# Phase coexistence in the Non-reciprocal Cahn-Hilliard model

Suropriya Saha<EMAIL_ADDRESS>Max Planck Institute for Dynamics and
Self-Organization (MPIDS), D-37077 Göttingen, Germany

###### Abstract

We establish the criterion for the phase coexistence in a mixture of
nonreciprocally interacting scalar densities. For an arbitrary number of
components the active pressure exists for a specific class of interactions,
and when the free energy receives no contribution from cross couplings between
spatial gradients of two different species. In this case, the pressure can be
used to determine phase equilibrium, i.e. to construct binodals, and the
active mixture can be mapped to a passive system with an effective free
energy. For general interfacial tension, the pressure changes discontinuously
across a flat interface which assumes the form of an active Laplace pressure
in two dimensions.

Introduction – The principles governing phase equilibrium form the cornerstone
of the physics of passive mixtures. The concepts first explored by Gibbs [13,
34] are relevant to this day several decades later in multiple areas of
research, from membrane-less organelles [14, 15, 16] to active phase
separation [Wittkowski2014, 27, 30, 18, 19, 17]. The concept of pressure
requires careful consideration when applied to active matter [9, 24] which
consists of units that are driven at the microscopic scale and collectively
operate away from equilibrium in a quantifiable [23, 25, 26] manner. The role
of an active pressure depends on details of the active driving– recent works
have argued the relevance [3] or otherwise [12] of a pressure like quantity in
different incarnations of activity.

In this paper we ask a similar question in a new class of active matter
systems, those driven by effective non-reciprocal interactions [28, 29]. The
theoretical model that we explore in this paper in the Non-Reciprocal Cahn-
Hilliard model (NRCH) [6, 5], that describes the evolution of a system of
multiple interacting conserved densities. The dynamical behaviour of NRCH,
which emerges naturally in scalar mixtures with more than one conservation law
[32], has been shown to be varied, ranging from travelling waves [6, 5],
moving lattices, spiral defects [1], Turing instabilities [33] to
spatiotemporal chaos [2]. In this work we explore the possibility that a
condition similar to pressure exists for non-reciprocally interacting
conserved scalar densities. We focus on steady states of NRCH that do not
break PT symmetry in contrast to [8, 6, 5], rather they evolve to a state with
microscopic or macroscopic phase separated states with zero flux at the steady
state, see Fig. 1 (a).

Figure 1: Pressure in the non-reciprocal Cahn-Hilliard model (a) Schematic
showing the compartmentalisation of a system of multiple non-reciprocally
interacting species. We find a quantity that plays the same role as pressure
in determining phase equilibrium. Panel (b) shows the analytic expression for
the active pressure matches very well with simulations as a function of the
non-reciprocal parameter $\alpha$. The red arrows indicate the exceptional
points corresponding to $|\alpha|<|\chi|$.

Model and main results – The NRCH model is represented by the set of $N$
equations $\partial_{t}\phi_{a}=\nabla^{2}\mu_{a}$ for $N$ conserved scalar
densities $\phi_{a}$, where $\mu_{a}$ is the chemical potential for the $a$-th
species. The system of equations represents an active mixture as $\mu_{a}$ is
not derivable from a free energy. For simplicity in calculations we assume a
polynomial form for $\mu_{a}$ as follows

$\displaystyle\mu_{a}=\frac{\mbox{d}f_{a}}{\mbox{d}\phi_{a}}+\sum_{b\neq
a}\chi_{ab}\frac{\partial
f_{ab}}{\partial\phi_{a}}-\sum_{b}{K}_{ab}\nabla^{2}\phi_{b},$ (1)

where $f_{a}$ is a function of $\phi_{a}$ only, and $f_{ab}$ is a function of
$\phi_{a}$ and $\phi_{b}$, where $a$ and $b$ are distinct from one other. The
coefficients of interaction $\chi_{ab}$ and the interfacial tension $K_{ab}$
are arbitrary constants. The system describes a passive mixture for
$\chi_{ab}=\chi_{ba}$ and $K_{ab}=K_{ba}$. Asymmetric coefficients both in the
bulk free energy and the interfacial tension arises naturally in systems with
nonreciprocal interactions, for example in a collection of phoretic Janus
colloids [22] or mixtures of quorum sensing particles [21]. For
$f_{ab}=\phi_{a}\phi_{b}$, Eq. (1), reduces to the form explored in [6]. Eq.
(1) represents a system with pair wise interactions between the species, an
exploration of the consequences with three species interactions permitting
terms of the form $\phi_{a}\phi_{b}\phi_{c}$ for example, is beyond the scope
of the present work.

We present a summary of our results before elucidating them in detail. The
interactions are chosen such that the system undergoes bulk phase separation
where multiple compartments, see Fig 1 (a) are formed within which the
concentrations of all species are uniform. The interfaces separating the
compartments are of vanishing width with a length-scale determined by the
interfacial tension. Within an interface the densities change rapidly from one
stationary composition to another. Integrating the product of the chemical
potential and the gradient of each species across a flat interface separating
two or more bulk phases, and summing over all the $N$ integrals so obtained,
we find a pressure like quantity $P$ that is constant throughout the system.
$P$ is used to determine phase coexistence as exactly one does in passive
systems. $P$ is related to a free energy that has exactly the same functional
form as the Lyapunov cost function of the minimal oscillator associated with
the dynamics [6]. The procedure can be extended to multi-component systems
with multiple conserved densities by constraining the interaction parameters
$\chi_{ab}$. $P$, however, changes discontinuously on introducing coupling
between the gradients of the interacting densities in the free energy, i.e.
for non vanishing $K_{ab}$, for $a\neq b$ implying that a Laplace pressure
develops even at flat interfaces.

Figure 2: Phase coexistence: Figure showing a comparison of the binodals
calculated using the conditions for equality of $P$ and the chemical
potentials and the phase composition obtained from numerical simulation of the
NRCH model. $f_{12}=\phi_{1}\phi_{2}$, and the interaction parameters are
indicated in the figure and $K_{12}=K_{21}=0$.

Conditions for phase coexistence– We first consider a binary system with
densities $\phi_{1}$ and $\phi_{2}$. We redefine the coefficient as
$\chi_{12}=\chi+\alpha$ and $\chi_{21}=\chi-\alpha$, using explicitly the
coefficient of reciprocal and nonreciprocal interactions $\chi$ and $\alpha$.
$\alpha$ is tuned at a value below the exceptional point, $|\alpha|<|\chi|$,
such that the system exhibits phase separation. We begin by consider a quasi
one-dimensional pattern, i.e. the density varies in only in a direction
perpendicular to the flat interface which we choose to be the $x$ direction.

We consider phase separation into two phases that we label $(1)$ and $(2)$. In
phase 1, the composition is
$(\phi_{a}^{(\mbox{\footnotesize{1}})},\phi_{b}^{(\mbox{\footnotesize{1}})})$,
while phase in 2 the composition is
$(\phi_{a}^{(\mbox{\footnotesize{2}})},\phi_{b}^{(\mbox{\footnotesize{2}})})$.
When the mixture reaches a steady state that is invariant under time
translation, the chemical potentials are constant throughout the system
$\mu_{a}^{(\mbox{\footnotesize{1}})}=\mu_{a}^{(\mbox{\footnotesize{2}})}\equiv\langle\mu_{a}\rangle$
for both species – taken together, these two conditions imposes chemical
equilibrium. To find a condition similar to mechanical balance in this system
out of equilibrium, we follow the procedure outlined in [3]. We proceed by
multiplying $\mu_{a}$ by $\partial_{x}\phi_{a}$ and integrate from a point
deep within one phase to another to obtain the following for species 1

$\displaystyle
I_{1}\equiv\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}x\,\mu_{1}\partial_{x}\phi_{1}=\langle\mu_{1}\rangle(\phi_{1}^{(\mbox{\footnotesize{2}})}-\phi_{1}^{(\mbox{\footnotesize{1}})}).$
(2)

Using Eq. (1) to evaluate $I_{1}$ we obtain an equivalent expression for the
integral

$\displaystyle
I_{1}=\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}x\,\frac{\mbox{d}f_{1}}{\mbox{d}\phi_{1}}\partial_{x}\phi_{1}-\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}xK_{11}(\partial^{2}_{x}\phi_{1})(\partial_{x}\phi_{1})$
$\displaystyle+\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}x\,[(\chi+\alpha)\frac{\partial
f_{12}}{\partial\phi_{1}}\partial_{x}\phi_{1}-K_{12}(\partial^{2}_{x}\phi_{2})(\partial_{x}\phi_{1})].$
(3)

The terms in the first line of Eq. (3) are total derivatives that depend only
on the bulk values of the fields, in contrast to the terms in the second line
with contributions from the interaction which are not. Defining a partial
pressure like quantity $P_{a}^{(i)}=\langle\mu_{a}\rangle\phi_{a}^{(i)}-f_{a}$
for species a in phase $(i)$, we can rewrite Eqs. (3) and (2) as for species 1
as

$\displaystyle
P_{1}^{(\mbox{\footnotesize{2}})}-P_{1}^{(\mbox{\footnotesize{1}})}=(\chi+\alpha)\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}x\,\frac{\partial
f_{12}}{\partial\phi_{1}}\partial_{x}\phi_{1}$
$\displaystyle-\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}xK_{12}(\partial^{2}_{x}\phi_{2})(\partial_{x}\phi_{1}).$
(4)

A similar relation holds for species 2 yielding an expression for
$P_{2}^{(\mbox{\footnotesize{2}})}-P_{2}^{(\mbox{\footnotesize{1}})}$.
Multiplying Eq. (4) by $\chi-\alpha$ and the equivalent one for species 2 by
$\chi+\alpha$, and adding up the contributions we obtain the following
relation that relates a quantity on the L.H.S. that depends only on the phase
composition with the quantity on the R.H.S. that depends on the detailed form
of the interface in the steady state.

$\displaystyle\left[P_{1}(\chi-\alpha)+P_{2}(\chi+\alpha)-(\chi^{2}-\alpha^{2})f_{ab})\right]^{(\mbox{\footnotesize{2}})}_{(\mbox{\footnotesize{1}})}$
$\displaystyle=-\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}x\left[(\chi-\alpha)K_{12}(\partial^{2}_{x}\phi_{2})(\partial_{x}\phi_{1})\right.$
$\displaystyle\left.+K_{21}(\chi+\alpha)(\partial^{2}_{x}\phi_{1})(\partial_{x}\phi_{2})\right]$
(5)

The integrand on the R.H.S. of Eq. (5) last line can be written as the a
complete derivative of the function
$\partial_{x}\phi_{1}\partial_{x}\phi_{2}$, for
$(\chi-\alpha)K_{12}=K_{21}(\chi+\alpha)$. We identify the active pressure as
the sum (difference) of two different terms

$\displaystyle P$ $\displaystyle=$ $\displaystyle
P_{r}-\frac{\alpha}{\chi}P_{nr},$ (6)

where the reciprocal and the non-reciprocal contributions are as follows

$\displaystyle P_{r}$ $\displaystyle=$ $\displaystyle P_{1}+P_{2}-\chi
f_{12}=\mu_{1}\phi_{1}+\mu_{2}\phi_{2}-f,$ $\displaystyle P_{nr}$
$\displaystyle=$ $\displaystyle P_{2}-P_{1}+\alpha f_{12}.$ (7)

The R.H.S. of Eq. (5) is the Laplace pressure difference across a flat
interface - clearly a signature of an active system. Notice that the
difference does not vanish when reciprocity is restored for the interfacial
tension, that is for $K_{12}=K_{21}$. The expression in (6), written such that
$P$ reduces to the passive case when $\alpha=0$, is invalid for $\chi=0$. If
interactions are completely non-reciprocal, the active pressure is
$P=P_{2}-P_{1}+\alpha f_{12}=\mbox{constant}$, if $K_{12}=-K_{21}$. At the
exceptional point $|\chi|=|\alpha|$, and for $\alpha>0$, the pressure reduces
to the form $P=2P_{2}$ and $P=2P_{1}$ and $\alpha<0$. This approach also
yields the condition for phase coexistence for $K_{11}\neq K_{22}$, where we
expect a conserved version of a Turing instability [33]. We also find that the
pressure reduces to the form derived in [20] when one of the species in the
binary mixture does not phase separate in the absence of interactions.

The limitation of the approach outlined for the binary system is that if we
introduce another interaction term $f^{\prime}_{12}$ in Eq. (1), with a
functional form that is different from $f_{12}$ and using a different non-
reciprocal coupling parameter $\alpha^{\prime}\neq\alpha$, then we cannot use
this method to obtain $P$. Strikingly, if the interactions are purely non-
reciprocal, then the active pressure in presence of $f_{ab}^{\prime}$ is given
by a straightforward generalisation of Eq. (6) and (7) - $P=P_{1}-P_{2}+\alpha
f_{12}+{\alpha}^{\prime}f^{\prime}_{12}$, with an effective free energy
$f_{1}-f_{2}+\alpha f_{12}+\alpha^{\prime}f^{\prime}_{12}$.

The existence of $P$ points at an underlying effective free energy. Consider
the following for an effective bulk free energy density

$f=(1-\alpha\chi^{-1})f_{1}+(1+\alpha\chi^{-1})f_{2}+\chi(1-\alpha^{2}\chi^{-2})f_{12}.$
(8)

Defining chemical potentials as derivatives of Eq. (8),
$\bar{\mu}_{1}\equiv\delta f/\delta\phi_{1}$, the pressure evaluated as
$P=\bar{\mu}_{1}\phi_{1}+\bar{\mu}_{2}\phi_{2}-\bar{f}$, is identical to that
in Eq. (6). $\bar{\mu}_{a}$ is related to $\langle\mu_{a}\rangle$ as
$\bar{\mu}_{1}=\langle\mu_{1}\rangle/(\chi+\alpha)$ and
$\bar{\mu}_{2}=\langle\mu_{2}\rangle/(\chi-\alpha)$, and are well defined at
all points other than the exceptional point $|\chi|=|\alpha|$. This implies
that we can infer the phase composition using a common tangent construction on
the effective free energy $\bar{f}$ at all values of $\alpha$ other than at
$\chi$. We find that the free energy Eq. (8) has the same functional form as
the Lyapunov function associated with the minimal oscillator defined as
$\dot{x}_{1}=-\mu_{1}({x_{1},x_{2}})$.

To illustrate the usefulness of (6) in determining the phase composition, we
present a few concrete examples. We first consider the case $K_{12}=K_{21}=0$.
We choose simple polynomial forms for the free energy
${f}_{a}=s_{a}{\phi_{a}^{2}}/{2}+{\phi_{a}^{4}}/{4c_{a}^{2}}$, where $s_{a}$
is constrained to be $\pm 1$. For $s_{a}=-1$, the system partitions into two
compartments with densities $\pm c_{a}$, which is chosen to be unity for all
numerical simulations presented here. Equivalent to the passive case, we can
use the equality of the active pressure and chemical potentials to determine
the composition in the phases for binary or ternary phase separation, for
details see SI. The results are shown in Fig. 2. For vanishing average density
$\bar{\phi}_{1}=\bar{\phi}_{2}=0$, the system partitions into two phases with
compositions that differ in sign. The compositions lie in the minima of the
potential $f$ in Eq. (8). At the EP the $f$ is quasi one-dimensional such that
$\mu_{2}=0$. At this point a saddle node bifurcation occurs where the minima
splits into a saddle point and an unstable maxima. At large enough $\alpha$,
$f$ is concave to the ($\phi_{1}-\phi_{2}$) plane. We also choose a nonlinear
form for the interactions $f_{12}=\phi_{1}\phi_{2}(1+\lambda\phi_{1}\phi_{2})$
to show the success of the common tangent composition in predicting the phase
composition.

For a vanishing $K_{ab}$, the non-reciprocal surface tension, the
discontinuity in pressure disappears in one dimension and the mapping to the
passive case is complete. The mapping to equilibrium phase separation fails
when $K_{ab}\neq 0$ and the discontinuous jump in the pressure becomes
relevant. We will also show later that interfacial effects in two are higher
dimensions is subtle as the non-reciprocal interaction modifies the
interfacial tension and thus the Laplace pressure within a droplet.

Figure 3: Active Laplace pressure (a) Density profiles $\phi_{1}$ and
$\phi_{2}$ in the steady state for $\chi$, $\alpha$ and $K_{12}=K_{21}=-5$
showing micro phase separation. (b) The active Laplace pressure is calculated
by varying $K_{12}$ in the range $(-5,5)$. The active Laplace pressure
difference between the pressure obtained from simulations and theory are
plotted showing excellent agreement.

Excess Laplace Pressure within a droplet– To calculate the change in pressure
across a curved interface, we consider a spherical droplet of size $R$.
Integrating $\mu_{a}\partial_{r}\phi_{a}$ from $r=0$ to $r=\infty$, i.e. from
inside a droplet to the outside we obtain the following expression for the
excess pressure $\Delta P$

$\displaystyle\Delta P=$
$\displaystyle\int_{0}^{\infty}\mbox{d}r\left(1-\alpha\chi^{-1}\right)\left[K_{11}(\phi^{\prime}_{1})^{2}{r^{-1}}+K_{12}{\phi^{\prime}_{1}\phi^{\prime}_{2}}{r^{-1}}\right]$
$\displaystyle+\left(1+{\alpha}{\chi^{-1}}\right)\left[K_{22}{(\phi^{\prime}_{2})^{2}}{r^{-1}}+K_{21}{\phi^{\prime}_{1}\phi^{\prime}_{2}}{r^{-1}}\right]$
$\displaystyle+\left(1-{\alpha}{\chi^{-1}}\right)K_{12}\phi^{\prime\prime}_{1}\phi^{\prime}_{2}+\left(1+{\alpha}{\chi^{-1}}\right)K_{21}\phi^{\prime\prime}_{2}\phi^{\prime}_{1}.$
(9)

The first two lines on the R.H.S. of Eq. (9) leads to contributions that scale
inversely with the droplet size $R$. At equilibrium, $\alpha=0$ and
$K_{ab}=K_{ba}$, $\Delta P$ is always a positive quantity, and it does not
depend explicitly on the interaction parameter $\chi$. It receives
contribution only from the first two lines which scale as $R^{-1}$. For
nonzero $\alpha/\chi$ appears explicitly in the expression for $\Delta P$. For
non-reciprocal surface tension, $\Delta P$ can be a negative thus signalling a
clear departure from equilibrium.

Figure 4: Active Laplace pressure in a droplet– (a) For $K_{12}=K_{21}=0$, the
excess pressure $\Delta P$ inside a droplet scales linearly with inverse of
the droplet size as it does in passive systems. Panel (a) shows a few instance
of the excess Laplace pressure $R^{-1}$, see the interaction parameters in the
inset and the parameters in the Appendix. (b) A heatmap showing a droplet at
steady state for $K_{12}=K_{21}=5$. The density profiles across a line
intersecting the centre of the droplet are shown below the heatmap. (c)
Laplace pressure for $K_{12}=K_{21}$ evaluated for values indicated. The
excess pressure $\Delta P$ as obtained from simulations is plotted against the
analytical expression in Eq. (9) showing good agreement. (d) Same for
$K_{12}=K_{21}=-0.5$, i.e. fully anti-symmetric interfacial tension
coefficients, showing that the theory fully captures the active Laplace
presssure measured in simulations.

Pressure in a multicomponent system– We now show that we can find conditions
for phase equilibrium for an arbitrary number of components following the
steps outlined for $N=2$. Consider, same as was done previously for the binary
system, that the system partitions into two compartments with compositions
labelled 1 or 2. By summing over all species $b$ we can write an expression
similar to Eq. (3)

$\displaystyle P_{a}^{(2)}-P_{a}^{(1)}=\sum_{b\neq
a}\chi_{ab}\int_{(\mbox{\footnotesize{1}})}^{(\mbox{\footnotesize{2}})}\mbox{d}x\,\frac{\partial
f_{ab}}{\partial\phi_{a}}\partial_{x}\phi_{a}$ (10)

Multiplying both sides of Eq. (10) by a constant $\lambda_{a}$ and summing
over all components we want to write Eq. (10) in the following form

$\displaystyle
P^{(2)}-P^{(1)}\equiv\left[\sum_{a}\lambda_{a}P_{a}-\sum_{\langle
ab\rangle}\lambda_{a}\chi_{ab}f_{ab}\right]^{(\mbox{\footnotesize{2}})}_{(\mbox{\footnotesize{1}})}=0,$
(11)

where $<ab>$ denotes distinct pairs, and thus obtain an expression for the
pressure $P$. In other words, the goal is to find a set of coefficients
$\\{\lambda_{a}\\}$ such that the integrals on the R.H.S. of Eq. (10) can be
directly evaluated and Eq. (11) is satisfied. This is possible strictly if,
given any pair of species $a$ and $b$ the following relation holds for each
distinct pair

$\displaystyle\frac{\lambda_{a}}{\lambda_{b}}=\frac{\chi_{ba}}{\chi_{ab}}.$
(12)

Eq. (12) represents $N(N-1)/2$ equations that are used to determine $N-1$ free
coefficients $\\{\lambda_{a}\\}$. The remaining $(N-1)(N-2)/2$ equations
constrain the interaction coefficients $\chi_{ab}$. A system with N components
can thus be mapped to the equilibrium case only in the restricted case where
the $N(N-1)$ parameters of the interaction matrix are constrained by
$(N-1)(N-2)/2$ equations, i.e. $(N+2)(N-1)/2$ of the parameters are free thus
implying that the mapping to equilibrium is possible only for a restricted set
of interaction coefficients $\chi_{ab}$.

For illustration we consider $N=3$ for which the interaction matrix is
characterised by 6 parameters. We will show that for a mapping to equilibrium,
we need to constrain the parameters with a single constraint consistent with
the conclusions for the general $N$. The three equations for $\lambda_{a}$ are

$\displaystyle\frac{\lambda_{1}}{\lambda_{2}}=\frac{\chi_{21}}{\chi_{12}},\,\frac{\lambda_{1}}{\lambda_{3}}=\frac{\chi_{31}}{\chi_{13}},\,\frac{\lambda_{2}}{\lambda_{3}}=\frac{\chi_{32}}{\chi_{23}}.$
(13)

Choosing $\lambda_{1}=1$, we have $\lambda_{2}={\chi_{12}}/{\chi_{21}}$ and
$\lambda_{3}={\chi_{13}}/{\chi_{31}}$ from the first two equations in (13).
The third equation constraints the coefficients as

$\displaystyle\chi_{12}\chi_{31}\chi_{23}\chi_{21}^{-1}\chi_{13}^{-1}\chi_{32}^{-1}=1.$
(14)

To verify Eq. (11), we have run simulations in one dimension varying the total
number of components from $3-15$ and for about 10 instances of the interaction
$\chi_{ab}$ and verify that at steady state, $P$ is indeed constant throughout
the system for all components, see Fig. 5 (a) with negligible variance as
shown in Fig. 5 (b-c). To show this we choose $K_{aa}=1$ for all $N$ species
and all cross-couplings $K_{ab}=0$. The results are shown in Fig. 5

Figure 5: Pressure in multicomponent systems– (a) The validity of the concept
of pressure is checked numerically in multi-component systems. The markers
denote $P$ calculated in the steady state for $10$ examples for all values of
$N$. (b) Shows the average value of $P$ as a function of $N$. (c) Spatial
deviation from a constant $P$ averaged over the $10$ realisations shows that
the deviations from a constant P is negligible.

Conclusion– To conclude, we have obtained an expression for pressure in
nonreciprocally interacting scalar densities which can be used to determine
phase coexistence in a broad class of systems with steady states comprising
both bulk phase separated and micro-phase separated states. Counter-
intuitively, a system with exclusively inter species nonreciprocal
interactions can always be mapped to a free energy, phase separation occurs if
the free energy allows a convex hull construction. The concepts explored in
this paper can be extended to other forms of active mixtures with
orientational degrees of freedom [8, 21].

Pressure plays a more fundamental role than being just a condition determining
phase coexistence in equilibrium physics. A question that arises naturally –
consider a collection of diffusiophoretic colloids [31] which generically have
nonreciprocal interactions between them and confine them to an enclosure, can
we relate the average force per unit area to the expression obtained here?
That is, does the pressure described here truly enforce mechanical
equilibrium?

##  I acknowledment

SS thanks B Mahault for multiple insights and recommendations, R Golestanian
for many discussions and encouragement, and the HPC facility and maintenance
team at LMP, MPIDS for their help and support. SS thanks V Novak for help with
the graphics.

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# Coloring Torus Knots by Conjugation Quandles

Filippo Spaggiari
<EMAIL_ADDRESS>
Department of Algebra, Faculty of Mathematics & Physics, Charles University
Prague, Czech Republic

###### Abstract

In the first part of this paper, we present general results concerning the
colorability of torus knots using conjugation quandles over any abstract
group. Subsequently, we offer a numerical characterization for the
colorability of torus knots using conjugation quandles over some particular
groups, such as the matrix groups $\mathsf{GL}(2,q)$ and $\mathsf{SL}(2,q)$,
the dihedral group, and the symmetric group.

###### Contents

  1. 1 Introduction
  2. 2 Fundamentals
  3. 3 Conjugation quandle coloring of torus knots
    1. 3.1 First characterization
    2. 3.2 Properties of conjugation quandle coloring
    3. 3.3 The main characterization
  4. 4 Coloring with particular groups
    1. 4.1 General linear group
    2. 4.2 Special linear group
    3. 4.3 Dihedral group
    4. 4.4 Symmetric group
  5. 5 Conclusions

##  1 Introduction

Coloring invariants serve as valuable computational tools in various contexts
(see [Cla+14], [Fis+16], and [EN15]). In addition, Kuperberg’s NP certificate
of knottedness can be effectively interpreted using coloring techniques (see
[Kup14]). A significant class of examples related to coloring is provided by
conjugation quandles. Notably, a nontrivial coloring achieved with a quandle
$Q$ implies the existence of a nontrivial coloring with $Q/\lambda$, where
$\lambda$ represents the Cayley kernel. This quotient quandle can be embedded
into $\operatorname{\mathsf{Conj}}(\operatorname{\mathsf{Aut}}(Q))$.

Matrix groups hold particular interest due to Kuperberg’s certificate
involving coloring by $\operatorname{\mathsf{Conj}}(\mathsf{GL}(2,q))$. His
proof also suggests that the problem of coloring by
$\operatorname{\mathsf{Conj}}(\mathsf{GL}(2,q))$ is challenging in general,
prompting us to begin with a simpler class: the torus knots. By investigating
coloring invariants in this specific context, we can gain valuable insights
and potentially extend our understanding to more complex classes of knots.

Coloring of torus knots has garnered considerable attention and has been
explored from various perspectives, as evidenced by the works [ZL23], [BC22],
[Iwa07], and [AK09]. Notably, the concept of coloring by Alexander Quandles
has been thoroughly examined in the work of Asami and Kuga ([AK09]). Building
upon these contributions, this research paper delves into the analysis of
coloring torus knots using conjugation quandles.

The organization of this work is structured as follows. In Section 2, we lay
the groundwork by introducing fundamental notions related to torus knots and
quandles, thereby providing a necessary foundation for subsequent discussions.
In Section 3, we present general results pertaining to the conjugation quandle
colorings of torus knots over abstract groups. This includes our main theorem
(see Theorem 3.19), which characterizes the colorability of torus knots, as
well as exploring various properties of colorings. Subsequently, in Section 4,
we apply the results obtained in Section 3 to establish characterization
theorems for specific groups, such as the matrix groups $\mathsf{GL}(2,q)$ and
$\mathsf{SL}(2,q)$, the dihedral group, and the symmetric group. The selection
of these particular groups is twofold in purpose: firstly, they provide a
manageable context for formulating elementary statements and characterization
theorems, although the proofs might be notably technical. Secondly, these
groups have been under investigation in other contexts and papers, as
evidenced by relevant references in the bibliography. Section 5 concludes this
work, by discussing and presenting some questions that remained open about
using conjugation quandles to understand the colorability of torus knots.

##  2 Fundamentals

We begin by introducing and reviewing the mathematical tool necessary for
developing the upcoming theory. Firstly, we provide the definition of a
quandle, which is the algebraic structure playing a fundamental role in this
paper. Subsequently, we proceed to the geometric counterpart, where we
introduce and construct torus knots, along with their representations and
colorings. Finally, we conclude by presenting some well-known knot-theoretical
results that serve as the basis for making additional assumptions on the
parameters. For a comprehensive understanding of Knot and Quandle Theory, we
recommend consulting the books by Murasugi [Mur96] and Elhamdadi [EN15].

######  Definition 2.1.

A (right) quandle is a binar $(Q,\rhd)$ satisfying the following axioms:

  1. (i)

$\forall\mkern 2.0mux,y,z\in Q:\ (x\rhd y)\rhd z=(x\rhd z)\rhd(y\rhd z)$.

  2. (ii)

$\forall\mkern 2.0mux,y\in Q\ \exists\mkern 2.0mu!z\in Q:\ z\rhd x=y$.

  3. (iii)

$\forall\mkern 2.0mux\in Q:\ x\rhd x=x$.

Out of a given group, we can construct an important class of quandles.

######  Definition 2.2.

Let $G$ be a group. The (right) conjugation quandle over $G$ is the quandle
obtained by taking $Q=G$ as the underlying set and $x\rhd y=yxy^{-1}$ as the
binary operation. We denote it by $\operatorname{\mathsf{Conj}}(G)$.

The quandle operation exhibits nice features in conjugation quandles. In the
following lemma, we introduce several properties that will be consistently
utilized throughout the entire paper, without mentioning them explicitly.

######  Lemma 2.3.

Let $G$ be a group. In $\operatorname{\mathsf{Conj}}(G)$, for every
$x,y,x_{i},y_{i}\in G$ and $k\in\mathbbm{N}$ we have

  1. (i)

$(x_{1}x_{2})\rhd y=(x_{1}\rhd y)(x_{2}\rhd y)$.

  2. (ii)

$x^{k}\rhd y=(x\rhd y)^{k}$.

  3. (iii)

$x\rhd(y_{1}y_{2})=(x\rhd y_{2})\rhd y_{1}$.

  4. (iv)

$x\rhd y^{k}=(\ldots((x\rhd y)\rhd y)\rhd\ldots)\rhd y$ ($k$ times).

###### Proof.

All of them are straightforward computations. ∎

We now proceed to introduce some fundamental notions of Knot Theory. All the
knots in this paper are intended to be oriented.

######  Definition 2.4.

Let $K$ be a regular diagram of an oriented knot, or simply, a knot. We denote
by $\operatorname{\mathsf{Arcs}}(K)$ the set of connected strands of (the
diagram) $K$. By a (positive) crossing in $K$ we mean a triple
$(x,y,z)\in\operatorname{\mathsf{Arcs}}(K)^{3}$ such that _$x$ passes under
$y$ producing $z$_ (see Figure 1). In this case, we denote $x\rhd y=z$ and we
call it crossing relation of $K$.

Figure 1: Crossing relation.

######  Definition 2.5.

Let $K$ be a knot and $(Q,\rhd)$ a quandle. A $Q$-coloring of $K$ is a mapping
$c\colon\operatorname{\mathsf{Arcs}}(K)\to Q$ such that for every crossing
$(x,y,z)$ of $K$ the equation $c(x)\rhd c(y)=c(z)$ holds in $Q$. If $c$ is the
constant function, we say that it is the trivial coloring, every other
coloring is called non-trivial. A knot $K$ is said to be $Q$-colorable if
there exists a non-trivial coloring of $K$.

######  Remark 2.6.

It is evident that every knot can be trivially colored; that is, a trivial
coloring always exists. Therefore, our focus lies solely on the existence of
non-trivial colorings. Consequently, we have chosen to define the $Q$-coloring
in the manner described earlier, excluding trivial colorings.

######  Definition 2.7.

Let $m,n$ be two positive integers and consider the braid with $n$ strands and
$m$ twists as in Figure 2. The $n$ leftmost strands are called initial arcs,
the $n$ rightmost strands are called terminal arcs, and the $m$ diagonal
strands are called bridges. The closure of such a braid, identifying each
initial arc with the corresponding terminal arc, is called $(m,n)$-torus knot
(link), and denoted by $\mathsf{K}(m,n)$ (see Figure 3).

Figure 2: Braid diagram for $\mathsf{K}(m,n)$.

######  Remark 2.8.

Every torus knot (torus link) has the remarkable property of being embeddable
on the surface of the trivial torus, without any points of self-intersection.
Conversely, any knot lying on the surface of the trivial torus can be shown to
be equivalent to $\mathsf{K}(m,n)$, for some integers $m$ and $n$. This
explains the terminology used in Knot Theory. The depiction of this knot
(link) on the trivial torus is shown in Figure 3. However, the diagram in
Figure 3 contains excessive information. Therefore, like many other authors,
we prefer to use a more concise and schematic braid representation, as shown
in Figure 2. This simplified representation is known as the _braid diagram_
(or _standard diagram_), and it disregards the specific identification
presented in the knot definition. Certainly, when one is working with a torus
knot, referring either to the initial arc or to the corresponding terminal arc
naturally imparts the same information. When coloring is involved, we find the
braid diagram particularly useful: it allows us to label the initial and
terminal arcs with their respective colors easily. This labeling simplifies
the coloring process and enhances our understanding of the knot’s properties.

Figure 3: Knot diagram for $\mathsf{K}(5,4)$.

There are two significant, well-known results that govern the overall behavior
of a torus knot. Specifically, we have precise knowledge regarding how to set
the parameters to achieve a pair of equivalent links or to cause the torus
link to collapse into a knot (proofs and details can be found in Murasugi’s
book [Mur96]).

######  Theorem 2.9 (Classification of torus links).

Let $m,n,r,s\geq 2$. Then $\mathsf{K}(m,n)$ is equivalent to $\mathsf{K}(r,s)$
if and only if $\\{m,n\\}=\\{r,s\\}$.

######  Theorem 2.10 (Torus knots and links).

Let $m,n\geq 1$. Then $\mathsf{K}(m,n)$ is a knot if and only if
$\operatorname{\mathsf{gcd}}(m,n)=1$.

##  3 Conjugation quandle coloring of torus knots

###  3.1 First characterization

The first result is a characterization of coloring a torus knot using a
conjugation quandle: coloring a knot with $\operatorname{\mathsf{Conj}}(G)$ is
equivalent to finding a tuple of elements in $G$ that satisfy certain group
term equations, or alternatively, certain quandle term equations.

######  Definition 3.1.

Let $m,n\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$, let
$G$ be a group, and let $x_{0},\dots,x_{n-1}\in G$. We say that the tuple
$(x_{0},\dots,x_{n-1})$ extends to a coloring of $\mathsf{K}(m,n)$ if there
exists a unique $\operatorname{\mathsf{Conj}}(G)$-coloring of
$\mathsf{K}(m,n)$ of which $(x_{0},\dots,x_{n-1})$ are the colors of the
initial arcs, respectively.

######  Remark 3.2.

Given a coloring, we may always extract the tuple of the colors of the initial
arcs. That tuple naturally extends to the given coloring. Moreover, observe
that the constant tuple extends to the trivial coloring.

######  Theorem 3.3.

Let $m,n\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$, let
$G$ be a group, and let $x_{0},\dots,x_{n-1}\in G$ be not all equal. The
following conditions are equivalent.

  1. (i)

The tuple $(x_{0},\dots,x_{n-1})$ extends to a (non-trivial) coloring for
$\mathsf{K}(m,n)$.

  2. (ii)

$|\\{x_{0+i\pmod{n}}x_{1+i\pmod{n}}\ldots x_{m-1+i\pmod{n}}\colon
i=0,\dots,n-1\\}|=1.$

  3. (iii)

For $u=\prod_{j=0}^{m-1}x_{n-m+j\pmod{n}}$, we have

$x_{i}\rhd u=x_{i-m\pmod{n}}\quad\quad\forall\mkern 2.0mui=0,\dots,n-1.$

Moreover, in this case, the element $u$ of (iii) is the common value in the
set of (ii).

###### Proof.

Prove the implications separately. All the indices of symbols $x$ are assumed
to be computed modulo $n$ and all the indices of symbols $y$ are assumed to be
computed modulo $m$.

$(i)\implies(iii)$

Denote by $y_{0},\dots,y_{m-1}$ the colors of the bridges, as in the Figure 4.

Figure 4: Colors in the proof of Theorem 3.3

Observe that $x_{0}=y_{0}$. By the definition of coloring, we have the
following relations

$(((y_{j}\rhd y_{j+1})\rhd y_{j+2})\rhd\ldots)\rhd
y_{m-1}=x_{j+m-n}\quad\quad\forall\mkern 2.0muj=0,\dots,m-1.$ (1)

Moreover, by the geometry of the torus knot, because of the $m$ twists, we
also have

$(((x_{i}\rhd y_{0})\rhd y_{1})\rhd\ldots)\rhd
y_{m-1}=x_{i-m}\quad\quad\forall\mkern 2.0mui=0,\dots,n-1.$ (2)

Now, if we expand equations (2), using the quandle axioms and equations (1),
we obtain

$\displaystyle x_{i-m}$ $\displaystyle=(((x_{i}\rhd y_{0})\rhd
y_{1})\rhd\ldots)\rhd y_{m-1}$ $\displaystyle=((((x_{i}\rhd y_{1})\rhd
y_{2})\rhd\ldots)\rhd y_{m-1})\rhd((((y_{0}\rhd y_{1})\rhd
y_{2})\rhd\ldots)\rhd y_{m-1})$ $\displaystyle=((((x_{i}\rhd y_{1})\rhd
y_{2})\rhd\ldots)\rhd y_{m-1})\rhd x_{n-m}$ $\displaystyle=(((((x_{i}\rhd
y_{2})\rhd y_{3})\rhd\ldots)\rhd y_{m-1})\rhd((((y_{1}\rhd y_{2})\rhd
y_{3})\rhd\ldots)\rhd y_{m-1}))\rhd x_{n-m}$ $\displaystyle=(((((x_{i}\rhd
y_{2})\rhd y_{3})\rhd\ldots)\rhd y_{m-1})\rhd x_{n-m+1})\rhd x_{n-m}$
$\displaystyle=\dots$ $\displaystyle=(((((x_{i}\rhd x_{n-1})\rhd
x_{n-2})\rhd\ldots)\rhd x_{n-m+2})\rhd x_{n-m+1})\rhd x_{n-m}$
$\displaystyle=(x_{n-m}x_{n-m+1}\ldots
x_{n-2}x_{n-1})x_{i}(x_{n-1}^{-1}x_{n-2}^{-1}\ldots
x_{n-m+1}^{-1}x_{n-m}^{-1})$ $\displaystyle=x_{i}\rhd u.$

$(iii)\implies(i)$

Associate the elements $x_{0},\dots,x_{n-1}$ to the initial arcs, and compute
the colors of the bridges $y_{0},\dots,y_{m-1}$ recursively as follows:

$\displaystyle y_{0}$ $\displaystyle=x_{0}$ $\displaystyle y_{j}$
$\displaystyle=(((x_{j}\rhd y_{0})\rhd y_{1})\rhd\ldots)\rhd
y_{j-1}\quad\quad\forall\mkern 2.0muj=0,\dots,m-1.$

This allows computing the color of all other elements in
$\operatorname{\mathsf{Arcs}}(\mathsf{K}(m,n))$ and the conditions $x_{i}\rhd
u=x_{i-m}$ guarantee the good definition of colors $x_{i}$’s for the diagram
of $\mathsf{K}(m,n)$.

$(ii)\implies(iii)$

Note that (ii) can be seen as a chain of equations, where the terms are the
product of the colors whose indices are consecutive and shifted by the same
constant, possibly reduced modulo $n$. Fix $i\in\\{0,\dots,n-1\\}$. Then,
because of (ii), we have

$\displaystyle x_{i}\rhd u=ux_{i}u^{-1}$
$\displaystyle=(x_{n-m}x_{n-m+1}\ldots
x_{n-2}x_{n-1})x_{i}(x_{n-m}x_{n-m+1}\ldots x_{n-2}x_{n-1})^{-1}$
$\displaystyle=(x_{n-m+i}x_{n-m+1+i}\ldots
x_{n-2+i}x_{n-1+i})x_{i}(x_{n-m+i+1}x_{n-m+1+i+1}\ldots
x_{n-2+i+1}x_{n-1+i+1})^{-1}$ $\displaystyle=(x_{i-m}x_{i-m+1}\ldots
x_{i-2}x_{i-1})x_{i}(x_{i}^{-1}x_{i-1}^{-1}\ldots
x_{i-m+2}^{-1}x_{i-m+1}^{-1})=x_{i-m}.$

$(iii)\implies(ii)$

Expand the equations as follows

$\displaystyle x_{i}\rhd u=x_{i-m}\iff(x_{n-m}x_{n-m+1}\ldots
x_{n-2}x_{n-1})x_{i}=x_{i-m}(x_{n-m}x_{n-m+1}\ldots x_{n-2}x_{n-1}).$ (3)

Set $i=0=n$ in (3), and cancel out the term $x_{n-m}$ on the left obtaining

$\displaystyle x_{n-m+1}x_{n-m+2}\ldots
x_{n-2}x_{n-1}x_{i}=x_{n-m}x_{n-m+1}\ldots x_{n-2}x_{n-1},$ (4)

which is one of the equations in (ii). Substitute (4) in (3), set $i=n-m+1$,
and cancel out the first term again to obtain another of the equations.
Proceeding this way, since $\operatorname{\mathsf{gcd}}(m,n)=1$ we obtain all
the equations in $(ii)$. ∎

######  Definition 3.4.

Let $m,n\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$, let
$G$ be a group, and let $x_{0},\dots,x_{n-1}\in G$. Let
$(x_{0},\dots,x_{n-1})$ extend to a coloring of $\mathsf{K}(m,n)$. We refer to
the element $u=x_{n-m}x_{n-m+1}\ldots x_{n-2}x_{n-1}\in G$ (as in Theorem 3.3)
as the harlequin of $(x_{0},\dots,x_{n-1})$.

###  3.2 Properties of conjugation quandle coloring

It is natural to inquire whether a given knot coloring can be used to create a
coloring for another knot. We observe that the answer to this question is
frequently affirmative, and it involves certain divisibility conditions on the
parameters of the torus knot. Throughout this subsection, $G$ denotes any
fixed group.

######  Proposition 3.5.

Let $m,n,t\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$
and $\operatorname{\mathsf{gcd}}(tm,n)=1$. If $\mathsf{K}(m,n)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable, then also $\mathsf{K}(tm,n)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable.

###### Proof.

The diagram of $\mathsf{K}(tm,n)$ can be obtained by gluing $t$ copies of the
braid diagram of $\mathsf{K}(m,n)$, so we may use the given non-trivial
coloring of $\mathsf{K}(m,n)$ to obtain a non-trivial coloring of
$\mathsf{K}(tm,n)$. ∎

######  Remark 3.6.

In the previous proposition, the condition
$\operatorname{\mathsf{gcd}}(tm,n)=1$ is required only for $\mathsf{K}(tm,n)$
not to be a link (see Theorem 2.10), and it is not directly required in the
proof. While many results presented here could be extended to links, we adhere
to the convention of exclusively focusing on knots throughout this paper. This
approach also involves assuming the greatest common condition divisor on the
parameters.

######  Proposition 3.7.

Let $m,n,t\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$
and $\operatorname{\mathsf{gcd}}(m,tn)=1$. If $\mathsf{K}(m,n)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable, then also $\mathsf{K}(m,tn)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable.

###### Proof.

From Theorem 2.9 and Proposition 3.5, we can infer that

$\mathsf{K}(m,n)\mbox{ is colorable }\implies\mathsf{K}(n,m)\mbox{ is
colorable }\implies\mathsf{K}(tn,m)\mbox{ is colorable
}\implies\mathsf{K}(m,tn)\mbox{ is colorable}.\qed$

######  Proposition 3.8.

Let $m,n,t\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$,
and $\operatorname{\mathsf{gcd}}(m,tn)=1$. Let $(y_{0},\dots,y_{tn-1})$ extend
to a coloring of $\mathsf{K}(m,tn)$, and define

$x_{i}=\prod_{j=0}^{t-1}y_{it+j},\quad\quad\mbox{for }i=0,\dots,n-1.$

Then $(x_{0},\dots,x_{n-1})$ extends to a (possibly trivial) coloring of
$\mathsf{K}(m,n)$.

###### Proof.

Let $v$ be the harlequin of $(y_{0},\dots,y_{tn-1})$ in $\mathsf{K}(m,tn)$,
and define $u=v^{t}$. We want to prove that $(x_{0},\dots,x_{n-1})$ extends to
a coloring of $\mathsf{K}(m,n)$ with harlequin $u$. Using the fact that
$y_{j}\rhd v=y_{j-m\pmod{tn}}$, we have

$x_{i}\rhd u=\left(\prod_{j=0}^{t-1}y_{it+j}\right)\rhd
v^{t}=\prod_{j=0}^{t-1}\left(y_{it+j}\rhd
v^{t}\right)=\prod_{j=0}^{t-1}y_{it+j-tm\pmod{tn}}=\prod_{j=0}^{t-1}y_{(i-m)t+j\pmod{tn}}=x_{i-m\pmod{n}},$

therefore Theorem 3.3 applies. ∎

######  Remark 3.9.

The construction in the proof of Proposition 3.8 has the possibility of
producing a trivial coloring. However, this is not considered a drawback. In
fact, we will utilize this feature in various proofs by contradiction in the
subsequent discussions. We show this behavior in the following example.

######  Example 3.10.

$\mathsf{K}(2,3)$ is $\operatorname{\mathsf{Conj}}(\mathsf{S}_{3})$-colorable.
In fact, $(x_{0},x_{1},x_{2})$ extends to a coloring of $\mathsf{K}(2,3)$,
where $x_{0}=(2\ 3),x_{1}=(1\ 2)$ and $x_{2}=(1\ 3)$. By Proposition 3.7,
define

$\displaystyle y_{0}$ $\displaystyle=y_{3}=y_{6}=y_{9}=y_{12}=x_{0}=(2\ 3)$
$\displaystyle y_{1}$ $\displaystyle=y_{4}=y_{7}=y_{10}=y_{13}=x_{1}=(1\ 2)$
$\displaystyle y_{2}$ $\displaystyle=y_{5}=y_{8}=y_{11}=y_{14}=x_{2}=(1\ 3)$

and observe that $(y_{0},\dots,y_{14})$ extends to a coloring of
$\mathsf{K}(2,15)$. However, if we apply Proposition 3.8 to the previous (non-
trivial) coloring of $\mathsf{K}(2,15)$, we obtain a tuple which extends to a
trivial coloring of $\mathsf{K}(2,5)$, indeed:

$\displaystyle z_{0}$ $\displaystyle=y_{0}y_{1}y_{2}=(1\ 2)$ $\displaystyle
z_{1}$ $\displaystyle=y_{3}y_{4}y_{5}=(1\ 2)$ $\displaystyle z_{2}$
$\displaystyle=y_{6}y_{7}y_{8}=(1\ 2)$ $\displaystyle z_{3}$
$\displaystyle=y_{9}y_{10}y_{11}=(1\ 2)$ $\displaystyle z_{4}$
$\displaystyle=y_{12}y_{13}y_{14}=(1\ 2).$

A direct computation with GAP shows, in fact, that $\mathsf{K}(2,5)$ is not
$\operatorname{\mathsf{Conj}}(\mathsf{S}_{3})$-colorable.

######  Lemma 3.11.

Let $m,n\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$, and
let $(x_{0},\dots,x_{n-1})$ extend to a coloring of $\mathsf{K}(m,n)$. If
$x_{i}=x_{j}$ for some colors $x_{i},x_{j}$ with
$\operatorname{\mathsf{gcd}}(j-i,n)=1$, then $(x_{0},\dots,x_{n-1})$ extends
the trivial coloring. In particular, we have $x_{0}=x_{1}=\dots=x_{n-1}$.

###### Proof.

Since $\operatorname{\mathsf{gcd}}(m,n)=\operatorname{\mathsf{gcd}}(j-i,n)=1$,
both $m$ and $j-i$ are invertible modulo $n$. Let $u$ be the harlequin of
$(x_{0},\dots,x_{n-1})$ and consider $k=m^{-1}(j-i)$. Then, for every
$t\in\mathbbm{Z}$, we have

$x_{i}=x_{j}\implies x_{i}\rhd u^{tk}=x_{j}\rhd u^{tk}\implies
x_{i-t(j-i)}=x_{j-t(j-i)}.$

In particular, $x_{i}=x_{i-t(j-i)}$ for all $t\in\mathbbm{Z}$. Let
$h\in\\{0,\dots,n-1\\}$ and define $\bar{t}=(j-i)^{-1}(i-h)$. Then
$x_{h}=x_{i-\bar{t}(j-i)}$. Now, the choice of $h$ was arbitrary, therefore
all the colors in the tuple are equal, hence, the coloring is trivial. ∎

######  Corollary 3.12.

Let $m\in\mathbbm{N}$ and $p$ be a prime such that $p\nmid m$. Let
$(x_{0},\dots,x_{p-1})$ extend to a coloring of $\mathsf{K}(m,p)$. Then,
either it extends to the trivial coloring, or the colors in the tuple
$(x_{0},\dots,x_{p-1})$ are all distinct.

###### Proof.

Let $(x_{0},\dots,x_{p-1})$ extend to a coloring of $\mathsf{K}(m,p)$, and
assume that $x_{i}=x_{j}$ for some indices $0\leq i<j\leq p-1$. Since $p$ is
prime, the condition $\operatorname{\mathsf{gcd}}(j-i,p)=1$ trivially holds,
so Lemma 3.11 applies. ∎

###  3.3 The main characterization

The importance of the divisors of the parameters becomes immediately apparent
in light of the propositions presented in the preceding subsection. In this
investigation, we shall first conclude our analysis of the relationship
between these divisors and coloring. Our inquiry reveals that the colorability
of a knot can be attributed to (at the very least) one of the prime divisors
of the parameters, as demonstrated in Theorem 3.17. Following this, we proceed
to establish a specific characterization (see Theorem 3.19) concerning the
coloring of $\mathsf{K}(m,n)$, which relies only on a single element in the
group, in contrast to the dependence on $n$ elements as proven in Theorem 3.3.
This simplifies the process of verifying the colorability of a specific torus
knot, owing to the involvement of fewer elements and the group-theoretical
nature of the provided equivalent condition. Throughout this subsection, $G$
denotes any fixed group. We start with the following arithmetic lemma.

######  Lemma 3.13.

Let $a,b\in\mathbbm{Z}$ be such that $\operatorname{\mathsf{gcd}}(a,b)=1$.
Then $\operatorname{\mathsf{gcd}}(a-b,ab)=1$.

###### Proof.

By contradiction, let $p$ be a prime such that $p\mid a-b$ and $p\mid ab$.
Then, without loss of generality, $p\mid b$, which, together with the
condition $p\mid a-b$, implies that $p\mid a$, against the assumption of
$\operatorname{\mathsf{gcd}}(a,b)=1$. ∎

######  Proposition 3.14.

Let $m,p,q\in\mathbbm{N}$ be such that
$\operatorname{\mathsf{gcd}}(m,pq)=\operatorname{\mathsf{gcd}}(p,q)=1$. Then,
$\mathsf{K}(m,pq)$ is $\operatorname{\mathsf{Conj}}(G)$-colorable if and only
if either $\mathsf{K}(m,p)$ or $\mathsf{K}(m,q)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable.

###### Proof.

One implication follows directly from Proposition 3.7. Conversely, let
$(x_{0},\dots,x_{pq-1})$ extend to a non-trivial coloring of
$\mathsf{K}(m,pq)$. By contradiction, assume that both $\mathsf{K}(m,p)$ and
$\mathsf{K}(m,q)$ are not non-trivially colorable, and define

$y_{i}=\prod_{k=0}^{q-1}x_{iq+k},\quad\quad\mbox{for }i=0,\dots,p-1,$
$z_{j}=\prod_{k=0}^{p-1}x_{jq+k},\quad\quad\mbox{for }j=0,\dots,q-1.$

By the assumption and Proposition 3.8, the tuples $(y_{0},\dots,y_{p-1})$ and
$(z_{0},\dots,z_{q-1})$ extend to trivial colorings of $\mathsf{K}(m,p)$ and
$\mathsf{K}(m,q)$, respectively, in particular this implies that
$y_{0}=y_{1}=\dots=y_{p-1}$ and $z_{0}=z_{1}=\dots=z_{q-1}$, or equivalently

$x_{0}x_{1}\dots x_{q-1}=x_{q}x_{q+1}\dots
x_{2q-1}=\dots=x_{(p-1)q}x_{(p-1)q+1}\dots x_{pq-1},$ (5) $x_{0}x_{1}\dots
x_{p-1}=x_{p}x_{p+1}\dots x_{2p-1}=\dots=x_{(q-1)p}x_{(q-1)p+1}\dots
x_{pq-1}.$ (6)

Proceed by case analysis, distinguishing among the possible values of $m$.

Case $m=2$:

From Theorem 3.3, we have that

$x_{0}x_{1}=x_{1}x_{2}=\dots=x_{pq-1}x_{0}.$ (7)

Thus, for every $i,j=0,\dots,p-1$, we have

$\displaystyle y_{i}=y_{j}\ $
$\displaystyle\implies\prod_{k=0}^{q-1}x_{iq+k}=\prod_{k=0}^{q-1}x_{jq+k}$
$\displaystyle\implies\begin{cases}\displaystyle{x_{iq+0}\prod_{k=1}^{q-1}x_{iq+k}=x_{jq+0}\prod_{k=1}^{q-1}x_{jq+k}}\\\
\displaystyle{\prod_{k=0}^{q-2}x_{iq+k}x_{(i+1)-1}=\prod_{k=0}^{q-2}x_{jq+k}x_{(j+1)-1}}\end{cases}\implies\begin{cases}x_{iq}=x_{jq}\\\
x_{(i+1)-1}=x_{(j+1)-1}\end{cases}$

where we have canceled the product out because it is made by an even number of
factors with consecutive indices, exploiting the equations of Theorem 3.3(ii),
namely equations (7). Now, deleting the first and the last term of each of the
equations (5) and iterating this procedure, we obtain that

$x_{iq+k}=x_{jq+k}\quad\quad\forall\mkern 2.0mui,j=0,\dots,p-1,\,\forall\mkern
2.0muk=0,\dots,q-1.$ (8)

Proceeding in the same way, using $(z_{0},\dots,z_{q-1})$, we get

$x_{sp+h}=x_{tp+h}\quad\quad\forall\mkern 2.0mus,t=0,\dots,q-1,\,\forall\mkern
2.0muh=0,\dots,p-1.$ (9)

We may rewrite equations (8) and (9) in a more compact and equivalent form

$\displaystyle x_{iq+k}$ $\displaystyle=x_{k}\quad\quad\forall\mkern
2.0mui=0,\dots,p-1,\,\forall\mkern 2.0muk=0,\dots,q-1,$ (10) $\displaystyle
x_{jq+h}$ $\displaystyle=x_{h}\quad\quad\forall\mkern
2.0muj=0,\dots,q-1,\,\forall\mkern 2.0muh=0,\dots,p-1.$ (11)

Now, consider any $c\in\\{0,\dots pq-1\\}$. By the conditions on $m,p$ and
$q$, we may assume that $2<p<q$. By the division with reminder theorem we have
$c=aq+t,t=bp+r$ for some unique $a,b,t,r\in\mathbb{Z}$ with $0\leq t<q$ and
$0\leq r\leq p-1<q-1$, hence $c=aq+bp+r$. Consider the Diophantine equation
$jp+iq=r$, which has solution $(i,j)\in\mathbb{Z}^{2}$ because
$1=\operatorname{\mathsf{gcd}}(p,q)\mid r$. Then we have $c=(b+j)p+(a+i)q$,
and, computing indices modulo $pq$ together with equations (10) and (11), we
have

$x_{c}=x_{(b+j)p+(a+i)q}=x_{(b+j)p+(a+i)q+0}=x_{(a+i)q+0}=x_{0}.$

Now, the choice of $c$ was arbitrary, therefore all the colors
$x_{0},\dots,x_{pq-1}$ are equal, which is a contradiction.

Case $m>2$:

Since $\operatorname{\mathsf{gcd}}(q,m)=1$, consider $t=q^{-1}\pmod{m}$. By
the equations (5), multiplying $t$ elements with consecutive indices (possibly
with repetitions), we have

$\displaystyle y_{0}y_{1}\dots y_{t-1}=y_{1}y_{2}\dots
y_{t}\implies(x_{0}x_{1}\dots x_{q-1})y_{1}\dots y_{t-1}=(x_{q}x_{q+1}\dots
x_{2q-1})y_{2}\dots y_{t}\implies x_{0}=x_{q},$

where the last implication holds because the products are made by $tq-1$
factors with consecutive indices, which is divisible by $m$, thus Theorem
3.3(ii) holds. Proceeding in the same way, using $(z_{0},\dots,z_{q-1})$, we
obtain that $x_{0}=x_{p}$. Thus, we have $x_{p}=x_{q}$ and
$\operatorname{\mathsf{gcd}}(p,q)=1$ by assumption. Because of Lemma 3.13 we
also get $\operatorname{\mathsf{gcd}}(p-q,pq)=1$, which implies that all the
colors are equal because of Lemma 3.11, and this is a contradiction. ∎

######  Proposition 3.15.

Let $m,p,e\in\mathbbm{N}$ such that $p$ is prime and $p\nmid m$. Then,
$\mathsf{K}(m,p^{e})$ is $\operatorname{\mathsf{Conj}}(G)$-colorable if and
only if $\mathsf{K}(m,p)$ is $\operatorname{\mathsf{Conj}}(G)$-colorable.

###### Proof.

One implication follows directly from Proposition 3.7. Conversely, let
$(x_{0},\dots,x_{p^{e}-1})$ extend to a non-trivial coloring of
$\mathsf{K}(m,p^{e})$. There are three possible cases:

  1. (a)

$x_{i}=x_{j}$ if and only if $i\equiv j\pmod{p}$.

  2. (b)

There are $i,j\in\\{0,\dots,p^{e-1}-1\\}$ such that $x_{i}=x_{j}$ if and only
if $i\not\equiv j\pmod{p}$.

  3. (c)

All the $x_{i}$’s are different colors.

Observe that case (a) provides a non-trivial coloring for $\mathsf{K}(m,p)$,
because the equations in Theorem 3.3 hold already. Case (b) is impossible
because it would contradict Lemma 3.11, being
$\operatorname{\mathsf{gcd}}(j-i,p^{e})=1$. Assume case (c). Let $u$ be the
harlequin of $(x_{0},\dots,x_{p^{e}-1})$, and define

$y_{i}=\prod_{k=0}^{p^{e-1}-1}x_{ip^{e-1}+k},\quad\quad\mbox{for
}i=0,\dots,p-1.$

From Proposition 3.8 we know that it extends to a coloring of
$\mathsf{K}(m,p)$ with harlequin $v=u^{p^{e-1}}$. Proceed by case analysis,
distinguishing among the possible values of $m$.

Case $m=2$:

If the colors $y_{0},\dots,y_{p-1}$ are all different, the proof is completed.
Assume that two of them are equal, say $y_{i}=y_{j}$ for some distinct
$i,j\in\\{0,\dots,p-1\\}$. Then

$\displaystyle y_{i}=y_{j}$
$\displaystyle\implies\prod_{k=0}^{p^{e-1}-1}x_{ip^{e-1}+k}=\prod_{k=0}^{p^{e-1}-1}x_{jp^{e-1}+k}$
$\displaystyle\implies
x_{ip^{e-1}}\prod_{k=1}^{p^{e-1}-1}x_{ip^{e-1}+k}=x_{jp^{e-1}}\prod_{k=1}^{p^{e-1}-1}x_{jp^{e-1}+k}\implies
x_{ip^{e-1}}=x_{jp^{e-1}}$

where we have canceled the product out because it is made by an even number of
factors with consecutive indices, exploiting the equations of Theorem 3.3(ii).
This leads to a contradiction because we are assuming that all the colors are
different, hence this sub-case is indeed impossible.

Case $m>2$:

From Corollary 3.12 we know that either the elements of the tuple
$(y_{0},\dots,y_{p-1})$ are all distinct, or $(y_{0},\dots,y_{p-1})$ extends
to the trivial coloring. In the first case, the proof is completed. Assume
that $y_{0}=y_{1}=\dots=y_{p-1}$. Since $\operatorname{\mathsf{gcd}}(m,p)=1$,
consider $t=p^{-1}\pmod{m}$. Multiplying $t$ elements with consecutive indices
(possibly with repetitions), we have

$\displaystyle y_{0}y_{1}\dots y_{t-1}=y_{1}y_{2}\dots
y_{t}\implies(x_{0}x_{1}\dots x_{p-1})y_{1}\dots y_{t-1}=(x_{p}x_{p+1}\dots
x_{2p-1})y_{2}\dots y_{t}\implies x_{0}=x_{p},$

where the last implication holds because the products are made by $tp-1$
factors, which is divisible by $m$, thus Theorem 3.3(ii) holds. This leads to
a contradiction because we are assuming that all the colors of
$(x_{0},\dots,x_{p^{e}-1})$ are different, hence also this sub-case is
impossible. ∎

######  Proposition 3.16.

Let $m,n\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$.
Then, $\mathsf{K}(m,n)$ is $\operatorname{\mathsf{Conj}}(G)$-colorable if and
only if there is a prime factor $q$ of $n$ such that $\mathsf{K}(m,q)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable.

###### Proof.

One direction follows from Proposition 3.7. For the other, write the prime
factorization of $n=\prod_{i=1}^{k}p_{i}^{e_{i}}$ and conclude with a simple
induction argument using Propositions 3.14 and 3.15. ∎

######  Theorem 3.17.

Let $m,n\in\mathbbm{N}$ be such that $\operatorname{\mathsf{gcd}}(m,n)=1$.
Then, $\mathsf{K}(m,n)$ is $\operatorname{\mathsf{Conj}}(G)$-colorable if and
only if there is a prime factor $p$ of $m$ and a prime factor $q$ of $n$ such
that $\mathsf{K}(p,q)$ is $\operatorname{\mathsf{Conj}}(G)$-colorable.

###### Proof.

One direction follows from Propositions 3.5 and 3.7. For the converse, we can
use Proposition 3.16, together with Theorem 2.9, to infer that

$\displaystyle\mathsf{K}(m,n)\mbox{ is colorable }$
$\displaystyle\implies\mathsf{K}(m,q)\mbox{ is colorable for some }q\mid n$
$\displaystyle\implies\mathsf{K}(q,m)\mbox{ is colorable for some }q\mid n$
$\displaystyle\implies\mathsf{K}(q,p)\mbox{ is colorable for some }q\mid
n\mbox{ and }p\mid m$ $\displaystyle\implies\mathsf{K}(p,q)\mbox{ is colorable
for some }p\mid m\mbox{ and }q\mid n.\qed$

######  Remark 3.18.

Due to Theorem 3.17, when examining the colorability of $\mathsf{K}(m,n)$ we
can assume that $n$ is prime and that $m\nmid n$. We will see that assuming
$m$ to be prime as well is inconsequential. It is important to note that if
$m=1$ then $\mathsf{K}(1,n)$ is only trivially colorable, so we can also
assume $m\neq 1$.

We present now the main results of this paper, related to the study of
conjugation quandle coloring of torus knots.

######  Theorem 3.19.

Let $G$ be a group, and let $m,p\in\mathbbm{N}$ be such that $m\geq 2$ and $p$
is prime with $p\nmid m$. Then $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable if and only if there is $u\in G$
such that the centralizers
$\mathsf{C}_{G}(u^{p})\setminus\mathsf{C}_{G}(u)\neq\emptyset$.

###### Proof.

We prove that $\mathsf{K}(m,p)$ is $\operatorname{\mathsf{Conj}}(G)$-colorable
if and only if there are $x_{0},u\in G$ such that

$\begin{cases}ux_{0}u^{-1}\neq x_{0}\\\ u^{p}x_{0}u^{-p}=x_{0},\end{cases}$

which is equivalent to the condition on the centralizers in the statement.

Let $(x_{0},\dots x_{p-1})$ extend to a non-trivial coloring of
$\mathsf{K}(m,p)$ with harlequin $u$. Because of Corollary 3.12, all the
colors must be distinct. Because of Theorem 3.3, for every
$t\in\\{0,\dots,p-1\\}$ we have $x_{0}\rhd u^{k}=x_{t}$ for
$k=-m^{-1}t\pmod{p}$, that is, all the colors can be obtained from $x_{0}$ and
(a suitable power of) $u$. Therefore, we have $x_{0},u\in G$ such that

$\begin{cases}ux_{0}u^{-1}=x_{-m}\\\ ux_{-m}u^{-1}=x_{-2m}\\\ \vdots\\\
ux_{-(p-1)}u^{-1}=x_{-p}=x_{0}\end{cases}$

which is equivalent to

$\begin{cases}ux_{0}u^{-1}=x_{-m}\neq x_{0}\\\ u^{p}x_{0}u^{-p}=x_{0}\\\
\end{cases}$

where the second equation is obtained by combining all the previous ones. Note
that, since all the colors are different, in this setting, we do not need to
require also $u^{i}x_{0}u^{-i}\neq x_{0}$ for all $i=0,\dots,p-1$, because if
we had both $u^{i},u^{p}\in\mathsf{C}_{G}(x_{0})$, being the centralizer a
subgroup of $G$, we would also have
$u^{\operatorname{\mathsf{gcd}}(i,p)}=u^{1}=u\in\mathsf{C}_{G}(x_{0})$, which
is forbidden by the first equation. Conversely, let $x_{0},u\in G$ as in the
statement, and define $x_{-im}=u^{i}x_{0}u^{-i}$ for $i=\\{0,\dots,p-1\\}$.
This is indeed a coloring because, by definition of conjugation quandle, we
have $x_{-im}\rhd u=x_{-im-m}$, and $x_{-pm}=x_{0}$. Moreover, since
$\operatorname{\mathsf{gcd}}(m,p)=1$, this is enough to define all the colors.
∎

######  Remark 3.20.

Observe that the colorability of $\mathsf{K}(m,p)$ is independent on $m$. This
allows us to designate the initial entry as any convenient number, provided
that the second parameter is assumed to be prime.

######  Remark 3.21.

In the notation of Theorem 3.19, note that if there is an element $u\in G$ of
order $p$ such that $u\not\in\mathsf{Z}(G)$, then $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(G)$-colorable.

##  4 Coloring with particular groups

We now apply the theorems mentioned above to specific cases. We initiate the
analysis by considering matrix groups, and then proceed with dihedral and
symmetric groups.

###  4.1 General linear group

In this subsection, with $p$ and $q$ we denote two prime numbers, and the
group $\mathsf{GL}(2,q)$ is denoted by $G$. Our objective is to derive a
numerical characterization for the colorability of $\mathsf{K}(m,p)$ solely in
terms of $p$ and $q$, as we have determined that the parameter $m$ is
irrelevant (see Remark 3.20).

######  Remark 4.1.

The following table displays the representatives of the conjugacy classes of
$\mathsf{GL}(2,q)$, together with their centralizers. In virtue of Theorem
3.19, for a representative $u$ of each conjugacy class, we want to compute
when the condition
$\mathsf{C}_{\mathsf{GL}(2,q)}(u^{p})\setminus\mathsf{C}_{\mathsf{GL}(2,q)}(u)\neq\emptyset$
holds, so we aim to recreate the table with $u^{p}$ instead of $u$ and compare
the results.

Type | $u$ | $\mathsf{C}_{\mathsf{GL}(2,q)}(u)$
---|---|---
Type 1 | $\begin{pmatrix}a&0\\\ 0&a\end{pmatrix}\quad a\neq 0$ |  $\mathsf{GL}(2,q)$
Type 2 | $\begin{pmatrix}a&0\\\ 0&b\end{pmatrix}\quad 0<a<b$ | $\left\\{\begin{pmatrix}u&0\\\ 0&v\end{pmatrix}\in\mathsf{GL}(2,q)\colon u,v\neq 0\right\\}$
Type 3 | $\begin{pmatrix}a&1\\\ 0&a\end{pmatrix}\quad a\neq 0$ | $\left\\{\begin{pmatrix}u&v\\\ 0&u\end{pmatrix}\in\mathsf{GL}(2,q)\colon u\neq 0\right\\}$
Type 4 | $\begin{pmatrix}0&1\\\ a&b\end{pmatrix}\quad x^{2}-bx-a\mbox{ irreducible}$ | $\left\\{\begin{pmatrix}u&v\\\ au&u+bv\end{pmatrix}\in\mathsf{GL}(2,q)\colon u\neq 0\mbox{ or }v\neq 0\right\\}$

######  Proposition 4.2.

Let $u\in G$ be a matrix of type 1. Then
$\mathsf{C}_{G}(u)=\mathsf{C}_{G}(u^{p})=G$.

###### Proof.

The matrix power $u^{p}$ is still a scalar matrix, hence its centralizer is
again maximal. ∎

######  Proposition 4.3.

Let $u\in G$ be a matrix of type 2. Then
$\mathsf{C}_{G}(u^{p})\setminus\mathsf{C}_{G}(u)\neq\emptyset$ if and only if
$p\mid q-1$.

###### Proof.

The matrix power $u^{p}=\big{(}\begin{smallmatrix}a^{p}&0\\\
0&b^{p}\end{smallmatrix}\big{)}$ is still diagonal, so its centralizer is
strictly larger if and only if $u^{p}$ is a scalar matrix, that is when
$a^{p}\equiv b^{p}\pmod{q}$. This happens if and only if $(ab^{-1})^{p}\equiv
1\pmod{q}$ that is when $\mathbb{F}_{q}^{\times}$ has an element $u$ of order
$p$ of the form $u=ab^{-1}$, or equivalently when $p\mid q-1$, by Cauchy’s
Theorem. ∎

######  Proposition 4.4.

Let $u\in G$ be a matrix of type 3. Then
$\mathsf{C}_{G}(u^{p})\setminus\mathsf{C}_{G}(u)\neq\emptyset$ if and only if
$p=q$.

###### Proof.

A direct computation shows that the matrix power

$u^{p}=\begin{pmatrix}a&1\\\
0&a\end{pmatrix}^{p}=\begin{pmatrix}a^{p}&pa^{p-1}\\\ 0&a^{p}\end{pmatrix}$

has the same centralizer as the matrix $u$, unless $u^{p}$ is a scalar matrix.
This happens when $pa^{p-1}\equiv 0\pmod{q}$ that is, when $p=q$, being both
primes. ∎

######  Lemma 4.5.

Let $u=\big{(}\begin{smallmatrix}0&1\\\ a&b\end{smallmatrix}\big{)}\in G$ be a
matrix of type 4. Then for every $n\geq 1$ we have

$\begin{pmatrix}0&1\\\ a&b\end{pmatrix}^{n}=\begin{pmatrix}x_{n-1}&y_{n-1}\\\
x_{n}&y_{n}\end{pmatrix}$

where

$\begin{cases}x_{0}=0\\\
y_{0}=1\end{cases},\quad\quad\begin{cases}x_{n}=ay_{n-1}\\\
y_{n}=x_{n-1}+by_{n-1}.\end{cases}\quad n\geq 1.$

###### Proof.

It follows easily by induction. ∎

######  Lemma 4.6.

In the notation of Lemma 4.5, assuming $q\neq 2$, we have

$y_{n}=\frac{d^{-1}}{2}\left((d+c)^{n+1}+(d-c)^{n+1}\right)$

where $c=\frac{b}{2}$ and $d=\frac{\sqrt{b^{2}+4a}}{2}$.

###### Proof.

By Lemma 4.5 we have for every $n\in\mathbb{N}$

$\begin{pmatrix}x_{n}\\\ y_{n}\end{pmatrix}=\begin{pmatrix}0&a\\\
1&b\end{pmatrix}^{n}\begin{pmatrix}0\\\ 1\end{pmatrix}.$

Let $A=\big{(}\begin{smallmatrix}0&a\\\ 1&b\end{smallmatrix}\big{)}$. Compute
the matrix power $A^{n}$ using the diagonalization technique. Let
$\lambda_{1}=c+d,\lambda_{2}=c-d\in\mathbb{F}_{q^{2}}$ be the eigenvalues of
$A$, and let $U\in\mathsf{GL}(2,q^{2})$ be the matrix such that
$U^{-1}AU=\big{(}\begin{smallmatrix}\lambda_{1}&0\\\
0&\lambda_{2}\end{smallmatrix}\big{)}$. Hence
$A^{n}=U\big{(}\begin{smallmatrix}\lambda_{1}^{n}&0\\\
0&\lambda_{2}^{n}\end{smallmatrix}\big{)}U^{-1}$ and

$\begin{pmatrix}x_{n}\\\ y_{n}\end{pmatrix}=\begin{pmatrix}k&l\\\
m&r\end{pmatrix}\begin{pmatrix}\lambda_{1}^{n}\\\
\lambda_{2}^{n}\end{pmatrix}$

for some $k,l,m,r\in\mathbb{F}_{q^{2}}$. Using the known conditions
$x_{0}=0,y_{0}=1,x_{1}=a,y_{1}=b$ we obtain

$k=\frac{a}{\lambda_{1}-\lambda_{2}}=\frac{a}{2d},\quad\quad
l=\frac{-a}{\lambda_{1}-\lambda_{2}}=\frac{-a}{2d},\quad\quad
m=\frac{b-\lambda_{2}}{\lambda_{1}-\lambda_{2}}=\frac{d+c}{2d},\quad\quad
r=\frac{\lambda_{1}-b}{\lambda_{1}-\lambda_{2}}=\frac{d-c}{2d}.$

Now compute the powers $\lambda_{1}^{n}$ and $\lambda_{2}^{n}$ using the
Binomial Theorem:

$\displaystyle\lambda_{1}^{n}$
$\displaystyle=(c+d)^{n}=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2k}d^{2k}c^{n-2k}+\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2k+1}d^{2k+1}c^{n-2k-1}$
$\displaystyle\lambda_{2}^{n}$
$\displaystyle=(c-d)^{n}=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2k}d^{2k}c^{n-2k}-\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2k+1}d^{2k+1}c^{n-2k-1}$

Therefore

$y_{n}=m\lambda_{1}^{n}+r\lambda_{2}^{n}=\ \dots\
=d^{-1}\sum_{\begin{subarray}{c}i=0\\\
i\,odd\end{subarray}}^{n+1}\binom{n+1}{i}d^{i}c^{(n+1)-i}=\frac{d^{-1}}{2}\left((d+c)^{n+1}+(d-c)^{n+1}\right).$

∎

######  Lemma 4.7.

The condition $p\mid q+1$ holds if and only if there are $p-1$ elements $u$ of
multiplicative order $p$ in $\mathbb{F}_{q^{2}}$, all belonging to
$\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$, and each of those can be
expressed as

$u=\frac{c+d}{c-d}$

where $c\in\mathbb{F}_{q}$ and
$d\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$.

###### Proof.

Assume that $p\mid q+1$, thus $p\mid q^{2}-1=|\mathbb{F}_{q^{2}}|$. Under this
assumption, it is known that in $\mathbb{F}_{q^{2}}$ there are exactly $p-1>0$
elements of order $p$. Assume, by contradiction, that one of those was in
$\mathbb{F}_{q}$. Then it would generate a multiplicative cyclic subgroup of
$\mathbb{F}_{q}^{\times}$ of order $p$ containing all such elements, and
implying that $p\mid q-1$ which is impossible, being $p\neq 2$. Then every
element of order $p$ must belong to
$\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$. Consider now, for every
$c\in\mathbb{F}_{q^{2}}^{\times}$ the map
$\varphi_{c}\colon\mathbb{F}_{q^{2}}\to\mathbb{F}_{q^{2}}$ defined by

$\varphi_{c}(x)=\begin{cases}\displaystyle{\frac{c+x}{c-x}}\quad x\neq c\\\
-1\quad\quad x=c\end{cases}.$

It is easy to see that $\varphi_{c}$ is bijective. Since $p>2$, the element
$-1$ does not have order $p$. Therefore all the $p-1$ elements of order $p$
are contained in $\varphi_{c}\left(\mathbb{F}_{q^{2}}\setminus\\{c\\}\right)$.
Note that, if for an element $u\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$
of order $p$ we have $u=\varphi_{c}(x)$, for some $c\in\mathbb{F}_{q}$, then,
necessarily, we need to have $x\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$,
otherwise, we would get that $u\in\mathbb{F}_{q}$.

Conversely, the existence of elements of order $p$ in
$u\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$ implies that $p\mid
q^{2}-1=|\mathbb{F}_{q^{2}}^{\times}|$. If we had $p\mid
q-1=|\mathbb{F}_{q}^{\times}|$ then by Cauchy’s theorem,
$\mathbb{F}_{q}^{\times}$ would contain one (hence, all) of them, which is
against the assumptions. It follows that $p\mid q+1$. ∎

######  Lemma 4.8.

In the notation of Lemma 4.5 and Lemma 4.6, we have $y_{p-1}\equiv 0\pmod{q}$
if and only if $p\mid q+1$.

###### Proof.

Analyze first when $q=2$. The only case in which the polynomial $x^{2}-bx-a$
is irreducible in $\mathbb{F}_{2}$ is when $a=b=1$, and its splitting field is
$\mathbb{F}_{2}[x]/\langle x^{2}+x+1\rangle\cong\mathbb{F}_{4}$. Assuming
$a=b=1$, a simple induction argument shows that $y_{n}\equiv 0\pmod{2}$ if and
only if $n\equiv 2\pmod{3}$, thus, $y_{p-1}\equiv{0}$ if and only if $p\equiv
0\pmod{3}$, that is $p=3$. Therefore, the claim holds for $q=2$.

Assume now $q\neq 2$, then Lemma 4.6 applies and we have

$\displaystyle y_{p-1}\equiv
0\pmod{q}\iff\frac{d^{-1}}{2}\left((d+c)^{n+1}+(d-c)^{n+1}\right)\equiv
0\pmod{q}\iff\left((c+d)(c-d)^{-1}\right)^{p}\equiv 1\pmod{q}$

which holds if and only if the equation $u^{p}=1$ has solution in
$\mathbb{F}_{q^{2}}$ for some $u$ of the form $u=\frac{c+d}{c-d}$ with
$c\in\mathbb{F}_{q}$ and $d\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q}$. The
conclusion now follows from Lemma 4.7. ∎

######  Proposition 4.9.

Let $u\in G$ be a matrix of type 4. Then
$\mathsf{C}_{G}(u^{p})\setminus\mathsf{C}_{G}(u)\neq\emptyset$ if and only if
$p\mid q+1$.

###### Proof.

A direct computation shows that the matrix power $u^{p}$ has the same
centralizer as the matrix $u$ unless $u^{p}$ is a scalar matrix, and this
happens if and only if $y_{p-1}\equiv 0\pmod{q}$. The conclusion now follows
from Lemma 4.8. ∎

We summarize Propositions 4.2, 4.3, 4.4, 4.9 in the following table.

Type | $u^{p}$ | $\mathsf{C}_{\mathsf{GL}(2,q)}(u^{p})\setminus\mathsf{C}_{\mathsf{GL}(2,q)}(u)\neq\emptyset$
---|---|---
Type 1 | $\begin{pmatrix}a^{p}&0\\\ 0&a^{p}\end{pmatrix}\quad a\neq 0$ | Never
Type 2 | $\begin{pmatrix}a^{p}&0\\\ 0&b^{p}\end{pmatrix}\quad 0<a<b$ | $p\mid q-1$
Type 3 | $\begin{pmatrix}a^{p}&pa^{p-1}\\\ 0&a^{p}\end{pmatrix}\quad a\neq 0$ | $p=q$
Type 4 | $\begin{pmatrix}x_{p-1}&y_{p-1}\\\ ay_{p-1}&x_{p-1}+by_{p-1}\end{pmatrix}\quad x^{2}-bx-a\mbox{ irreducible}$ | $p\mid q+1$

In conclusion, by combining all the information acquired from the previous
results with the fact that conjugacy classes form a partition of $G$, we
obtain the main result of this subsection.

######  Theorem 4.10
($\operatorname{\mathsf{Conj}}(\mathsf{GL}(2,q))$-coloring of Torus Knots).

Let $m,p\in\mathbbm{N}$ be such that $m\geq 2$ and $p$ is prime with $p\nmid
m$. The torus knot $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(\mathsf{GL}(2,q))$-colorable if and only if
$p\mid q(q+1)(q-1)$.

###  4.2 Special linear group

We proceed in a similar manner as in the case of $\mathsf{GL}(2,q)$,
distinguishing among representatives of conjugacy classes and their
centralizers. Once again, throughout this subsection, with $p$ and $q$ we
denote two prime numbers, and the special linear group $\mathsf{SL}(2,q)$ is
denoted by $G$. Our objective is to obtain a numerical characterization for
the colorability of $\mathsf{K}(m,p)$ solely in terms of $p$ and $q$.

Type | $u$ | $\mathsf{C}_{\mathsf{SL}(2,q)}(u)$
---|---|---
Type 1 | $\begin{pmatrix}a&0\\\ 0&a\end{pmatrix}\quad a^{2}=1$ |  $\mathsf{SL}(2,q)$
Type 2 | $\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\quad a\neq 0$ | $\left\\{\begin{pmatrix}u&0\\\ 0&u^{-1}\end{pmatrix}\in\mathsf{SL}(2,q)\colon u\neq 0\right\\}$
Type 3 | $\begin{pmatrix}a&b\\\ 0&a\end{pmatrix}\quad a^{2}=1,b=1\mbox{ or }b\mbox{ non-square}$ | $\left\\{\begin{pmatrix}u&v\\\ 0&u\end{pmatrix}\in\mathsf{SL}(2,q)\colon u^{2}=1\right\\}$
Type 4 | $\begin{pmatrix}0&1\\\ -1&a\end{pmatrix}\quad a=r+r^{q},r\in\mathbb{F}_{q^{2}}\setminus\mathbb{F}_{q},r^{q+1}=1$ | $\left\\{\begin{pmatrix}u&v\\\ -v&u+av\end{pmatrix}\in\mathsf{SL}(2,q)\colon u(u+av)+v^{2}=1\right\\}$

######  Remark 4.11.

The subsequent propositions can be proven using the same techniques presented
for the case of $\mathsf{GL}(2,q)$ above. In many cases, the computations turn
out to be exactly the same; however, in some instances, we encounter certain
refinements due to the fewer parameters involved. The proofs are substantially
identical to those presented in Section 4.1. Therefore, in the following
discussion, we simply state the results for the case where
$G=\mathsf{SL}(2,q)$.

######  Proposition 4.12.

Let $u\in G$ be a matrix of type 1. Then
$\mathsf{C}_{G}(u)=\mathsf{C}_{G}(u^{p})=G$.

######  Proposition 4.13.

Let $u\in G$ be a matrix of type 2. Then
$\mathsf{C}_{G}(u^{p})\setminus\mathsf{C}_{G}(u)\neq\emptyset$ if and only if
$p\mid q-1$.

######  Proposition 4.14.

Let $u\in G$ be a matrix of type 3. Then
$\mathsf{C}_{G}(u^{p})\setminus\mathsf{C}_{G}(u)\neq\emptyset$ if and only if
$p=q$.

######  Proposition 4.15.

Let $u\in G$ be a matrix of type 4. Then
$\mathsf{C}_{G}(u^{p})\setminus\mathsf{C}_{G}(u)\neq\emptyset$ if and only if
$p\mid q+1$.

Combining the previous results, we obtain exactly the same numeric condition
as in Theorem 4.10. We may therefore strengthen the statement, adjoining the
claim related to the special linear group.

######  Theorem 4.16.

Let $m,p\in\mathbbm{N}$ be such that $m\geq 2$ and $p$ is prime with $p\nmid
m$. The following are equivalent.

  1. (i)

The torus knot $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(\mathsf{GL}(2,q))$-colorable.

  2. (ii)

The torus knot $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(\mathsf{SL}(2,q))$-colorable.

  3. (iii)

$p\mid q(q+1)(q-1)$.

###  4.3 Dihedral group

We now proceed to discuss the dihedral groups. We denote the dihedral group of
the $n$-gon as ${\mathsf{D}_{n}}$, and employ the following presentation:

$\mathsf{D}_{n}=\left\langle r,s\colon r^{n}=s^{2}=1,\
srs=r^{-1}\right\rangle.$

Our objective is to derive a numerical characterization for the colorability
of $\mathsf{K}(m,p)$ solely in terms of $p$ and $n$.

######  Theorem 4.17 ($\operatorname{\mathsf{Conj}}(\mathsf{D}_{n})$-coloring
of Torus Knots).

Let $m,p\in\mathbbm{N}$ be such that $m\geq 2$ and $p$ is prime with $p\nmid
m$. The torus knot $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(\mathsf{D}_{n})$-colorable if and only if $p\mid
n$.

###### Proof.

Assume $p\nmid n$. Then there’s an element
$u=r^{\frac{n}{p}}\in\mathsf{D}_{n}$ of order $p$. It is well known that

$\mathsf{Z}(\mathsf{D}_{n})=\left\langle\begin{array}[]{lr}\\{1\\}&n\mbox{
odd}\\\ \\{1,r^{\frac{n}{2}}\\}&n\mbox{ even}\end{array}\right..$

If $n$ is odd, then trivially $u\not\in\mathsf{Z}(\mathsf{D}_{n})$. If $n$ is
even, we also have $u\not\in\mathsf{Z}(\mathsf{D}_{n})$, because if that was
the case, we would have $\frac{n}{2}=\frac{n}{p}$, implying $p=2$, which is
excluded. By Remark 3.21, we conclude that $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(\mathsf{D}_{n})$-colorable.

Conversely, we assume that $p\nmid n$ and prove that $\mathsf{K}(m,p)$ is not
$\operatorname{\mathsf{Conj}}(\mathsf{D}_{n})$-colorable. By Theorem 3.19, it
is enough to check that the condition
$\mathsf{C}_{\mathsf{D}_{n}}(u^{p})\setminus\mathsf{C}_{\mathsf{D}_{n}}(u)\neq\emptyset$
never holds. Recall that every element of $u\in\mathsf{D}_{n}$ may be uniquely
expressed as $u=s^{t}r^{k}$ for some $t\in\\{0,1\\}$ and
$k\in\\{0,\dots,n-1\\}$. For $t=1$, then $u$ is an involution, hence
$\mathsf{C}_{\mathsf{D}_{n}}(u^{p})=\mathsf{C}_{\mathsf{D}_{n}}(u)$. Assume
$t=0$, that is $u=r^{k}$. A direct computation shows that

$\mathsf{C}_{\mathsf{D}_{n}}(r^{k})=\mathsf{C}_{\mathsf{D}_{n}}(r^{pk})=\left\langle\begin{array}[]{lr}\langle
r\rangle&k\neq\frac{n}{2}\\\ \mathsf{D}_{n}&k=\frac{n}{2}\end{array}\right.$

therefore, also in this case,
$\mathsf{C}_{\mathsf{D}_{n}}(u^{p})\setminus\mathsf{C}_{\mathsf{D}_{n}}(u)\neq\emptyset$
does not hold. ∎

###  4.4 Symmetric group

We conclude this section and the paper with a discussion on the symmetric
groups. We denote the symmetric group over $n$ letters as ${\mathsf{S}_{n}}$.
Our goal is, again, to derive a numerical characterization for the
colorability of $\mathsf{K}(m,p)$ solely in terms of $p$ and $n$.

######  Theorem 4.18 ($\operatorname{\mathsf{Conj}}(\mathsf{S}_{n})$-coloring
of Torus Knots).

Let $m,p\in\mathbbm{N}$ be such that $m\geq 2$ and $p$ is prime with $p\nmid
m$. The torus knot $\mathsf{K}(m,p)$ is
$\operatorname{\mathsf{Conj}}(\mathsf{S}_{n})$-colorable if and only if $p\leq
n$.

###### Proof.

If $p\leq n$, then $p\mid n!=|\mathsf{S}_{n}|$, hence $\mathsf{S}_{n}$ has an
element of order $p$, which is not in the centre because
$\mathsf{Z}(\mathsf{S}_{n})=\\{1\\}$. Remark 3.21 allows us to conclude that
$\mathsf{K}(m,p)$ is $\operatorname{\mathsf{Conj}}(\mathsf{S}_{n})$-colorable.

Conversely, we assume that $p<n$ and prove that $\mathsf{K}(m,p)$ is not
$\operatorname{\mathsf{Conj}}(\mathsf{D}_{n})$-colorable. By Theorem 3.19, it
is enough to check that the condition
$\mathsf{C}_{\mathsf{S}_{n}}(u^{p})\setminus\mathsf{C}_{\mathsf{S}_{n}}(u)\neq\emptyset$
never holds. Let $u\in\mathsf{S}_{n}$ and consider its complete factorization
in disjoint cycles $u=\sigma_{1}\dots\sigma_{t}$, for some $t\geq 1$. Since
disjoint cycles commute, we have $u^{p}=\sigma_{1}^{p}\dots\sigma_{t}^{p}$. In
particular, since $p<n$ prime, if $\sigma$ is an $r$-cycle, then also
$\sigma^{p}$ is an $r$-cycle. This implies that $u$ and $u^{p}$ have the same
cycle structure, hence they are conjugate in $\mathsf{S}_{n}$. Consider
$\tau\in\sigma_{n}$ such that $u^{p}=\tau u\tau^{-1}$. Then
$\mathsf{C}_{\mathsf{S}_{n}}(u^{p})=\mathsf{C}_{\mathsf{S}_{n}}(\tau
u\tau^{-1})=\tau\mathsf{C}_{\mathsf{S}_{n}}(u)\tau^{-1}$, hence
$|\mathsf{C}_{\mathsf{S}_{n}}(u^{p})|=|\mathsf{C}_{\mathsf{S}_{n}}(u)|$. Since
it always hold that
$\mathsf{C}_{\mathsf{S}_{n}}(u)\leq\mathsf{C}_{\mathsf{S}_{n}}(u^{p})$, the
two centralisers must be equal, hence $\mathsf{K}(m,p)$ is not
$\operatorname{\mathsf{Conj}}(\mathsf{S}_{n})$-colorable if $p<n$. ∎

##  5 Conclusions

We end this paper by posing a few questions for potential future research.
Through applying the broad description of torus knots, we have derived
characterization theorems for coloring torus knots by employing conjugation
quandles with specific groups. Is it possible to extend this approach to
additional small groups?

######  Problem 1.

Characterize the conjugation quandle coloring of $\mathsf{K}(m,n)$ using other
small groups.

Moreover, there exists a knot-theoretical tool that enables the association of
a polynomial with any given knot, encoding certain properties. This technique
is known as the Alexander polynomial (see [Mur96]). The Alexander polynomials
of torus knots are well-understood and easy to manipulate, involving specific
divisibility conditions on their parameters. It’s only natural to inquire
whether there is a correlation between Alexander polynomials and colorings.

######  Problem 2.

What are the relations (if any) between the conjugation quandle coloring of
$\mathsf{K}(m,n)$ and its Alexander polynomial?

Furthermore, a well-known family of satellite knots is the one consisting of
Whitehead doubles. For a given knot $K$, its Whitehead double $\mathsf{W}(K)$
is constructed by duplicating its arcs and introducing two additional
crossings (see [JL12]). Is it feasible to formulate a characterization theorem
for the quandle colorability of $\mathsf{W}(\mathsf{K}(m,n))$ using a strategy
akin to what we achieved in Theorem 3.19? Ideally, this approach would begin
by simplifying the task, initially omitting divisors and following a similar
pattern as seen in Theorem 3.17.

######  Problem 3.

Characterize the conjugation quandle colorability of the Whitehead double of
$\mathsf{K}(m,n)$.

Acknowledgements

> This paper is built upon research carried out during the Ph.D. studies of
> the author, who received partial support from both the GAUK grant
> (301-10/252012) and Z. Patáková’s Primus grant (301-45/247107). Furthermore,
> the author wishes to express gratitude to their Ph.D. advisor, David
> Stanovský, for his patient guidance and inquisitive approach. Additionally,
> appreciation is extended to Petr Vojtěchovský for offering valuable insights
> that have enriched specific prior results.

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