# Non-equilibrium Ising Model on a 2D Additive Small-World Network

R. A. Dumer  Instituto de Física - Universidade Federal de Mato Grosso,
78060-900, Cuiabá MT, Brazil.  M. Godoy  Instituto de Física - Universidade
Federal de Mato Grosso, 78060-900, Cuiabá MT, Brazil.

###### Abstract

In this work, we have studied the Ising model with one- and two-spin flip
competing dynamics on a two-dimensional _additive_ small-world network
(A-SWN). The system model consists of a $L\times L$ square lattice where each
site of the lattice is occupied by a spin variable that interacts with the
nearest neighbor spins and it has a certain probability $p$ of being
additionally connected at random to one of its farther neighbors. The dynamics
present in the system can be defined by the probability $q$ of being in
contact with a heat bath at a given temperature $T$ and, at the same time,
with a probability of $1-q$ the system is subjected to an external flux of
energy into the system. The contact with the heat bath is simulated by one-
spin flip according to the Metropolis prescription, while the input of energy
is mimicked by the two-spin flip process, involving a simultaneous flipping of
a pair of neighboring spins. We have employed Monte Carlo simulations to
obtain the thermodynamic quantities of the system, such as, the total
$\textrm{m}_{\textrm{L}}^{\textrm{F}}$ and staggered
$\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ magnetizations per spin, the
susceptibility $\chi_{\textrm{L}}$, and the reduced fourth-order Binder
cumulant $\textrm{U}_{\textrm{L}}$. We have built the phase diagram for the
stationary states of the model in the plane $T$ versus $q$, showing the
existence of two continuous transition lines for each value of $p$: one line
between the ferromagnetic $F$ and paramagnetic $P$ phases, and the other line
between the $P$ and antiferromagnetic $AF$ phases. Therefore, we have shown
that the phase diagram topology changes when $p$ increases. Using the finite-
size scaling analysis, we also obtained the critical exponents for the system,
where varying the parameter $p$, we have observed a different universality
class from the Ising model in the regular square lattice to the A-SWN.

Competing dynamics; Small-world network; Stationary state; Phase transitions;

##  .1 Introduction

Figure 1: Schematic representation of the system and the A-SWN. The blue
square dots indicate the sites on one of the sublattices, the white square
dots are the sites on the other sublattice, the solid black lines are the
nearest-neighbor interactions $J$ between pairs of spins, and the blue-dashed
lines are long-range interaction $J_{ik}$ added to the network with a certain
probability $p$. (a) for $p=0$, (b) for $p=0.5$ and (c) for $p=1$.

In the 1960s, the dynamic behavior of the Ising model was successfully
described by Glauber [1] and Kawasaki [2] mechanisms. This instigated interest
in the competition between the Glauber and Kawasaki stochastic process, one-
spin flip and two-spin exchange, respectively, in this model. This competition
can be simulated by the Glauber process with probability $q$ simulating the
system in contact with a heat bath at a temperature $T$, and at the same time,
with probability $1-q$, the Kawasaki process mimics an input of energy into
the system [3]. Each of these dynamical processes singly satisfies the
detailed balance condition, which drives the system toward equilibrium.
However, when both act simultaneously, the detailed balance is no longer
satisfied and the system is forced out of equilibrium.

The Ising model on a regular square lattice has the critical temperature and
universality class are given by the critical exponents well-known exactly at
the equilibrium state [4]. Therefore, the stationary non-equilibrium states
were obtained by the two competing dynamic processes described above, and a
self-organization is observed by the disappearance of the ordered
ferromagnetic $F$ phase in the transition to the paramagnetic $P$ phase, and
identification of the ordered antiferromagnetic $AF$ phase, as we increase the
flow of energy into the system [5]. However, through the Monte Carlo
simulations (MC) the critical exponents of the system have been obtained, and
because it is a system with the same symmetry, spatial dimension, and range
interactions, the exponents are the same as at the equilibrium state model,
and known exactly [6]. In the same way, Godoy and Figueredo investigated the
mixed-spin Ising model, which does not admit spin exchanges between the spin
sublattices, consequently do not admit to utilizing the Kawasaki dynamic.
Thus, the competing dynamic was made by the one- and two-spin flip mechanisms,
and even with that, they have also obtained the self-organization phenomena
[7], and based on the critical behavior of the system, the universality class
of the system is the same that the Ising model with only spin-1/2 [8].
Therefore, in the non-equilibrium models, the universality class of the
stationary critical behavior is the same as in the equilibrium models. All of
these works were studied on regular square lattices.

By using graph theory, Watts and Strogatz quantify the properties of Small-
World phenomena as demonstrated in Milgram’s 1967 study [9]. As an underlying
assumption of the Watts-Strogatz model (WS-model) [10], vertices of graphs are
sites of networks, and edges are connections between sites of the networks. By
introducing a disorder parameter $p$, as the probability of randomly rewriting
each one of the connections in a regular lattice, we can obtain the SWN in
specific regions in the interval $0<p\leq 1$. The SWN regime is identified in
regions of $p$ where the network possesses local clustering, $C(p)$, of a
regular lattice, but at the same time has an average distance between any two
sites, $l(p)$, characteristic of a random lattice. In addition to the WS-
model, some variants of this model were also developed to describe the
properties of an SWN. One of these variants [11] uses a regular square
lattice, and we can add a long-range interaction to each site with a certain
probability $p$. This leads to a small typical separation, preserving the
clustering property of a regular lattice. While we have described the rewiring
SWN (R-SWN) in the WS-model [10], this last form is known as additive SWN
(A-SWN) [11].

These networks have been used in numerous physical models since the initial
SWN model was put forth [12, 13, 14, 15, 16], including the Ising model in 1D,
2D, and 3D for the investigation of the critical phenomena at equilibrium
system [17, 18, 19, 20, 21, 22, 23, 24]. According to these findings for the
Ising model, an order to the disorder phase transition is established for
$T\neq 0$ with $0<p\leq 1$, and it is seen that the addition of long-range
interactions changes the critical behavior of the system.

The interesting behavior of the Ising model at the equilibrium SWN, its
investigation was also carried out about the non-equilibrium phase transitions
by the competing dynamics: analytically in 1D [25], by MC simulations in 2D
[26], and by the Gaussian model in 3D [27]. In all of these works they have
been using the competition between the Glauber and Kawasaki dynamics, and have
no conclusions about the mean-field critical behavior observed at the
equilibrium Ising model on an SWN [18, 20, 21, 22, 24, 28]. However, in 2D and
3D systems, is obtained the $AF-P$ and $F-P$ phase transitions, characteristic
of the self-organization phenomena, and observed in all of the other systems
at the non-equilibrium state by the competing dynamics.

In the present work, we have investigated the Ising model in a two-dimensional
A-SWN, where each site of the network is occupied by a spin variable spin-1/2
that can assume values $\pm 1$. We limit by one the number of long-range
interactions that each site can receive with probability $p$, and divide the
network into two sublattices, each new interaction created should connect
these sublattices. The system is in a non-equilibrium regime by competing
between two dynamic processes that do not conserve the order parameter: with
competition probability $q$, the one-spin flip process simulates the system in
contact with a heat bath at temperature $T$, and with competition probability
$1-q$ the two-spin flip process mimics the system subjected to an external
energy flux into it. Therefore, the system is studied at the non-equilibrium
regime due to competing dynamics. We verified the phase transition between the
$AF$ and $F$ ordered phases to the $P$ disordered phase, and if the system is
in this A-SWN regime, it exhibit the same mean-field critical behavior
observed at equilibrium systems with long-range interactions by the A-SWN, see
Ref. [28]. The behavior of the phase transitions, phases diagrams and critical
exponents by FSS analysis also are described and compared with those of Ref.
[28].

This work is organized as follows: In Section .2, we describe the model, the
network, and the motion equations for the non-equilibrium Ising model. In
Section .3, we present the MC simulation method used. The behavior of the
phase transitions, phase diagrams, and critical exponents by FSS analysis is
described in Section .4. Finally, in Section .5, we present our conclusions.

##  .2 Model

The Ising model with $N=L^{2}$ spins $\sigma_{i}=\pm 1$ on a regular square
lattice $L\times L$, periodic boundary conditions, and a nearest-neighbor
ferromagnetic interaction of strength $J$ has been studied in this work (see
Fig. 1(a)). On the other hand, with a certain probability $p$, we can add one
long-range interaction $J_{ik}$ to each site of that regular square lattice.
We divided the system into two sublattices to add the long-range interactions
$J_{ik}$, in which one sublattice plays the role of central spins, while the
other sublattice contains the spins in which the central spins can connect, to
beyond their nearest neighbors. Thus, to choose a long-range interaction
$J_{ik}$ for a site $i$, the sublattice of $i$ will be the sublattice of the
central spins, then, we choose randomly a site $k$ from another sublattice. If
the site $k$ does not be one of its nearest neighbors already naturally
coupled with $i$, we picked a random number $0<r<1$, and if $r\leq p$ (with
$p$ predefined), then we couple the site $k$ to the neighbors of site $i$, and
for the site $k$ we couple the site $i$ to its neighbors. The attempt to add a
long-range interaction $J_{ik}$ is made once to each site that does not have a
long-range interaction $J_{ik}$ in the network, and as result, we have a
network with an average coordination number $z=4+p$. Therefore, we can think
as an example of some situations: i) for $p=0$, i.e., the probability of
adding a long-range interaction $J_{ik}$ to any site on the lattice is zero,
therefore, we have a regular square lattice, see Fig 1(a); ii) for $p=0.5$, we
are in the A-SWN regime because in addition to the conservation of $C(p)$, and
we also have an average short path length between network sites, through the
shortcuts created by the long-range interaction $J_{ik}$ added between the
sublattices, see Fig. 1(b); finally, for $p=1$, all sites on the network have
a long-range interaction $J_{ik}$ connecting the two sublattices, and
consequently, it is the network with the shortest typical separation between
the sites on the network, see Fig. 1(c).

Thus, as the regular structure in $p=0$ keeps unaltered (Fig. 1(a)), we have a
high local clustering for any value of $p$, and conform we increase $p$, the
long-range interaction $J_{ik}$ is added to the network, creating shortcuts
between the sites that before in the simple regular lattice would be more
distant, consequently decreasing the typical distance $l(p)$ of the network.
The $l(p)$ scales linearly $l(p\to 0)\sim L/2$ and logarithmically $l(p\to
1)\sim\ln(L^{1.77})$, being these regimes referred to as the “large-world” and
“small-world” respectively. The cross-over between these regimes occurs when
the average number of shortcuts is about one, or in the other words, we can
say in the SWN regime when $p\apprge 2L^{-2}$ [5]. Versed on this, our study
is based on $p\geq 0.25$ values, where the A-SWN is found and the decay of $l$
as a function of $p$ undergoes less, i.e., having approximately the same value
of $l$.

The ferromagnetic Ising spin energy is described by the Hamiltonian of the
form:

${\cal H}=-J\sum_{\left\langle
i,j\right\rangle}\sigma_{i}\sigma_{j}-\sum_{\left\langle
i,k\right\rangle}J_{ik}\sigma_{i}\sigma_{k},$ (1)

where $J$ is the nearest-neighbor ferromagnetic interaction, and $J_{ik}$ is
the long-range interaction on the A-SWN. The first sum is over all the pair of
nearest-neighbor spins on the regular square lattice and the second sum is
made over all the pairs of spins $(i,k)$ connected through long-range
interaction on the A-SWN. Here, we always are considering $J_{ik}=J=1$.

We are dealing with the non-equilibrium Ising model and in an SWN, being the
time evolution of the states of the system governed by two competing dynamical
processes: one simulating the contact of the system with a heat bath at
temperature $T$, with the one-spin flip process and probability $q$ to occur,
and at the same time but with probability $(1-q)$ to occur, the system is
subjected to an external flux of energy into the system with the two-spin flip
process, where in addition to flipping the chosen spin, it simultaneously
flips one of its randomly chosen neighbors.

Let us call $p(\\{\sigma\\},t)$ the probability of finding the system in the
state
$\\{\sigma\\}=\\{\sigma_{1},...,\sigma_{i},...,\sigma_{j},...\sigma_{N}\\}$ at
time $t$, the motion equation for the probability states evolve in time
according to the master equation

$\frac{d}{dt}p(\\{\sigma\\},t)=qG+(1-q)V,$ (2)

where $qG$ represents the process of relaxation of the spins in contact with a
heat bath at temperature $T$, favoring the lowest energy in the system, and
$(1-q)V$ represents the process independent of the temperature, where the
energy of the system increases by one external flow of energy into it. $G$ and
$V$ are described by

$\begin{array}[]{ccc}G=&\sum_{i,\\{\sigma^{\prime}\\}}\left[W(\sigma_{i}\to\sigma_{i}^{\prime})p(\\{\sigma\\},t)+\right.\\\
&\left.-W(\sigma_{i}^{\prime}\to\sigma_{i})p(\\{\sigma^{\prime}\\},t)\right]&,\end{array}$
(3)

$\begin{array}[]{ccc}V=&\sum_{i,j,\\{\sigma^{\prime}\\}}\left[W(\sigma_{i}\sigma_{j}\to\sigma_{i}^{\prime}\sigma_{j}^{\prime})p(\\{\sigma\\},t)+\right.\\\
&\left.-W(\sigma_{i}^{\prime}\sigma_{j}^{\prime}\to\sigma_{i}\sigma_{j})p(\\{\sigma^{\prime}\\},t)\right]&,\end{array}$
(4)

where $\\{\sigma^{\prime}\\}$ denotes the spin configurations after the spin
flipping, $W(\sigma_{i}\to\sigma_{i}^{\prime})$ is the transition rate between
states in the one-spin flip process, and
$W(\sigma_{i}\sigma_{j}\to\sigma_{i}^{\prime}\sigma_{j}^{\prime})$ the
transition rate between the states in the two-spin flip process, with the
order parameter being conserved in none of the dynamic processes.

If $0<q<1$, we have two dynamics processes acting simultaneously, the detailed
balance is not satisfied and the system is forced out of equilibrium. As these
processes favor the states of higher and lower energy of the system, with the
competition it is possible to find stationary states for the order parameter
in the $AF$, $F$, and $P$ phases. It is worth noting that to reach the
stationary state in the $AF$ phase was of fundamental importance to use the
$J_{ik}$ between the sublattices, because of the antiparallel ordering in
which this phase is characterized.

##  .3 Monte Carlo simulations

Let $(k,l)$ and $(k^{\prime},l^{\prime})$ be the coordinates of a site in our
two-dimensional SWN and one of your neighbors respectively. The periodic
boundary conditions were used in all our simulations. Starting the initial
state of the system with all spins aligned in the same direction, a new
configuration is generated by the following the Markov process: for a given
temperature $T$, competition probability $q$, and additive probability $p$, we
choose a random spin from the lattice, i.e., we choose a coordinate $k$ and
$l$ at random. Then we generate a random number $\xi$ between zero and one,
and if $\xi\leq q$ we choose the one-spin flip process. In this process, the
flipping probability is dependent on $W(\sigma_{kl}\to\sigma_{kl}^{\prime})$,
which is given by the Metropolis prescription as follows:

$W(\sigma_{kl}\to\sigma_{kl}^{\prime})=\left\\{\begin{array}[]{cccc}\exp(-\Delta
E_{kl}/k_{B}T)&\textrm{if}&\Delta E_{kl}>0\\\ 1&\textrm{if}&\Delta E_{kl}\leq
0&,\end{array}\right.$ (5)

where $\Delta E_{kl}$ is the change in the energy after flipping the spin
$\sigma_{kl}\to\sigma_{kl}^{\prime}$, $k_{B}$ is the Boltzmann constant, and
$T$ is the absolute temperature, thus, the new state is accepted if $\Delta
E_{kl}\leq 0$, and in the case of $\Delta E_{kl}>0$ we choose another random
number $1<\xi_{1}<0$ and if $\xi_{1}\leq\exp(-\Delta E_{kl}/k_{B}T)$ the new
state is also accepted, but if none of the conditions are satisfied, we do not
change the state of the system. On the other hand, if $\xi>q$, the two-spin
flip process is chosen. In this case, in addition to the spin $\sigma_{kl}$,
we also randomly choose one of its neighbors $\sigma_{k^{\prime}l^{\prime}}$,
which can be either the nearest neighbor or the farthest neighbor coming from
a $J_{ik}$. In this process, the two spins chosen are flipping simultaneously,
and for that, the two-spin flip probability is dependent on
$W(\sigma_{kl}\sigma_{k^{\prime}l^{\prime}}\to\sigma_{kl}^{\prime}\sigma_{k^{\prime}l^{\prime}}^{\prime})$,
which is given by

$W(\sigma_{kl}\sigma_{k^{\prime}l^{\prime}}\to\sigma_{kl}^{\prime}\sigma_{k^{\prime}l^{\prime}}^{\prime})=\left\\{\begin{array}[]{c}0\\\
1\end{array}\begin{array}[]{c}\textrm{if}\\\
\textrm{if}\end{array}\begin{array}[]{cc}\Delta
E_{kl,k^{\prime}l^{\prime}}\leq 0\\\ \Delta
E_{kl,k^{\prime}l^{\prime}}>0&,\end{array}\right.$ (6)

where $\Delta E_{kl,k^{\prime}l^{\prime}}$ is the change in the energy after
flipping the spins $\sigma_{kl}$ and $\sigma_{k^{\prime}l^{\prime}}$. Thus, in
this process, the new state is just accepted if $\Delta
E_{kl,k^{\prime}l^{\prime}}>0$.

Repeating the Markov process $N$ times, we have one Monte Carlo Step (MCS). In
our simulations, for $p\neq 0$, we have waited for $2\times 10^{4}$ MCS for
the system to reach the stationary state, for all the lattice sizes. We used
more $5\times 10^{3}$ MCS to calculate the thermal averages of the quantities
of interest. The average over the samples was done using $25$ independent
samples for any lattice. On the other hand, for the case $p=0$, we needed to
wait for $5\times 10^{5}$ MCS to reach the equilibrium state, and after
$3\times 10^{5}$ MCS to calculate the thermal average, only over one sample.

The measured thermodynamic quantities in our simulations are: the total
magnetization per spin $\textrm{m}_{\textrm{L}}^{\textrm{F}}$, the staggered
magnetization per spin $\textrm{m}_{\textrm{L}}^{\textrm{AF}}$, the magnetic
susceptibility $\chi_{\textrm{L}}$ and the reduced fourth-order Binder
cumulant $\textrm{U}_{\textrm{L}}$:

$\textrm{{\hbox{{m}_{{L}}^{{F}}}}}=\frac{1}{N}\left[\left\langle\sum_{kl}\sigma_{kl}\right\rangle\right],$
(7)

$\textrm{m}_{\textrm{L}}^{\textrm{AF}}=\frac{1}{N}\left[\left\langle\sum_{kl}(-1)^{(k+l)}\sigma_{kl}\right\rangle\right],$
(8)

$\chi_{\textrm{L}}=\frac{N}{k_{B}T}\left[\left\langle
m^{2}\right\rangle-\left\langle m\right\rangle^{2}\right],$ (9)

$\textrm{U}_{\textrm{L}}=1-\frac{\left[\left\langle
m^{4}\right\rangle\right]}{3\left[\left\langle
m^{2}\right\rangle^{2}\right]},$ (10)

where $\left[\ldots\right]$ denotes the average over the samples,
$\left\langle\ldots\right\rangle$ is the thermal average over the MCS in the
stationary state, and $m$ can be $\textrm{m}_{\textrm{L}}^{\textrm{F}}$ or
$\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ in Eq. (9) and (10), respectively. The
lattice sizes from $L=24$ to $L=256$ are simulated and the data are analyzed
via finite-size scaling theory (FSS). These Eqs. (7), (8), (9) and (10) obey
the following FSS relations in the neighborhood of the stationary critical
point $\lambda_{C}$:

$m=L^{-\beta/\nu}m_{0}(L^{1/\nu}\varepsilon),$ (11)

$\chi_{\textrm{L}}=L^{\gamma/\nu}\mathcal{X}_{0}(L^{1/\nu}\varepsilon),$ (12)

$\textrm{U}_{\textrm{L}}=U_{0}(L^{1/\nu}\varepsilon),$ (13)

where $\varepsilon=(\lambda-\lambda_{C})/\lambda_{C}$, $\lambda$ can be $T$ or
$q$. Here $m_{0}$, $\chi_{0}$ and $\textrm{U}_{0}$ are scaling functions,
where $\beta$, $\gamma$, and $\nu$ are the critical exponents related to
magnetization, susceptibility, and the length correlation, respectively. The
derivative of Eq. (13) with respect to the parameter $\lambda$ gives us the
following scaling relation:

$\textrm{U}^{\prime}_{\textrm{L}}=\frac{L^{1/\nu}}{\lambda_{C}}U^{\prime}_{0}(L^{1/\nu}\varepsilon).$
(14)

We have determined the critical exponent relations $\beta/\nu$, $\gamma/\nu$
and $\nu$ from slope of a log-log plot of
$\textrm{m}_{\textrm{L}}(\lambda_{C})$,
$\mathcal{X}_{\textrm{L}}(\lambda_{C})$ or
$\textrm{U}^{\prime}_{\textrm{L}}(\lambda_{C})$ versus lattice size $L$
respectively. We also have used another alternative method to estimate the
values of the critical exponents, the data collapse from the scaling
functions.

Figure 2: Thermodynamic quantities in the phase transitions of the non-
equilibrium system, with $p=0.75$, $T=1$, and lattice sizes $L$ shown in the
figures. (a) Staggered magnetization $\textrm{m}_{\textrm{L}}^{\textrm{AF}}$,
in (b) and (c) we have respectively the Binder cumulant
$\textrm{U}_{\textrm{L}}^{\textrm{AF}}$ and the staggered susceptibility
$\chi_{\textrm{L}}^{\textrm{ AF}}$. (d) Total magnetization
$\textrm{m}_{\textrm{L}}^{\textrm{F}}$ of the system is represented, the
Binder cumulant $\textrm{U}_{\textrm{L}}^{\textrm{F}}$ in (e), and the total
susceptibility $\chi_{\textrm{L }}^{\textrm{F}}$ in (f). The error bars in the
magnetization are smaller than the size of the symbols, so for a better
interpretation of the results, these were omitted.

##  .4 Results and Discussions

In this section, we illustrate and discuss the results of the magnetic
properties of the Ising model on a 2D A-SWN at the non-equilibrium regime by
the two competing dynamics. For the study about the critical behavior and
phase transitions at the non-equilibrium system, it was convenient to fix the
temperature $T$, additive probability $p$, and to use the competition
parameter $q$ as a variable to transit between the ordered to disordered
phases in the regions of $T$ and $p$ of the phase diagram. It is convenient
because the two-spin flip mechanism is independent of the temperature $T$, and
in the present work we do not have used $p$ as a variable to identify the
phase transitions.

Before studying the thermal phase diagrams, we will present the best results
for the behavior of thermodynamic quantities and critical point values. These
results were obtained where most sites have the same coordination number
$z=5$. Therefore, in Fig. 2, we have shown one of the best results for the
thermodynamic quantities obtained in the stationary state as a function of
$q$, for fixed $p=0.75$ and $T=1$. We can see the self-organization in the
system, by finding an $AF$ phase, being represented in the staggered
magnetization $\textrm{m}_{\textrm{L}}^{\textrm{AF}}$. These because in high
values of $q$ we have the transition between the $F$ to $P$ phase (see Fig.
2(d)) and from this $P$ phase to the ordered $AF$ phase (see Fig. 2(a)) as we
increase the flow of energy into the system ($q\rightarrow 0)$. For these
magnetizations, we also have their respective reduced fourth-order Binder
cumulants, $\textrm{U}_{\textrm{L}}^{\textrm{AF}}$ (Fig. 2(b)) and
$\textrm{U}_{\textrm{L}}^{\textrm{F}}$ (Fig. 2(e)) beyond the magnetic
susceptibilities $\chi_{\textrm{L}}^{\textrm{AF}}$ (Figs. 2(c)) and
$\chi_{\textrm{L}}^{\textrm{F}}$ (Fig. 2(f)).

Figure 3: Thermodynamic quantities as a function of $q$, for different values
of $p$ as indicated in the figures. We fixed the value of $T=1$, and for
lattice sizes $L=256\,\,(\triangle)$ and $L=24\,\,(\times)$. (a) Staggered
magnetization $\textrm{m}_{\textrm{L}}^{\textrm{AF}}$, in (b) and (c) we have
respectively the staggered susceptibility $\chi_{\textrm{L}}^{\textrm{ AF}}$
and Binder cumulative $\textrm{U}_{\textrm{L}}^{\textrm{AF}}$. (d) Total
magnetization $\textrm{m}_{\textrm{L}}^{\textrm{F}}$ of the system is
represented, the Binder cumulant $\textrm{U}_{\textrm{L}}^{\textrm{F}}$ in
(e), and the total susceptibility $\chi_{\textrm{L }}^{\textrm{F}}$ in (f).

The thermodynamic quantities for the other $p$ values in the A-SWN regime,
such as $p=0.25$, $p=0.5$, $p=0.75$, and $p=1$, have also been computed. In
the order to compare the behavior during phase transitions, we also exhibited
the same thermodynamic quantities in the conventional square lattice Ising
model, $p=0$, in Fig. 3. These result can see in details for the
$\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ and
$\textrm{m}_{\textrm{L}}^{\textrm{F}}$ in Figs. 3(a) and 3(d), respectively,
$\textrm{U}_{\textrm{L}}^{\textrm{AF}}$ in Fig. 3(b) and
$\textrm{U}_{\textrm{L}}^{\textrm{F}}$ in Fig. 3(e), in addition to
$\chi_{\textrm{L}}^{\textrm{AF}}$ in Fig. 3(c) and
$\chi_{\textrm{L}}^{\textrm{F}}$ in Fig. 3(f). We have presented only the
smaller $(L=24)$ and the larger $(L=256)$ linear lattice size and they are
enough so that we can observe the finite-size behavior and the critical point
change $q_{c}$ as we increase $p$. On the other hand, for the calculation of
$q_{c}$, we have used all six lattice sizes of the system.

Figure 4: Extrapolation of the critical transition probability $q_{c}$
obtained for linear lattice sizes $24\leq L\leq 256$, and for different values
of $p$ as indicated in the figures. (a) The points in the transition between
$AF-P$ phases are represented, and (b) the points in the transition between
the $P-F$ phases. The values of critical transition probabilities
$q_{c}(L\rightarrow\infty)$ can be seen in Table 1 and Table 2, respectively.

To evaluate the $q_{c}$, we have employed two methods. Firstly, we obtained by
extrapolating the susceptibility discontinuity to when $L\to\infty$, which
returns $q_{c}(\infty)$, using finite lattice sizes $24\leq L\leq 256$, in the
plot of maximum susceptibility as a function of $1/L$. Secondly, we obtained
by the crossing of the Binder cumulant curves for the different lattice sizes
$L$. In Fig. 4 the values of $q$ are displayed where the susceptibility has
its maximum value, $\chi_{\textrm{L}}^{max}$, as a function of $1/L$ for the
values of $p$ selected. We also have the best fit of the points, which is a
linear fit, and for the extrapolation, when $L\to\infty$, we have the
estimated of $q_{c}$ by using the linear coefficient, i.e., we have made the
infinite-size extrapolation in according to
$q(\chi_{\textrm{L}}^{max})-q_{c}(\infty)=\alpha L^{-1}$. By extrapolation,
the critical points $q_{c}(\infty)$ in the transition between the $AF-P$
phases are represented in Fig. 4(a) and the transition between the $F-P$
phases are represented in Fig. 4(b).

The critical point values using magnetic susceptibility data, $q_{c}^{\chi}$,
and their respective errors are exhibited in Table 1 for the transition
between the $AF-P$ phases. In this transition, we also have used the crossing
of the Binder cumulant curves in the selected lattice sizes $24\leq L\leq
256$, to obtain the another estimate for the critical points, $q_{c}^{U}$,
which are shown in Table 3, and the characterization of the second-order phase
transition in the system [29, 30, 31, 32]. For the transition between $F-P$
phases, the values of $q_{c}^{\chi}$ obtained are exhibited in Table 2, and
the critical points obtained through the crossing of the Binder cumulant
curves $q_{c}^{U}$, can be seen in Table 4. The critical points obtained in
both methods are equivalent.

Figure 5: Phase diagrams with different background colors for the phase
transitions between $AF-P$ and $P-F$ phases, and for different values of $p$
as indicated in the figures. The black lines are just a guide for the eyes in
the second-order phase transitions and the black square dots are critical
points calculated. The color map refers to the magnetization of the system,
$\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ and
$\textrm{m}_{\textrm{L}}^{\textrm{F}}$, as indicated in the color bars.

Table 1: Critical competition probability $q_{c}$, based on the extrapolating of the susceptibility discontinuity and in the $AF-P$ phase transition, for $T=1$ $p$ | $q_{c}^{\chi}$ | $\beta/\nu$ | $\gamma/\nu$ | $\nu$
---|---|---|---|---
$0$ | $0.00149\pm 0.0001$ | $0.16\pm 0.05$ | $1.77\pm 0.03$ | $1.04\pm 0.09$
$0.25$ | $0.03966\pm 0.002$ | $0.46\pm 0.07$ | $1.07\pm 0.06$ | $1.07\pm 0.03$
$0.5$ | $0.07176\pm 0.003$ | $0.45\pm 0.04$ | $1.06\pm 0.04$ | $0.96\pm 0.09$
$0.75$ | $0.09062\pm 0.002$ | $0.46\pm 0.03$ | $1.04\pm 0.02$ | $1.02\pm 0.05$
$1.0$ | $0.09774\pm 0.002$ | $0.48\pm 0.02$ | $1.05\pm 0.02$ | $1.06\pm 0.08$
Table 2: Critical competition probability $q_{c}$, based on the extrapolating of the susceptibility discontinuity and in the $F-P$ phase transition, for $T=1$. $p$ | $q_{c}^{\chi}$ | $\beta/\nu$ | $\gamma/\nu$ | $\nu$
---|---|---|---|---
$0$ | $0.964\pm 0.002$ | $0.11\pm 0.02$ | $1.67\pm 0.09$ | $1.06\pm 0.09$
$0.25$ | $0.917\pm 0.002$ | $0.47\pm 0.06$ | $1.01\pm 0.06$ | $0.96\pm 0.09$
$0.5$ | $0.895\pm 0.001$ | $0.46\pm 0.02$ | $1.04\pm 0.02$ | $0.96\pm 0.09$
$0.75$ | $0.883\pm 0.002$ | $0.46\pm 0.01$ | $1.06\pm 0.01$ | $0.99\pm 0.06$
$1.0$ | $0.880\pm 0.001$ | $0.49\pm 0.01$ | $1.0\pm 0.02$ | $1.03\pm 0.04$

With the critical point values, we built the phase diagram which shows the
regions on the plane of $T$ versus $q$, where the $F$, $P$, and $AF$ phases
are found. The phase diagrams are presented in Fig. 5, for different values of
$p$, where we can see the greater the probability of adding $J_{ik}$, the
greater the region where we find the ordered phases.

Now, in order to better understand the behavior of these phases (see Fig. 5),
we can relate these ordered phases to dynamics used in the competition. The
$AF$ phase, observing the $q$-axis, is found when $q\to 0$ and the order
parameter $\textrm{m}_{\textrm{L}}^{\textrm{AF}}\rightarrow 1$ (see the
figures on the left side in Fig. 5), i.e., when the two-spin flip dynamic
prevails in the competition. This is because, in the dynamic that simulates
the system with an external energy flow into it, the change in the spin states
is only accepted if it increases the energy of the system. Considering the
Hamiltonian model, Eq. (1), the state of the highest energy to which the
dynamics lead the system is the one where the spins are aligned antiparallel.
The antiparallel order also can be achieved through the A-SWN, because if we
analyze locally, the antiferromagnetic phase occurs when a central spin is in
the up (down) state, and its neighbors, to whom it is connected, are in the
down (up) state. Extending this analysis to the entire network, an ordering of
this type only occurs when we have well-defined what are the central spins and
what sites they can connect to, otherwise, completely random long-range
interactions can connect two distant sites in the network that the highest
local energy configuration of one of these is unfavorable to the local
antiparallel ordering of the other site, thus, making it impossible to obtain
the stationary state with an $AF$ phase in the system. In this context, the
$F$ phase is found in the limit that $q\to 1$ and the order parameter
$\textrm{m}_{\textrm{L}}^{\textrm{F}}\rightarrow 1$ (see the figures of the
right side in Fig. 5), i.e., when the one-spin flip dynamic prevails. This
dynamic is responsible to simulate the system in contact with the heat bath at
temperature $T$, and favors the lowest energy state through the thermal
equilibrium, in which all spins have the same state following the Hamiltonian
system, so, if we wanted to, we could treat them without the sublattices in
the A-SWN regime. On the other hand, when none of the dynamics prevails, i.e.,
between the extremes of the probability $q$-value, no one of the expected
order phase types is found in the system. Thus, we have most of the values of
$q$, the $P$ phase in the system is found, where both
$\textrm{m}_{\textrm{L}}^{\textrm{AF}}\rightarrow 0$ and
$\textrm{m}_{\textrm{L}}^{\textrm{F}}\rightarrow 0$. Another important
observation is that the phase diagram topology changes when $p$ increases, but
the phases do not disappear.

Table 3: Critical competition probability $q_{c}$, based on the crossing of the fourth-order Binder cumulant curves and in the $AF-P$ phase transition, for $T=1$. The effective dimension is given by hyperscaling relation $d_{\textrm{eff}}=2\beta/\nu+\gamma/\nu$.  $p$ | $q_{c}^{U}$ | $\beta$ | $\nu\left(\textrm{$\textrm{m}^{\textrm{AF}}$}\right)$ | $\gamma$ | $\nu\left(\chi^{\textrm{AF}}\right)$ | $d_{\textrm{eff}}$
---|---|---|---|---|---|---
$0$ | $0.00145\pm 0.00004$ | $0.125\pm 0.03$ | $1.0\pm 0.05$ | $1.75\pm 0.04$ | $1.0\pm 0.05$ | $2.00\pm 0.21$
$0.25$ | $0.0390\pm 0.0009$ | $0.46\pm 0.06$ | $0.96\pm 0.06$ | $1.06\pm 0.05$ | $1.0\pm 0.05$ | $2.02\pm 0.29$
$0.5$ | $0.0715\pm 0.0006$ | $0.44\pm 0.05$ | $1.0\pm 0.05$ | $1.05\pm 0.03$ | $1.0\pm 0.04$ | $1.93\pm 0.24$
$0.75$ | $0.0921\pm 0.0007$ | $0.48\pm 0.04$ | $0.98\pm 0.06$ | $1.05\pm 0.06$ | $1.0\pm 0.04$ | $2.03\pm 0.31$
$1.0$ | $0.0978\pm 0.0004$ | $0.48\pm 0.03$ | $1.02\pm 0.06$ | $1.06\pm 0.06$ | $0.98\pm 0.04$ | $2.02\pm 0.23$

Table 4: Critical competition probability $q_{c}$, based on the crossing of the fourth-order Binder cumulant curves and in the $F-P$ phase transition, for $T=1$. The effective dimension is given by hyperscaling relation $d_{\textrm{eff}}=2\beta/\nu+\gamma/\nu$.  $p$ | $q_{c}^{U}$ | $\beta$ | $\nu\left(\textrm{$\textrm{m}^{\textrm{F}}$}\right)$ | $\gamma$ | $\nu\left(\chi^{\textrm{F}}\right)$ | $d_{\textrm{eff}}$
---|---|---|---|---|---|---
$0$ | $0.965\pm 0.0004$ | $0.125\pm 0.02$ | $1.0\pm 0.05$ | $1.70\pm 0.07$ | $1.01\pm 0.03$ | $1.95\pm 0.20$
$0.25$ | $0.917\pm 0.0009$ | $0.45\pm 0.05$ | $1.0\pm 0.05$ | $1.05\pm 0.05$ | $1.0\pm 0.04$ | $1.95\pm 0.26$
$0.5$ | $0.895\pm 0.0008$ | $0.46\pm 0.04$ | $0.95\pm 0.07$ | $1.05\pm 0.04$ | $1.0\pm 0.04$ | $2.02\pm 0.12$
$0.75$ | $0.883\pm 0.0005$ | $0.47\pm 0.03$ | $1.0\pm 0.06$ | $1.04\pm 0.06$ | $1.0\pm 0.05$ | $1.98\pm 0.23$
$1.0$ | $0.879\pm 0.0006$ | $0.49\pm 0.04$ | $1.0\pm 0.07$ | $1.02\pm 0.05$ | $0.99\pm 0.06$ | $2.00\pm 0.26$

Figure 6: (a) and (d) contains the best fit of the staggered
$\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ and total
$\textrm{m}_{\textrm{L}}^{\textrm{F}}$ magnetization curves in the critical
point as a function of the different values of $L$ in the log-log plot,
respectively, for the $F$ and $AF$ phases, in which the slopes we obtained the
ratio $\beta/\nu$ between the critical exponents. (b) and (e) in the $F$ and
$AF$ phases, we have the best fit of the staggered
$\chi_{\textrm{L}}^{\textrm{AF}}$ and total
$\textrm{$\chi$}_{\textrm{L}}^{\textrm{F}}$ susceptibility curves at the
critical point as a function of $L$, and the slopes have given us the ratio
$\gamma/\nu$. The critical exponent $\nu$, we have obtained by the slope of
the best fit of the Binder cumulant derivative
$\textrm{U'}_{\textrm{L}}^{\textrm{AF}}$ and
$\textrm{U'}_{\textrm{L}}^{\textrm{F}}$ at the critical point, as represented
in (c) and (f) for the $F$ and $AF$ phases, respectively. The critical
exponents obtained by this method for the $AF$ phase can be seen in Table 1,
and in Table 2 for the $F$ phase.

After the presentation of the phase diagrams by exploiting the thermal
variations of the order parameters, the Binder cumulant and the magnetic
susceptibility, we can now study the critical behavior of these quantities in
the vicinity of the phase transitions using the FSS method to evaluate some
critical exponents of the model. Therefore, to obtain the critical exponents,
we also used two methods, both referring to the FSS method, using the scale
relations of Eqs. (11), (12), and (14). One of the methods refers to the value
of the thermodynamic quantities at the critical point, in which when we make a
log-log plot of the value of these quantities as a function of $L$. Using the
scale relations, we obtain ratios between the critical exponents through the
slope of the line of best fit of those points. In Fig. 6, the behavior of
thermodynamic quantities near the critical point can be seen as a function of
$L$ in the log-log plot. In Figs. 6(a) and 6(d) we were able to find the ratio
$-\beta/\nu$ in the $AF-P$ and $F-P$ phase transitions, respectively, using
the scaling relation of the Eq. (11), through the slope in the linear fit of
the points for each selected value of $p$, as indicated in the figures. The
same can be done using the scaling relation of the Eq. (12), however, the
critical exponent ratio is $\gamma/\nu$ and obtained by the slope of the
linear fits of Figs. 6(b) and 6(e), for the different values of $p$ and in the
$AF-P$ and $F-P$ phase transitions, respectively. Finally, the ratios between
the exponents obtained previously, it is useful to use the scaling relation of
Eq. (14), which we have used the data of the Binder cumulant derivative to
obtain information related to the critical exponent of correlation length,
$\nu$. Here, they were obtained from the linear fit of the curves of Figs.
6(c) and 6(f) for the different $p$ values and $AF-P$ and $F-P$ phase
transitions, respectively. It is worth noting that as our interest is in the
slope of the log-log plot, we changed the linear coefficients of the straight
lines to separate the lines and thus making it easier for the reader to see
the fits. All the ratios between the values of the critical exponents obtained
by the log-log plot of the scaling relations can be seen in Table 1 for the
$AF-P$ phase transitions, and in Table 2 for the $F-P$ phase transitions.

Figure 7: Data collapse by FSS analysis for (a) staggered magnetization
$\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ and (c) susceptibility
$\textrm{$\chi$}_{\textrm{L}}^{\textrm{AF}}$ for different values of $L$ as
indicated in the figure, and fixed $p=0.75$. In (b) and (d) we have the data
collapse for $\textrm{m}_{\textrm{L}}^{\textrm{AF}}$ and
$\textrm{$\chi$}_{\textrm{L}}^{\textrm{AF}}$, respectively, but for all values
of $p$, in which the critical exponents were obtained using the best data
collapse for all lattice sizes, and here we only display the lattice sizes
$L=24\>(+)$ and $L=256\>(\square)$. The $\varepsilon$ parameter is set to
$\varepsilon=(q-q_{c})/q_{c}$. The dashed lines represent the asymptotic
behavior of the scale functions. The values of the critical exponents $\beta$,
$\gamma$, $\nu\left(\textrm{$\textrm{m}^{\textrm{AF}}$}\right)$, and
$\nu\left(\chi^{\textrm{AF}}\right)$ of the best data collapse can be seen in
Table 3.

Another method used to obtain the critical exponents is through the scaling
functions in Eqs. (11), (12) and (14), in the around of the critical point.
For this, we isolate the scale function and plot it in a log-log plot through
the curves of $\textrm{$\textrm{m}^{\textrm{F}}$}L^{\beta/\nu}$ and
$\textrm{$\textrm{m}^{\textrm{AF}}$}L^{\beta/\nu}$ as a function of
$\left|\varepsilon\right|L^{1/\nu}$, resulting in a single curve for all
lattice sizes $L$ if we have the correct critical exponents and critical
points adjusted in the scaling relations. In this method, the data collapse
can also be obtained in a plot that does not have the axes on the logarithmic
scale, but the asymptotic behavior that relates to the critical exponents are
not present. We can obtain the critical exponents because the data collapse in
the vicinity of the critical point, depends on the correct critical exponents
of the system to occur, in this way, we adjust them to obtain the best data
collapse in the criticality, and consequently, the exponents involved in this
data collapses are the critical exponents of the system. All values of the
critical exponents obtained by data collapse of the scaling relations can be
seen in Table 3 for the $AF-P$ phase transitions
$\left(\beta,\nu\left(\textrm{$\textrm{m}^{\textrm{AF}}$}\right),\nu\left(\chi^{\textrm{AF}}\right),\gamma\right)$,
and in Table 4 for the $F-P$ phase transitions
$\left(\beta,\nu\left(\textrm{$\textrm{m}^{\textrm{F}}$}\right),\nu\left(\chi^{\textrm{F}}\right),\gamma\right)$.

In Fig. 7, we have shown the data collapse for the scaling functions of
magnetization, Fig. 7(a), and for magnetic susceptibility, Fig. 7(c), for the
$AF-P$ phase transition, with $p=0.75$, which was the best data collapse
obtained. In this phase transition, we also plotted for all values of $0\leq
p\leq 1.0$, as can be seen in Fig. 7(b) and 7(d), the scaling function of
magnetization and magnetic susceptibility, respectively, showing the best data
collapse for the selected $p$ values, but displaying only the lattice sizes
$L=24\>(+)$ and $L=256\>(\square)$ for the best differentiation between the
collapsed curves.

Figure 8: Data collapse by FSS analysis for (a) total magnetization
$\textrm{m}_{\textrm{L}}^{\textrm{F}}$ and (c) susceptibility
$\textrm{$\chi$}_{\textrm{L}}^{\textrm{F}}$ for different values of $L$ as
indicated in the figure, and fixed $p=0.75$. In (b) and (d) we have the data
collapse for $\textrm{m}_{\textrm{L}}^{\textrm{F}}$ and
$\textrm{$\chi$}_{\textrm{L}}^{\textrm{F}}$, respectively, but for all values
of $p$, in which the critical exponents were obtained using the best data
collapse for all lattice sizes, and here we only display the lattice sizes
$L=24\>(+)$ and $L=256\>(\square)$. The $\varepsilon$ parameter is set to
$\varepsilon=(q-q_{c})/q_{c}$. The dashed lines represent the asymptotic
behavior of the scale functions. The values of the critical exponents $\beta$,
$\gamma$, $\nu\left(\textrm{$\textrm{m}^{\textrm{F}}$}\right)$, and
$\nu\left(\chi^{\textrm{F}}\right)$ of the best data collapse can be seen in
Table 4.

However, in the $F-P$ phase transition, the figures that present the best data
collapse, for $p=0.75$, are Fig. 8(a) for the magnetization scaling function,
and Fig. 8(c) for the magnetic susceptibility scaling function. We also have
calculated for all values of $0\leq p\leq 1.0$, but also only displaying here
two lattice sizes, $L=24\>(+)$ and $L=256\>(\square)$, and they can be found
in Figs. 8(b) and 8(d) for the magnetization and the magnetic susceptibility
scaling function, respectively. The critical exponents obtained by this
method, data collapse, can be found in Table 3 for $AF-P$ phase transitions,
and in Table 4 for phase transitions from $F$ to $P$.

In both methods, we obtained very approximate values for the critical
exponents for the selected $p$ values. The best results shown here are based
on data collapse, this is due to the fact that for $p=0$ the values of the
critical exponents are closer to the values of the Ising model on the regular
square lattice, which they are known by exact solution and MC simulation,
$\beta=1/8$, $\gamma=7/4$, and $\nu=1$. On the other hand, when we increase
the additive probability $p$, we also increase the number of $J_{ik}$ added to
the system, thus, it is convenient to use the scaling relations of systems
that can have mean-field critical behavior, by the prediction that we have a
system above the Ising model critical dimension, $d=4$, in the A-SWN regime.
To do that, it is enough in the scaling relations, Eqs. (11), (12), and (14),
to substitute the linear length $L$ of the lattice by the total number of
spins in the system, $L^{2}=N$. By doing this, as predicted, are obtained
approximately the mean-field critical exponents $\beta=1/2$, $\gamma=1$ and
$\nu=1/2$. The behavior of both the critical exponents for $p=0$ and the A-SWN
regime $(0<p\leq 1)$ were represented in Fig. 9(a) for the $AF-P$ phase
transition and in Fig. 9(b) for the $F-P$ phase transition.

Figure 9: (a) Representation of the critical exponents presented in Tables 3
and 1 for the critical behavior of the system in the $AF-P$ phase transition,
taking into account the mean-field scale relationships for the A-SWN regime
$(0<p\leq 1)$. (b) Representation of the critical exponents presented in
Tables 4 and 2 for the system in the $F-P$ phase transition, also using the
mean-field scale relations in the A-SWN regime $(0<p\leq 1)$. In both figures,
we have the comparison with the mean-field critical exponents (MF) by the
dotted lines, $\gamma=1.0$ and $\beta=\nu=0.5$.

The critical exponents are not independent one each other, but related by
simple scaling laws, as is the case with the hyperscaling law
$d_{\textrm{eff}}=2\beta/\nu+\gamma/\nu$, in which we have as a result the
effective dimension $d_{\textrm{eff}}$ of the system. With this law, we see
that the system has approximately the same critical exponents in the A-SWN
regime, as we are returned that $d_{\textrm{eff}}\cong 4.0$ with the mean
field critical exponents, and $d_{\textrm{eff}}\cong 2.0$ following the data
in the tables for $p=0$, obtained with the scale relationships of Eqs. (11),
(12) and (14).

The universality class can be defined by the set of exponents in the phase
transition, as in the case of the second-order phase transitions, in which
systems very different from each other can share the same set of critical
exponents. In general, these systems share the same spatial dimension,
symmetries, and range of interactions. Here, following the set of critical
exponents obtained at $p=0$, we have the same universality class of the
equilibrium Ising model in the regular square lattice. However, in the A-SWN
regime $(0<p\leq 1)$, we have long-range interactions in the system, and, due
to its consequent set of critical exponents, the system belongs mean-field
universality class. By comparing with the results obtained for the Ising model
in the two-dimensional A-SWN at the thermodynamic equilibrium regime [24, 28],
we see that both the non-equilibrium model and the equilibrium model have the
same universality class, mean-field universality class, in stationary critical
behavior.

##  .5 Conclusions

In this work, we have developed MC simulations to study the thermodynamic
quantities and the critical behavior of the non-equilibrium Ising model on a
2D A-SWN. By using the one- and two- spin flip competing dynamics we reach the
stationary state of the system at the non-equilibrium regime. We have found
two types of phase transitions, from $P$ to $AF$ and from $P$ to $F$ phases,
when the two-spin flip dynamic prevails in the system, and when the one-spin
flip prevails in the system, respectively. To found the phases we have used
the total and staggered magnetizations per spin, and its respective
susceptibility and reduced fourth-order Binder cumulant, both as a function of
the competition parameter $q$. With the last two quantities are obtained the
critical points of the system and we built the phase diagrams of the system.
We have observed that increasing the coordination number of the network by
adding long-range interactions to our A-SWN, with addition probability $p$, we
also increase the regions of the ordered phases on the diagram. Through the
FSS arguments, we calculated the critical exponents $\beta$, $\gamma$, and
$\nu$, of the system, and for the A-SWN regime $(0<p\leq 1)$ we obtained the
same exponents of a system with mean-field critical behavior, except for $p=0$
as expected, we obtained the critical exponents of the Ising model in a
regular square lattice. Thus, in the A-SWN regime, we have concluded that the
non-equilibrium system is in the mean-field universality class, as the
equilibrium system in the A-SWN [24, 28]. That equivalence between the
critical behavior of the equilibrium and non-equilibrium models were already
predicted and observed in other systems [4, 5, 6, 7, 8]. It is also important
to specify that our results concerning the regions of the phase diagram based
on the changes in the critical points, and the mean-field behavior is in
agreement with the observed behavior of disorder with shortcuts added to the
Ising model in an SWN [12, 20, 18, 21, 22, 24, 28].

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###### Acknowledgements.

This work was partially supported by the Brazilian Agencies CNPq, UFMT and
FAPEMAT


# Crossingless sheaves and their classes in equivariant K-theory

Galyna Dobrovolska  Department of Mathematics, Ariel University, Ariel 40700
Israel<EMAIL_ADDRESS>
###### Abstract.

We introduce crossingless sheaves in certain equivariant derived categories
which are analogous to the Bezrukavnikov-Mirkovic exotic sheaves for two-block
nilpotents. We calculate the classes of crossingless sheaves in equivariant
K-theory of Cautis-Kamnitzer varieties.

##  1\. Introduction

The exotic t-structure was defined by Bezrukavnikov and Mirkovic in [3] in
order to study modular representation theory of Lie algebras. The exotic
t-structure is defined using an action of the affine braid group on derived
categories of varieties related to the Springer resolution defined by
Bezrukavnikov and Riche in [4]. The irreducible objects in the heart of an
exotic t-structure are called simple exotic sheaves.

More precisely, let $\bf k$ be an algebraically closed field of characteristic
$p\gg 0$ and let $\mathfrak{g}$ be a reductive Lie algebra defined over $\bf
k$. Let $\lambda$ be a regular integral weight and let $e$ be a nilpotent. Let
${\rm Mod}_{e}^{fg,\lambda}$ be the category of modules with genralized
central character $(\lambda,e)$. Let $\mathcal{B}_{e,\bf k}$ be the Springer
fiber corresponding to $e$. Then Theorem 5.3.1 of [2] states that

$D^{b}(Coh_{\mathcal{B}_{e,\bf k}}(\widetilde{g}_{\bf k}))\simeq D^{b}({\rm
Mod}_{e}^{fg,\lambda}).$

It is also known that the tautological t-structure on the derived category of
modules corresponds to the exotic t-structure on the derived category of
coherent sheaves.

Exotic sheaves for two-block nilpotents were studied by Anno and Nandakumar in
[1]. Let $\mathcal{D}_{n}^{\prime}$ be the derived category of coherent
sheaves on a two-block Springer fibre (where the nilpotent is in type A and
has two blocks of sizes $m+n$ and $n$). Anno and Nandakumar defined the
objects $\Psi^{\prime}_{\alpha}\in D_{n}^{\prime}$ for any
$(m,n)$-crossingless matching $\alpha$ and proved that these objects are the
simple exotic sheaves for two-block nilpotents. Cautis-Kamnitzer varieties
$Y_{k}$ which contain Springer fibers and Slodowy slices were introduced in
[5] (see Definition 3.1). The equivariant K-theory of $Y_{k}$ is a free
$\mathbb{C}[q,q^{-1}]$-module on $2^{k}$ generators (See Proposition 3.3).
Classes of $\Psi^{\prime}_{\alpha}$ in K-theory of Cautis-Kamnitzer varieties
were computed in [7] and subsequently applied to the calculation of dimensions
of modular representations of $\mathfrak{sl}_{n}$ with a two-block central
character.

Cautis and Kamnitzer studied categorification of Reshetikhin-Turaev invariants
of linear tangles in [5]. In the course of their study they defined functors
$\Psi(\alpha)$ for linear tangles $\alpha$. We complete their representation
to a representation of affine tangles so that we have functors $\Psi(\alpha)$
for any affine tangle $\alpha$. Using these functors, we define an analogous
concept to Anno-Nandakumar’s exotic sheaves for a two-block nilpotent in
certain equivariant derived category. We call the objects we defined
crossingless sheaves (see Definition 3.11). We compute the classes of
crossingless sheaves in equivaraint K-theory of Cautis-Kamnitzer varieties
(see Theorem 3.23, which is our main theorem).

This paper introduces techniques which can be used for defining
representations of affine tangles on equivariant derived category of Cautis-
Kamnitzer varieties with respect to the ${\mathbb{C}}^{*}$ in the centralizer
of the two-block nilpotent and calculating characters of modular
representations with central character equal to a two-block nilpotent (note
that dimension formulas for these modular representations have been computed
in [7]). These goals will be achieved in subsequent publications by the
author.

### Acknowledgements

I thank Vinoth Nandakumar for sharing his ideas and for many useful
conversations. I also thank Joel Kamnitzer for useful correspondence and Roman
Travkin for helpful conversations.

##  2\. Preliminaries

###  2.1. Affine tangles

The combinatorics of affine tangles will be necessary for what follows; we
recall the definitions here (see Section 3 of [1] for more details).

######  Definition 2.1.

If $q\equiv r\pmod{2}$, a $(q,r)$ affine tangle is an embedding (up to
isotopy) of $\frac{q+r}{2}$ arcs and a finite number of circles into the
region $\\{(x,y)\in\mathbb{C}\times\mathbb{R}|1\leq|x|\leq 2\\}$, such that
the end-points of the arcs are
$(1,0),(\zeta_{q},0),\cdots,(\zeta_{q}^{q-1},0),(2,0),(2\zeta_{r},0),\cdots,(2\zeta_{r}^{r-1},0)$
in some order; here $\zeta_{k}=e^{\frac{2\pi i}{k}}$.

Figure 1 illustrates the composition of affine tangles.

Figure 1.

######  Definition 2.2.

If $q\equiv r\pmod{2}$, a framed $(q,r)$ affine tangle is an embedding (up to
isotopy) of $\frac{q+r}{2}$ “rectangular arcs” and a finite number of circles
into the region $\\{(x,y)\in\mathbb{C}\times\mathbb{R}|1\leq|x|\leq 2\\}$.
Here a “rectangular arc” is an injective map from $[0,1]\times[0,1]$ to
$\\{(x,y)\in\mathbb{C}\times\mathbb{R}|1\leq|x|\leq 2\\}$ where the segments
$[0,1]\times 0$ and $[0,1]\times 1$ are mapped to the segments
$[x]\times[0,1]$ where $x=\zeta_{p}^{i}$ or $x=2\zeta_{q}^{j}$.

######  Remark 2.3.

Framed tangles are necessary for the validity of the Reidemeister I move in
the functorial tangle categorification statement from Proposition 3.9. Below
we will often abbreviate “$(q,r)$-affine tangle” to “$(q,r)$-tangle”. In [1],
we used $\;\widehat{}\;$ notation to denote framed tangles; we omit that here.
Note that given a $(q,r)$ framed affine tangle $\alpha$, and a $(r,s)$ framed
affine tangle $\beta$, we can compose them using scaling and concatenation to
obtain a $(q,s)$ affine tangle $\beta\circ\alpha$. This composition is
associative up to isotopy.

######  Definition 2.4.

Given $1\leq i\leq n$, define the following affine tangles:

  * •

Let $g_{n}^{i}$ denote the $(n-2,n)$ tangle with an arc connecting
$(2\zeta_{n}^{i},0)$ to $(2\zeta_{n}^{i+1},0)$.

  * •

Let $f_{n}^{i}$ denote the $(n,n-2)$ tangle with an arc connecting
$(\zeta_{n}^{i},0)$ and $(\zeta_{n}^{i+1},0)$.

  * •

Let $t_{n}^{i}(1)$ (respectively, $t^{i}_{n}(2)$) denote the $(n,n)$ tangle in
which a strand connecting $(\zeta_{n}^{i},0)$ to $(2\zeta_{n}^{i+1},0)$ passes
above (respectively, beneath) a strand connecting $(\zeta_{n}^{i+1},0)$ to
$(2\zeta_{n}^{i},0)$.

  * •

Let $r_{n}$ denote the $(n,n)$ tangle connecting $(\zeta_{n}^{j},0)$ to
$(2\zeta_{n}^{j-1},0)$ for each $1\leq j\leq n$ (clockwise rotation of all
strands).

  * •

Let $s_{n}^{i}$ denote the $(n,n)$-tangle with a strand connecting
$(\zeta_{j},0)$ to $(2\zeta_{j},0)$ for each $j$, and a strand connecting
$(\zeta_{i},0)$ to $(2\zeta_{i},0)$ passing clockwise around the circle,
beneath all the other strands.

We also introduce new generators $w_{n}^{i}(1)$ and $w_{n}^{i}(2)$, which
correspond to positive and negative twists of framing of the $i$-th strand of
an $(n,n)$ identity tangle.

The below figures depict some of these elementary tangles. Here the label $i$
in the inner (resp. outer) circle denotes the point $(\zeta^{i},0)$ (resp.
$(2\zeta^{i},0)$).

These are called “elementary” tangles because any framed tangle can be
expressed as (or rather, is isotopic to) a product of the tangles
$g_{n}^{i},,f_{n}^{i},t_{n}^{i}(1),t_{n}^{i}(2),w_{n}^{i}(1),w_{n}^{i}(2)$ and
$r_{n}$. Alternatively, this statement is true if we replace $r_{n}$ by
$s_{n}^{n}$ (to see this, one uses the fact that $r_{n}=s_{n}^{n}\circ
t_{n}^{n-1}(2)\circ\cdots\circ t_{n}^{1}(2)$).

######  Definition 2.5.

A $(q,r)$-tangle is a“linear tangle” if it can be expressed in terms of the
generators
$g_{n}^{i},f_{n}^{i},t_{n}^{i}(1),t_{n}^{i}(2),w_{n}^{i}(1),w_{n}^{i}(2)$
(i.e. without using $r_{n}$).

The above definition is motivated that the fact any such tangle can be
represented with endpoints on a line segment (as opposed to an annulus); see
Section 4 of [5] for a pictorial depiction.

Below is the list of relations which the generators
$g_{n}^{i},f_{n}^{i},t_{n}^{i}(1),t_{n}^{i}(2),w_{n}^{i}(1),w_{n}^{i}(2),r_{n}$
satisfy. Note that $r^{\prime}_{n}$ is the inverse of $r_{n}$.

  1. (1)

${f}_{n}^{i}\circ{g}_{n}^{i+1}=id={f}_{n}^{i+1}\circ{g}_{n}^{i}$

  2. (2)

(Reidemeister 1) ${f}_{n}^{i}\circ{t}_{n}^{i\pm
1}(l)\circ{g}_{n}^{i}={w}_{n}^{i}(l)$

  3. (3)

${t}_{n}^{i}(2)\circ{t}_{n}^{i}(1)=id={t}_{n}^{i}(1)\circ{t}_{n}^{i}(2)$

  4. (4)

${t}_{n}^{i}(l)\circ{t}_{n}^{i+1}(l)\circ{t}_{n}^{i}(l)={t}_{n}^{i+1}(l)\circ{t}_{n}^{i}(l)\circ{t}_{n}^{i+1}(l)$

  5. (5)

${g}_{n+2}^{i+k}\circ{g}_{n}^{i}={g}_{n+2}^{i}\circ{g}_{n}^{i+k-2}$

  6. (6)

${f}_{n}^{i+k-2}\circ{f}_{n+2}^{i}={f}_{n}^{i}\circ{f}_{n+2}^{i+k}$

  7. (7)

${g}_{n}^{i+k-2}\circ{f}_{n}^{i}={f}_{n+2}^{i}\circ{g}_{n+2}^{i+k},\quad{g}_{n}^{i}\circ{f}_{n}^{i+k-2}={f}_{n+2}^{i+k}\circ{g}_{n+2}^{i}$

  8. (8)

${g}_{n}^{i}\circ{t}_{n-2}^{i+k-2}(l)={t}_{n}^{i+k}(l)\circ{g}_{n}^{i},\quad{g}_{n}^{i+k}\circ{t}_{n-2}^{i}(l)={t}_{n}^{i}(l)\circ{g}_{n}^{i+k}$

  9. (9)

${f}_{n}^{i}\circ{t}_{n}^{i+k}(l)={t}_{n-2}^{i+k-2}(l)\circ{f}_{n}^{i},\quad{f}_{n}^{i+k}\circ{t}_{n}^{i}(l)={t}_{n-2}^{i}(l)\circ{f}_{n}^{i+k}$

  10. (10)

${t}_{n}^{i}(l)\circ{t}_{n}^{i+k}(m)={t}_{n}^{i+k}(m)\circ{t}_{n}^{i}(l)$

  11. (11)

${t}_{n}^{i}(1)\circ{g}_{n}^{i+1}={t}_{n}^{i+1}(2)\circ{g}_{n}^{i},\quad{t}_{n}^{i}(2)\circ{g}_{n}^{i+1}={t}_{n}^{i+1}(1)\circ{g}_{n}^{i}$

  12. (12)

${r}_{n}\circ{r}_{n}^{\prime}=id={r}_{n}^{\prime}\circ{r}_{n}$

  13. (13)

${r}_{n-2}^{\prime}\circ{f}_{n}^{i}\circ{r}_{n}={f}_{n}^{i+1},\
i=1,\ldots,n-2;\quad{f}_{n}^{n-1}\circ({r}_{n})^{2}={f}_{n}^{1}$

  14. (14)

${r}_{n}^{\prime}\circ{g}_{n}^{i}\circ{r}_{n-2}={g}_{n}^{i+1},\
i=1,\ldots,n-2;\quad({r}_{n}^{\prime})^{2}\circ{g}_{n}^{n-1}={g}_{n}^{1}$

  15. (15)

${r}_{n}^{\prime}\circ{t}_{n}^{i}(l)\circ{r}_{n}={t}_{n}^{i+1}(l);\quad({r}_{n}^{\prime})^{2}\circ{t}_{n}^{n-1}(l)\circ({r}_{n})^{2}={t}_{n}^{1}(l)$

We have the following additional relations for twists:

  16. (16)

${w}_{n}^{i}(1)\circ{w}_{n}^{i}(2)=id,\quad{w}_{n}^{i}(l)\circ{w}_{n}^{j}(k)={w}_{n}^{j}(k)\circ{w}_{n}^{i}(l),\
i\neq j$

  17. (17)

$\hat{w}_{n}^{i}(k)\circ{g}_{n}^{i}={w}_{n}^{i+1}(k)\circ{g}_{n}^{i},\quad{w}_{n}^{i}(k)\circ{g}_{n}^{j}={g}_{n}^{j}\circ{w}_{n}^{i+1\pm
1}(k),\ i\neq j,j+1$

  18. (18)

${f}_{n}^{i}\circ{w}_{n}^{i}(k)={f}_{n}^{i}\circ{w}_{n}^{i+1}(k),\quad{w}_{n}^{i}(k)\circ{f}_{n}^{j}={f}_{n}^{j}\circ{w}_{n}^{i-1\pm
1}(k),\ i\neq j,j+1$

  19. (19)

${w}_{n}^{i}(k)\circ{t}_{n}^{i}(l)={w}_{n}^{i+1}(k)\circ{t}_{n}^{i}(l),\quad{w}_{n}^{i}(k)\circ{t}_{n}^{j}(l)={t}_{n}^{j}(l)\circ{w}_{n}^{i}(k),\
i\neq j,j+1$

  20. (20)

${t}_{n}^{i}\circ{w}_{n}^{i}(k)={f}_{n}^{i}\circ{w}_{n}^{i+1}(k),\quad{w}_{n}^{i}(k)\circ{f}_{n}^{j}={t}_{n}^{j}\circ{w}_{n}^{i}(k),\
i\neq j,j+1$

  21. (21)

${w}_{n}^{i}(k)\circ{r}_{n}={r}_{n}\circ{w}_{n}^{i-1}(k),\quad{w}_{n}^{i}(k)\circ{r}^{\prime}_{n}={r}^{\prime}_{n}\circ{w}_{n}^{i+1}(k)$

###  2.2. Exotic t-structures for two-block nilpotents

In this section we summarize the results of [1] for motivational purposes. Let
$\mathfrak{g}=\mathfrak{sl}_{m+2n}$, and $\mathcal{B}$ be the flag variety.
Let $e\in\mathfrak{g}^{*}$ be a nilpotent linear functional with Jordan type
$(m+n,n)$, and $\lambda\in\mathfrak{h}^{*}//W$ be a regular Harish-Chandra
character.

Let $S_{m,n}\subset\mathfrak{g}$ be the corresponding Mirkovic-Vybornov
transverse slice (see [8] for a definition, and see below for an explicit
matrix realization). Let
$U_{m+n,n}\subset\tilde{\mathfrak{g}}=T^{*}\mathcal{B}$ be its preimage under
the Grothendieck-Springer resolution map.

$\displaystyle S_{m,n}$
$\displaystyle=\\{\left(\begin{array}[]{ccccccccc}&1&&&&&&&\\\
&&&\ddots&&&&&\\\ &&&&1&&&&\\\
a_{1}&a_{2}&&\cdots&a_{m+n}&b_{1}&b_{2}&\cdots&b_{n}\\\ &&&&&&1&&\\\
&&&&&&&\ddots&\\\ &&&&&&&&1\\\
c_{1}&\cdots&c_{n}\;0&\cdots&0&d_{1}&d_{2}&\cdots&d_{n}\end{array}\right)\\}$

######  Definition 2.6.

Let $\langle\cdot,\cdot\rangle$ denote the Killing form.

$\displaystyle S^{*}_{m,n}$ $\displaystyle=\\{x\in\mathfrak{g}\;|\;\langle
x,z-e\rangle=0\;\forall\;z\in S_{m,n}\\}$
$\displaystyle\textbf{U}_{\hat{e}}\mathfrak{g}$
$\displaystyle=\textbf{U}\mathfrak{g}/\langle x^{p}-x^{[p]}-\langle
e,x\rangle^{p}\;|\;x\in S^{*}_{m,n}\rangle$

Denote by $\text{Mod}_{\hat{e},\lambda}^{fg}(\textbf{U}\mathfrak{g})$ the
subcategory of $\text{Mod}_{e,\lambda}^{fg}(\textbf{U}\mathfrak{g})$
consisting of those modules which are defined over the quotient
$\textbf{U}_{\hat{e}}\mathfrak{g}$.

[BMR] localization theory gives the equivalence in the below Proposition; see
Section 5.1 of [1] for more details. Under this equivalence, the tautological
t-structure on the LHS is mapped to the exotic t-structure on the RHS. We
refer to the images of the irreducible objects in the LHS as simple exotic
sheaves.

######  Proposition 2.7.

$\mathcal{C}_{m,n}:=D^{b}\text{Mod}^{\text{fg}}_{\hat{e},\lambda}(U\mathfrak{g})\simeq
D^{b}\text{Coh}_{\mathcal{B}_{e}}(U_{e})$

######  Proposition 2.8.

Let $\alpha$ be an $(m+2p,m+2q)$-tangle. Then we have a functor
$\Psi^{\prime}(\alpha):\mathcal{C}_{m,p}\rightarrow\mathcal{C}_{m,q}$; this
collection of functors satisfy the functorial tangle relations
$\Psi^{\prime}(\alpha)\circ\Psi^{\prime}(\beta)\simeq\Psi^{\prime}(\alpha\circ\beta)$.

These are defined by Fourier-Mukai transforms on the categories of coherent
sheaves; see Section $4$ of [1] for more details of the proof. These functors
can be used to describe the simple exotic sheaves as follows.

######  Definition 2.9.

Let $\text{Cross}(m,n)$ be the set of all matchings on an annulus with $m$
unlabelled points on the inner circle, $m+2n$ labelled points on the outer
circle (with labels $1,\cdots,m+2n$), such that there are no crossings or
loops, and each point on the inner circle is connected to a point on the outer
circle.

######  Example 2.10.

See Figure 2 for an example of an element $\alpha\in\text{Cross}(2,2)$.

Figure 2.

######  Definition 2.11.

Each $\alpha\in\text{Cross}(m,n)$ can be viewed as a framed tangle, with the
trivial framing. Thus using Proposition 2.8 we have a functor
$\Psi^{\prime}(\alpha):\mathcal{C}_{m,0}\rightarrow\mathcal{C}_{m,n}$. Since
$\mathcal{B}_{e}$ is a point when $n=0$, it follows that
$\mathcal{C}_{m,0}\simeq D^{b}(\text{Vect})$.Given
$\alpha\in\text{Cross}(m,n)$, let
$\Psi^{\prime}_{\alpha}=\Psi^{\prime}(\alpha)(\textbf{k})$.

######  Theorem 2.12.

The objects $\\{\Psi^{\prime}_{\alpha}\;|\;\alpha\in\text{Cross}(m,n)\\}$ are
the simple exotic sheaves.

See Section 5.3 of [1] for a proof of the above theorem.

##  3\. Coherent sheaves on Cautis-Kamnitzer varieties and the main theorem

We will need Cautis-Kamnitzer’s framework for much of what follows. The
Cautis-Kamnitzer varieties $Y_{k}$ are constructed as follows (see Section 2.1
of [5] for more details):

######  Definition 3.1.

Consider a $2k$-dimensional vector space $V_{k}$ over $\mathbb{C}$ with basis
$e_{1},\ldots,e_{k},f_{1},\ldots,f_{k}$ and a nilpotent $z$ such that
$ze_{i+1}=e_{i}$, $zf_{i+1}=f_{i}$ for $i\geq 1$, and $ze_{1}=zf_{1}=0$. Let
$L_{0}=0$. Then:

$\displaystyle Y_{k}=\\{(L_{1}\subset\cdots\subset L_{k}\subset
V_{k})\;|\;\text{ dim}(L_{i})=i,\ zL_{i}\subset L_{i-1}\\}$

Let $\mathbb{C}^{*}$ act on $V_{k}$ by $t\cdot e_{i}=t^{-2i}e_{i}$ and $t\cdot
f_{i}=t^{-2i}f_{i}$. This $\mathbb{C}^{*}$-action induces a
$\mathbb{C}^{*}$-action on $Y_{k}$ by $t\cdot(L_{1},...,L_{k})=(t\cdot
L_{1},...,t\cdot L_{k})$. Denote by $K^{0}_{\mathbb{C}^{*}}(Y_{k})$ the
equivariant $K$-theory of $Y_{k}$ with respect to the above action of
$\mathbb{C}^{*}$.

We have a natural forgetful map $p_{k}:Y_{k}\rightarrow Y_{k-1}$ which is a
$\mathbb{P}^{1}$-bundle. To see this, suppose that we have
$(L_{1},...,L_{k-1})\in Y_{k-1}$ and are considering possible choices of
$L_{k}$. It is easy to see that we must have $L_{k-1}\subset L_{k}\subset
z^{-1}(L_{k})$. Since $z^{-1}(L_{k-1})/L_{k-1}$ is always two dimensional,
this fibre is a ${\mathbb{P}}^{1}$. It follows that $Y_{k}$ is an iterated
$\mathbb{P}^{1}$-bundle of dimension $k$, since $Y_{1}\simeq\mathbb{P}^{1}$.

######  Definition 3.2.

Let $\mathcal{V}_{s}$ be the tautological vector bundle on $Y_{k}$
corresponding to $L_{s}$. For each $1\leq s\leq k$, we have a line bundle
$\Lambda_{s}=\mathcal{V}_{s}/\mathcal{V}_{s-1}$ on this space (note that non-
equivariantly we have $\Lambda_{k}=\mathcal{O}(-1)_{Y_{k}/Y_{k-1}}$, i.e. the
relative $\mathcal{O}(-1)$ on the $\mathbb{P}^{1}$-fibration $Y_{k}\rightarrow
Y_{k-1}$). Given $i_{s}\in\\{0,1\\}$ for each $1\leq s\leq k$, define also the
following line bundle:

$\Lambda_{i_{1},i_{2},\cdots,i_{k}}:=\bigotimes_{1\leq s\leq
k}\Lambda_{s}^{i_{s}}$

Recall from Section $6.2$ of [5] that:

######  Proposition 3.3.

We have that $K^{0}_{\mathbb{C}^{*}}(Y_{k})$ is a free
$\mathbb{C}[q,q^{-1}]$-module with generators given by the classes of the line
bundles $[\Lambda_{i_{1},i_{2},\cdots,i_{k}}]$, where $i_{s}\in\\{0,1\\}$ for
each $1\leq s\leq k$.

######  Definition 3.4.

Let $V$ denote the standard representation of $U_{q}(\mathfrak{sl}_{2})$. It
is a free ${\mathbb{C}}[q,q^{-1}]$-module with basis $v_{0}$, $v_{1}$.
Identify $K^{0}_{\mathbb{C}^{*}}(Y_{k})$ with $V^{\otimes k}$ as
$\mathbb{C}[q,q^{-1}]$-modules, so that

$[\Lambda_{i_{1},\cdots,i_{k}}]=q^{-\sum_{s}si_{s}}v_{i_{1}}\otimes\cdots\otimes
v_{i_{k}}$

######  Definition 3.5.

Denote the monomial basis for $V^{\otimes k}$ as follows. Let
$I\subset\\{1,\cdots,k\\}$ be a subset, and for $1\leq a\leq n$ let $I(a)=1$
if $a\in I$, and $I(a)=0$ if $a\notin I$. Define:

$v_{I}=v_{I(1)}\otimes v_{I(2)}\otimes\cdots\otimes v_{I(k)}$

######  Definition 3.6.

Let $\mathcal{D}_{k}=D^{b}_{\mathbb{C}^{*}}\text{Coh}(Y_{k})$ be the
equivariant derived category of coherent sheaves on $Y_{k}$.

######  Definition 3.7.

For each linear $(r,s)$-tangle $\alpha$, we define the map
$\psi(\alpha):V^{\otimes r}\rightarrow V^{\otimes s}$ of
$U_{q}({\mathfrak{s}l}_{2})$-modules as follows. It suffices to define the map
on the elementary linear tangles. Below $k\in\\{1,2\\}$ and $i,j\in\\{0,1\\}$.

$\displaystyle\psi(f_{2}^{1}):$ $\displaystyle V^{\otimes
2}\rightarrow\mathbb{C}[q,q^{-1}];\qquad\psi(f_{2}^{1})(v_{i}\otimes
v_{i})=0,\psi(f_{2}^{1})(v_{0}\otimes v_{1})=q,\psi(f_{2}^{1})(v_{1}\otimes
v_{0})=-1.$ $\displaystyle\psi(g_{2}^{1}):$
$\displaystyle\mathbb{C}[q,q^{-1}]\rightarrow V^{\otimes
2};\qquad\psi(g_{2}^{1})(1)=q^{-1}v_{1}\otimes v_{0}-v_{0}\otimes v_{1}.$
$\displaystyle\psi(t_{2}^{1}(1)):$ $\displaystyle V^{\otimes 2}\to V^{\otimes
2};\qquad\psi(t_{2}^{1}(1))(v_{0}\otimes v_{0})=v_{0}\otimes
v_{0},\psi(t_{2}^{1}(1))(v_{0}\otimes v_{1})=(1-q^{2})v_{0}\otimes
v_{1}+qv_{1}\otimes v_{0},$ $\displaystyle\psi(t_{2}^{1}(1))(v_{1}\otimes
v_{0})=v_{0}\otimes v_{1},\psi(t_{2}^{1}(1))(v_{1}\otimes v_{1})=v_{1}\otimes
v_{1}.$ $\displaystyle\psi(t_{2}^{1}(2)):$ $\displaystyle V^{\otimes 2}\to
V^{\otimes 2};\qquad\psi(t_{2}^{1}(2))(v_{0}\otimes v_{0})=v_{0}\otimes
v_{0},\psi(t_{2}^{1}(2))(v_{0}\otimes v_{1})=q^{-1}v_{1}\otimes v_{0},$
$\displaystyle\psi(t_{2}^{1}(2))(v_{1}\otimes v_{0})=(1-q^{-2})v_{1}\otimes
v_{0}+q^{-1}v_{0}\otimes v_{1},\psi(t_{2}^{1}(2))(v_{1}\otimes
v_{1})=v_{1}\otimes v_{1}.$

Now we define $\psi(g_{n}^{i})$ as follows, and similarly for
$\psi(f_{n}^{i})$ and $\psi(t_{n}^{i}(k))$.

$\displaystyle\psi(g_{n}^{i}):$ $\displaystyle V^{\otimes n-2}\rightarrow
V^{\otimes n};\psi(g_{n}^{i})=\text{id}_{V^{\otimes
i-1}}\otimes\psi(g_{2}^{1})\otimes\text{id}_{V^{\otimes n-2-i+1}}$

Note that $\psi$ is well-defined due to [9] and the introduction to [5].

######  Remark 3.8.

We can identify $\psi(\alpha):V^{\otimes r}\rightarrow V^{\otimes s}$ with the
Reshetikhin-Turaev invariant of $\alpha$ from [9] up to a change of notation.

The below proposition is a strengthening of the results in Section 6 of [5],
the difference being that we use affine tangles (and not linear tangles).

######  Proposition 3.9.

For each $(q,r)$-tangle $\alpha$, there exist a functor
$\Psi(\alpha):\mathcal{D}_{q}\rightarrow\mathcal{D}_{r}$, and these are
compatible under composition, i.e. given an $(r,s)$-angle $\beta$,
$\Psi(\alpha\circ\beta)\simeq\Psi(\alpha)\circ\Psi(\beta)$. When $\alpha$ is a
linear tangle, the image of $\Psi(\alpha)$ in the Grothendieck group is the
Reshetikhin-Turaev invariant $\psi(\alpha)$.

###### Proof.

When $\alpha$ is a linear tangle, the functors are certain Fourier-Mukai
kernels which are described in Section 4 of [1]; the functorial relations are
proven in Section 5. Define the functors $\Psi(t^{i}_{m+2n}(1))$ and
$\Psi(t^{i}_{m+2n}(2))$ via the distinguished triangles:

$\Psi(g^{i}_{m+2n})\Psi(f^{i}_{m+2n})[-1]\\{1\\}\to\text{id}\to\Psi(t^{i}_{m+2n}(1)),$
$\Psi(t^{i}_{m+2n}(2))\to\text{id}\to\Psi(g^{i}_{m+2n})\Psi(f^{i}_{m+2n})[1]\\{-1\\}.$

We have $\psi(t_{2}^{1}(1))=1+q\psi(g_{2}^{1})\psi(f_{2}^{1})$ and
$\psi(t_{2}^{1}(2))=1+q^{-1}\psi(g_{2}^{1})\psi(f_{2}^{1})$. From these
formulas the above formulas for $\psi(t_{2}^{1}(1)),\psi(t_{2}^{1}(2))$
follow. Theorem 6.5 of [5] proves that, on the level of Grothendieck groups,
$\Psi(\alpha)$ corresponds to $\psi(\alpha)$. It remains to construct the
functors $\Psi(\alpha)$ when $\alpha$ is an affine tangle; this follows from
the techniques used in Section 4.5 of [1]. It suffices to construct a functor
$\Psi(s_{k}^{k})$, and check that it satisfies the necessary functorial tangle
relations (13) and (14) on page 7. Define $\Psi(s_{k}^{k})$ to be the functor
of tensoring with the line bundle $\Lambda_{k}^{-1}$; to check the functorial
tangle relations, the same argument used in Theorem 4.27 and Proposition 4.25
of [1] works verbatim.

Alternatively, see Proposition 6.4 in [6] ∎

######  Definition 3.10.

Let $W_{m,n}\subset V_{m+2n}$ denote the vector subspace with basis
$e_{1},\ldots,e_{m+n},f_{1},\cdots,f_{n}$, so that $z|_{W_{m,n}}$ has Jordan
type $(m+n,n)$. Identify $\mathfrak{sl}_{m+2n}$ with $\mathfrak{sl}(W_{m,n})$.
Denote by $\mathcal{B}_{m+n,n}$ the Springer fiber for
$z\in\mathfrak{sl}(W_{m,n})$:

$\mathcal{B}_{m+n,n}=\\{(0\subset U_{1}\subset\cdots\subset U_{m+2n-1}\subset
U_{m+2n}=W_{m,n})\;|\;zU_{i}\subseteq U_{i-1}\\}$

By analogy with Definition 2.11 (cf. Theorem 2.12) we define:

######  Definition 3.11.

For any $\alpha\in\text{Cross}(m,n)$, the crossingless sheaf $\Psi_{\alpha}$
is defined as the object $\Psi({\mathcal{O}}_{\mathcal{B}_{m,0}})$.

Note that $\mathcal{B}_{m,0}$ is a point and
$\Psi({\mathcal{O}}_{\mathcal{B}_{m,0}})$ is the skyscraper sheaf supported on
the point $\mathcal{B}_{m,0}$ in $Y_{m}$.

In the remaining part of the paper we compute the classes of $\Psi_{\alpha}$
in equivariant K-theory of Cautis-Kamnitzer varieties.

For our computations, we will need the following lemma which tells us what the
effect of tensoring by $\Lambda_{i}^{-1}$ on $K^{0}(Y_{k})$ is, when $i\leq
k$. It more or less follows from the argument used to prove Theorem $6.2$ of
[5] (see equation (20) in particular).

######  Lemma 3.12.

We have the following identity.

$[\Lambda_{i}^{-1}]=2q^{-2}[\mathcal{O}]-q^{-4}[\Lambda_{i}]$

###### Proof.

Consider the two exact sequences below. Note here that
$z^{-1}\mathcal{V}_{i-1}$ denotes the vector bundle whose fiber at a point is
the pre-image of the corresponding fiber of $\mathcal{V}_{i-1}$ at that point,
under the map $z$.

$\displaystyle 0\rightarrow\text{ker}(z)\rightarrow z^{-1}$
$\displaystyle\mathcal{V}_{i-1}\rightarrow\mathcal{V}_{i-1}\rightarrow 0$
$\displaystyle 0\rightarrow\mathcal{V}_{i}/\mathcal{V}_{i-1}\rightarrow
z^{-1}\mathcal{V}_{i-1}$ $\displaystyle/\mathcal{V}_{i-1}\rightarrow
z^{-1}\mathcal{V}_{i-1}/\mathcal{V}_{i}\rightarrow 0$

Taking determinants, we obtain the following (noting that
$\text{ker}(z)\simeq\mathcal{O}^{\oplus
2}\\{2\\},\text{det}(\text{ker}(z))\simeq\mathcal{O}\\{4\\}$):

$\displaystyle\text{det}(z^{-1}\mathcal{V}_{i-1})$
$\displaystyle\simeq\text{det}(\mathcal{V}_{i-1})\otimes\text{det}(\mathcal{O}^{\oplus
2}\\{2\\})$ $\displaystyle\Lambda_{i}\otimes
z^{-1}\mathcal{V}_{i-1}/\mathcal{V}_{i}$
$\displaystyle\simeq\text{det}(z^{-1}\mathcal{V}_{i-1}/\mathcal{V}_{i-1})$
$\displaystyle\simeq\text{det}(z^{-1}\mathcal{V}_{i-1})\otimes{\text{det}(\mathcal{V}_{i-1})}^{-1}\simeq\text{det}(\mathcal{O}^{\oplus
2}\\{2\\})$ $\displaystyle z^{-1}\mathcal{V}_{i-1}/\mathcal{V}_{i}$
$\displaystyle\simeq\Lambda_{i}^{-1}\otimes\text{det}(\mathcal{O}^{\oplus
2}\\{2\\})$

Using the first exact sequence,
$[z^{-1}\mathcal{V}_{i-1}/\mathcal{V}_{i-1}]=2[\mathcal{O}\\{2\\}]$; so
$[\Lambda_{i}^{-1}\otimes\text{det}(\mathcal{O}^{\oplus
2}\\{2\\})]=2[\mathcal{O}\\{2\\}]-[\Lambda_{i}]$. Therefore
$[\Lambda_{i}^{-1}\otimes\mathcal{O}\\{4\\}]=2[\mathcal{O}\\{2\\}]-[\Lambda_{i}]$,
or $q^{4}[\Lambda_{i}^{-1}]=2q^{2}[\mathcal{O}]-[\Lambda_{i}]$. ∎

######  Definition 3.13.

Given an $(m+2p,m+2q)$-tangle $\alpha$, denote by
$\overline{\alpha}:(\mathbb{C}^{2})^{\otimes
m+2p}\rightarrow(\mathbb{C}^{2})^{\otimes m+2q}$ the map induced on the
Grothendieck group by the functor
$\Psi(\alpha):\mathcal{D}_{m+2p}\rightarrow\mathcal{D}_{m+2q}$. Given an
element $\alpha\in\text{Cross}(m,n)$, denote by
$\underline{\alpha}\in(\mathbb{C}^{2})^{\otimes m+2n}$ the image of the
irreducible object $\Psi_{\alpha}$ in the Grothendieck group.

######  Lemma 3.14.

Let $i_{s}\in\\{0,1\\}$ and $l_{s}\in\mathbb{Z}$ for $1\leq s\leq k$. Then we
have:

$\displaystyle\overline{r_{k}}(v_{0}\otimes v_{i_{2}}\otimes\cdots\otimes
v_{i_{k}})$ $\displaystyle=(1-q^{2})(\sum_{2\leq j\leq
k}q^{\\{\\#s<j,i_{s}=1\\}}q^{k}\delta_{i_{j},1}v_{i_{2}}\otimes\cdots\otimes
v_{i_{j-1}}\otimes v_{0}\otimes v_{i_{j+1}}\otimes\cdots v_{i_{k}}\otimes
v_{0})+$ $\displaystyle\ \ \ \ \
+q^{\\{\\#s,i_{s}=1\\}}v_{i_{2}}\otimes\cdots\otimes
v_{i_{k}}\otimes(2q^{-2}v_{0}-q^{-4}q^{-k}v_{1})$
$\displaystyle\overline{r_{k}}(v_{1}\otimes v_{i_{2}}\otimes\cdots\otimes
v_{i_{k}})$ $\displaystyle=q^{k}v_{i_{2}}\otimes\cdots\otimes v_{i_{k}}\otimes
v_{0}$
$\displaystyle[\Lambda_{1}^{l_{1}}\otimes\Lambda_{2}^{l_{2}}\otimes\cdots\otimes\Lambda_{k}^{l_{k}}]$
$\displaystyle=\bigotimes_{s=1}^{k}(l_{s}q^{2l_{s}-2}q^{-s}v_{1}-(l_{s}-1)q^{2l_{s}}v_{0})$

###### Proof.

Using Proposition 3.9, we compute that:

$\displaystyle\overline{r_{k}}(v_{i_{1}}\otimes\cdots\otimes v_{i_{k}})$
$\displaystyle=\overline{s_{k}^{k}}\circ\overline{t_{k}^{k-1}(1)}\circ\cdots\circ\overline{t_{k}^{1}(1)}(v_{i_{1}}\otimes\cdots\otimes
v_{i_{k}})$

Using Definition 3.7 and straightforward induction, one shows that

$\overline{t_{k}^{k-1}(1)}\circ\cdots\circ\overline{t_{k}^{1}(1)}(v_{0}\otimes
v_{i_{2}}\otimes\cdots\otimes v_{i_{k}})=$ $=(1-q^{2})(\sum_{2\leq j\leq
k}q^{\\{\\#s<j,i_{s}=1\\}}\delta_{i_{j},1}v_{i_{2}}\otimes\cdots\otimes
v_{i_{j-1}}\otimes v_{0}\otimes v_{i_{j+1}}\otimes\cdots v_{i_{k}}\otimes
v_{1})+q^{\\{\\#s,i_{s}=1\\}}v_{i_{2}}\otimes\cdots\otimes v_{i_{k}}\otimes
v_{0},$
$\overline{t_{k}^{k-1}(1)}\circ\cdots\circ\overline{t_{k}^{1}(1)}(v_{1}\otimes
v_{i_{2}}\otimes\cdots\otimes v_{i_{k}})=v_{i_{2}}\otimes\cdots\otimes
v_{i_{k}}\otimes v_{1}$

The conclusion follows by noting that for any $j_{s}\in\\{0,1\\},s=1,\dots,k$

$\overline{s_{k}^{k}}([\Lambda_{1}^{j_{1}}]\otimes\cdots\otimes[\Lambda_{k}^{j_{k}}])=[\Lambda_{1}^{j_{1}}]\otimes\cdots\otimes[\Lambda_{k}^{j_{k}-1}],$

and using Definition 3.4.

It follows from Lemma 3.12 that
$[\Lambda_{i}^{l}]=lq^{2l-2}[\Lambda_{i}]-(l-1)q^{2l}[\mathcal{O}]$ for all
$l\in\mathbb{Z}$. From Definition 3.4, it follows that:

$\displaystyle[\Lambda_{1}^{l_{1}}\otimes\Lambda_{2}^{l_{2}}\otimes\cdots\otimes\Lambda_{k}^{l_{k}}]$
$\displaystyle=[\Lambda_{1}^{l_{1}}]\otimes[\Lambda_{2}^{l_{2}}]\otimes\cdots\otimes[\Lambda_{k}^{l_{k}}]$
$\displaystyle=(l_{1}q^{2l_{1}-2}[\Lambda_{1}]-(l_{1}-1)q^{2l_{1}}[\mathcal{O}])\otimes\cdots(l_{k}q^{2l_{k}-2}[\Lambda_{k}]-(l_{k}-1)q^{2l_{k}}[\mathcal{O}])$
$\displaystyle=\bigotimes_{s=1}^{k}(l_{s}q^{2l_{s}-2}q^{-s}v_{1}-(l_{s}-1)q^{2l_{s}}v_{0}).$

Note that in the last equality we used Definition 3.4. ∎

######  Definition 3.15.

For any $k$ denote by $A_{r,k}$ the $2^{k}\times 2^{k}$ matrix which is the
matrix of the linear operator $\overline{r_{k}}$ in the basis $v_{I}$,
$I\subset\\{1,\cdots,k\\}$, i.e. such that
$A_{r,k}v_{I}=\overline{r_{k}}v_{I}$. Note that the matrix $A_{r,k}$ has been
calculated in Lemma 3.14.

######  Definition 3.16.

Given $\alpha\in\text{Cross}(m,n)$, define the set $C(\alpha)$ to consist of
the following pairs:

$C(\alpha)=\\{(i,j)\;|\;1\leq i<j\leq m+2n;i\text{ and }j\text{ are connected
in the outer circle on the diagram }\alpha\\}$

######  Definition 3.17.

Let $T(\alpha)$ consist of all $n$-element subsets
$S\subset\\{1,\cdots,m+2n\\}$ such that $|S\cap C|=1$ for each $C\in
C(\alpha)$; clearly $|T(\alpha)|=2^{n}$. Let $\text{max}(C)$ be the maximal
element in $C$, and for each $S\in T(\alpha)$ define:

$\text{sgn}_{q}(S)=(-q)^{c(S)}\text{ where }c(S)=\\#\\{C:C\in
C(\alpha),|C|=2,\;S\cap C=\text{max}(C)\\}.$

Let $D(\alpha)$ be the set of points on the outer circle which are connected
to points on the inner circle by arcs. Let $D(\alpha)=\\{d_{1},\dots,d_{m}\\}$
where $d_{1}<\cdots<d_{m}$. For a subset $J\subset D(\alpha)$ and
$i\in\\{1,\dots,m\\}$ we define $J_{i}=1$ if $d_{i}\in J$ and $J_{i}=0$ if
$d_{i}\notin J$.

######  Example 3.18.

Let $\alpha\in\text{Cross}(2,2)$ be as in Example 2.10. Then we have:

$\displaystyle T(\alpha)$ $\displaystyle=\\{(1,3),(1,6),(2,3),(2,6)\\}$
$\displaystyle\text{sgn}(1,3)$
$\displaystyle=1,\text{sgn}(2,6)=q^{2},\text{sgn}(1,6)=\text{sgn}(2,3)=-q$

######  Lemma 3.19.

Let $\alpha\in\text{Cross}(m,0)$ (ie. $\alpha$ is the “identity”
$(m,m)$-tangle). Then:

$\underline{\alpha}=(v_{0}-q^{-1}v_{1})\otimes\cdots\otimes(v_{0}-q^{-m}v_{1})=\bigotimes_{i=1}^{m}(v_{0}-q^{-i}v_{1})\in
V^{\otimes m}$

###### Proof.

The Springer fiber $y_{m}=\mathcal{B}_{m,0}$ is a point; we will prove the
statement that the class $[\mathcal{O}_{y_{m}}]\in K^{0}(Y_{m})$ is
$(v_{0}-q^{-1}v_{1})\otimes\cdots\otimes(v_{0}-q^{-m}v_{1})$ by induction.
First note that when $m=1$, $Y_{1}\simeq\mathbb{P}^{1}$, we have the following
equivariant short exact sequence:

$\displaystyle 0\rightarrow\Lambda_{1}\rightarrow$
$\displaystyle\mathcal{O}\rightarrow i_{*}\mathcal{O}_{y_{1}}\rightarrow 0,$

and hence
$\underline{\alpha}=[\Lambda_{1}^{0}]-[\Lambda_{1}^{1}]=v_{0}-q^{-1}v_{1}$.

For the induction step, consider the $\mathbb{P}^{1}$-bundle
$p_{m}:Y_{m}\rightarrow Y_{m-1}$. We have $p_{m}(y_{m})=y_{m-1}$, and for
every $m$ denote the inclusion of the point by $j_{m}:\\{y_{m}\\}\rightarrow
Y_{m}$. Taking a relative version of the above exact sequence on the fiber
gives:

$\displaystyle 0\rightarrow$
$\displaystyle\;\mathcal{O}_{p_{m}^{-1}(y_{m-1})}\otimes\Lambda_{0,\cdots,0,1}\rightarrow\mathcal{O}_{p_{m}^{-1}(y_{m-1})}\rightarrow
j_{m*}\mathcal{O}_{y_{m}}\rightarrow 0$

By the inductive hypothesis,
$[\mathcal{O}_{y_{m-1}}]=(v_{0}-q^{-1}v_{1})\otimes\cdots\otimes(v_{0}-q^{-m+1}v_{1})$.
Since
$p_{m}^{*}\Lambda_{i_{1},\cdots,i_{m-1}}=\Lambda_{i_{1},\cdots,i_{m-1},0}$,
using Definition 3.4 we compute that:

$\displaystyle[p_{m}^{*}]v_{i_{1}}\otimes\cdots\otimes v_{i_{m-1}}$
$\displaystyle=v_{i_{1}}\otimes\cdots\otimes v_{i_{m-1}}\otimes v_{0}$
$\displaystyle[\mathcal{O}_{p_{m}^{-1}(y_{m-1})}]$
$\displaystyle=[p_{m}^{*}\mathcal{O}_{y_{m-1}}]=(v_{0}-q^{-1})v_{1}\otimes\cdots\otimes(v_{0}-q^{-m+1}v_{1})\otimes
v_{0}.$

We also deduce that
$[\mathcal{O}_{p_{m}^{-1}(y_{m-1})}\otimes\Lambda_{0,\cdots,0,1}]=(v_{0}-q^{-1})v_{1}\otimes\cdots\otimes(v_{0}-q^{-m+1}v_{1})\otimes
q^{-m}v_{1}$ (using Definition 3.4). This now completes the proof:

$[j_{m*}\mathcal{O}_{y_{m}}]=[\mathcal{O}_{p_{m}^{-1}(y_{m-1})}]-[\mathcal{O}_{p_{m}^{-1}(y_{m-1})}\otimes\Lambda_{0,\cdots,0,1}]=\bigotimes_{i=1}^{m}(v_{0}-q^{-i}v_{1}).$

∎

######  Definition 3.20.

We say that a matching $\alpha\in\text{Cross}(m,n)$ is “good” if
$C_{1}(\alpha)=\emptyset$.

######  Lemma 3.21.

If the matching $\alpha$ is good, then in the notation of Definition 3.17 we
have:

$\displaystyle\underline{\alpha}$
$\displaystyle=q^{-n}\sum_{\begin{subarray}{c}I\cup J,\\\ I\in T(\alpha),\\\
J\subset D(\alpha)\end{subarray}}\text{sgn}_{q}(I)(-q)^{\sum iJ_{i}}v_{I\cup
J}\qquad\qquad$

###### Proof of Lemma 3.21.

Proof by induction. For the induction step, note that if
$\alpha\in\text{Cross}(m,n)$ is a good matching, we can pick $i$ such that
$1<i<m+2n$ and $(i,i+1)\in C(\alpha)$, express $\alpha=g_{m+2n}^{i}\beta$, for
some $\beta\in\text{Cross}(m,n-1)$. Then we can complete the proof using
notation from Definition 3.17 and Proposition 3.9:

$\displaystyle\underline{\alpha}$
$\displaystyle=\psi(g_{m+2n}^{i})\underline{\beta}$
$\displaystyle=q^{-n+1}\sum_{\begin{subarray}{c}I\cup J,\\\ I\in T(\beta),\\\
J\subset D(\beta)\end{subarray}}(-q)^{\sum
iJ_{i}}\text{sgn}_{q}(I)\psi(g_{m+2n}^{i})v_{I\cup J}$
$\displaystyle=q^{-n}\sum_{\begin{subarray}{c}I\cup J,\\\ I\in T(\alpha),\\\
J\subset D(\alpha)\end{subarray}}(-q)^{\sum iJ_{i}}\text{sgn}_{q}(I)v_{I\cup
J}$

∎

######  Definition 3.22.

For a good matching $\alpha\in{\rm Cross}(m,n)$, denote by $b_{\alpha}=\sum
b_{\alpha,I}v_{I}$ the vector in the vector space spanned by $v_{I}$,
$I\in\\{1,\cdots,m+2n\\}$ such that $b_{\alpha}=\underline{\alpha}$. Note that
for any good matching $\alpha$, the vector $b_{\alpha}$ was calculated in
Lemma 3.21.

Let $\alpha=r_{m+2n}^{k}\beta$ where $\beta$ is a good matching and $k\geq 0$
(to see the existence of such an expression, simply consider the arc $(i,j)\in
C(\alpha)$ with maximal distance between its endpoints; then $k$ corresponds
to one of the endpoints). Recalling Definitions 3.15 and 3.22, we have:

######  Theorem 3.23.

For a crossingless matching $\alpha\in{\rm Cross}(m,n)$, the class in
equivariant K-theory of $Y_{m+2n}$ of the sheaf $\Psi_{\alpha}$ is equal to
$A_{r,m+2n}^{k}b_{\beta}$.

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