Swampland dS conjecture in Mimetic $f(R,T)$ gravity
S. Noori Gashtia111Email<EMAIL_ADDRESS>J.
Sadeghia222Email<EMAIL_ADDRESS>S. Upadhyayb,c,d,e333Email:
<EMAIL_ADDRESS>M. R. Alipoura444Email<EMAIL_ADDRESS>
aDepartment of Physics, Faculty of Basic Sciences,
University of Mazandaran P. O. Box 47416-95447, Babolsar, Iran
bDepartment of Physics, K.L.S. College, Nawada, Bihar 805110, India.
cDepartment of Physics, Magadh University, Bodh Gaya, Bihar 824234, India
dInter-University Centre for Astronomy and Astrophysics (IUCAA),
Pune, Maharashtra 411007, India.
eSchool of Physics, Damghan University, Damghan 3671641167, Iran.

###### Abstract

In this paper, we study a theory of gravity called mimetic $f(R,T)$ in the
presence of swampland dS conjecture. For this purpose, we introduce several
inflation solutions of the Hubble parameter H(N) from $f(R,T)=R+\delta T$
gravity model, in which R is Ricci scalar, and T denotes the trace of the
energy-momentum tensor. Also, $\delta$ and $N$ are the free parameter and a
number of e-fold, respectively. Then we calculate quantities such as
potential, Lagrange multiplier, slow-roll, and some cosmological parameters
such as $n_{s}$ and $r$. Then we challenge the mentioned inflationary model
from the swampland dS conjecture. We discuss the stability of the model and
investigate the compatibility or incompatibility of this inflationary scenario
with the latest Planck observable data.
Keywords: Mimetic $f(R,T)$ Gravity, Lagrange Multiplier, Swampland Conjecture

###### Contents

  1. 1 Introduction
  2. 2 Mimetic $f(R,T)$ Gravity
  3. 3 Inflation Setup
  4. 4 Several inflation solutions of Hubble parameters
    1. 4.1 Model I
    2. 4.2 Model II
    3. 4.3 Model III
  5. 5  Mimetic $f(R,T)$ in RSC Perspective
    1. 5.1 further refining swampland conjecture
      1. 5.1.1 Model I
      2. 5.1.2 Model II
      3. 5.1.3 Model III
  6. 6 Stability
  7. 7 Conclusion
  8. 8 Appendix A: The slow-roll and spectral parameters (model I)
  9. 9 Appendix B: The slow-roll and spectral parameters (model II)
  10. 10 Appendix C: The swampland conjecture structure

##  1 Introduction

The inflationary scenario was proposed to study the expansion of the universe.
It expresses how the formation of large scale structures and tries to answer
problems such as fine-tuning, horizon, and flatness [1, 2, 3] concerning
different conditions and conjectures such as slow-roll, constant-roll, etc.
Large-scale and coherent structures of the universe and CMB anisotropies are
explained in such a way that the exponential growth of pre-inflationary
density fluctuations over a brief period, i.e., approximately $(10^{-33}s)$
can lead to the possibility of creating this structures[4, 5, 6, 7]. It has
recently been proven that a scalar field called $"Inflaton"$ is responsible
for inflation[1]. Of course, many potentials for inflation have been studied
consistently with observable data. Their cosmological applications have been
discussed in detail[8, 9, 10, 11, 12]. Modified theories of gravity have
always been of interest to researchers in inflation because they do not need
to consider the dark matter and dark energy in the face of several
cosmological issues such as late-time cosmic acceleration and have been very
useful[13, 14]. We also remark that in addition to the scalar field in the
cosmological inflation scenario, other types of modified gravitational
theories have been examined that are consistent with observable data and have
discussed various kinds of cosmological consequences such as f(R), f(R, T ),
f(G )gravity, also in braneworld models and so on[15, 16, 17, 18, 19, 20]. We
noted that mimetic gravity is an organized Weyl symmetric augmentation of
Einstein gravity connected by a singular disformal transformation and explains
the evolution of the universe, so that Integrated study from primordial
inflation to the late-time acceleration. This mimetic gravity has been
investigated in the inflation and solutions of black holes, late-time
acceleration, cosmology, and the rotation curves of wormholes, so for further
study, you can see in Ref.s.[21, 22, 23, 24, 25, 26, 27, 28]. Considering the
above concepts, the main purpose of this article is to investigate the
solutions of cosmic inflation in the framework of mimetic f (R, T ) gravity
with Lagrange multiplier and the potential for the monitoring of swampland
conjectures. We first introduce some inflation solutions of the Hubble
parameter H(N) and calculate quantities such as potential and Lagrange
multiplier and two important cosmological parameters such as $n_{s}$ and $r$,
which are called the scalar spectral index and tensor to scalar ratio,
respectively. Then by considering one inflation solution of the Hubble
parameter, the compatibility or incompatibility of this model of the latest
observable data concerning swampland conjectures and analyze the results. This
study’s exciting and unique point is the combination of mimetic f (R, T)
gravity with swampland conjectures in inflation that will have interesting
cosmological results and implications. In general, f(R, T) gravity is
introduced in [29] has had many successes in various cosmological studies such
as cosmology, Big bang nucleosynthesis, temporally varying physical constants,
density perturbations, viscous cosmology, inflation, late-time acceleration,
baryogenesis, and redshift drift. [30, 31, 32, 33, 34, 35, 36, 37, 38].
Researchers recently studied the swampland conjectures. The swampland program
and the weak gravity conjecture have been studied for many cosmological
concepts such as inflation, black hole physics, energy and dark matter, brane
structure, and other cosmology concepts. These conjectures have been used with
other concepts and cosmological conditions, such as various effective
theories. The cosmological applications of these studies are well articulated.
In the swampland program, we are faced with concepts such as swampland (a set
of theories incompatible with quantum gravity) and landscape (a set of
theories compatible with quantum gravity). This article also considers the
refined swampland conjecture, which we will introduce in detail in this
article’s continuation. You can see Refe.s.[39, 40, 41, 42, 43, 44, 45, 46,
47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 65, 66, 67, 68, 69]. This
paper is organized as follows.
In Sec 2, we briefly explain the mimetic $f(R,T)$ gravity. In Sec 3, we
overview mimetic $f(R,T)$ gravity in an inflationary scenario. In Sec 4, we
study several inflation solutions of the Hubble parameters such as $H(N)$ are
$(XN^{a}+YN\exp(bN))^{d}$, $(Xc^{N}+YN^{a})^{d}$ and $(X+YN)^{d}$ from
$f(R,T)$ gravity and calculate quantities such as potential, Lagrange
multiplier, slow-roll, and other cosmological parameters. In Sec 5, we
consider one of the inflationary solutions of the Hubble parameter and study
the mimetic $f(R,T)$ inflationary model from the swampland conjecture point of
view. Also, we employ some exciting figures and investigate the compatibility
or incompatibility of the inflationary scenario with the latest observable
data, i.e., BICEP2/Keck Array data, Planck data, and further refining the
swampland dS conjecture. In Sec 6, we discuss the stability of the model and
finally, we summarize the results with future remarks in Sec 7.

##  2 Mimetic $f(R,T)$ Gravity

This section will explain the mimetic $f(R,T)$ gravity. This section’s main
purpose is to examine inflation from the point of view of mimetic $f(R,T)$
gravity. By parameterizing the metric using the new degrees of freedom, a wide
class of solutions can be created[70, 71, 72, 73, 74, 75, 76, 77]. one can
introduce the metric$g_{ij}$ as an auxiliary metric $\widetilde{g}_{ij}$ with
an auxiliary scalar field that is expressed in the following form, as
$g_{ij}=-\widetilde{g}_{ij}\partial_{\alpha}\partial_{\beta}\phi\widetilde{g}^{\alpha\beta}$.
We can also give an explanation with respect to the relation expressed as
follows, $-(\widetilde{g}_{ij},\phi)\partial_{i}\phi\partial_{j}\phi
g^{ij}=1$. With respect to this inflationary model, flat FRW space, and
according to the mentioned above and a Lagrange multiplier with a mimetic
potential, this model’s action is expressed in the following form[77].

$S=\int\sqrt{-g}d^{4}x\bigg{(}\mathcal{L}_{m}-V(\phi)+\lambda(g_{ij}\partial_{i}\phi\partial_{j}\phi+1)+f\big{(}R(g^{ij},T)\big{)}\bigg{)},$
(1)

where $R$, $T$, $\mathcal{L}_{m}$ are Ricci scalar, the trace of the energy-
momentum tensor and matter Lagrangian, respectively. The field equation with
respect to equation (1) and the metric $g_{ij}$ which is given by,

$\begin{split}&\frac{1}{2}g_{ij}(\lambda(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+1)-V(\phi))+\frac{1}{2}g_{ij}f(R,T)+\nabla_{i}\nabla_{j}f_{R}-R_{ij}f_{R}\\\
&-\lambda\partial_{i}\phi\partial_{j}\phi+\frac{1}{2}T_{ij}-f_{T}(T_{ij}+\Pi_{ij})-g_{ij}\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}f_{R}=0,\end{split}$ (2)

where $\Pi_{ij}=g^{\alpha\beta}\frac{\delta T_{\alpha\beta}}{\delta g^{ij}}$
and $\hbox{$\hbox to0.0pt{$\sqcup$\hss}\sqcap$}=\nabla^{i}\nabla_{i}$,
$\nabla_{i}$, are the d’Alembertian operator and covariant derivative, also
$f_{R}$ and $f_{T}$ denote partial derivatives of $f(R,T)$ with respect to $R$
and $T$, respectively[77]. The energy-momentum tensor for the cosmos filled
with perfect fluid is expressed in the following form.

$\begin{split}T_{ij}=p_{m}g_{ij}+(p_{m}+\rho_{m})v_{i}v_{j},\end{split}$ (3)

where $p_{m}$, $\rho_{m}$ and $v_{i}$ are the pressure and energy density and
the four velocity, respectively. Also the scalar curvature $R$ can be
expressed as $R=6(2H^{2}+\dot{H})$ and $T=\rho-3p$, If assumed
$\mathcal{L}_{m}=-p_{m}$, one can obtain[77],

$\begin{split}&\frac{1}{2}g_{ij}\big{(}\lambda(g^{\alpha\beta}\partial_{\alpha}\phi\partial_{\beta}\phi+1)-V(\phi)\big{)}+\frac{1}{2}g_{ij}f(R,T)+\nabla_{i}\nabla_{j}f_{R}-R_{ij}f_{R}\\\
&-\lambda\partial_{i}\phi\partial_{j}\phi+\frac{1}{2}T_{ij}-f_{T}(T_{ij}+p_{m}g_{ij})-g_{ij}\hbox{$\hbox
to0.0pt{$\sqcup$\hss}\sqcap$}f_{R}=0.\end{split}$ (4)

With respect to equations (1), (2) and the auxiliary scalar field $\phi$, we
will have,

$\begin{split}-V^{\prime}(\phi)-2\nabla^{i}(\lambda\partial_{i}\phi)=0,\end{split}$
(5)

prime represents the derivative concerning $\phi$, and with respect to
equations (1), (2), and the Lagrange multiplier $\lambda$, we arrive,

$\begin{split}g^{ij}\partial_{i}\phi\partial_{j}\phi=-1.\end{split}$ (6)

The Friedmann equations for the FRW background with the presumption that the
scalar field depends on t is given by the following equation[77].

$\begin{split}6(\dot{H}+H^{2})f_{R}-f(R,T)-6H\frac{\textrm{d}f_{R}}{\textrm{d}t}-\lambda(\dot{\phi}^{2}+1)-2f_{T}p_{m}+V(\phi)+\rho_{m}(2f_{T}+1)=0,\end{split}$
(7)

$\begin{split}f(R,T)-2(\dot{H}+3H^{2})f_{R}+p_{m}(4f_{T}+1+4H\frac{\textrm{d}f_{R}}{\textrm{d}t})-\lambda(\dot{\phi}^{2}+1)+2\frac{\textrm{d}^{2}f_{R}}{\textrm{d}t^{2}}-V(\phi)=0,\end{split}$
(8)

$\begin{split}6H\lambda\dot{\phi}-V^{\prime}(\phi)+2\frac{\textrm{d}(\lambda\dot{\phi})}{\textrm{d}t}=0,\end{split}$
(9)

$\begin{split}\dot{\phi}^{2}-1=0,\end{split}$ (10)

where the dot represents the time derivative. By assuming the $\phi=\phi(t)$
the equation (7) convert to,

$\begin{split}-2(\dot{H}+3H^{2})f_{R}+f(R,T)+4H\frac{\textrm{d}f_{R}}{\textrm{d}t}+2\frac{\textrm{d}^{2}f_{R}}{\textrm{d}t^{2}}-V(t)+p_{m}(4f_{T}+1)=0.\end{split}$
(11)

So the mimetic potential in $f(R,T)$ which is given by

$\begin{split}V(t)=f(R,T)+4H\frac{\textrm{d}f_{R}}{\textrm{d}t}+2\frac{\textrm{d}^{2}f_{R}}{\textrm{d}t^{2}}+p_{m}(4f_{T}+1)-2(\dot{H}+3H^{2})f_{R}.\end{split}$
(12)

According to the above concepts, by selecting appropriate models from $f(R,T)$
gravity, one can reconstruct the mimetic potential in equation (12). We can
also solve equation (7) with respect to the Lagrange multiplier that we will
have.

$\begin{split}\lambda(t)=3(\dot{H}+H^{2})f_{R}-\frac{1}{2}f(R,T)-3H\frac{\textrm{d}f_{R}}{\textrm{d}t}+\rho_{m}(f_{T}+\frac{1}{2})+\frac{1}{2}V(t)-f_{T}p_{m}.\end{split}$
(13)

##  3 Inflation Setup

After an overview of mimetic $f(R,T)$ gravity, now in this section, we want to
study the effect of several different inflation models within the mimetic
$f(R,T)$ gravity structure and compare it with the latest Planck observable
data[79, J]. Hence we choose the simplest form of the $f(R,T)$ gravity model,
i.e., $f(R,T)=R+\delta T$. Then we introduce $n_{s}$, $r$ and slow-roll
parameters in this structure[77]. these parameters can be easily expressed in
terms of the number of e-folds using the following equations, so

$\begin{split}\frac{\textrm{d}}{\textrm{d}t}=H(N)\frac{\textrm{d}}{\textrm{d}N},\hskip
28.45274pt\frac{\textrm{d}^{2}}{\textrm{d}t^{2}}=H(N)\frac{\textrm{d}H}{\textrm{d}N}+H^{2}(N)\frac{\textrm{d}^{2}}{\textrm{d}N^{2}}.\end{split}$
(14)

So the slow-roll parameters in terms of $N$, which is given by,

$\begin{split}\epsilon=-\frac{H(N)}{4H^{\prime}(N)}(\frac{6\frac{H^{\prime}(N)}{H(N)}+\frac{H^{\prime\prime}(N)}{H(N)}+(\frac{H^{\prime}(N)}{H(N)})^{2}}{\frac{H^{\prime}(N)}{H(N)}+3})^{2},\end{split}$
(15)

$\begin{split}\eta=-(\frac{\frac{1}{2}(\frac{H^{\prime}(N)}{H(N)})^{2}+9\frac{H^{\prime}(N)}{H(N)}+3\frac{H^{\prime\prime}(N)}{H(N)}+3\frac{H^{\prime\prime}(N)}{H^{\prime}(N)}-\frac{1}{2}(\frac{H^{\prime\prime}(N)}{H^{\prime}(N)})^{2}+\frac{H^{\prime\prime}(N)}{H^{\prime}(N)}}{2(\frac{H^{\prime}(N)}{H(N)}+3)}),\end{split}$
(16)

where primes represent a derivative of N. The scalar spectral index and
tensor-to-scalar ratio which is given by,

$\begin{split}n_{s}=1+2\eta-6\epsilon,\hskip
28.45274ptr=16\epsilon.\end{split}$ (17)

One can consider the $n_{s}$ and $r$ restrictions resulted from the
marginalized joint 68% and 95% CL regions of the Planck 2018 in compound with
BK14+BAO data, viz $n_{s}=0.9649\pm 0.0042$ at 68% Cl, and $r<0.1$ at 95% CL
and from BICEP2/Keck Array BK14 recent data $r<0.056$ at 95% CL [J]. The exact
value of the mimetic potential and Lagrange multiplier can also be determined
according to equation (14) and employ the mimetic $f(R,T)$ gravity. In the
following, we introduce several inflation solutions of the Hubble parameter
H(N) and use the method expressed in these two sections. We obtain the exact
values for each cosmological parameter, mimetic potential, and Lagrange
multiplier. Then, we examine these results from the swampland conjecture
perspective and compare them with observable data.

##  4 Several inflation solutions of Hubble parameters

This section considers three impressive solutions of Hubble parameters named
models I, II, and III, respectively.

###  4.1 Model I

For the first inflation model, the Hubble rate is given by the following
function of the number of e-folds:

$\begin{split}H(N)=(Xc^{N}+YN^{a})^{d},\end{split}$ (18)

where $X,Y,c,a$ and $d$ are free parameters and $N$ is the number of
e-folding. the $\epsilon$, $\eta$, $n_{s}$ and $r$ are calculated as shown in
Appendix A (Eq.s(53-56)).

By determining each of these free parameters’ values, we confirm the
correspondence of each of these inflation scenarios with the observable data.
According to the mentioned point, we consider the values of each of these free
parameters in this form.

$Y=50,\ X=-1,\ c=1,\ a=-0.0005,\ d=1.$

Hence, the two cosmological parameters, i.e., the scalar spectral index and
the tensor-to-scalar ratio for these free parameters,

$r=0.000152005,\hskip 28.45274ptn_{s}=0.966221,$

which correspond to the observable data [79]. It can be noted that after
examining each of these cosmological parameters and comparing them with
observable data, the values of mimetic potential and the Lagrange multiplier
responsible for producing the inflationary model (18) can also be given
according to the equations (18), (12), (13) and (14) in mimetic f(R, T)
gravity. So we will have

$\begin{split}V=&-\frac{(-2+3(-1+\delta)\delta)d(YN^{a}+Xc^{N})^{-1+d}(YN^{a}a+XNc^{N}\log(c))(YN^{a}+Xc^{N})^{d}}{N(1+\delta)(1+3\delta)}\\\
&+\frac{3(1+\delta)(YN^{a}+Xc^{N})^{2d}}{1+3\delta}-\delta(-3+\rho),\end{split}$
(19)

$\begin{split}\lambda=&\frac{1}{2}\bigg{(}-\frac{3d\delta^{2}(YN^{a}+Xc^{N})^{-1+d}(YN^{a}a+XNc^{N}\log(c))(YN^{a}+Xc^{N})^{d}}{N(1+4\delta+3\delta^{2})}\\\
&-\frac{3(2+5\delta)(YN^{a}+Xc^{N})^{2d}}{1+3\delta}-3+\delta+(2+\delta)\rho\bigg{)}.\end{split}$
(20)

###  4.2 Model II

This part follows a similar process to the last detail; hence we consider
another inflationary model in which the Hubble rate describes the cosmological
evolution.

$\begin{split}H(N)=(XN^{a}+YN\textrm{e}^{bN})^{d},\end{split}$ (21)

where $X,Y,b,a$ and $d$ are free parameters. The slow-roll and spectral
parameters of model II are obtained as shown in Appendix B (Eq.s(57-60)). Now,
we consider the free parameters for the second inflation model as

$Y=25,\ X=-1.5,\ b=-0.025,\ a=-0.09,\ d=0.1.$

Hence, the scalar spectral index and the tensor-to-scalar ratio obtain

$n_{s}=0.965577,\hskip 28.45274ptr=0.0155542,$

which correspond to the observable data [79]. like model I, the values of
mimetic potential and the Lagrange multiplier are calculated as

$\begin{split}V=&-\frac{1}{N(1+\delta)(1+3\delta)}\bigg{(}XN^{a}(YN\textrm{e}^{Nb}+XN^{a})^{-1+2d}\bigg{[}9N(-1+\delta)(1+2\delta)(1+\delta)\\\
&+a\Big{(}-4+\delta\big{(}-5+\delta(11+6\delta)\big{)}\Big{)}d\bigg{]}+YN\textrm{e}^{bN}(Y\textrm{e}^{bN}+XN^{a})^{-1+2d}\bigg{(}9N(-1+\delta)(1+\delta)\\\
&\times(1+2\delta)+(1+Nb)\Big{(}-4+\delta\big{(}-5+\delta(11+6\delta)\big{)}\Big{)}d\bigg{)}+N\delta(1+\delta)(1+3\delta)\rho\bigg{)},\end{split}$
(22)

$\begin{split}\lambda=&-3Y^{2}N^{2}\textrm{e}^{2Nb}(-1+\delta)-3X^{2}N^{2a}(-1+\delta)-6AY\textrm{e}^{Nb}N^{1+a}(-1+\delta)\\\
&-{(YNe^{bN}+XN^{a})^{d}\Big{(}XaN^{a-1}+Y\textrm{e}^{Nb}(1+Nb)\Big{)}(-3+\delta)d}\\\
&-\frac{1}{2N(1+\delta)(1+3\delta)}(Y\textrm{e}^{Nb}N+XN^{a})^{-1+2d}(-2+5\delta)\\\
&\times\bigg{[}XN^{a}(3N(1+\delta)+a(2+3\delta)d)+YN\textrm{e}^{Nb}\Big{(}3N(1+\delta)+(1+Nb)(2+3\delta)d\Big{)}\bigg{]}\\\
&+\rho+\frac{\delta\rho}{2}.\end{split}$ (23)

###  4.3 Model III

In the final inflation model, we consider a subclass of the cases studied
above, where the following Hubble parameter describes the cosmological
evolution. In this model, $X$, $Y$, and $d$ are constants parameters. the
important point, i.e., the constants parameter are $Y<0$, $X>0$. So, we have
$|\dot{H}|\ll H^{2}$ during the inflationary duration, which causes
$\frac{X}{Y}\gg N$, such that the Hubble parameter is roughly constant (de
Sitter) during the inflation. After inflation finishes, the second sentence
evolves important, and the Hubble rate decays.

$\begin{split}H(N)=(X+YN)^{d}.\end{split}$ (24)

We can investigate The slow-roll parameter in equations (15) and (16) in the
following form.

$\begin{split}\epsilon=-\frac{Yd(6X+Y(-1+6N+2d))^{2}}{4(X+YN)(3X+Y(3N+d))^{2}},\end{split}$
(25)

$\begin{split}\eta=\frac{Y\Big{(}Y+8YN+X(8-26d)-2Yd(-2+13N+3d)\Big{)}}{4(X+NY)(3X+Y(3N+d))}.\end{split}$
(26)

The scalar spectral index and the tensor-to-scalar ratio in equations (17) are
obtained as,

$\begin{split}n_{s}=&1+\frac{3Yd(6X+Y(-1+6N+2d))^{2}}{2(X+YN)(3X+Y(3N+d))^{2}}\\\
&+\frac{Y\Big{(}Y+8YN+X(8-26d)-2Yd(-2+13N+3d)\Big{)}}{2(X+YN)(3X+Y(3N+d))},\end{split}$
(27)

$\begin{split}r=-16\frac{Yd(6X+Y(-1+6N+2d))^{2}}{4(X+YN)(3X+Y(3N+d))^{2}}.\end{split}$
(28)

Like the previous part, we determine these free parameters, i.e.,

$X=0.75,\ Y=-0.00699999,\ d=0.2.$

So, the scalar spectral index and the tensor-to-scalar ratio obtain respect to
these free parameters.

$n_{s}=0.964569,\hskip 28.45274ptr=0.06836,$

where correspond to the observable data [79]. Also, the mimetic potential and
the Lagrange multiplier are calculated as

$\begin{split}V=&\frac{-9(X+YN)^{-1+2d}(X+YN)(-1+\delta)(1+2\delta)(1+\delta)+Y\Big{(}4+\delta(5-\delta(11+6\delta))\Big{)}d}{(1+\delta)(1+3\delta)}-\delta\rho,\end{split}$
(29)

$\begin{split}\lambda=&\frac{(X+YN)^{-1+2d}(-3(X+YN)(1+\delta)\Big{(}-4+\delta+6\delta^{2}\Big{)}+Y\Big{(}10+\delta\big{(}18-\delta(5+6\delta)\big{)}\Big{)}d}{2(1+\delta)(1+3\delta)}\\\
&+(2+\delta)\rho.\end{split}$ (30)

##  5  Mimetic $f(R,T)$ in RSC Perspective

In this section, we examine one of the inflation solutions of the Hubble
parameter H(N),i.e., model (iii) from the swampland conjecture point of view
and compare their results with observable data[79]. Researchers have recently
studied swampland conjectures as an exciting subject for examining
cosmological structures such as black holes, dark matter, dark energy,
inflation, etc. Also their cosmological implications have been thoroughly
reviewed [55, 56, 57, 58, 59, 65, 66, 67, 68, 69]. But a significant point to
note about swampland conjectures is that it still struggles with many
problems, For example, the discrepancy between single-field slow-roll
inflation and swampland dS conjecture.[65] Of course, solutions have always
been proposed to solve the existing problems and contradictions, such as
considering a Gauss-Bonnet term to calculate the slow-roll parameters, which
leads to resolving the mentioned contradiction[65]. Therefore, they are always
looking for a deeper study and matching the mentioned conjectures with
different concepts of cosmology with other approaches. One of these techniques
is, in fact, the correction of conjectures themselves. For example, a new
conjecture called further refined swampland dS conjecture has recently been
introduced, which has been able to answer many contradictions in various
studies of cosmology[60, 61]. Of course, swampland conjectures, in many cases,
are in good agreement with the concepts, and the latest observable data,
including the warm inflation[62]. Also, recently, many efforts have been made
to generalize the swampland program. Many conjectures have been added to it,
including gravitino swampland conjecture[63]. Of course, one of the most
important motivations in the study of different concepts of cosmology
according to the swampland conjectures can be pointed out in the inflation,
for example, by examining different types of inflation models in various
frameworks and also other types of modified theories of gravity such as f(R),
f(R, T), f(G), etc., according to the structure of swampland conjectures and
analysis of results and their comparison with the latest observable data can
achieve a new classification of inflation models. It is even possible to
determine the best model from many models through these adaptations. But
despite all the above explanations, this statement is still incomplete, and as
we mentioned, it still faces serious problems and challenges[40, 41, 42, 43,
44, 45, 46, 47, 48, 49]. Therefore, this article intends to challenge a kind
of inflation model with recent conjectures. Of course, the swampland program,
which has recently emerged from string theory, has been introduced to answer a
long-standing problem, quantum gravity. Among the conjectures used in this
program, we can mention distance conjecture, dS conjecture, weak gravity
conjecture, and TCC[50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 65]. In this
statement, we also use concepts such as the landscape, that is, a set of
theories consistent with quantum gravity, and a broader area surrounding the
landscape, which expresses a set of approaches that are not compatible with
quantum gravity and considered as swampland. Therefore, one introduced a
series of mathematical formulations to present these conjectures, such as the
dS swampland conjecture. The satisfaction of these equations shows the
compatibility of the models with the introduced conjectures. In other words,
the swampland program expresses a series of conventions and criteria that the
adherence of different theories to these criteria leads to their compatibility
with quantum gravity. These concepts provide very powerful tools for examining
the compatibility of different cosmological concepts and structures with the
ideas mentioned and the latest observable data[64] The dS swampland conjecture
mentioned in the following equation is always used in many cosmic concepts,
different types of phenomenological models, string theory, etc., by expressing
the constraint on potential[50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 65]. Also,
these conjectures about different types of the most important cosmology
parameters, such as scalar spectrum index and tensor to scalar ratio, which
are among the most important cosmological parameters in studying different
inflation models, have had interesting results. In this article, we intend to
challenge them with new restrictions. One of the most important motivations in
these studies could be to find models consistent with these conjectures, which
in some way leads to compatibility with quantum gravity, which may be used to
consider new solutions to study quantum gravity. Of course, these arguments
are theoretical, profoundly consistent with some concepts, and related to the
latest observable data. This article considers the refined swampland
conjecture, which is expressed in the following form.

$|\nabla V|\geq\frac{C_{1}}{M_{p}}V,\hskip 12.0pt\frac{min\nabla\partial
V\leq}{V}-\frac{C_{2}}{M_{pl}^{2}}V,$ (31)

The above equations for the $V>0$, which is given by

$\sqrt{2\epsilon_{V}}\geq C_{1},\hskip 12.0ptor\hskip 12.0pt\eta_{V}\leq-
C_{2},$

where $C_{1}$ and $C_{2}$ are both positive constants of the order of
$\mathcal{O}(1)$. We consider the mimetic potential of inflation solution of
the Hubble parameter H(N) in equation (24), and with respect to the mimetic
potential (29) and the scalar field as a function of $N$ that is defined as
$\frac{\textrm{d}N}{\textrm{d}t}=H(N)$ and $\phi=t$, one can obtain,

$\begin{split}V(\phi)=&-\delta\rho+\frac{1}{(1+\delta)(1+3\delta)}\bigg{(}\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi^{\frac{1}{1-d}}\Big{)}^{-1+2d}\\\
&\times\bigg{[}Y\Big{(}4+\delta\big{(}5-\delta(11+6\delta)\big{)}\Big{)}d-9(-1+\delta)(1+\delta)(1+2\delta)\\\
&\times\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{]}\bigg{)}.\end{split}$
(32)

Other cosmological quantities and parameters can also be reconstructed in this
way. According to equation (32), the first and second derivatives of the
potential and swampland dS conjecture are calculated as shown in Appendix C
(Eq.s(61-64)). We use a specific process to examine a series of constraints
and compare them to observable data [79]. Using the condition as
$\frac{\textrm{d}N}{\textrm{d}t}=H(N)$ and $\phi=t$, we rewrite equations (27)
and (28) and then we invert these equations as $(\phi-n_{s})$ and $(\phi-r)$.
The descriptions and reconstructed values of these two cosmological parameters
are expressed as

$\begin{split}n_{s}&=1+\frac{24Yd\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{3}{-1+d}}\bigg{[}Y-2Yd-6\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{]}^{2}}{(3+Yd\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{-1+d}})^{2}}\\\
&+\frac{Y\Big{(}(X+Y)^{1-d}-Y(d-1)\phi\Big{)}^{\frac{1}{-1+d}}\bigg{(}8-26d+Y(1+4d-6d^{2})\Big{(}(X+Y)^{1-d}-Y(d-1)\phi\Big{)}^{\frac{1}{d-1}}\bigg{)}}{2\bigg{(}3+Yd\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{d-1}}\bigg{)}},\end{split}$
(33)

and

$r=-\frac{4Yd\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{3}{-1+d}}\Big{(}Y-2Yd-6\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\Big{)}^{2}}{\Big{(}3+Yb\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{-1+d}}\Big{)}^{2}}.$
(34)

Here, we examined one of the inflation solutions of the Hubble parameter
$H(N)$ and studied the mimetic $f(R,T)$ inflationary model from the swampland
conjectures point of view. We investigated this inflationary scenario’s
compatibility or incompatibility with the latest observable data, i.e., Planck
data and BICEP2/Keck Array data. For this purpose, the limitations of the
swampland conjecture were determined in terms of two cosmological parameters.
$n_{s}$ and $r$ are obtained for all the considered inflation models according
to the selected values for free parameters. They were consistent with Planck’s
observable data [79]). We obtained these two cosmological parameters in terms
of $\phi$. By placing each of these structures as $(\phi-n_{s})$ and
$(\phi-r)$ in equations (63) and (64), new structures are formed, i.e.,
$(C_{1}-n_{s})$, $(C_{2}-n_{s})$, $(C_{1}-r)$ and $(C_{2}-r)$. Then, we draw
some figures for these structures with respect to Planck’s observable data and
validated values of free parameters for (model III). The plots of $C_{1}$ and
$C_{2}$ in terms of $n_{s}$ are depicted in FIGS. 1 and 2.

Figure 1: The plot of $C_{1}$ and $C_{2}$ in terms of $n_{s}$

Figure 2: The plot of $C_{1}$ and $C_{2}$ in terms of $r$

As shown in figure (1), the restriction of swampland conjectures is determined
in terms of the scalar spectral index, which used the validated values for the
free parameters for this inflation model obtained in the previous section. As
you can see, the acceptable values are specified for the scalar spectral index
and the constant coefficient of swampland conjectures ($C_{1}$) and ($C_{2}$).
In literature, the constants swampland conjecture, ($C_{1}$) and ($C_{2}$) are
the positive value of unit order. You can see these issues in the figures. The
validated values for the ($C_{1}$) and ($C_{2}$) are shown in the figures.
($C_{2}$) values is less than the ($C_{1}$). Similarly, the swampland
conjecture’s limitations for this inflation model in terms of the tensor-to-
scalar ratio are also shown in figure (2). The validated values for this
cosmological parameter and the constant coefficients of the swampland
conjecture are well specified. We also benefited from the accepted values of
free parameters for this inflation model. Other restrictions can also be
considered below. Another type of constraint is the study of changes in two
cosmological parameters, i.e., the tensor-to-scalar ratio in terms of the
scalar spectral index with respect to mimetic $f(R,T)$ gravity.

Figure 3: The plot of $r$ in term of $n_{s}$

Another limitation is shown in figure (3) about the changes in the two
cosmological parameters, namely the tensor-to-scalar ratio in terms of the
scalar spectral index. The allowable range of both parameters is well
specified in this plot. The critical point is to examine these changes
according to the validated values for the mentioned inflation model and
compare them with observable data. One can discuss all these limitations
concerning other mimetic $f(R,T)$ gravity and other models as $f(G)$ gravity
and compare the results of these studies. It can lead to exciting developments
and even determine which theories are more concrete and more in line with
observable data. Also, to confirm our arguments about the swampland conjecture
and the calculations performed, we consider the observable data such as $r$,
$n_{s}$,$C_{1}$, $C_{2}$ and the constant parameters that are determined for
each model in this paper. We study new restrictions by forming a function as
$f\geq C_{1}^{2}C_{2}^{2}$ and determine the allowable range of the swampland
component, and specify the $C_{i}s$ that can be acceptable for the model.

Figure 4: The resulting constraints for $C_{1}$ and $C_{2}$ with respect to
$r$, and $n_{s}$ and constant parameters. According to the calculations
performed in the text, the allowable range for this model pick for the
vertical axis contains values smaller than $1.26911\times 10^{-6}$ in terms of
a permissible range for the horizontal axis, i.e., the interval [0, 1]. This
allowable area is determined as a small window in the figure4

In general, as shown in figure 4, the allowable range for the swampland
conjecture is determined according to the observable data $r$, $n_{s}$ and the
constant parameters. That is, it can be concluded that $C_{i}\leq
1.26911\times 10^{-6}$(the allowable range for these calculations is in a
small window in figure 4). can be the allowable values for this model, which
can be another confirmation of the alignment of the model about the swamp
conjectures.

###  5.1 further refining swampland conjecture

According to the explanations we provided about recent conjectures and the
equations mentioned in the previous section, we can use further refining
swampland conjecture conditions to examine any effective low-energy theory of
gravity concerning $M_{p}=1$. [61, 82]. A new conjecture that a combination of
two relations in equation (31) called further refining swampland conjecture
was introduced by Andriot, and Roupec, which is expressed as.[60].

$\bigg{(}M_{p}\frac{\nabla
V}{V}\bigg{)}^{q}-aM_{p}^{2}\frac{min\nabla_{j}\nabla_{j}V}{V}\geq b\hskip
12.0pta+b=1,\hskip 12.0pta,b>0,\hskip 12.0ptq>2.$

The above equations can also be rewritten in terms of slow-roll parameters in
the following form.

$\big{(}2\epsilon_{V}\big{)}^{q/2}-a\eta_{V}\geq b.$

In connection with the recent conjecture, some works have been done in the
literature, which you can see for more review in [61, 82]. With respect to the
above equations, we will have,

$(F_{1})^{\frac{q}{2}}-aF_{2}\geq b,$ (35)

where the parameters $a,b$ and $q$ are free parameters. Also $F_{1}$ and
$F_{2}$ are defined as follows

$F_{1}=\frac{|\textrm{d}V(\phi)/\textrm{d}\phi|}{V(\phi)}=\sqrt{2\epsilon_{V}}\hskip
42.67912pt,F_{2}=\frac{\textrm{d}^{2}V(\phi)/\textrm{d}\phi^{2}}{V(\phi)}=\eta_{V}.$
(36)

Now, we rewrite $F_{1}$ and $F_{2}$ in the slow-roll region according to the
parameters $r$ and $n_{s}$, the tensor-to-scalar ratio, and the spectral index
of the primordial curvature power spectrum, respectively.

$F_{1}=\sqrt{2\epsilon_{V}}=\sqrt{\frac{r}{8}},\hskip
42.67912ptF_{2}=\eta_{V}=\frac{1}{2}(n_{s}-1+\frac{3r}{8}).$ (37)

Here, we examine whether these three inflation models within the mimetic f(R;
T) gravity structure can satisfy further refining swampland dS conjecture.

####  5.1.1 Model I

For the first model The scalar spectral index and the tensor-to-scalar ratio,
are obtained $n_{s}=0.966221$ and $r=0.000152005$. By placing these values in
Equation (37), we find

$F_{1}=\sqrt{2\epsilon_{V}}=\sqrt{\frac{r}{8}}=0.00436,\hskip
42.67912ptF_{2}=\eta_{V}=\frac{1}{2}(n_{s}-1+\frac{3r}{8})=-0.01686.$ (38)

Considering the refined swampland conjecture (31), we find

$C_{1}\leq 0.00436,\hskip 42.67912ptC_{2}\leq 0.01686,$ (39)

since $C_{1}$ and $C_{2}$ are not unit orders
($C_{1}=C_{2}\neq\mathcal{O}(1)$), they contradict the refined swampland
conjecture. We now examine the first model by further refining the swampland
conjecture. By placing the equation (38) into (35), we obtained

$(0.00436)^{q}+0.01686a\geq 1-a\hskip 28.45274pt,or\hskip
28.45274pt(0.00436)^{q}\geq 1-1.01686a.$ (40)

We can find $a$ to satisfy the condition

$\frac{1}{1.01686}[1-(0.00436)^{q}]\leq a<1,\hskip 28.45274ptq>2,$ (41)

therefore, further refining the swampland conjecture can be satisfying.
Equation (40) holds for all values of $q>2$ with respect to
$a<\frac{1}{1.01686}=0.98342$. We can give an example for $a,b,q$ variables
that met for this model. By selecting $q=2.1$, and using relations (40) and
(41) we have $0.98340\leq a<0.98342$ and by investigating the $a=0.98341$, we
obtain $b=1-a=0.01659$.

####  5.1.2 Model II

For the second model, the scalar spectral index and the tensor-to-scalar
ratio, are obtained $n_{s}=0.965577$ and $r=0.0155542$. By placing these
values in equation (37), we find

$F_{1}=\sqrt{2\epsilon_{V}}=\sqrt{\frac{r}{8}}=0.044094,\hskip
42.67912ptF_{2}=\eta_{V}=\frac{1}{2}(n_{s}-1+\frac{3r}{8})=-0.014295.$ (42)

Considering the refined swampland conjecture (31), we find

$C_{1}\leq 0.044094,\hskip 42.67912ptC_{2}\leq 0.014295,$ (43)

$C_{1}$ and $C_{2}$ are not constants of the order of $\mathcal{O}(1)$, so
they contradict the refined swampland conjecture. We now examine model II by
the further refining swampland conjecture. By placing the equation (42) into
(35), we obtained

$(0.044094)^{q}+0.014295a\geq 1-a\hskip 28.45274ptor\hskip
28.45274pt(0.00436)^{q}\geq 1-1.014295a.$ (44)

We can find $a$ to satisfy the recent further conjecture

$\frac{1}{1.014295}[1-(0.044094)^{q}]\leq a<1,\hskip 28.45274ptq>2.$ (45)

Like the previous model, further refining swampland conjecture can be
satisfying met for model II. Equation (44) holds for all values of $q>2$ with
respect to $a<\frac{1}{1.014295}=0.985906$. We can give an example for $a,b,q$
variables that met for this model. By selecting $q=2.1$, and using relations
(44) and (45) we have $0.984503\leq a<0.985906$. By choosing the $a=0.985013$,
we will have $b=1-a=0.014987$.

####  5.1.3 Model III

We follow the similar route, so with respect to $n_{s}=0.964569$, $r=0.06836$
and by placing these values in equation (37), we find

$F_{1}=\sqrt{2\epsilon_{V}}=\sqrt{\frac{r}{8}}=0.092439,\hskip
42.67912ptF_{2}=\eta_{V}=\frac{1}{2}(n_{s}-1+\frac{3r}{8})=-0.004898.$ (46)

So, we obtain

$C_{1}\leq 0.092439,\hskip 42.67912ptC_{2}\leq 0.004898,$ (47)

where $C_{1}=C_{2}\neq\mathcal{O}(1)$, so they are in contradiction with the
refined swampland conjecture. We now examine the model by the further refining
swampland conjecture. By placing the equation (46) into (35), we obtained

$(0.092439)^{q}+0.004898a\geq 1-a\hskip 42.67912ptor\hskip
42.67912pt(0.092439)^{q}\geq 1-1.004898a.$ (48)

To satisfy the conjecture we obtain the $a$

$\frac{1}{1.004898}(1-(0.092439)^{q})\leq a<1,\hskip 28.45274ptq>2,$ (49)

here, further refining swampland conjecture can be satisfying. Equation (48)
holds for all values of $q>2$ with respect to $a<\frac{1}{1.01686}=0.995126$.
We can give an example for the $a,b,q$ variables. By selecting $q=2.1$, and
using relations (48) and (49) we have $0.988424\leq a<0.995126$. Also by
obtaining the $a=0.992156$, we will have the $b=1-a=0.007844$.

##  6 Stability

Here, we check the stability of the model for one case of mimetic gravity,
namely, model III. However, following the same analysis, one can check the
stability of the rest two cases as well. In general, there are different ways
to study the stability of a model. In cosmology, the stability of different
methods is examined, e.g., Refs. [17, 80, 81]. However, we use the speed of
sound method to evaluate the stability of our considered model. Generally,
stability studied by the method of sound speed is described by

$C_{s}^{2}=\frac{\textrm{d}P}{\textrm{d}\rho}>0.$ (50)

It means the stability conditions occur when $C_{s}^{2}$ is positive. To plot
the diagram, we need to obtain the energy density and pressure so that we will
have:

$\begin{split}\rho=&\frac{\bigg{(}X+Y\Big{(}(X+Y)^{1-d}+Yt(1-d)\Big{)}^{\frac{d}{1-d}}\bigg{)}^{2d}}{1+4\delta+3\delta^{2}}\bigg{(}3+3\delta-\frac{Yd\delta\Big{(}(X+Y)^{1-d}-Yt(-1+d)\Big{)}^{\frac{1-d}{d}}}{(X+Y)}\bigg{)},\end{split}$
(51)

and

$\begin{split}P=&-\frac{1}{-\delta^{2}+(1+2\delta)^{2}}\bigg{[}(3+3\delta)\bigg{(}X+Y\Big{(}\big{(}(X+Y)^{1-d}-Yt(-1+d)\big{)}^{\frac{1}{1-d}}\Big{)}^{d}\bigg{)}^{2d}\\\
&+Y(2+3\delta)\bigg{(}X+Y\Big{(}\big{(}(X+Y)^{1-d}-Yt(-1+d)\big{)}^{\frac{1}{1-d}}\Big{)}^{d}\bigg{)}^{-1+2d}d\bigg{]}.\end{split}$
(52)

Figure 5: square of sound speed versus cosmic time.

As shown in figure 5, the stability of our model is evident throughout the
evolution of the universe, and this stability is apparent at all times. In the
stability figures, the unit of cosmic time is usually referred to as billion
years. In fact, if we consider the cosmic time coordinates as billion years,
we find that our model as a whole is a stable model at all times. For a
physically viable model, the sound speed should be positive and less than the
speed of light $(C=1)$. Thus the condition $0\leq C_{s}^{2}\leq 1$ validates
the acceptability of the model. Figure 5 shows that a specific constant value
is obtained for large values of t

##  7 Conclusion

The mimetic $f(R,T)$ gravity model is examined from the swampland conjecture
point of view. For this purpose, we introduced several inflation solutions of
the Hubble parameter such as $(XN^{a}+YN\exp(bN))^{d}$, $(Xc^{N}+YN^{a})^{d}$
and $(X+YN)^{d}$ in $f=R+\delta T$ gravity model. Then, we calculated
quantities such as potential and Lagrange multiplier, slow-roll, and other
cosmological parameters. We also studied the scalar spectrum index and the
tensor to the scalar ratio for each model separately. Also, by adjusting the
free parameters of each model, acceptable values for these two crucial
cosmological parameters, namely $n_{s}$ and $r$, were determined, which also
corresponded to the latest observable data such as Planck data and BICEP2/Keck
Array data. Then, we selected the third model to examine the compatibility of
the mimetic f (R, T) inflationary model with the recently introduced refined
swampland dS conjecture. We calculated the refined swampland dS conjecture
concerning the potential and its first and second-order derivatives. Since the
study of each of the first and second components of the swampland conjecture
viz $C_{1}$ and $C_{2}$ in terms of two important cosmological parameters,
namely $n_{s}$ and $r$, can be significant, so according to the mentioned
equations, we created structures such as $C_{1,2}-n_{s}$ and $C_{1,2}-r$. By
plotting some figures, we determined the allowable range related to each of
these cosmological parameters and the components of the swampland conjecture.
We also studied a new constraint that is formed according to the combination
of two components of the swampland as a function $f\geq C_{1}^{2}C_{2}^{2}$
and we determined the allowable range of the swampland component that
specifies the $C_{i}$ that can be acceptable for the model. Thus the virtual
range is specified as $C_{i}\leq 1.26911\times 10^{-6}$. The allowable range
for these calculation of model III was in a small window in figure 4. Then, we
challenged the future refining swampland conjecture with these models. We
determined all models contradict with swampland ds conjuncture since their
coefficients are not unite order,i.e., $C_{1}=C_{2}\neq\mathcal{O}(1)$. Then
by adjusting the free parameters of recent farther refining swampland
conjecture viz $a,b,q$, we showed compatibility of all models with it. In
addition to the above calculations, we also had to check the stability of the
model. Therefore, we studied the stability of model III for the mimetic f(R,
T) gravity. We concluded that the stability of our model is evident throughout
the evolution of the universe, and this stability is apparent at all times.
Given the above concepts, such a construction of inflation models can be
challenged with other swampland conjectures such as TCC. one can study
additional limitations, which could be an idea for a more in-depth study of
future work. Of course, this will be interesting to study perturbations of
these models in full detail, which is the subject of future work.

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##  8 Appendix A: The slow-roll and spectral parameters (model I)

By placing equation (18) into the equations (15) and (16), we will have

$\begin{split}\epsilon&=-\bigg{\\{}\bigg{(}Nd(YN^{a}+Xc^{N})^{1-d}(YN^{a}+Xc^{N})^{-3d}\bigg{(}d(YN^{a}+Xc^{N})^{-2+2d}(YN^{-1+a}a+Xc^{N}\log(c))^{2}\\\
&+(YN^{a}+Xc^{N})^{-2+d}\bigg{[}(-1+d)(YN^{-1+a}a+Xc^{N}\log(c))^{2}\\\
&+(YN^{a}+Xc^{N})\Big{(}YN^{-2+a}a(-1+a)+Xc^{N}\log(c)^{2}\Big{)}\bigg{]}(YN^{a}+Xc^{N})^{d}\\\
&+(YN^{a}+Xc^{N})^{-1+d}(YN^{-1+a}a+Xc^{N}\log(c))(YN^{a}+Xc^{N})^{d}\bigg{)}^{2}\bigg{)}\bigg{/}\bigg{(}4(YN^{a}a+XNc^{N}\log(c))\\\
&\bigg{[}3+d(YN^{a}+Xc^{N})^{-1+d}(YN^{-1+a}a+Xc^{N}\log(c))(YN^{a}+Xc^{N})^{-d}\bigg{]}^{2}\bigg{)}\bigg{\\}},\end{split}$
(53)

$\begin{split}\eta=&-\bigg{\\{}\bigg{(}-\bigg{[}\bigg{(}(YN^{a}+Xc^{N})^{-4+2d}(YN^{a}+Xc^{N})^{2-2d}\bigg{[}(-1+d)(YN^{-1+a}a+Xc^{N}\log(c))^{2}\\\
&+(YN^{a}+Xc^{N})\Big{(}YN^{-2+a}(-1+a)a+Xc^{N}\log(d)^{2}\Big{)}\bigg{]}^{2}\bigg{)}\Big{/}\Big{(}2\big{(}YN^{-1+a}a+Xc^{N}\log(c)\big{)}^{2}\Big{)}\bigg{]}\\\
&+4(YN^{a}+Xc^{N})^{-2+d}(YN^{a}+Xc^{N})^{1-d}\Big{/}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}\\\
&\times\bigg{[}(-1+d)\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}+(YN^{a}+Xc^{N})\Big{(}YN^{-2+a}(-1+a)a+Xc^{N}\log(c)^{2}\Big{)}\bigg{]}\\\
&+\frac{1}{2}d^{2}(YN^{a}+Xc^{N})^{-2+2d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}(YN^{a}+Xc^{N})^{-2d}+3d(YN^{a}+Xc^{N})^{-2d}\\\
&\times\bigg{[}(-1+d)\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}+(YN^{a}+Xc^{N})\Big{(}YN^{-2+a}(-1+a)a+Xc^{N}\log(c)^{2}\Big{)}\bigg{]}\\\
&\times(YN^{a}+Xc^{N})^{-d}+9d(YN^{a}+Xc^{N})^{-1+d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{-d}\bigg{)}\bigg{/}\\\
&2\bigg{[}3+d(YN^{a}+Xc^{N})^{-1+d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{-d}\bigg{]}\bigg{\\}}.\end{split}$
(54)

With respect to equations (19), (20), and (17), the scalar spectral index and
tensor-to-scalar ratio are calculated in the following form.

$\begin{split}n_{s}-1=&-2\bigg{\\{}\bigg{(}-\bigg{[}\bigg{(}(YN^{a}+Xc^{N})^{-4+2d}(YN^{a}+Xc^{N})^{2-2d}\bigg{[}(-1+d)\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}\\\
&+(YN^{a}+Xc^{N})\Big{(}YN^{-2+a}(-1+a)a+Xc^{N}\log(c)^{2}\Big{)}\bigg{]}^{2}\bigg{)}\Big{/}\Big{(}2\big{(}YN^{-1+a}a+Xc^{N}\log(c)\big{)}^{2}\Big{)}\bigg{]}\\\
&+4(YN^{a}+Xc^{N})^{-2+d}(YN^{a}+Xc^{N})^{1-d}\Big{/}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}\\\
&\times\bigg{[}(-1+d)\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}+(YN^{a}+Xc^{N})\Big{(}YN^{-2+a}(-1+a)a+Xc^{N}\log(c)^{2}\Big{)}\bigg{]}\\\
&+\frac{1}{2}d^{2}(YN^{a}+Xc^{N})^{-2+2d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}(YN^{a}+Xc^{N})^{-2d}+3d(YN^{a}+Xc^{N})^{-2d}\\\
&\times\bigg{[}(-1+d)\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}+(YN^{a}+Xc^{N})\Big{(}YN^{-2+a}(-1+a)a+Xc^{N}\log(c)^{2}\Big{)}\bigg{]}\\\
&\times(YN^{a}+Xc^{N})^{-d}+9d(YN^{a}+Xc^{N})^{-1+d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{-d}\bigg{)}\Big{/}\\\
&\bigg{[}2\bigg{(}3+d(YN^{a}+Xc^{N})^{-1+d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{-d}\bigg{)}\bigg{]}\bigg{\\}}\\\
&+6\bigg{\\{}\bigg{(}Nd(YN^{a}+Xc^{N})^{1-d}(YN^{a}+Xc^{N})^{-3d}\bigg{(}d(YN^{a}+Xc^{N})^{-2+2d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}\\\
&+(YN^{a}+Xc^{N})^{-2+d}\bigg{[}(-1+d)\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}+(YN^{a}+Xc^{N})\\\
&\times\Big{(}YN^{-2+a}a(-1+a)+Xc^{N}\log(c)^{2}\Big{)}\bigg{]}(YN^{a}+Xc^{N})^{d}+(YN^{a}+Xc^{N})^{-1+d}\\\
&\times\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{d}\bigg{)}^{2}\bigg{)}\bigg{[}4\Big{(}YN^{a}a+XNc^{N}\log(c)\Big{)}^{-1}/\\\
&\bigg{(}3+d(YN^{a}+Xc^{N})^{-1+d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{-d}\bigg{)}^{2}\bigg{]}\bigg{\\}},\end{split}$
(55)

$\begin{split}r=&-16\bigg{\\{}\bigg{(}Nd(YN^{a}+Xc^{N})^{1-d}(YN^{a}+Xc^{N})^{-3d}\bigg{[}d(YN^{a}+Xc^{N})^{-2+2d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}\\\
&+(YN^{a}+Xc^{N})^{-2+d}\bigg{(}(-1+d)\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}^{2}+(YN^{a}+Xc^{N})\\\
&\times\Big{(}YN^{-2+a}a(-1+a)+Xc^{N}\log(c)^{2}\Big{)}\bigg{)}(YN^{a}+Xc^{N})^{d}+(YN^{a}+Xc^{N})^{-1+d}\\\
&\times\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{d}\bigg{]}^{2}\bigg{)}\bigg{[}4\Big{(}YN^{a}a+XNc^{N}\log(c)\Big{)}^{-1}\Big{/}\\\
&\bigg{(}3+d(YN^{a}+Xc^{N})^{-1+d}\Big{(}YN^{-1+a}a+Xc^{N}\log(c)\Big{)}(YN^{a}+Xc^{N})^{-d}\bigg{)}^{2}\bigg{]}\bigg{\\}}.\end{split}$
(56)

##  9 Appendix B: The slow-roll and spectral parameters (model II)

With respect to equation (21), (15) and (16), one can obtain

$\begin{split}\epsilon=&-d\bigg{[}(X^{2}N^{2a})a(-1+6N+2ad)+Y^{2}\textrm{e}^{2Nb}N^{2}\Big{(}-1+6N+6N^{2}b+2(1+Nb)^{2}d\Big{)}XY\textrm{e}^{Nb}N^{1+a}\\\
&\times\bigg{[}N^{2}b(6+b)+a(-3+a+4d)+2N\Big{(}3+b+a(3-b+2bd)\Big{)}\bigg{]}\bigg{]}^{2}\Big{/}4N(Y\textrm{e}^{Nb}N+XN^{a})\\\
&\times\bigg{[}XN^{a}a+Y\textrm{e}^{Nb}N(1+Nb)(XN^{a}(3N+ad))+YN\textrm{e}^{Nb}(d+N(3+bd))\bigg{]}^{2},\end{split}$
(57)

$\begin{split}\eta=&-\frac{1}{4N^{2}(YN\textrm{e}^{Nb}+XN^{a})^{2}\bigg{(}3+\frac{ad}{N}+\frac{(1-a+Nb)d}{N\textrm{e}^{XN^{a}}}\bigg{)}}\bigg{\\{}18N(YN\textrm{e}^{Nb}+XN^{a})\\\
&\times\Big{(}YNe^{Nb}(1+Nb)+XN^{a}a\Big{)}d+\Big{(}YN\textrm{e}^{Nb}(1+Nb)+XN^{a}a\Big{)}^{2}d^{2}\\\
&+8N(YNe^{Nb}+XN^{a})\bigg{(}X^{2}N^{2a}a(-1+ad)+Y^{2}\textrm{e}^{2Nb}N^{2}(-1+(1+Nb)^{2}d)\bigg{)}\\\
&+XY\textrm{e}^{Nb}N^{1+a}\Big{(}a^{2}+Nb(2+Nb)+a(-3+2Nb(-1+d)+2d)\Big{)}\bigg{\\}}\bigg{/}\\\
&\bigg{\\{}XN^{a}a+YNe^{Nb}(1+Nb)+6d\bigg{[}X^{2}N^{2a}a(-1+ad)+Y^{2}N^{2}\textit{e}^{2Nb}(-1+(1+Nb)^{2}d)\bigg{]}\\\
&+XY\textit{e}^{Nb}N^{1+a}\Big{(}a^{2}+Nb(2+Nb)+a(-3+2Nb(-1+d)+2d)\Big{)}-\bigg{[}X^{2}N^{2a}a(-1+ad)\\\
&+Y^{2}N^{2}\textit{e}^{2Nb}(-1+(1+Nb)^{2}d)+XY\textrm{e}^{Nb}N^{1+a}\bigg{(}a^{2}+Nb(2+Nb)\\\
&+a\Big{(}-3+2Nb(-1+d)+2d\Big{)}\bigg{)}\bigg{]}^{2}\Big{/}\Big{(}XN^{a}a+YN\textrm{e}^{Nb}(1+Nb)\Big{)}^{2}\bigg{\\}}.\end{split}$
(58)

Also, $n_{s}$ and r, with respect to the above equations, calculate in the
following form

$\begin{split}n_{s}-1=&-\frac{1}{2N^{2}(YN\textrm{e}^{Nb}+XN^{a})^{2}\Big{(}3+\frac{ad}{N}+\frac{(1-a+Nb)d}{N\textrm{e}^{XN^{a}}}\Big{)}}\bigg{(}18N(YN\textrm{e}^{Nb}+XN^{a})\\\
&\times\Big{(}YN\textrm{e}^{Nb}(1+Nb)+XN^{a}a\Big{)}d+\Big{(}YN\textrm{e}^{Nb}(1+Nb)+XN^{a}a\Big{)}^{2}d^{2}\\\
&+\bigg{[}8N(YN\textrm{e}^{Nb}+XN^{a})\bigg{(}X^{2}N^{2a}a(-1+ad)+Y^{2}\textrm{e}^{2Nb}N^{2}(-1+(1+Nb)^{2}d)\\\
&+XY\textrm{e}^{Nb}N^{1+a}\Big{(}a^{2}+NY(2+Nb)+a\big{(}-3+2NY(-1+d)+2d\big{)}\Big{)}\bigg{)}\bigg{]}\Big{/}\\\
&\Big{(}XN^{a}a+YN\textrm{e}^{Nb}(1+Nb)\Big{)}+6d\bigg{(}X^{2}N^{2a}a(-1+ad)+Y^{2}N^{2}\textit{e}^{2Nb}(-1+(1+Nb)^{2}d)\\\
&+XY\textrm{e}^{Nb}N^{1+a}\Big{(}a^{2}+NY(2+Nb)+a(-3+2Nb(-1+d)+2d)\Big{)}\bigg{)}-\bigg{[}X^{2}N^{2a}a(-1+ad)\\\
&+Y^{2}N^{2}\textrm{e}^{2Nb}(-1+(1+Nb)^{2}d)+XY\textrm{e}^{Nb}N^{1+a}\bigg{(}a^{2}+NY(2+Nb)\\\
&+a(-3+2NY(-1+d)+2d)\bigg{)}\bigg{]}^{2}\Big{/}\Big{(}XN^{a}a+YN\textrm{e}^{Nb}(1+Nb)\Big{)}^{2}\bigg{)}\\\
&+6\bigg{(}\bigg{[}d\bigg{[}X^{2}N^{2a}a(-1+6N+2ad)\\\
&+Y^{2}\textrm{e}^{2Nb}N^{2}\Big{(}-1+6N+6N^{2}Y+2(1+Nb)^{2}d\Big{)}XY\textrm{e}^{Nb}N^{1+a}\bigg{(}N^{2}Y(6+b)\\\
&+a(-3+a+4d)+2N\Big{(}3+b+a(3-b+2bd)\Big{)}\bigg{)}\big{]}^{2}\bigg{]}\Big{/}\bigg{[}4N(Y\textrm{e}^{Nb}N+XN^{a})\\\
&\Big{(}XN^{a}a+Y\textrm{e}^{Nb}N(1+Nb)\Big{)}\Big{(}XN^{a}(3N+ad)+YN\textrm{e}^{Nb}(d+N(3+bd))\Big{)}^{2}\bigg{]}\bigg{)},\end{split}$
(59)

$\begin{split}r=&-4\bigg{(}d\bigg{(}X^{2}N^{2a}a(-1+6N+2ad)+Y^{2}\textrm{e}^{2Nb}N^{2}\Big{(}-1+6N+6N^{2}b+2(1+Nb)^{2}d\Big{)}XY\textrm{e}^{Nb}N^{1+a}\\\
&\bigg{[}N^{2}b(6+b)+a(-3+a+4d)+2N\Big{(}3+b+a(3-b+2bd)\Big{)}\bigg{]}\bigg{)}^{2}\bigg{)}\bigg{/}\bigg{[}N(Y\textrm{e}^{Nb}N+XN^{a})\\\
&\Big{(}XN^{a}a+Y\textrm{e}^{Nb}N(1+Nb)\Big{)}\Big{(}XN^{a}(3N+ad)+YN\textrm{e}^{Nb}\big{(}d+N(3+bd)\big{)}\Big{)}^{2}\bigg{]}.\end{split}$
(60)

##  10 Appendix C: The swampland conjecture structure

According to the potential of equation (32), its first and second derivatives
are calculated as follows.

$\begin{split}V^{\prime}(\phi)=&-\bigg{[}Y(X+Y)^{d}d\Big{(}\big{(}(X+Y)^{1-d}-Y(-1+d)\phi\big{)}^{\frac{1}{1-d}}\Big{)}^{-1+2d}\bigg{(}Y\Big{(}-4+\delta\big{(}-5+\delta(11+6\delta)\big{)}\Big{)}\\\
&(-1+2d)+18(-1+\delta)(1+\delta)(1+2\delta)\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{)}]\bigg{/}\\\
&\bigg{[}(1+\delta)(1+3\delta)\Big{(}X+Y-Y(X+Y)^{d}(-1+d)\phi\Big{)}\bigg{]},\end{split}$
(61)

and

$\begin{split}V^{\prime\prime}(\phi)=&-\bigg{[}Y^{2}(X+Y)^{2d}d\Big{(}\big{(}(X+Y)^{1-d}-Y(-1+d)\phi\big{)}^{\frac{1}{1-d}}\Big{)}^{-1+2d}\\\
&\bigg{(}Y\Big{(}-4+\delta\big{(}-5+\delta(11+6\delta)\big{)}\Big{)}(2+d(-7+6d))+18(-1+\delta)(1+2\delta)(1+\delta)\\\
&\times(-1+3d)\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{)}\bigg{]}\bigg{/}\\\
&\bigg{[}(1+\delta)(1+3\delta)\Big{(}X+Y-Y(X+Y)^{d}(-1+d)\phi\Big{)}^{2}\bigg{]},\end{split}$
(62)

by placing equations (32), (C1), and (C2) in swampland dS conjectures in
equation (31), we will have.

$\begin{split}-&\bigg{[}\bigg{(}Y(X+Y)^{d}d\Big{(}\big{(}(X+Y)^{1-d}-Y(-1+d)\phi\big{)}^{\frac{1}{1-d}}\Big{)}^{-1+2d}\bigg{(}Y\Big{(}-4+\delta\big{(}-5+\delta(11+6\delta)\big{)}\Big{)}\\\
&(-1+2d)+18(-1+\delta)(1+\delta)(1+2\delta)\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{)}\bigg{)}\Big{/}\\\
&\bigg{[}(1+\delta)(1+3\delta)\Big{(}X+Y-Y(X+Y)^{d}(-1+d)\phi\Big{)}\bigg{]}\bigg{]}\Big{/}\\\
&-\delta\rho+\frac{1}{(1+\delta)(1+3\delta)}\bigg{(}\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi^{\frac{1}{1-d}}\Big{)}^{-1+2d}\\\
&\times\bigg{[}Y\Big{(}4+\delta\big{(}5-\delta(11+6\delta)\big{)}\Big{)}d-9(-1+\delta)(1+\delta)(1+2\delta)\\\
&\times\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{]}\bigg{)}>C_{1},\end{split}$
(63)

$\begin{split}-&\bigg{[}\bigg{(}Y^{2}(X+Y)^{2d}d\Big{(}\big{(}(X+Y)^{1-d}-Y(-1+d)\phi\big{)}^{\frac{1}{1-d}}\Big{)}^{-1+2d}\\\
&\bigg{[}Y\Big{(}-4+\delta\big{(}-5+\delta(11+6\delta)\big{)}\Big{)}\big{(}2+d(-7+6d)\big{)}+18(-1+\delta)(1+2\delta)(1+\delta)\\\
&(-1+3d)\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{]}\bigg{)}\Big{/}\\\
&\bigg{(}(1+\delta)(1+3\delta)\Big{(}X+Y-Y(X+Y)^{d}(-1+d)\phi\Big{)}^{2}\bigg{)}\bigg{]}\bigg{/}\\\
&-\delta\rho+\frac{1}{(1+\delta)(1+3\delta)}\bigg{(}\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi^{\frac{1}{1-d}}\Big{)}^{-1+2d}\\\
&\times\bigg{[}Y\Big{(}4+\delta\big{(}5-\delta(11+6\delta)\big{)}\Big{)}d-9(-1+\delta)(1+\delta)(1+2\delta)\\\
&\times\Big{(}(X+Y)^{1-d}-Y(-1+d)\phi\Big{)}^{\frac{1}{1-d}}\bigg{]}\bigg{)}<-C_{2}\end{split}$
(64)


# A characterization of quasipositive two-bridge knots

Burak Ozbagci
(with an appendix by Stepan Orevkov)  Institut de Mathématiques de Toulouse,
Université Paul Sabatier, 118 route de Narbonne, 31062 Toulouse, France
Steklov Mathematical Institute, Gubkina 8, Moscow, Russia
<EMAIL_ADDRESS>Department of Mathematics, Koç
University, 34450, Istanbul,Turkey<EMAIL_ADDRESS>
###### Abstract.

We prove a simple necessary and sufficient condition for a two-bridge knot
$K(p,q)$ to be quasipositive, based on the continued fraction expansion of
$p/q$. As an application, coupled with some classification results in contact
and symplectic topology, we give a new proof of the fact that smoothly slice
two-bridge knots are non-quasipositive. Another proof of this fact using
methods within the scope of knot theory is presented in the Appendix.

##  1\. Introduction

Notions of quasipositivity and strong quasipositivity for links were
introduced and explored by Rudolph in a series of papers (see, for example,
[28, 29, 30, 31, 32, 33]). Let $\sigma_{1},\ldots,\sigma_{n-1}$ denote the
standard generators of the braid group $B_{n}$, and let
$\sigma_{i,j}=(\sigma_{i}\ldots\sigma_{j-2})(\sigma_{j-1})(\sigma_{i}\ldots\sigma_{j-2})^{-1}.$
Strongly quasipositive links are the links which can be realized as the
closure of braids of the form $\prod_{k=1}^{m}\sigma_{i_{k},j_{k}}$. The
weaker notion of quasipositive link is any link which can be realized as the
closure of a braid of the form $\prod_{k=1}^{m}w_{k}\sigma_{i_{k}}w_{k}^{-1}$,
where $w_{k}\in B_{n}$ for all $1\leq k\leq m$. An oriented link is called
positive if it has a positive diagram, i.e., a diagram in which all crossings
are positive.

Throughout this paper, we assume that $p>q$ are relatively prime positive
integers. The oriented lens space $L(p,q)$ is defined by $-p/q$ surgery on the
unknot in $S^{3}$ and it is well-known that $L(p,q)$ is the double cover of
$S^{3}$ branched along the two-bridge link $K(p,q)$, which we depicted in
Figure 1. When $p$ is odd, $K(p,q)$ is a knot, and otherwise it has two
components.

The negative continued fraction of $p/q$ is defined by

$\frac{p}{q}=[a_{1},a_{2},\ldots,a_{k}]^{-}=a_{1}-\cfrac{1}{a_{2}-\cfrac{1}{\ddots-\cfrac{1}{a_{k}}}}\;,\qquad
a_{i}\geq 2\text{ for all $1\leq i\leq k,$}$

where the coefficients $a_{1},a_{2},\ldots,a_{k}$ are uniquely determined by
$p/q$. We say that the negative continued fraction of $p/q$ is even if $a_{i}$
is even for all $1\leq i\leq k$.

######  Theorem 1.1.

If the two-bridge link $K(p,q)$ is quasipositive then the negative continued
fraction of $p/q$ is even.

If the negative continued fraction of $p/q$ is even, then $pq$ must be even.
Corollary 1.2 immediately follows from this simple observation coupled with
Theorem 1.1.

######  Corollary 1.2.

If $pq$ is odd, then the two-bridge knot $K(p,q)$ is non-quasipositive.

Conversely, we can express Tanaka’s quasipositivity criterion [34, Proposition
3.1] for two-bridge knots in terms of negative continued fractions as follows:

######  Proposition 1.3 (Tanaka [34]).

The two-bridge knot $K(p,q)$ is strongly quasipositive, provided that the
negative continued fraction of $p/q$ is even.

We would like to point out that Proposition 1.3 cannot possibly hold for
arbitrary two-bridge links. For example, $16/3=[6,2,2]^{-}$ and
$16/13=[2,2,2,2,4]^{-}$, but the two-bridge link $K(16,3)$ and its mirror
$K(16,13)$ cannot both be quasipositive since Hayden [15, Corollary 1.5]
proved that if a link and its mirror are both quasipositive, then the link is
an unlink.

Combining Theorem 1.1 and Proposition 1.3, we have the following
characterization of quasipositive two-bridge knots.

######  Corollary 1.4.

The two-bridge knot $K(p,q)$ is quasipositive if and only if the negative
continued fraction of $p/q$ is even.

######  Remark 1.5.

It is clear that strong quasipositivity implies quasipositivity by definition.
Conversely, Boileau and Rudolph [4, Proposition 3.6 and Corollary 3.7] proved
for a family of alternating arborescent links, including 2-bridge links, that
quasipositivity implies strong quasipositivity. On the other hand, by the work
of Rudolph [32] and Nakamura [24, Lemma 4.1], positive diagrams represent
strongly quasipositive links. Moreover, strongly quasipositive two-bridge
links are positive, since any two-bridge link can be obtained as the boundary
of plumbings of annuli and strong quasipositivity behaves natural with respect
to the plumbing operation [31]. The upshot is that, positivity,
quasipositivity and strong quasipositivity are in fact equivalent for two-
bridge links.

The proof of Theorem 1.1 is based on the work of Plamenevskaya [26], coupled
with the following result on contact topology.

######  Proposition 1.6.

The lens space $L(p,q)$ admits a tight contact structure with trivial first
Chern class if and only if the negative continued fraction of $p/q$ is even.

Recall that a knot in $S^{3}$ is called smoothly slice if it bounds a smooth
properly embedded disk in $B^{4}$. Rudolph [30, Proposition 2] showed that the
only smoothly slice strongly quasipositive knot is the unknot, as a corollary
to the celebrated work of Kronheimer and Mrowka [17]. As we pointed out in
Remark 1.5, quasipositivity and strong quasipositivity are equivalent for a
two-bridge link. Therefore, we conclude that smoothly slice two-bridge knots
are non-quasipositive. We will provide a new proof of this fact as an
application of Corollary 1.4 coupled with the following result on symplectic
topology.

######  Proposition 1.7.

Let $(Y,\xi)$ be the contact double cover of the standard contact $3$-sphere
$(S^{3},\xi_{st})$ branched along the knot $K$ which we assume to be
transverse to $\xi_{st}$. If $K$ is smoothly slice and quasipositive, then
$(Y,\xi)$ admits a rational homology ball Stein filling.

After we finished a first draft of this paper, it came to our attention that
in [25, Remark 2], Orevkov observed that quasipositivity implies (by results
of [25] combined with [7]) strong quasipositivity for two-bridge links, and
for a larger class of links including all alternating Montesinos links—which
does not directly follow from the arguments in [4, Proposition 3.6 and
Corollary 3.7]. Since we convinced him that his result would be of interest to
knot enthusiasts, he kindly agreed to write the details of his [25, Remark 2]
in the Appendix (especially in the two-bridge case where [21] can be used
instead of [7]), which in turn, gives yet another proof of the fact that
smoothly slice two-bridge knots are non-quasipositive.

##  2\. Applications of contact and symplectic topology

We begin with the proof of Proposition 1.6.

###### Proof of Proposition 1.6.

Suppose that $p/q=[a_{1},a_{2},\ldots,a_{k}]^{-}$, where $a_{i}\geq 2$ is even
for all $1\leq i\leq k$. Note that $L(p,q)$ can be obtained by surgery on a
chain of unknots with framings $-a_{1},-a_{2},\ldots,-a_{k}$, respectively.
Since $a_{i}\geq 2$ is even for all $1\leq i\leq k$, we can Legendrian realize
each unknot in the chain, with respect to the standard contact structure, so
that the rotation number of each Legendrian unknot is zero. It follows by [14,
Proposition 2.3] that the first Chern class of the Stein fillable (and hence
tight) contact structure on $L(p,q)$ obtained by Legendrian surgery on the
resulting Legendrian link is zero.

To prove the only if direction, suppose that
$p/q=[a_{1},a_{2},\ldots,a_{k}]^{-}$, where $a_{i}\geq 2$ for all $1\leq i\leq
k$ and $a_{j}$ is odd for some $1\leq j\leq k$. Let $\xi$ be the Stein
fillable contact structure on $L(p,q)$ obtained by Legendrian surgery along an
arbitrary Legendrian realization $\mathcal{L}$ of the chain of unknots with
smooth framings $-a_{1},-a_{2},\ldots,-a_{k}$, respectively. Let
$\overline{\mathcal{L}}$ be the Legendrian link obtained from $\mathcal{L}$ by
taking the mirror image of each component and let $\overline{\xi}$ be the
Stein fillable contact structure on $L(p,q)$ obtained by Legendrian surgery
along $\overline{\mathcal{L}}$. It follows that rotation number of each
component of $\overline{\mathcal{L}}$ is the negative of the rotation number
of the corresponding component of $\mathcal{L}$ and hence $\overline{\xi}$ is
obtained from $\xi$ by reversing the orientation of the contact planes. By
Honda’s classification [16, Theorem 2.1] of tight contact structures on
$L(p,q)$, the contact structures $\xi$ and $\overline{\xi}$ are not isotopic
because of the assumption that $a_{j}$ is odd for some $1\leq j\leq k$. Note
that non-isotopic tight contact structures on $L(p,q)$ are non-homotopic [16,
Proposition 4.24]. This implies that $c_{1}(\xi)\neq 0$ since according to
Gompf [14, Corollary 4.10], an oriented plane field in any closed oriented
$3$-manifold is homotopic to itself with reversed orientation if and only if
it has trivial first Chern class. ∎

We now give a proof of Theorem 1.1, based on Proposition 1.6.

###### Proof of Theorem 1.1.

The double cover of $S^{3}$ branched along $K(p,q)$ is $L(p,q)$. We can make
$K(p,q)$ transverse to the standard contact structure $\xi_{st}$ by isotopy.
Let $(L(p,q),\xi)$ be the contact double cover of $(S^{3},\xi_{st})$ branched
along the transverse link $K(p,q)$. Suppose that $K(p,q)$ is quasipositive. By
the work of Plamenevskaya [26, Proposition 1.4 and Lemma 5.1], we conclude
that $\xi$ is Stein fillable and moreover $c_{1}(\xi)=0$. It follows by
Proposition 1.6 that the negative continued fraction of $p/q$ must be even. ∎

The proof of Proposition 1.3 is essentially obtained by rephrasing Proposition
3.1 in Tanaka’s paper [34], where he uses regular continued fractions to
describe a sufficient condition for a two-bridge knot to be strongly
quasipositive.

###### Proof of Proposition 1.3.

A regular continued fraction of $p/q$ is defined by

$\frac{p}{q}=[c_{1},c_{2},\ldots,c_{2m+1}]=c_{1}+\cfrac{1}{c_{2}+\cfrac{1}{\ddots+\cfrac{1}{c_{2m+1}}}}\;,\qquad
c_{i}>0\text{ for all $1\leq i\leq 2m+1$}.$

Note that there is always a regular continued fraction of $p/q$ of odd length,
as above. To see this, suppose that $p/q$ has a regular continued fraction of
even length, i.e. $\frac{p}{q}=[c_{1},c_{2},\ldots,c_{2m}]$ with $c_{i}>0$. If
$c_{2m}=1$, then

$\frac{p}{q}=[c_{1},c_{2},\ldots,c_{2m}]=[c_{1},c_{2},\ldots,c_{2m-2},1+c_{2m-1}]$

and otherwise,

$\frac{p}{q}=[c_{1},c_{2},\ldots,c_{2m}]=[c_{1},c_{2},\ldots,c_{2m-1},-1+c_{2m},1].$

Using an odd length regular continued fraction
$\frac{p}{q}=[c_{1},c_{2},\ldots,c_{2m+1}]$, we define the two bridge link
$K(p,q)$ as depicted in Figure 1, where the integer inside each box denotes
the signed number of half twists to be inserted.

Figure 1. Two-bridge link $K(p,q)$, where $p/q=[c_{1},c_{2},\ldots,c_{2m+1}].$

In [34], Tanaka uses the notation $\mathcal{C}(c_{1},c_{2},\ldots,c_{2m+1})$,
with each $c_{i}>0$, to describe a two-bridge knot. By comparing [34, Figure
6], for example, with our Figure 1, we see that our definition of $K(p,q)$ is
the mirror image of the one described by Tanaka, but agrees with the one
described by Lisca [18].

According to [34, Proposition 3.1], the two-bridge knot
$\mathcal{C}(c_{1},c_{2},\ldots,c_{2m+1})$, which is the mirror image of our
$K(p,q)$ defined above, is strongly quasipositive as long as $c_{i}$ is even
for each even index $2\leq i\leq 2m$. To express this condition in terms of
negative continued fractions, we just describe how to convert a given regular
continued fraction of any $p/q$ to the negative continued fraction of
$p/(p-q)$ and vice versa in Lemma 2.1. This finishes the proof of Theorem 1.1,
since by Lemma 2.1, we can easily deduce that the negative continued fraction
of $p/(p-q)$ is even if and only if $p/q$ has a regular continued fraction of
odd length where each even indexed coefficient is even. ∎

######  Lemma 2.1.

Suppose that $p/q=[c_{1},c_{2},\ldots,c_{2m+1}]$ with $c_{i}>0$ for all $1\leq
i\leq 2m+1$. Then

$\frac{p}{q}=[1+c_{1},\underbrace{2,\ldots,2}_{c_{2}-1},2+c_{3},\underbrace{2,\ldots,2}_{c_{4}-1},\ldots,2+c_{2m-1},\underbrace{2,\ldots,2}_{c_{2m}-1},1+c_{2m+1}]^{-}$

and

$\frac{p}{p-q}=[\underbrace{2,\ldots,2}_{c_{1}-1},2+c_{2},\underbrace{2,\ldots,2}_{c_{3}-1},2+c_{4},\ldots,\underbrace{2,\ldots,2}_{c_{2m-1}-1},2+c_{2m},\underbrace{2,\ldots,2}_{c_{2m+1}-1}]^{-}$

###### Proof.

The negative continued fraction of $p/q$ can be obtained from a given regular
continued fraction of $p/q$ by a straightforward induction argument, whereas
the negative continued fraction of $p/(p-q)$ is obtained from that of $p/q$ by
the Riemenschneider’s point diagram method [27]. ∎

Using Lemma 2.1, we can rephrase Corollary 1.4 as follows:

######  Corollary 2.2.

The two-bridge knot $K(p,q)$ is quasipositive if and only if $p/(p-q)$ has a
regular continued fraction of odd length where each even indexed coefficient
is even.

Finally, we turn our attention to Proposition 1.7.

###### Proof of Proposition 1.7.

Rudolph [28] showed that quasipositive links arise as the transverse
intersection of $S^{3}\subset\mathbb{C}^{2}$, with a complex curve. Therefore,
the quasipositivity assumption implies that $K$ bounds a complex curve in
$B^{4}$. Since $K$ is assumed to be smoothly slice as well, there is a smooth
disk in $B^{4}$ with boundary $K$. But by the ”local Thom conjecture” [17,
Corollary 1.3] , the complex curve minimizes genus, so the slice disk can be
assumed to be complex. Hence, the analytic double cover of $B^{4}$ equipped
with its standard complex structure, branched along this complex slice disk,
is a rational homology ball Stein filling of $(Y,\xi)$. It is well-known that
the double cover is a rational homology ball and it is in fact Stein by [19,
Theorem 3].∎

##  3\. Smoothly slice two-bridge knots are non-quasipositive

As we pointed in the Introduction, the fact that smoothly slice two-bridge
knots are non-quasipositive follows by combining [30, Proposition 2] with [4,
Proposition 3.6 and Corollary 3.7]. Here we provide an alternate proof based
on contact and symplectic topology.

######  Corollary 3.1.

Smoothly slice two-bridge knots are non-quasipositive.

###### Proof.

Lisca [18] showed that $L(p,q)$ bounds a rational homology ball if and only if
$p/q$ belongs to a certain subset $\mathcal{R}$ of the set of positive
rational numbers. Lisca also showed that, for odd $p$, any two-bridge knot
$K(p,q)$ is smoothly slice if and only if $p/q\in\mathcal{R}$. By definition,
the set $\mathcal{R}$ contains a subset, denoted by $\mathcal{O}$ here, which
consists of $p/q>0$ such that $p=m^{2}$ (for odd $m$), and $q=mh-1$ where
$0<h<m$ and $(m,h)=1$.

Suppose that $p/q\in\mathcal{R}\setminus\mathcal{O}$ and $K(p,q)$ is a
smoothly slice, quasipositive two-bridge knot. We can isotope $K(p,q)$ to be
transverse to the standard contact structure $\xi_{st}$ in $S^{3}$. Let
$(L(p,q),\xi)$ be the double cover of $(S^{3},\xi_{st})$ branched along the
transverse knot $K$. According to Proposition 1.7, $(L(p,q),\xi)$ admits a
rational homology ball Stein filling, which in turn, implies that $\xi$ is
isomorphic to the canonical contact structure $\xi_{can}$, because no
virtually overtwisted lens space has a rational homology ball symplectic
filling by the work of Golla and Starkston [13]. (See also [10, Lemma 1.5],
[11, Proposition 11]). This gives a contradiction to the fact that
$(L(p,q),\xi_{can})$ admits a rational homology ball symplectic filling if and
only if $p/q\in\mathcal{O}$, as shown by Lisca [18, Corollary 1.2(c)].

Now suppose that $p/q\in\mathcal{O}$, i.e. $p=m^{2}$, for odd $m>1$ and
$q=mh-1$ where $0<h<m$ and $(m,h)=1$. If $h$ is even, then $q=mh-1$ is odd and
$K(p,q)$ is non-quasipositive by Corollary 1.4. On the other hand, if $h$ is
odd, then $q=mh-1$ is even but $q^{\prime}=m(m-h)-1$ is odd and
$qq^{\prime}\equiv 1(\mbox{mod}\;m^{2}).$ Since $K(p,q)$ is isotopic to
$K(p,q^{\prime})$ [5, Chapter 12], we conclude that $K(p,q^{\prime})$ and
hence $K(p,q)$ is non-quasipositive, again by Corollary 1.4. ∎

Acknowledgment: We would like to thank Sebastian Baader, Peter Feller, Marco
Golla, and András Stipsicz for their useful comments on a draft of this note.

##  4\. Appendix by Stepan Orevkov

The Seifert graph of a connected link diagram $D$ is the graph $G_{D}$ whose
vertices correspond to Seifert circles and the edges correspond to the
crossings. Each edge is equipped with the sign of the corresponding crossing.
We say that a diagram $D$ is reduced if $G_{D}$ does not have any edge whose
removal disconnects $G_{D}$. Let $w(D)$ denote the writhe of $D$, which is the
sum of the signs of all crossings of $D$.

Suppose that $D$ is an alternating diagram of a link $L$. Let $b=b(L)$ denote
the braid index of $L$ and $s=s(D)$ denote the number of Seifert circles of
$D$. We define $d^{\pm}=d^{\pm}(D)$ as the number of edges of $\pm$ sign in a
spanning tree of $G_{D}$.

Suppose that $\beta$ is a braid with $b$ strands realizing $L$. Due to
Dynnikov–Prasolov Theorem [9], $w(\beta)$ does not depend on the choice of
$b$-strand $\beta$ realizing $L$, which allows one to define the numbers
$\mathbb{r}^{\pm}=\mathbb{r}^{\pm}(D)$ from the system of equations

$\mathbb{r}^{+}+\mathbb{r}^{-}=s-b\quad\mbox{ and
}\quad\mathbb{r}^{+}-\mathbb{r}^{-}=w(D)-w(\beta).$

By the work of Rudolph [32] and Nakamura [24, Lemma 4.1], positive diagrams
represent strongly quasipositive links. Conversely, Baader [2] showed that
homogeneous strongly quasipositive knots are positive. Note that the class of
homogeneous links, introduced by Cromwell [6], includes all alternating links.
Moreover, alternating strongly quasipositive links have positive alternating
diagrams by Boileau, Boyer and Gordon [3, Corollary 7.3]. Therefore, the
following question appears naturally (see [2], for example):

######  Question 4.1.

Is it true that alternating quasipositive links have positive diagrams?

A partial answer to this question was provided as follows:

######  Theorem 4.2.

([25, Theorem 4]). If $D$ is a reduced alternating diagram of a quasipositive
link $L$, which satisfies the inequality

$2\mathbb{r}^{-}(D)\leq d^{-}(D)$ (4.1)

then $D$ is positive, and hence $L$ is strongly quasipositive.

######  Proposition 4.3.

([25, Remark 2]). The inequality (4.1) is satisfied by a standard alternating
diagram of any two-bridge link or any alternating Montesinos link.

######  Remark 4.4.

As pointed out in [25, Remark 2], inequality (4.1) is actually proven in [7]
for the diagrams from [7, Thm. 4.10, 4.12, 4.14] (in particular, for those in
Proposition 4.3) even though it is not formulated in [7] explicitly. Since it
is not so easy to recognize the proof of this fact without carefully reading
the whole paper [7], one of our goals here is to help the reader to extract
this proof from [7].

######  Remark 4.5.

Note that by Proposition 4.3, if a two-bridge link or an alternating
Montesinos link is quasipositive, then it is positive.

The statement in Proposition 4.6 was claimed in [25, Remark 2] and as
mentioned there, it follows from [7]. For rational links, however, it can also
be easily derived from Murasugi’s work [21, Section 3].

######  Proposition 4.6.

Every oriented two-bridge (aka rational) link admits an alternating diagram
$D$ satisfying the inequalities $2\mathbb{r}^{-}(D)\leq d^{-}(D)$ and
$2\mathbb{r}^{+}(D)\leq d^{+}(D)$.

###### Proof.

For rational links we follow the orientation convention used in Murasugi’s
book [22]. Let $L$ be a rational oriented link of type $(p,q)$. For the mirror
image $D^{*}$ of $D$, we have $d^{\pm}(D^{*})=d^{\mp}(D)$ and
$\mathbb{r}^{\pm}(D^{*})=\mathbb{r}^{\mp}(D)$. Therefore, without loss of
generality we may assume that $q$ is odd and $0<q<p$. Using the notation
introduced in [21, Section 3], let

$\frac{p}{p-q}=[2n_{1,1},\ldots,2n_{1,k_{1}},\;-2n_{2,1},\ldots,-2n_{2,n_{2}},\;\ldots\;,(-1)^{t-1}2n_{t,1},\ldots,(-1)^{t-1}2n_{t,k_{t}}]^{-},$

where we assume $n_{i,j}>0$.

Let $b$ be the braid index of $L$ and $e=w(\beta)$ be the exponent sum of a
$b$-braid $\beta$ representing $L$. By [21, Prop. 4.2 and Thm. 4.3] we have

$b=t+1+\sum_{i=1}^{t}\sum_{j=1}^{k_{i}}(n_{i,j}-1),\qquad
e=\frac{1-(-1)^{t}}{2}+\sum_{i=1}^{t}(-1)^{i-1}\sum_{j=1}^{k_{i}}n_{i,j}.$
(4.2)

Using the standard properties of rational links and continued fractions, one
easily checks that $L$ admits an alternating diagram $D$ shown in Figure 1
where $T_{i}$ are the tangles defined by the braids

$T_{i}=\begin{cases}\sigma_{1}^{1-2n_{i,1}}\Big{(}\prod_{j=2}^{k_{i}}\sigma_{2}\sigma_{1}^{2-2n_{i,j}}\Big{)}\sigma_{1}^{-1},&\text{$i$
is odd,}\\\
\sigma_{2}^{2n_{i,1}-1}\Big{(}\prod_{j=2}^{k_{i}}\sigma_{1}^{-1}\sigma_{2}^{2n_{i,j}-2}\Big{)}\sigma_{2},&\text{$i$
is even,}\end{cases}$

as depicted in Figure 2 for odd $i$. Note that, for odd (resp. even) $i$, all
crossings of $T_{i}$ are positive (resp. negative).

$t$ is odd $t$ is even

Figure 2. Alternating diagram $D$ for the rational link $L$.

$2n_{i,1}-1$ $2n_{i,2}-2$ $2n_{i,k_{i}}-1$

Figure 3. The tangle $T_{i}$ for an odd $i$.

Let $w=w(D)$ and $s=s(D)$ be the writhe and the number of Seifert circles of
$D$. Each tangle $T_{i}$ contributes $(-1)^{i-1}w_{i}$ to $w$, where
$w_{i}=1+\sum_{j}(2n_{i,j}-1)$. If $i$ is odd (resp. even), then $T_{i}$
contributes $w_{i}-k_{i}$ to $d^{+}$ (resp. to $d^{-}$). We have
$s-(d^{+}+d^{-})=1$ (the Euler characteristic of a spanning tree).

Hence

$s=t+1+\sum_{i=1}^{t}\sum_{j=1}^{k_{i}}2(n_{i,j}-1),\quad
w=\frac{1-(-1)^{t}}{2}+\sum_{i=1}^{t}(-1)^{i-1}\sum_{j=1}^{k_{i}}(2n_{i,j}-1),$
(4.3) $d^{+}=\lceil t/2\rceil+\sum_{\text{$i$
odd}}\;\sum_{j=1}^{k_{i}}2(n_{i,j}-1),\qquad d^{-}=\lfloor
t/2\rfloor+\sum_{\text{$i$ even}}\;\sum_{j=1}^{k_{i}}2(n_{i,j}-1).$ (4.4)

By combining (4.2) and (4.3) we obtain

$s-b=\sum_{i=1}^{t}\sum_{j=1}^{k_{i}}(n_{i,j}-1),\qquad
w-e=\sum_{i=1}^{t}(-1)^{i-1}\sum_{j=1}^{k_{i}}(n_{i,j}-1).$ (4.5)

Recall that $\mathbb{r}^{\pm}=\mathbb{r}^{\pm}(D)$ are defined by

$\mathbb{r}^{+}+\mathbb{r}^{-}=s-b,\qquad\mathbb{r}^{+}-\mathbb{r}^{-}=w-e.$
(4.6)

By combining (4.4), (4.5), and (4.6) we obtain

$2\mathbb{r}^{+}=d^{+}-\lceil t/2\rceil\leq d^{+},\qquad
2\mathbb{r}^{-}=d^{-}-\lfloor t/2\rfloor\leq d^{-}.$

∎

###  4.1. How to extract the proof of Proposition 4.3 from [7]

The paper [7] is devoted to a computation of the braid index of alternating
links presented by link diagrams of some specific forms. An upper bound for
the braid index is the number of Seifert circles. A lower bound is given by
Morton-Franks-Williams (MFW) inequality ([12, 20]). In many cases (those
indicated in Theorem 4.2) it is shown in [7] that the MFW bound is sharp. In
each case, this is done in [7] as follows. Given a reduced alternating diagram
$D$ of a link $L$, one chooses a certain collection $C$ of lone crossings, and
applies the Murasugi-Przytycki move [23] (MP-move) to each of them (MP-moves
are also depicted in [34, Fig. 2] and [8, Fig. 11]). The number of Seifert
surfaces of the resulting diagram $D^{\prime}$ (in general, non-alternating)
is equal to the braid index of $L$ and, moreover, the number of the performed
MP-moves at positive (negative) crossings is equal to $\mathbb{r}^{+}(D)$
(resp. $\mathbb{r}^{-}(D)$).

A constant-sign path of length $n$ in $G_{D}$ is a sequence
$v_{1},\dots,v_{n}$ of pairwise distinct vertices of $G_{D}$ such that each
pair of consecutive vertices $(v_{i},v_{i+1})$ is connected by an edge, and
all these edges are of the same sign. One can check that the vertices of
$G_{D}$ corresponding to $C$ are always chosen in [7] in some pairwise
disjoint constant-sign paths. Moreover, at most $\lfloor(n-1)/2\rfloor$
crossings is chosen in each of the paths of length $n$. It is clear that any
collection of pairwise disjoint paths is contained in some spanning tree. Thus
we obtain (4.1) for all diagrams mentioned in Remark 4.4, in particular, this
gives a proof of Proposition 4.3.

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