mathbf{w}}_{r}+\psi\dot{\mathbf{w}}_{r}=\Lambda_{r}\mathbf{w}_{r}\quad\mbox{where}\quad\mathbf{w}_{r}=V_{r}^{\top}D\mathbf{u}$
(18)

Analogously to the procedure by which we obtained equation (11) from (10), we
can build

$\mathbf{\dot{p}}_{r}(t)=H_{r}\mathbf{p}_{r}(t)\quad\mbox{with}\quad
H_{r}=\begin{pmatrix}0&I_{r}\\\ \Lambda_{r}&-\psi
I_{r}\end{pmatrix}\in\bbbr^{2r\times 2r}$ (19)

and
$\mathbf{p}_{r}=(\mathbf{w}_{r},\dot{\mathbf{w}}_{r})^{\top}\in\bbbr^{2r}$.

Now, we want to discuss the transformation of the initial conditions. Remember
that $\mathbf{u}(x_{i},0)=(0,\dots,0,|\Omega_{i}|^{-1},0,\dots,0)^{\top}$ and
together with the second part of (18) we obtain
$\mathbf{w}_{r}(x_{i},0)=V_{r}^{\top}\mathbf{e}_{i}$. Nothing will change for
the initial velocity $\mathbf{\dot{u}}(0)=\mathbf{0}$, because we set it zero
and so will $\mathbf{\dot{w}}_{r}(0)=\mathbf{0}$.

##  4 Time Discretisation

For discretisation of the explored time interval we divide
$\left[0,t_{M}\right]$ into intervals $I_{k}=\left[t_{k-1},t_{k}\right]$ with
$\tau=t_{k}-t_{k-1}$ as time increment, and we set $t_{0}=0$.

Let us now turn to time derivatives. The classic method of second order in
time accuracy is the Crank-Nicolson method [11]. We conjecture at this point
that this method may not be beneficial in our shape analysis framework.
Illustrating this, we calculate the first two iterations of a heat equation
with the ($l_{0}$ unstable) Crank-Nicolson method in comparison to the
$l_{0}$-stable implicit Euler method, cf. Figure 2. For the solution of the
heat equation we expect a smooth propagation away from the peak. However, the
Crank-Nicolson method causes significant oscillations in the solution, even
negative values appear.

Figure 2: This first two iterations of a heat equation solved with the Crank-
Nicolson method (left) and with the $l_{0}$-stable implicit Euler scheme
(right), respectively.

We conjecture at this point that such oscillations may spoil correspondence
quality, when performing numerical shape correspondence. This example
illustrates why we aim to explore $l_{0}$ stability of schemes.

### Second Order $l_{0}$-stable Scheme

In this section, we briefly explain the idea and approach of the method of
[16]. We have chosen this method because it does not depend on complicated
Butcher tableaus, but offers an intuitive approach.

Looking at equation (11) or (19), we find these are of the form

$\dot{\mathbf{x}}(t)=A\mathbf{x}(t)$ (20)

The solution of this is $\mathbf{x}(t)=\exp(tA)$ or for a time step further

$\mathbf{x}(t+\tau)=\exp((t+\tau)A)=\exp(\tau A)\mathbf{x}(t)$ (21)

Calculating $\exp(\tau A)$ is not a trivial task, so we use an approximation

$R(\tau A)\approx\exp(\tau A)$ (22)

instead. With the approximation $R(\tau H)=(I-a\tau H)^{-1}(I+(1-a)\tau H)$ we
obtain a first-order method. Choosing $a$ will lead to different well-known
schemes. For $a=0$ we receive the explicit method and for $a=1$ we obtain the
implicit method, like it is used in [2, 3] for time discretisation. The Crank-
Nicolson scheme can be obtained by setting $a=0.5$.

However, two out of three are not of second order and, in case of the explicit
and Crank-Nicolson method, are not $l_{0}$-stable. Twizell and co-authors [16]
present with

$R(\tau A)=(I-r_{1}\tau A)^{-1}(I-r_{2}\tau A)^{-1}(I+(1-a)\tau A)$ (23)

with $r_{1,2}=\frac{1}{2}\left(a\mp\sqrt{a^{2}-4a+2}\right)$ a method of
second order and choosing $a=2-\sqrt{2}-\varepsilon$, with $\varepsilon$ an
arbitrarily small positive number, will provide an $l_{0}$-stable method.

These considerations lead to the following discrete procedure for the wave
equation:

$\mathbf{p}^{k+1}=(I_{2r}-r_{1}\tau H_{r})^{-1}(I_{2r}-r_{2}\tau
H_{r})^{-1}(I_{2r}+(1-a)\tau H_{r})\mathbf{p}^{k}$ (24)

with $\mathbf{p}^{k}=\mathbf{p}(t_{k})$ and
$\mathbf{p}^{k+1}=\mathbf{p}(t_{k+1})=\mathbf{p}(t+\tau)$. Analogously for the
heat equation, we receive

$\mathbf{w}^{k+1}=(I_{r}-r_{1}\tau\Lambda_{r})^{-1}(I_{r}-r_{2}\tau\Lambda_{r})^{-1}(I_{r}+(1-a)\tau\Lambda_{r})\mathbf{w}^{k}$
(25)

As discussed in [3] we adopt the temporal domain $\left[0,t_{M}\right]$ for
the heat equation to $\left[0,t^{*}_{\text{h}}\right]$ and for wave and damped
wave equation to $\left[0,t^{*}_{\text{w}}\right]$ by

$t^{*}_{\text{h}}(\lambda_{r})=\frac{t_{M}\sqrt{\lvert\lambda_{N}\rvert}}{\sqrt{\lvert\lambda_{r}\rvert}}\qquad\text{resp.}\qquad
t^{*}_{\text{w}}(\lambda_{r})=\frac{t_{M}\sqrt[4]{\lvert\lambda_{N}\rvert}}{\sqrt[4]{\lvert\lambda_{r}\rvert}}$
(26)

The computational parameters are thus $\tau=\frac{t^{*}}{M}$, with $M$ being
the number of iterations.

##  5 Experiments

### Hit Rate and Geodesic Error

The percentage Hit Rate is defined as $TP/(TP+FP)$, where $TP$ is the number
of true positives and $FP$ is the number of false positives events.

The Princeton benchmark protocol [9] delivers the important key points for the
evaluation of the correspondence quality. It defines the quality of a matching
by the normalised intrinsic distance
$d_{\mathcal{M}}(x_{i},x^{*})/\sqrt{A_{\mathcal{M}}}$ from the calculated
matching $x_{i}$ to the ground-truth correspondence $x^{*}$. A matching is
accepted to be true if the normalised intrinsic distance is smaller then the
threshold $0.25$.

### Dataset

For the experiments we used a selction of shapes from the TOSCA dataset [4]
containing a total of 80 shapes of animals and humans in different poses. The
range of shape resolution in TOSCA goes from 4 344 points for the wolf shape
to 52 565 points for the david and michael shape.

### Experimental Settings

Figure 3:  The figure displays mean hit rates in % at geodesic error of 0.25
for different shapes and schemes in dependence of the scaling factor $c$. The
shape sets over which we average are cat, centaur, dog, horse, and wolf, taken
from the TOSCA Dataset [4].

As discussed, we use three methods to solve the heat and wave equation on
different shapes. As a reference method we employ the $l_{0}$-stable implicit
Euler method of first order in time [3]. We compare it with the discussed
second order methods, i.e. Crank-Nicolson and the $l_{0}$-stable method (23)
from [16].

Let us recall that we aim to explore the possibility to increase the time step
size $\tau$. The point with this is, that an implicit scheme should
theoretically perform stable independently on the time step size [11]. Thus we
expect that a rescaling of time step size may reveal if the $l_{0}$ stability
is consistently a useful property.

The time step $\tau$ and the number of iterations depend directly on each
other through $\tau M=t^{*}$. Since we fixed the value for $t^{*}$ with the
calculations from (26) an increase of $\tau$ leads to a decrease of $M$. We
denote the factor by which we have increased the time step size $\tau$ by $c$.
We have chosen $c=\left\\{1,5,10\right\\}$ for our experiments. With these
values for $c$ the number of iterations changed to
$M=\left\\{100,20,10\right\\}$.

### Results for the Heat Equation

A plot of results can be found in the left figure within Figure 3. We see that
all three methods lose some accuracy when the time step size is increased. The
Crank-Nicolson method gives the worst hit rates. For $c=1$ it can still keep
up with the other methods (73.84%), but it loses significantly more accuracy
(64.56%) when the time step size is increased, compared to the other two
methods. The second order $l_{0}$-stable method and the implicit Euler method
are very similar. Generally speaking, if we compare the hit rates from the
$l_{0}$-stable methods (first and second order), we observe nearly no
differences.

A look at the left plots of Figure 4 shows that the feature descriptor for the
individual methods is not very different. However, the beneficial aspect of an
$l_{0}$-stable method becomes evident, since for an increase of the time step
the Crank-Nicolson method produces clear oscillations, like in the simple
motivation example in Figure 2.

The results for the heat equation emphasize the usefulness of the $l_{0}$
stability as an important notion of practical importance in numerical shape
analysis.

### Results for Wave Equation

The right figure within Figure 3 is providing the plots of the mean hit rates.
Interesting is the fact, that with choosing a second order method we lose
about $8\%$ accuracy in the hit rate for $c=1$ compared to the Euler
integrator. At this point, further investigation is needed. However, we can
notice an increase in hit rate again by increasing the time step size. Again
we observe here that $l_{0}$ stability is a beneficial property.

Looking at the right plots in Figure 4, we see that a much more detailed
feature descriptor can be produced by the second order methods. However, this
does not increase the hit rate.

Figure 4:  The feature descriptors of the wolf shape in dependencies of the
time. The feature descriptor is received from the heat (left) or wave (right)
equation solved with a first order method [3], the Crank-Nicolson method and a
$l_{0}$-stable second order method [16]. From Top to Bottom we see the
different time scaling factor $c$. Starting with $c=1$ (top) going to $c=5$
(middle) and end up with $c=10$ (bottom). In the top plots the second-order
method are close to each other so they appear as one curve.

Summarising the findings, we observe again that the $l_{0}$ stable schemes
perform better than the classical Crank-Nicolson method.

##  6 Conclusion and Further Work

Summarising the experiments, we have validated that $l_{0}$ stability is an
important property for numerical integration in shape analysis. This is surely
a useful aspect for possible future developments.

There are some other indirect findings, e.g. that the use of higher order
schemes for time integration may not be of highly beneficial impact, since the
$l_{0}$-stable second order time integrator did not perform significantly
better than the implicit Euler scheme. However, this may be partly due to the
fact of the observed drop in accuracy in case of the wave equation, a subject
we aim to resolve in future work.

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# A restricted model for the bounded derived category of gentle algebras

Esha Gupta

###### Abstract.

We present a restricted model for the bounded derived category of gentle
algebras that encodes the indecomposable objects and positive extensions
between them. The model is then used to count the number of $d$-term silting
objects for linearly oriented $A_{n}$, recovering the result that they are
counted by the Pfaff-Fuss-Catalan numbers.

###### Contents

  1. 1 Introduction
  2. 2 Marked surfaces and admissible dissections
  3. 3 Admissible dissections and gentle algebras
  4. 4 Counting $d$-term silting objects in linearly oriented $A_{n}$

##  1\. Introduction

In recent years, (graded) gentle algebras have come to gather a lot of
attention because of their ubiquity in several areas of mathematics. These
include cluster theory, where they appear as cluster-tilted algebras and
Jacobian algebras associated to triangulations of marked surfaces [3, 9], as
well as geometry, where they appear as endomorphism rings of formal generators
of some partially wrapped Fukaya categories [6]. A geometric model for the
module categories of all gentle algebras was provided in [4], where the
authors realised them as tiling algebras associated to partial triangulations
of unpunctured surfaces with marked points on the boundary. A geometric model
for the $m$-cluster category of type $A_{n-1}$ was provided in [5] using
diagonals of a regular polygon. In [10], the authors provided a complete model
for the bounded derived category of a gentle algebra, which encoded
information such as the indecomposable objects, morphisms, mapping cones,
Auslander-Reiten translation for perfect objects, and Auslander–Reiten
triangles. A model for studying support $\tau$-tilting modules, or
equivalently $2$-term silting complexes, of all locally gentle algebras was
provided in [11]. For an acyclic quiver $Q$, it was shown in [8, Theorem 5.14]
that the hearts of $\mathcal{D}^{b}(\operatorname{mod}kQ)$ lying between the
canonical heart $\mathcal{H}_{Q}=\operatorname{mod}kQ$ and
$\mathcal{H}_{Q}[d-1]$ are in bijection with basic $d$-term silting complexes
as well as $d$-cluster tilting sets.

In this work, we modify the model of [10] to give a model for the bounded
derived category of a gentle algebra that encodes the indecomposable objects
and their positive extensions as (ungraded) arcs on a surface and their
intersections (Proposition 3.11). We can then restrict this model to study the
truncated homotopy categories
$\mathrm{K}^{[-d+1,0]}(\operatorname{proj}\Lambda)$ of $d$-term complexes of
projectives for a gentle algebra $\Lambda$. Since the model encodes positive
extensions, it is particularly suited for the study of $d$-term silting
complexes generalizing the case of $2$-term silting complexes studied in [11]
(Corollary 3.15, Remark 3.17). Applying the model to the case of linearly
oriented $A_{n}$ quiver, we recover the result of [12] that the number of
$d$-term silting complexes in $\mathrm{K}^{b}(\operatorname{proj}kA_{n})$ is
given by the Pfaff-Fuss-Catalan number $C^{d}_{n+1}$ (§ 4).

##  2\. Marked surfaces and admissible dissections

Throughout this work, $k$ will denote an algebraically closed field. We recall
here the special definition of marked surfaces as used in [2], the admissible
dissections of which are in bijection with gentle algebras.

######  Definition 2.1

[2, Definition 1.7] A _marked surface_ is a triple $(S,M,P)$, where

  1. (1)

$S$ is an oriented closed smooth surface with non-empty boundary $\partial S$;

  2. (2)

$M=M_{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ}\cup
M_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
is a finite set of marked points on $\partial S$. The elements of
$M_{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ}$
and
$M_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
will be represented by symbols
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$
and
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$,
respectively. They are required to alternate on each connected component of
$\partial S$, and each such component is required to contain at least one
marked point;

  3. (3)

$P=P_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
is a finite set of marked points in the interior of $S$, called punctures. The
elements of
$P_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
will also be represented by
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$.

######  Definition 2.2

[2, Definition 1.8] A
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$_-arc_
(or
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$_-arc_)
is a smooth map $\gamma$ from $[0,1]$ to $S\setminus P$ such that its
endpoints $\gamma(0)$ and $\gamma(1)$ are in
$M_{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ}$
(or in
$M_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$,
respectively). The curve $\gamma$ is required not to be contractible to a
point in
$M_{\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ}$
(or
$M_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$,
respectively).

We will usually consider arcs up to homotopy and inverses. Two arcs are said
to _intersect_ if any choice of homotopic representatives intersect.

######  Definition 2.3

[2, Definition 1.9] A collection of pairwise non-intersecting and pairwise
different
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arcs
$\\{\gamma_{1},\cdots,\gamma_{r}\\}$ on the surface $(S,M,P)$ is called
_admissible_ if the arcs $\\{\gamma_{1},\cdots,\gamma_{r}\\}$ do not enclose a
subsurface containing no punctures of
$P_{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$
and with no boundary segment on its boundary. A maximal admissible collection
of
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arcs
is called an _admissible
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-dissection_.

The notion of admissible
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-dissection
is defined similarly.

To any admissible
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-dissection,
we can associate a dual
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-dissection
in the following sense.

######  Proposition 2.4

[2, Proposition 1.13] Let $(S,M,P)$ be a marked surface, and let $\Delta$ be
an admissible
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-dissection.
There exists a unique admissible
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-dissection
$\Delta^{*}$ (up to homotopy) such that each arc of $\Delta^{*}$ intersects
exactly one arc of $\Delta$.

######  Definition 2.5

[2, Definition 1.14] The dissections $\Delta$ and $\Delta^{*}$ are called
_dual dissections_.

##  3\. Admissible dissections and gentle algebras

######  Definition 3.1

[2, Definition 2.1] Let $\Delta$ be an admissible
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-dissection
of a marked surface $(S,M,P)$. The $k$-algebra $A(\Delta)$ is the quotient of
the path algebra of the quiver $Q(\Delta)$ by the ideal $I(\Delta)$ defined as
follows:

  * •

the vertices of $Q(\Delta)$ are in bijection with the
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arcs
in $\Delta$;

  * •

there is an arrow $i\to j$ in $Q(\Delta)$ whenever the
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arcs
$i$ and $j$ meet at a marked point
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$,
with $i$ preceding $j$ in the counter-clockwise order around
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$,
and with no other arc coming to
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$
between $i$ and $j$.

  * •

the ideal $I(\Delta)$ is generated by the following relations: whenever $i$
and $j$ meet at a marked point as above, and the other end of $j$ meets $k$ at
a marked point as above, then the composition of the corresponding arrows
$i\to j$ and $j\to k$ is a relation.

######  Example 3.2

Let $(S,M,P)$ be the marked surface in Figure 1(a), and $\Delta$ the
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-dissection
dual to the depicted
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-dissection.
Then $A(\Delta)$ is the path algebra of the quiver with relations shown in
Figure 1(b).

(a)

(b)

Figure 1.

######  Theorem 3.3

[2, Theorem 2.3] The assignment $((S,M,P),\Delta)\to A(\Delta)$ defines a
bijection from the set of homeomorphism classes of marked surfaces $(S,M,P)$
with an admissible dissection to the set of isomorphism classes of gentle
algebras.

Let $\Lambda$ be a gentle algebra of rank $n$. Let $((S,M,P),\Delta)$ be the
marked surface with an admissible dissection obtained from the previous
theorem. We will give another model of the bounded derived category of
$\Lambda$ using this surface.

We start by choosing a labelling of the
${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ}$-points
in $M$. We replace each
${\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ}$-point
$v$ on a boundary component between two
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-points
with a collection of $\mathbb{Z}$-indexed
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-points
$T=\\{T_{(i,v)}\mid i\in\mathbb{Z}\\}$ arranged in descending order along the
orientation of the component. Moreover, we do so in such a way that the set
$T$ has precisely two limit points given by the two
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-points.
For a point $T_{(i,v)}$, we define $\epsilon(T_{(i,v)}):=i$ and
$\sigma(T_{(i,v)}):=v$. We consider special arcs joining these
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-points
which we call ‘slaloms’.

######  Example 3.4

The above process transforms the marked surface in Figure 2(a) to the surface
in Figure 2(b).

(a)

(b)

Figure 2.

######  Definition 3.5

A
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-arc
is a smooth map $\gamma$ from the interval $[0,1]$ to $S\setminus P$ such that
its endpoints $\gamma(0)$ and $\gamma(1)$ are
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-points.
The curve $\gamma$ is required not to be contractible to a
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-point.

######  Definition 3.6

Let $\gamma$ be a
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-arc.
We assume $\gamma$ intersects the arcs of the dual
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-dissection
$\Delta^{*}$ at a finite number of points, minimally and transversally. Define

$f_{\gamma}:\gamma\cap\Delta^{*}\to\mathbb{Z}$

as follows. The orientation of $\gamma$ induces a total order on the points in
$\gamma\cap\Delta^{*}$. Let $p_{0}$ be the initial point in this order. Then
$f(p_{0}):=\epsilon(s(\gamma))$. Next, if $p$ and $q$ are in
$\gamma\cap\Delta^{*}$ and $q$ is the successor of $p$, then $\gamma$ enters a
polygon enclosed by
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-arcs
of $\Delta^{*}$ via $p$ and leaves it via $q$. If the
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-points
in this polygon are to the left of $\gamma$, then $f(q):=f(p)+1$; otherwise,
$f(q):=f(p)-1$.

A
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-arc
$\gamma$ is called a _slalom_ if $f(q_{0})=\epsilon(t(\gamma))$, where $q_{0}$
is the last point of $\gamma\cap\Delta^{*}$.

######  Example 3.7

Let $\gamma_{1}$ be the slalom in Figure 3(a). Then the corresponding function
$f_{\gamma_{1}}$ is given as

$f_{\gamma_{1}}(A)=0,\ f_{\gamma_{1}}(B)=-1,\ f_{\gamma_{1}}(C)=0.$

Let $\gamma$ be a slalom connecting $T_{(i,v_{1})}$ to $T_{(j,v_{2})}$. We
consider the homotopy class of $\gamma$ where the homotopies are allowed to
move the endpoints of $\gamma$ along the boundary without crossing a
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-point.
Then this homotopy class contains a unique (up to homotopy)
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arc
connecting $v_{1}$ and $v_{2}$. We denote this
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arc
by $\sigma_{\gamma}$. Moreover, the function $f_{\gamma}$ naturally defines a
grading on the arc $\sigma_{\gamma}$. We will denote the graded
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arc
$(\sigma_{\gamma},f_{\gamma})$ as $\Sigma(\gamma)$.

######  Example 3.8

Let $\gamma_{1}$ be the slalom in Figure 3(a). Then using the procedure
described above, $\gamma_{1}$ is transformed into the graded arc
$\Sigma(\gamma_{1})$ shown in Figure 3(b).

(a)

(b)

Figure 3.

The map $\Sigma$ defines a bijection between the set of homotopy classes of
slaloms with the set of homotopy classes of graded
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arcs,
as defined in [2, Definition 2.4], which are, in turn, in bijection with
certain indecomposable objects of $K^{-,b}(\operatorname{proj}\Lambda)$ called
(finite) string objects. We denote by $P^{\bullet}_{\gamma}$ the object
associated to $\Sigma(\gamma)$. For two graded
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-arcs
$(\mu_{1},f_{1})$ and $(\mu_{2},f_{2})$, we will use the description of
$\mathrm{Hom}(P^{\bullet}_{(\mu_{1},f_{1})},P^{\bullet}_{(\mu_{2},f_{2})})$ in
terms of the ‘graded intersection points’ of $(\mu_{1},f_{1})$ and
$(\mu_{2},f_{2})$, as given in [10].

######  Definition 3.9

Let $\gamma_{1}$ and $\gamma_{2}$ be two slaloms and $T$ an intersection point
of $\gamma_{1}$ and $\gamma_{2}$ lying in the interior of $S$. Then $T$ is
called contractible if (one of) the region(s) bounded by the following three
segments is contractible (Figure 4):

  1. (1)

the part of $\gamma_{1}$ between $T$ and the boundary.

  2. (2)

the part of $\gamma_{2}$ between $T$ and the boundary.

  3. (3)

the part of the boundary between the endpoints of $\gamma_{1}$ and
$\gamma_{2}$.

Figure 4. $S$ is a contractible intersection point of $\gamma_{1}$ and
$\gamma_{2}$

######  Theorem 3.10

Let $\gamma_{1}$ and $\gamma_{2}$ be two slaloms. Then we have the following:

  1. (1)

There is a bijection between the set of interior intersection points of
$\Sigma({\gamma_{1}})$ and $\Sigma({\gamma_{2}})$, and the set of non-
contractible intersection points of $\gamma_{1}$ and $\gamma_{2}$.

  2. (2)

There is a bijection between the set of boundary intersection points of
$\Sigma({\gamma_{1}})$ and $\Sigma({\gamma_{2}})$ of positive degree, and the
set of contractible intersection points of $\gamma_{1}$ and $\gamma_{2}$.

  3. (3)

There is a bijection between the set of boundary intersection points of
$\Sigma({\gamma_{1}})$ and $\Sigma({\gamma_{2}})$ of degree $0$, and the
boundary intersection points of $\gamma_{1}$ and $\gamma_{2}$.

###### Proof.

Let $(\mu_{1},f_{1}):=\Sigma(\gamma_{1})$ and
$(\mu_{2},f_{2}):=\Sigma(\gamma_{2})$. Assume that we have chosen homotopy
representatives such that $\mu_{1}$ and $\mu_{2}$ intersect minimally, and so
do $\gamma_{1}$ and $\gamma_{2}$. We prove the three statements one by one.

  1. (1)

Let $p$ be an interior intersection point of $\Sigma(\gamma_{1})$ and
$\Sigma(\gamma_{2})$. It is possible to choose homotopy representatives of
$\gamma_{1}$ and $\gamma_{2}$ such that $p$ is also an interior intersection
point of $\gamma_{1}$ and $\gamma_{2}$. Moreover, $p$ is non-contractible
because, otherwise, $\mu_{1}$ and $\mu_{2}$ do not intersect minimally as we
can choose homotopy representatives that do not intersect at $p$ (Figure 5).

Figure 5. $p$ has to be a non-contractible intersection point of $\gamma_{1}$
and $\gamma_{2}$

Conversely, if $p$ is a non-contractible intersection point of $\gamma_{1}$
and $\gamma_{2}$, then it is still an interior intersection point of
$\Sigma(\gamma_{1})$ and $\Sigma(\gamma_{2})$ when they intersect minimally.

  2. (2)

Suppose $p$ is a boundary intersection point of $\Sigma(\gamma_{1})$ and
$\Sigma(\gamma_{2})$ of degree $>0$. Then, without loss of generality, we can
assume that we have the configuration around $p$ as shown in Figure 6(a) with
$f_{2}(p_{2})>f_{1}(p_{1})$.

(a)

(b)

Figure 6.

This implies that $\epsilon(q^{\prime})>\epsilon(q)$, where $q$ (resp.
$q^{\prime}$) is the endpoint of $\gamma_{1}$ (resp. $\gamma_{2}$) such that
$\sigma(q)=p$ (resp. $\sigma(q^{\prime})=p$). Thus, the configuration of
$\gamma_{1}$ and $\gamma_{2}$ is as shown in Figure 6(b). Since $\mu_{1}$ and
$\mu_{2}$ are homotopic to $\gamma_{1}$ and $\gamma_{2}$ respectively, and
they intersect on the boundary, the point $T$ has to be a contractible
intersection point of $\gamma_{1}$ and $\gamma_{2}$.

Conversely, let $T$ be a contractible intersection point of $\gamma_{1}$ and
$\gamma_{2}$. Then, without loss of generality, we can assume that
$\gamma_{1}$ and $\gamma_{2}$ are as shown in Figure 6(b). In particular, the
part of the boundary between $q$ and $q^{\prime}$ does not contain any
$\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet$-points,
which implies that $\sigma(q)=\sigma(q^{\prime})$, and hence
$\Sigma({\gamma_{1}})$ and $\Sigma({\gamma_{2}})$ intersect on the boundary.
Moreover, this intersection point has a positive degree because
$f_{2}(p_{2})=\epsilon(q^{\prime})>\epsilon(q)=f_{1}(p_{1})$.

  3. (3)

Suppose $p$ is a boundary intersection point of $\Sigma(\gamma_{1})$ and
$\Sigma(\gamma_{2})$ of degree $0$. Then we can assume that the configuration
around $p$ is as shown in Figure 6(a) such that $f_{1}(p_{1})=f_{2}(p_{2})$.
This implies that $\epsilon(q)=\epsilon(q^{\prime})=f_{1}(p_{1})$, where $q$
(resp. $q^{\prime}$) is the endpoint of $\gamma_{1}$ (resp. $\gamma_{2}$) such
that $\sigma(q)=p$ (resp. $\sigma(q^{\prime})=p$). Thus, $q=q^{\prime}$ is a
boundary intersection point of $\gamma_{1}$ and $\gamma_{2}$. Conversely, if
$q$ is a boundary intersection point of $\gamma_{1}$ and $\gamma_{2}$, then,
by definition, $\Sigma(\gamma_{1})$ and $\Sigma(\gamma_{2})$ have an
intersection point on the boundary of degree $0$.

∎

######  Proposition 3.11

Two slaloms $\gamma_{1},\gamma_{2}$ intersect in the interior of $S$ if and
only if either
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{1}},P^{\bullet}_{\gamma_{2}})\neq 0$
for some $i>0$ or
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{2}},P^{\bullet}_{\gamma_{1}})\neq 0$
for some $i>0$.

###### Proof.

Suppose $\gamma_{1}$ and $\gamma_{2}$ intersect in the interior of $S$ at a
point $T$. If $T$ is a contractible intersection point, then using Theorem
3.10, it corresponds to a boundary intersection point of $\Sigma(\gamma_{1})$
and $\Sigma(\gamma_{2})$ of positive degree. This implies that either
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{1}},P^{\bullet}_{\gamma_{2}})\neq 0$
for some $i>0$ or
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{2}},P^{\bullet}_{\gamma_{1}})\neq 0$
for some $i>0$. On the other hand, if $T$ is a non-contractible intersection
point, then it corresponds to an interior intersection point of
$\Sigma(\gamma_{1})$ and $\Sigma(\gamma_{2})$. Using [2, Lemma 3.5], we get
that either
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{1}},P^{\bullet}_{\gamma_{2}})\neq 0$
for some $i>0$ or
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{2}},P^{\bullet}_{\gamma_{1}})\neq 0$
for some $i>0$.

Conversely, suppose $\gamma_{1}$ and $\gamma_{2}$ do not intersect in the
interior of $S$. Then either they do not intersect at all or they intersect on
the boundary. In both cases, $\Sigma(\gamma_{1})$ and $\Sigma(\gamma_{2})$ can
only have zero or negative degree boundary intersection points using Theorem
3.10. This implies that
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{1}},P^{\bullet}_{\gamma_{2}})=0$ for
all $i>0$ and
$\mathrm{Ext}^{i}(P^{\bullet}_{\gamma_{2}},P^{\bullet}_{\gamma_{1}})=0$ for
all $i>0$. ∎

The above proposition helps us to give a characterization of presilting and
silting objects in $K^{-,b}(\operatorname{proj}\Lambda)$.

######  Definition 3.12

Say that a collection $\gamma_{1},\gamma_{2},\ldots,\gamma_{m}$ of slaloms is
mutually non-intersecting if $\gamma_{i}$ does not intersect $\gamma_{j}$ in
the interior of $\S$ for all $1\leq i,j\leq n$.

######  Corollary 3.13

There is a bijection between basic presilting objects in
$K^{-,b}(\operatorname{proj}\Lambda)$ and mutually non-intersecting
collections of slaloms in $S_{\Lambda}$.

###### Proof.

Suppose $T=\oplus_{i=1}^{m}T_{i}$ is a basic presilting object in
$K^{-,b}(\operatorname{proj}\Lambda)$ with $T_{i}$ indecomposable. Then, using
[2, Lemma 3.4], we get that $T_{i}$ is of the form $P^{\bullet}_{\gamma_{i}}$
for some slalom $\gamma_{i}$. Since $T$ is presilting, we get that
$\mathrm{Ext}^{j}(T_{i_{1}},T_{i_{2}})=0$ for all $j>0$ and for all $1\leq
i_{1},i_{2}\leq m$. Using the previous proposition, this gives that
$\\{\gamma_{1},\cdots,\gamma_{m}\\}$ is a mutually non-intersecting collection
of slaloms in $S_{\Lambda}$.

Conversely, given a mutually non-intersecting collection of slaloms
$\\{\gamma_{1},\cdots,\gamma_{m}\\}$ in $S_{\Lambda}$, the previous
proposition immediately implies that
$T=\oplus_{i=1}^{m}P^{\bullet}_{\gamma_{i}}$ is a basic presilting object in
$K^{-,b}(\operatorname{proj}\Lambda)$. ∎

Using [2, Proposition 5.7], we know that for a gentle algebra $\Lambda$, a
basic presilting object $X$ is silting if and only if it has $n=|\Lambda|$
many indecomposable summands. This gives us the following corollary.

######  Corollary 3.14

There is a bijection between basic silting objects in
$K^{-,b}(\operatorname{proj}\Lambda)$ and mutually non-intersecting
collections of $n$ slaloms in $S_{\Lambda}$.

We can also restrict the above model to obtain a characterization of $d$-term
presilting objects of $\Lambda$. To do this, we keep only the
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-points
labelled from $0$ to $-d+1$ between each pair of neighbouring
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-points.
A slalom in the restricted model is a slalom $\gamma$ in the original model
for which $f_{\gamma}(\gamma\cap\Delta^{*})\subseteq[-d+1,0]$, as this ensures
that $P^{\bullet}_{\gamma}$ is concentrated in $[-d+1,0]$. We will denote this
restricted model by $S_{\Lambda}^{d}$. It is easy to see that Theorem 3.10 and
Proposition 3.11 still hold in this restricted model, which gives us the
following corollary.

######  Corollary 3.15

There is a bijection between basic $d$-term silting objects of $\Lambda$ and
mutually non-intersecting collections of $n$ slaloms in $S^{d}_{\Lambda}$.

######  Remark 3.16

Note that in the full model $S_{\Lambda}$, for every finite arc starting at a
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-point,
there exists a unique choice for the other endpoint which makes it a slalom.
However, this is not the case in the restricted model $S^{d}_{\Lambda}$,
because of the additional condition that
${\gamma}(\gamma\cap\Delta^{*})\subseteq[-d+1,0]$.

######  Remark 3.17

In [11], the authors introduced a model for basic support $\tau$-tilting
modules by considering blossom points on the boundary of a marked surface and
slaloms connecting these points. Viewing these blossom points as
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-points,
one precisely recovers the model $S^{2}_{\Lambda}$ described above, which is
consistent with the fact that basic support $\tau$-tilting modules are in
bijection with basic $2$-term silting complexes [1].

##  4\. Counting $d$-term silting objects in linearly oriented $A_{n}$

Let $kA_{n}$ be the path algebra of the linearly oriented quiver of type
$A_{n}$, that is, $kA_{n}=k(1\to 2\to\cdots\to n)$. In this section, we will
give a recursive formula for the number of $d$-term silting objects in
$kA_{n}$ using the above corollary. This will recover the result of [12] that
these are counted by the Pfaff-Fuss-Catalan numbers.

The marked surface $(S,M,P)$ obtained from Theorem 3.3 for $\Lambda=kA_{n}$ is
shown in Figure 7(a). By replacing each
$\color[rgb]{0,.5,.5}\definecolor[named]{pgfstrokecolor}{rgb}{0,.5,.5}\circ$-point
in this figure with $d$
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-points,
we obtain the surface $S^{d}_{kA_{n}}$ defined above (Fig 7(b)).

(a)

(b)

Figure 7.

Let $B^{d}_{n}$ denote the number of basic silting objects in $kA_{n}$. Using
Corollary 3.15, we get that $B^{d}_{n}$ is also the number of collections of
$n$ mutually non-intersecting slaloms in Figure 7(b). To calculate
$B^{d}_{n}$, we first count the number of such collections containing some
fixed slalom $\gamma$. This is done by cutting the disc along this slalom, and
relabelling (one of) the parts thus obtained to get $S^{d}_{kA_{m}}$ for some
$m<n$. This allows us to build a recursive formula for $B^{d}_{n}$. We explain
the detailed process below.

Let $\Gamma$ be a collection of $n$ mutually non-intersecting slaloms. Such a
collection will be maximal with respect to the property of mutual non-
intersection as the number of indecomposable summands of any presilting object
is less than or equal to $n$ [13]. Our first claim is that in such a
collection of $n$ slaloms, at least one of
$(0,0),(0,-1),(0,-2),\cdots(0,-d+1)$ has to be an endpoint of some slalom.
This is because otherwise, the slalom in Figure 8(a) will contradict the
maximality of the collection.

(a)

(b)

Figure 8.

######  Lemma 4.1

Let $\gamma$ be a slalom in $S^{d}_{n}$ as shown in Figure 9(a). Then a
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-arc
lying in the disc $D^{\prime}$ is a slalom in $S^{d}_{n}$ if and only if it is
a slalom in $S^{d}_{i_{2}-i_{1}-1}$ obtained by relabelling the points of
$D^{\prime}$ as shown in Figure 9(b).

(a)

(b)

Figure 9.

###### Proof.

Since there are no slaloms in $S^{d}_{n}$ starting at $(i_{1},0)$ and lying in
$D^{\prime}$, we can remove it. Moreover, since the only slaloms in
$D^{\prime}$ starting at $(i_{1},m)$ for some $j\leq\leavevmode\nobreak\
m\leq-1$ end at $(l,m+1)$ for some $i_{1}+1\leq l\leq i_{2}-1$, removing the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-arc
$i_{1}$ and relabelling as in Fig 9(b) gives a bijection between the set of
slaloms starting at these points. Similarly, since the only slaloms in
$D^{\prime}$ ending at $(i_{2},m)$ for some $-d+1\leq m\leq j+1$ start at
$(l,m-1)$ for some $i_{1}+1\leq l\leq i_{2}-1$, removing the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-arc
$i_{2}$, the point $(i_{2},-d+1)$, and relabelling as in Fig 9(b) gives a
bijection between the set of slaloms ending at these points. Combining this
with the map that sends the slalom connecting $(s_{1},t)$ to $(s_{2},t+1)$ for
some $i_{1}<s_{1}<s_{2}<i_{2}$ in $D^{\prime}$ to the slalom connecting
$(s_{1}-i_{1},t)$ to $(s_{2}-i_{1},t+1)$ in the relabelled figure, we get the
required bijection. ∎

######  Lemma 4.2

Let $\gamma$ be a slalom in $S^{d}_{n}$ as shown in Figure 10(a). Then a
$\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\times$-arc
lying in the disc $D^{\prime}$ is a slalom in $S^{d}_{n}$ if and only if it is
a slalom in $S^{d}_{n-i}$ obtained by relabelling the points of $D^{\prime}$
as shown in Figure 10(b).

(a)

(b)

Figure 10.

###### Proof.

Since there are no slaloms in $S^{d}_{n}$ starting at $(i,0)$ and lying in
$D^{\prime}$ (other than possibly $\gamma$ when $j=0$), we can remove it.
Moreover, since the only slaloms in $D^{\prime}$ starting at $(i,m)$ for some
$j\leq m\leq-1$ end at $(l,m+1)$ for some $i+1\leq l\leq n$, removing the
${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\bullet}$-arc
$i$ and relabelling as in Fig 10(b) gives a bijection between the set of
slaloms starting at these points. Combining this with the map that sends the
slalom connecting $(s_{1},t)$ to $(s_{2},t+1)$ for some $i<s_{1}<s_{2}\leq n$
in $D^{\prime}$ to the slalom connecting $(s_{1}-i,t)$ to $(s_{2}-i,t+1)$ in
the relabelled figure, and the slalom connecting $(s,t)$ to $(0,t)$, for some
$i<s_{1}\leq n$ and $-d+1\leq t\leq j$, in $D^{\prime}$ to the slalom
connecting $(s-i,t)$ to $(0,t)$ in the relabelled figure, we get the required
bijection. ∎

######  Corollary 4.3

The number of mutually non-intersecting collections of $i_{2}-i_{1}-1$ slaloms
in $S^{d}_{n}$ lying in $D^{\prime}$ is $B^{d}_{i_{2}-i_{1}-1}$.

###### Proof.

It is easy to see that a collection of $i_{2}-i_{1}-1$ slaloms in $S^{d}_{n}$
lying in $D^{\prime}$ is mutually non-intersecting if and only if it is
mutually non-intersecting in $S^{d}_{i_{2}-i_{1}-1}$ under the above
bijection. Using Corollary 3.15, the number of such collections is equal to
$B^{d}_{i_{2}-i_{1}-1}$. ∎

We can now divide the problem into two cases.

  1. (1)

$(0,0)$ is the endpoint of a slalom: Let $\gamma_{d}$ be the rightmost such
slalom. Since this is the rightmost slalom connected to $(0,0)$, we need to
block the slaloms connecting $(0,0)$ to some $(j,0)$ for $j<i_{d-1}$. This
gives that at least one of $(1,-1),\cdots(1,-d+1)$ is the endpoint of some
slalom in $\Gamma$. Suppose $(1,-d+1)$ is the endpoint of some slalom in
$\Gamma$ and let $\gamma_{1}$ be the leftmost such slalom. To block the slalom
connecting $(0,0)$ to $(i_{1},0)$, we need that at least one of
$(i_{1},0),\cdots(i_{1},-d+2)$ is the endpoint of some slalom in $\Gamma$.
Repeating this argument with the assumption that at each step the point with
the lowest index is the endpoint of some slalom in $\Gamma$, we get a
collection of $d$ slaloms in $\Gamma$ arranged as shown in Figure 8(b).

Figure 11. The bigger disc can be divided into $d$ smaller pieces

Using Lemmas 4.1 and 4.2, each of the parts of the discs cut by these slaloms
can be relabelled to obtain $S^{d}_{m}$ for some $m<n$. The exact value of $m$
for each of the parts is shown in Figure 11. Thus the number of collections of
non-mutually intersecting $n$ slaloms containing the above collection of $d$
slaloms is given by the sum of the product of the number of collections of
non-mutually intersecting $a_{1},\cdots,a_{d}$ slaloms in
$S^{d}_{kA_{i_{1}-2}},S^{d}_{kA_{i_{2}-i_{1}-1}},\cdots,S^{d}_{kA_{i_{d-1}-i_{d-2}-1}},S^{d}_{kA_{n-i_{d-1}}}$
respectively. Now, since we know that $a_{1}\leq i_{1}-2$, $a_{l}\leq
i_{l}-i_{l-1}-1$ for $2\leq l\leq d-1$, $a_{d}\leq n-i_{d-1}$, and
$\Sigma_{m=1}^{d}a_{d}=n-d$, we get that $a_{1}=i_{1}-2$,
$a_{l}=i_{l}-i_{l-1}-1$ for $2\leq l\leq d-1$, and $a_{d}=n-i_{d-1}$. This
means that the above number is equal to the product
$B^{d}_{i_{1}-2}B^{d}_{i_{2}-i_{1}-1}\cdots
B^{d}_{i_{d-1}-i_{d-2}-1}B^{d}_{n-i_{d-1}}$. Taking into account all possible
values of $i_{1},\cdots i_{d-1}$, we get the sum

$\Sigma_{i_{d-1}=2}^{n}\Sigma_{i_{d-2}=2}^{i_{d-1}-1}\cdots\Sigma_{i_{1}=2}^{i_{2}-1}B^{d}_{i_{1}-2}B^{d}_{i_{2}-i_{1}-1}\cdots
B^{d}_{i_{d-1}-i_{d-2}-1}B^{d}_{n-i_{d-1}}.$

Now in the penultimate step of the above argument, when one of $(i_{d-2},0)$
or $(i_{d-2},-1)$ has to be connected to something in $\Gamma$, suppose
$(i_{d-2},-1)$ is not connected to anything. Then $(i_{d-2},0)$ has to be
connected to something, which is only possible if $i_{d-2}=i_{d-1}$ (Figure
12(a)).

(a)

(b)

Figure 12.

Again, in this case, the parts of the disc cut by the $d-1$ slaloms can be
relabelled by Lemmas 4.1 and 4.2 to get $S^{d}_{m}$ for some $m<n$. And we
will get that the number of collections $\Gamma$ containing this set of $d-1$
slaloms is given by the product of $d-1$ terms
$B^{d}_{i_{1}-2},B^{d}_{i_{2}-i_{1}-1},\cdots,B^{d}_{i-i_{d-3}-1},B^{d}_{n-i}$.
Repeating this argument at each step, we get that the total number of
collections $\Gamma$ in which $(0,0)$ is connected to something is given by

$\Sigma_{k=0}^{d-1}\binom{d-1}{k}\Sigma_{i_{k}=2}^{n}\Sigma_{i_{k-1}=2}^{i_{k}-1}\cdots\Sigma_{i_{1}=2}^{i_{2}-1}B^{d}_{i_{1}-2}B^{d}_{i_{2}-i_{1}-1}\cdots
B^{d}_{i_{k}-i_{k-1}-1}B^{d}_{n-i_{k}},$

where the sum over $0$ elements is taken to be $B^{d}_{n-1}$, corresponding to
the boundary case in Figure 12(b).

  2. (2)

$(0,0)$ is not the endpoint of a slalom: First suppose that $(0,-1)$ is
connected to something. It can be easily seen that the above cutting and
relabelling procedure works in this case as well, except that the maximum
number of parts in which the disc is cut is now $d-1$. This gives that the
number of collections $\Gamma$ in this case is

$\Sigma_{k=0}^{d-2}\prescript{d-2}{}{C}_{k}\Sigma_{i_{k}=2}^{n}\Sigma_{i_{k-1}=2}^{i_{k}-1}\cdots\Sigma_{i_{1}=2}^{i_{2}-1}B^{d}_{i_{1}-2}B^{d}_{i_{2}-i_{1}-1}\cdots
B^{d}_{i_{k}-i_{k-1}-1}B^{d}_{n-i_{k}}.$

In the general case, let $m$ be the smallest integer such that $(0,-m)$ is
connected to something. Then the number of collections is given by

$\Sigma_{k=0}^{d-1-m}\binom{d-1-m}{k}\Sigma_{i_{k}=2}^{n}\Sigma_{i_{k-1}=2}^{i_{k}-1}\cdots\Sigma_{i_{1}=2}^{i_{2}-1}B^{d}_{i_{1}-2}B^{d}_{i_{2}-i_{1}-1}\cdots
B^{d}_{i_{k}-i_{k-1}-1}B^{d}_{n-i_{k}}.$

Thus, we get that the total number of $d$-term silting objects in $kA_{n}$ is
given by the recursive formula

$B^{d}_{n}=\Sigma_{k=0}^{d-1}\Sigma_{m=k}^{d-1}\binom{m}{k}\Sigma_{i_{k}=2}^{n}\Sigma_{i_{k-1}=2}^{i_{k}-1}\cdots\Sigma_{i_{1}=2}^{i_{2}-1}B^{d}_{i_{1}-2}B^{d}_{i_{2}-i_{1}-1}\cdots
B^{d}_{i_{k}-i_{k-1}-1}B^{d}_{n-i_{k}},$

where $B_{0}^{d}=1$, and $B_{1}^{d}=d$.

Since the above recursive formula also counts the number of complete ordered
$d$-ary trees with $n+1$ internal nodes, which are known to be counted by the
Pfaff-Fuss-Catalan numbers [7], we recover the result that the number of
$d$-term silting objects in $kA_{n}$ is given by the Pfaff-Fuss-Catalan number
$C^{d}_{n+1}$.

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E. Gupta, Université Paris-Saclay, UVSQ, CNRS, Laboratoire de Mathématiques de
Versailles, 78000, Versailles, France.

E-mail address<EMAIL_ADDRESS>