# A combinatorial algebraic approach for the modified second-generation time-
delay interferometry

Zhang-Qi Wu1 Pan-Pan Wang1<EMAIL_ADDRESS>Wei-Liang Qian2,3,4
<EMAIL_ADDRESS>Cheng-Gang Shao1<EMAIL_ADDRESS>1 MOE Key Laboratory of
Fundamental Physical Quantities Measurement, Hubei Key Laboratory of
Gravitation and Quantum Physics, PGMF, and School of Physics, Huazhong
University of Science and Technology, Wuhan 430074, China  2 Escola de
Engenharia de Lorena, Universidade de São Paulo, 12602-810, Lorena, SP, Brazil
3 Faculdade de Engenharia de Guaratinguetá, Universidade Estadual Paulista,
12516-410, Guaratinguetá, SP, Brazil  4 Center for Gravitation and Cosmology,
College of Physical Science and Technology, Yangzhou University, Yangzhou
225009, China

(Oct. 11th, 2022)

###### Abstract

We generalize the combinatorial algebraic approach first proposed by
Dhurandhar et al. to construct various classes of modified second-generation
time-delay interferometry (TDI) solutions. The main idea behind the algorithm
is to enumerate, in a given order, a specific type of commutator between two
monomials defined by the products of particular time-displacement operators.
On the one hand, the above commutators can be systematically rewritten as the
elements of a left ideal, defined by the l.h.s. of the relevant equation for
the TDI solution. On the other hand, these commutators are shown to vanish if
we only keep up the first-order contributions regarding the rate of change of
armlengths. In other words, each commutator furnishes a valid TDI solution
pertaining to the given type of modified second-generation combinations. In
this work, the original algorithm, which only involves time-delay operators,
is extended by introducing the time-advance ones and then utilized to seek
solutions of the Beacon, Relay, Monitor, Sagnac, and fully symmetric Sagnac
types. We discuss the relation between the present scheme’s solutions and
those obtained by the geometric TDI approach, a well-known method of
exhaustion of virtual optical paths. In particular, we report the results on
novel Sagnac-inspired solutions that cannot be straightforwardly obtained
using the geometric TDI algorithm. The average response functions, floor noise
power spectral densities, and sensitivity functions are evaluated for the
obtained solutions.

##  I Introduction

In terms of ground-based laser interferometry, the first ever direct detection
of gravitational waves was accomplished by LIGO and Virgo collaborations in
2015 [1]. The observation has been widely recognized as the inauguration of an
era of gravitational-wave astronomy, as a new window to the universe had
become available besides that via the electromagnetic spectrum. Nonetheless,
ground-based gravitational detections are constrained by a couple of crucial
factors, such as the baseline length, Earth’s seismic vibrations, and gravity-
gradient noise. To this end, the measurement is only feasible for the
frequency band typically above 10Hz [2]. On the other hand, abundant wave
sources, potentially for more robust and durable signals, are associated with
a lower frequency band of 0.1mHz to 1Hz. The latter is aimed at by the ongoing
space-based gravitational wave detector projects, which include, notably, the
LISA [3], TianQin [4], Taiji [5], and DECIGO [6]. A space-borne detector
typically consists of three identical spacecraft that form a giant, almost
equilateral triangle configuration. Encoded as Doppler frequency shifts, the
information on the gravitational wave is embedded in the resultant beat notes
between the laser beams exchanged among the spacecraft. Unlike their ground-
based counterpart, the relative frequency fluctuations primarily come from the
laser’s intrinsic phase instability. In particular, the laser frequency noise
cannot be simply canceled out using an equal-arm Michelson configuration. This
is because the distances between spacecraft vary in time according to the
orbital dynamics, which gives rise to armlengths mismatch, where the rate of
change of the armlength is between $5$ and $10\mathrm{m/s}$ [3]. Typically,
the dominant laser frequency noise is about $7-8$ orders of magnitude higher
than those from other sources [7].

In this regard, the TDI algorithm, first introduced by Tinto _et al._ in 1999
[8], aims to construct a virtual equal-arm interferometer by a proper
combination of the delayed science data streams in order to suppress the laser
frequency noise [7]. The existing TDI solutions can be divided into four
categories: first-generation, modified first-generation, second-generation,
and modified second-generation. The first-generation TDI [9] approximates the
entire triangular configuration as a rigid constellation without rotation, and
subsequently, the detector arms are treated as static. The modified first-
generation TDI considers the Sagnac effect caused by a rigid rotation of the
entire constellation [10]. For both cases, mathematically, the delay
operations involved are commutative. Therefore, the solution space is a
polynomial ring $\mathscr{R}$ in three [11] and six [12] variables over the
rational numbers. Subsequently, the problem can be reformulated [11, 12] to
solve for the first module of syzygies of a relevant left ideal, whose
generators can be obtained through the Groebner basis [13]. The second-
generation TDI further considers nonrigid rotation so that the armlengths vary
independently but slowly in time [14, 15]. The residual noise is therefore
evaluated as an expansion in terms of time-derivatives of the armlengths
truncated at the second-order contributions. The modified second-generation
TDI discriminates between and enforces independent cancelation of distinct
cyclic directions of the detector arms, leading to a more restrictive
truncation scheme [15]. Owing to the non-commutative nature, the second and
modified second-generation TDI solutions are not straightforward. Vallisneri
proposed the geometric TDI [16], a method of exhaustion to seek TDI
combinations by enumerating close trajectories in the space-time diagram. The
approach is intuitive because it can be shown that a geometric TDI solution
corresponds to virtual equal-arm interferometry. It has been extensively
utilized in the literature to solve the second and modified second-generation
TDI solutions [17, 15]. Nonetheless, since the solution space of the geometric
TDI method grows by $3^{n}$ where $n$ is the number of links, searching for
TDI combinations is often computationally expensive. Moreover, by definition,
the algorithm demands that successive links must be “connected”. Thus the
solution space is somewhat restrictive. In particular, the well-known fully
symmetric Sagnac solutions of both generations lie beyond such a solution
space.

Since the TDI algorithm was first proposed, it has flourished experimentally
and theoretically in the literature. Regarding ground-based experiments, Vine
_et al._ verified the noise cancellation performance of the Sagnac TDI
combination [18], and more recently, Vinckier _et al._ implemented the optical
comb TDI scheme [19] by using the acousto-optic modulators and optical combs.
These results demonstrated TDI’s viability in effectively suppressing laser
frequency noise. From the statistical inference perspective, Romano _et al._
[20] pointed out that the principal component analysis of the noise covariance
matrix effectively furnishes a feasible TDI solution, and subsequently,
further development of Bayesian TDI [21, 22] was proposed. Recently,
Dhurandhar _et al._ showed [23] a close relationship between two matrix-based
TDI approaches that were independently proposed by Vallisneri [24] _et al._
and Tinto [25] _et al._. In addition, efforts have been devoted to related
topics such as eliminating residual clock noise using optical combs [26]. The
clock noise cancellation scheme has also been generalized to the second-
generation TDI combination by using sideband measurement [27, 28]. Notably,
Dhurandhar _et al._ elaborated a combinatorial algebraic algorithm [29] for
the modified second-generation TDI solutions in the case of one arm being
dysfunctional. In this simplified case, the polynomials that furnish a valid
TDI solution are specific elements of a polynomial ring in four variables that
corresponds to the kernel of the homomorphism

$\varphi:\mathscr{R}^{2}\to\mathscr{R},$ (1)

namely, the first module of syzygies. The proposed algorithm enumerates, in a
given order, a specific type of commutator between two monomials defined by
the products of particular time-displacement operators. The authors showed
that the commutators in question are the elements of the above left ideal
$\mathscr{R}^{2}$. Moreover, these commutators manifestly vanish if we only
keep up the first-order contributions regarding the rate of change of
armlengths. In other words, these commutators correspond to valid TDI
solutions. It was shown how such solutions could be systematically
constructed, and then the algorithm was applied to solve for the Michelson-
type TDI combinations of the modified second-generation.

Based on Dhurandhar et al.’s results [29], the present paper further develops
and explores the algorithm. By inspecting the procedure, we argue that the
interchange operation introduced in the original work is not an indispensable
element. Besides, the original algorithm, which only involves time-delay
operators, is extended by introducing the time-advance ones. As will be
discussed below, the essence of the algorithm is the introduction of
additional constraints to the TDI equation. Consequently, the solution space
simplifies so that the TDI solution can be constructed in terms of particular
commutators. Moreover, we show that the above generalizations expand the
solution space, giving rise to a more flexible and robust algorithm. Then, it
is utilized to seek TDI solutions of the Beacon, Relay, Monitor, Sagnac, and
fully symmetric Sagnac types. In particular, we report a novel set of Sagnac-
inspired solutions that cannot be straightforwardly derived using the
geometric TDI. We also discuss the relationship between the present scheme’s
solutions and those obtained by the geometric TDI approach. The average
response functions, floor noise power spectral densities, and sensitivity
functions are evaluated for the obtained solutions.

The remainder of the paper is organized as follows. In Sec. II, we briefly
present the problem of the TDI algorithm, together with the notations and
conventions used in this paper. The combinatorial approach proposed by
Dhurandhar et al. is discussed in Sec. III. The original application to
Michelson-type TDI solutions is revisited. In Sec. IV, the approach is
elaborated further by including time-advance operators. Subsequently, the
properties of the commutator are derived and discussed. A more general version
of the algorithm is formulated. Subsequently, in Sec. V, the main results are
presented by taking the Monitor type TDI solutions as an example. We show that
the nine sixteen-link modified second-generation TDI combinations, initially
obtained by employing the geometric TDI, can be readily retrieved.
Furthermore, we apply the method to the case of fully symmetric Sagnac
combinations. Besides the well-known solutions in the literature, we present a
few novel Sagnac-inspired solutions which cannot be straightforwardly obtained
using the geometric TDI. The average response functions, residual noise power
spectral densities, and the sensitivity curves of the obtained novel solutions
are evaluated. Further discussions and concluding remarks are given in the
last section. Some complementary derivations and discussions are relegated to
the Appendices of the paper. In Appendix A, we give proof of some of the
mathematical relations utilized in the main text. A few other classes of
modified second-generation TDI solutions, namely, the Beacon, Relay, and
Sagnac TDI combinations, are derived and presented in Appendix B using the
method proposed in the present paper. In Appendix C, we enumerate the lower-
order commutators that furnish the TDI solutions using the proposed algorithm.
We also show that two classes of higher-order TDI solutions can be induced
using the lower-order ones, and the results are presented in Appendix D. The
explicit expressions of noise power spectral densities and the average
response functions of the detector for the Sagnac-type combinations are given
in Appendix E.

##  II Definitions, notations, and conventions of the TDI algorithm

Figure 1:  A schematic diagram of the three-spacecraft constellation for the
space-borne gravitational wave detector.

As illustrated in Fig. 1, the experimental layout of a space-based
gravitational wave detector consists of three identical spacecraft, denoted as
SC$i$ (with $i=1,2,3$) [7]. The armlengths sitting on the opposite side of
SC$i$ are denoted as $L_{i}$ (and $L_{i^{\prime}}$) in the counterclockwise
(and clockwise) direction. Two optical benches, labeled by $i$ and
$i^{\prime}$, are installed on each spacecraft. The phasemeters of the optical
benches perform the measurements of three types of data streams, namely, the
science data streams $s_{i(i^{\prime})}$, test mass data streams
$\epsilon_{i(i^{\prime})}$, and reference data streams $\tau_{i(i^{\prime})}$.
The science data streams carry the essential information on the gravitational
waves embedded in the beat notes between the laser beams from the distant and
local spacecraft. The test mass and reference data streams are formed by the
interference between the two local laser beams from the adjacent optical
benches $i$ and $i^{\prime}$, whereas for the test mass data stream, one of
the laser beams is reflected from the test mass. By considering the laser
frequency noise $p_{i(i^{\prime})}$, optical bench motion noise
$\dot{\vec{\Delta}}_{i(i^{\prime})}$, test mass noise
$\dot{\vec{\delta}}_{i,(i^{\prime})}$, shot noise $N_{i(i^{\prime})}^{opt}$
associated with the optical benches $i(i^{\prime})$. The relevant data streams
recorded at the optical bench possess the following forms

$\displaystyle s_{i}(t)$
$\displaystyle=D_{i-1}p_{{(i+1)}^{{}^{\prime}}}(t)-p_{i}(t)+\nu_{(i+1)^{{}^{\prime}}}[\vec{n}_{i-1}\cdot
D_{i-1}\dot{\vec{\Delta}}_{(i+1)^{{}^{\prime}}}(t)+\vec{n}_{(i-1)^{{}^{\prime}}}\cdot\dot{\vec{\Delta}}_{i}(t)]+H_{i}(t)+N_{i}^{opt}(t),$
$\displaystyle\epsilon_{i}(t)$
$\displaystyle=p_{{i}^{{}^{\prime}}}(t)-p_{{i}}(t)-2\nu_{i^{{}^{\prime}}}[\vec{n}_{(i-1)^{{}^{\prime}}}\cdot\dot{\vec{\delta}}_{i}(t)-\vec{n}_{(i-1)^{{}^{\prime}}}\cdot\dot{\vec{\Delta}}_{i}(t)],$
$\displaystyle\tau_{i}(t)$ $\displaystyle=p_{i^{{}^{\prime}}}(t)-p_{i}(t),$
(2)

and

$\displaystyle s_{i^{{}^{\prime}}}(t)$
$\displaystyle=D_{{(i+1)}^{{}^{\prime}}}p_{i-1}(t)-p_{i^{{}^{\prime}}}(t)+\nu_{i-1}[\vec{n}_{i+1}\cdot\dot{\vec{\Delta}}_{i^{{}^{\prime}}}(t)+\vec{n}_{(i+1)^{{}^{\prime}}}\cdot
D_{(i+1)^{{}^{\prime}}}\dot{\vec{\Delta}}_{i-1}(t)]+H_{i^{{}^{\prime}}}(t)+N_{i^{{}^{\prime}}}^{opt}(t),$
$\displaystyle\epsilon_{i^{{}^{\prime}}}(t)$
$\displaystyle=p_{{i}}(t)-p_{{i}^{{}^{\prime}}}(t)-2\nu_{i}[\vec{n}_{i+1}\cdot\dot{\vec{\delta}}_{i^{{}^{\prime}}}(t)-\vec{n}_{i+1}\cdot\dot{\vec{\Delta}}_{i^{{}^{\prime}}}(t)],$
$\displaystyle\tau_{{i}^{{}^{\prime}}}(t)$
$\displaystyle=p_{i}(t)-p_{i^{{}^{\prime}}}(t),$ (3)

where $H_{i}(t),H_{i^{{}^{\prime}}}(t)$ represent the gravitational wave
signals, $D_{i(i^{\prime})}$ are the time-delay operators along the related
armlengths, $\nu_{i(i^{\prime})}$ are the laser’s frequency, and
$\vec{n}_{i(i^{{}^{\prime}})}$ are unit vectors along the armlengths.
Following the standard procedure of the TDI algorithm, the optical bench
motion noise can be eliminated by introducing intermediate variables [27].
Subsequently, the two local lasers are effectively connected by intra-
spacecraft phase locking, and the resultant observables read

$\displaystyle{\eta_{i}}(t)$
$\displaystyle={H_{i}}(t)+{{D}_{i-1}}{p_{i+1}}(t)-{p_{i}}(t)+\nu_{(i+1)^{{}^{\prime}}}{{\vec{n}}_{i-1}}\left[{{{D}_{i-1}}{{\dot{\vec{\delta}}}_{(i+1)^{\prime}}}(t)-{{\dot{\vec{\delta}}}_{i}}(t)}\right]+N_{i}^{{opt}}(t),$
$\displaystyle{\eta_{i^{\prime}}}(t)$
$\displaystyle={H_{i^{\prime}}}(t)+{{D}_{(i+1)^{\prime}}}{p_{i-1}}(t)-{p_{i}}(t)+\nu_{i-1}{{\vec{n}}_{i+1}}\cdot\left[{{{\dot{\vec{\delta}}}_{i^{\prime}}}(t)-{{D}_{(i+1)^{\prime}}}{{\dot{\vec{\delta}}}_{i-1}}(t)}\right]+N_{i^{\prime}}^{{opt}}(t).$
(4)

In terms of the above observables, a valid TDI solution [7] is aimed to
eliminate the three remaining independent laser frequency noise $p_{i}$ by a
combination of the form

$\mathrm{TDI}=\sum_{i=1,2,3}(q_{i}\eta_{i}+q_{i^{{}^{\prime}}}\eta_{i^{\prime}}),$
(5)

where $q_{i}$ and $q_{i^{\prime}}$ are polynomials in the six time-delay
operators. By focusing on the laser frequency noise and explicitly demanding
the coefficients before individual $p_{i}$ vanish, the above equation gives

$\displaystyle
q_{1}+q_{1^{\prime}}-q_{2^{\prime}}{D}_{3^{\prime}}-q_{3}{D}_{2}=0,$ (6)
$\displaystyle
q_{2}+q_{2^{\prime}}-q_{3^{\prime}}{D}_{1^{\prime}}-q_{1}{D}_{3}=0,$
$\displaystyle
q_{3}+q_{3^{\prime}}-q_{1^{\prime}}{D}_{2^{\prime}}-q_{2}{D}_{1}=0.$

Using two equations, one may choose to remove two of the six coefficients. For
instance, by eliminating $q_{1}$ and $q_{2}$, one finds

$q_{3}(1-D_{231})+q_{1^{\prime}}(D_{31}-D_{2^{\prime}})+q_{2^{\prime}}(D_{1}-D_{3^{\prime}31})+q_{3^{\prime}}(1-D_{1^{\prime}1})=0.$
(7)

From the geometric TDI perspective [15], different generations of the TDI
solutions can be classified by how the delayed laser frequency noise, as an
expansion using time derivatives of the armlengths, are truncated. To be
specific, for the first-generation TDI, one considers

$L_{i}(t)=L_{i}=\mathrm{const.},\ \ L_{i}=L_{i^{{}^{\prime}}}.$ (8)

For the modified first-generation TDI, one distinguishes armlengths of
different cyclic orders, namely,

$L_{i}(t)=L_{i}=\mathrm{const.},\ \ L_{i}\neq L_{i^{{}^{\prime}}}.$ (9)

Following the literature [14], one refers to the case where

$L_{i}(t)=L_{i}+t\dot{L}_{i},\ \ L_{i}\neq L_{i^{{}^{\prime}}},\ \
\dot{L}_{i}=\dot{L}_{i^{{}^{\prime}}}$ (10)

as the second-generation TDI. Lastly, the solutions satisfying

$L_{i}(t)=L_{i}+t\dot{L}_{i},\ \ L_{i}\neq L_{i^{{}^{\prime}}},\ \
\dot{L}_{i}\neq\dot{L}_{i^{{}^{\prime}}},$ (11)

as investigated by the studies [15] will be dubbed as the modified second-
generation TDI 111It is noted that the last case was referred to as the
second-generation TDI in some literature [35]..

For the first-generation TDI defined in Eqs. (8) and (9), the polynomials
$q_{i(i^{\prime})}$ form a commutative ring which allows us to solve for the
TDI combination using the Groebner basis [13]. On the other hand, the delay
operators in Eq. (6) can no longer be viewed as commutative in the context of
the second generation TDI defined in Eqs. (10) and (11). To be more precise,
the solution space of Eq. (6) is a left module over a non-commutative ring
$\mathscr{R}=\mathbb{Q}(D_{1},D_{2},D_{3},D_{1^{{}^{\prime}}},D_{2^{{}^{\prime}}},D_{3^{{}^{\prime}}})$,
where $\mathbb{Q}$ is the field of rational numbers. However, the non-
commutative nature of the problem implies that it is not straightforward to
solve for the general and exhaustive form of the TDI combinations using an
algebraic approach. In the next section, the properties of the commutators
between polynomials of the delay operator will be studied, which eventually
leads to a combinatorial algebraic approach.

##  III The original combinatorial algebraic algorithm

###  III.1 A specific type of commutators and the vanishing condition

By applying the delay operators $D_{i}$ and $D_{i^{{}^{\prime}}}$ on an
arbitrary time-dependent variable $\phi(t)$, we have

$\displaystyle D_{i}\phi(t)$ $\displaystyle=\phi(t-L_{i}(t)),$ (12)
$\displaystyle D_{ji}\phi(t)$
$\displaystyle=\phi(t-L_{j}(t)-L_{i}(t-L_{j}(t))),$

where, for convenience, one has denoted successive applications of the time-
delay operators by

$D_{j}D_{i}\phi(t)\equiv D_{ji}\phi(t).$ (13)

By making use of the expansion

$L_{i}(t)=L_{i}+t\dot{L}_{i}+\frac{1}{2}t^{2}\ddot{L}_{i}+\cdots,$ (14)

where the speed of light in the vacuum is taken as unit $c=1$, to the first
order, Eqs. (12) give,

$D_{i}\phi(t)\simeq\phi(t-L_{i})-\dot{\phi}(t-L_{i})t\dot{L}_{i},$ (15)

and

$\displaystyle D_{ji}\phi(t)$
$\displaystyle\simeq\phi(t-L_{j}(t)-L_{i}(t)+\dot{L}_{i}L_{j}(t))$ (16)
$\displaystyle\simeq\phi(t-L_{i}-L_{j})+\dot{\phi}(t-L_{i}-L_{j})\dot{L}_{i}L_{j}$
$\displaystyle-\dot{\phi}(t-L_{i}-L_{j})t(\dot{L}_{i}+\dot{L}_{j}).$

It is straightforward to generalize the above results to higher orders. For
instance, by applying three time-delay operators, one finds

$\displaystyle D_{kji}\phi(t)$ $\displaystyle\simeq\phi(t-L_{i}-L_{j}-L_{k})$
(17)
$\displaystyle+\dot{\phi}(t-L_{i}-L_{j}-L_{k})\big{(}\dot{L}_{i}(L_{j}+L_{k})+\dot{L}_{j}L_{k}\big{)}$
$\displaystyle-\dot{\phi}(t-L_{i}-L_{j}-L_{k})t(\dot{L}_{i}+\dot{L}_{j}+\dot{L}_{k}).$

It was first pointed out in [29] that many TDI combinations can be attributed
to a specific type of commutator whose form is related to the residual of the
laser frequency noise. For convenience, in what follows, we will use the
subscript $x$ to denote the first element of a commutator, and the subscript
$y$ will be reserved for the second element of the commutator. For instance, a
commutator will be formally written as $[D_{x_{1}x_{2}\dots
x_{n}},D_{y_{1}y_{2}\dots y_{n}}]$. Let us denote the commutator associated
with the residual laser noise of the TDI solution as

$\mathrm{TDI}^{p}=[D_{x_{1}x_{2}\dots x_{n}},D_{y_{1}y_{2}\dots
y_{n}}]p_{i}(t),$ (18)

where $x_{i}$ and $y_{i}$ are one of time-delay operators $D_{i}$ or
$D_{i^{{}^{\prime}}}$.

We illustrate the above statement with two examples. Let us take the modified
first-generation Michelson combination $X_{1}$ as the first example. We have

$\displaystyle{X}_{1}(t)$
$\displaystyle=\eta_{1}+D_{3}\eta_{2^{\prime}}+D_{33^{\prime}}\eta_{1^{\prime}}+D_{33^{\prime}2^{\prime}}\eta_{3}$
(19)
$\displaystyle-\left(\eta_{1^{\prime}}+D_{2^{\prime}}\eta_{3}+D_{2^{\prime}2}\eta_{1}+D_{2^{\prime}23}\eta_{2^{\prime}}\right).$

It is straightforward to show that the residual of the laser frequency noise
is

$X^{p}_{1}(t)=[D_{33^{{}^{\prime}}},D_{2^{{}^{\prime}}2}]p_{1}(t).$ (20)

For the second example, we consider the modified second-generation Michelson
combination $X_{2}$,

$\displaystyle X_{2}(t)$
$\displaystyle=\eta_{1}+{D}_{3}\eta_{2^{\prime}}+{D}_{33^{\prime}}\eta_{1^{\prime}}+{D}_{33^{\prime}2^{\prime}}\eta_{3}+{D}_{33^{\prime}2^{\prime}2}\eta_{1^{\prime}}$
(21)
$\displaystyle+{D}_{33^{\prime}2^{\prime}22^{\prime}}\eta_{3}+{D}_{33^{\prime}2^{\prime}22^{\prime}2}\eta_{1}+{D}_{33^{\prime}2^{\prime}22^{\prime}23}\eta_{2^{\prime}}$
$\displaystyle-\left(\eta_{1^{\prime}}+{D}_{2^{\prime}}\eta_{3}+{D}_{2^{\prime}2}\eta_{1}+{D}_{2^{\prime}23}\eta_{2^{\prime}}+{D}_{2^{\prime}233^{\prime}}\eta_{1}\right.$
$\displaystyle\left.+{D}_{2^{\prime}233^{\prime}3}\eta_{2^{\prime}}+{D}_{2^{\prime}233^{\prime}33^{\prime}}\eta_{1^{\prime}}+{D}_{2^{\prime}233^{\prime}332^{\prime}}\eta_{3}\right),$

the residual reads

$X^{p}_{2}(t)=[D_{33^{{}^{\prime}}2^{{}^{\prime}}2},D_{2^{{}^{\prime}}233^{{}^{\prime}}}]p_{1}(t).$
(22)

From the two Michelson combinations given by Eqs. (20) and (22), it is
observed that the residuals can always be written as commutators composed of
the products of two components $D_{33^{{}^{\prime}}}$ and
$D_{2^{{}^{\prime}}2}$. This indicates that one may construct TDI solutions
from specific commutators satisfying some rules. Indeed, modified second-
generation Michelson-type combinations of higher orders can be obtained from
the following residuals:

$[D_{33^{{}^{\prime}}2^{{}^{\prime}}22^{{}^{\prime}}233^{{}^{\prime}}},D_{2^{{}^{\prime}}233^{{}^{\prime}}33^{{}^{\prime}}2^{{}^{\prime}}2}],$
(23)
$[D_{33^{{}^{\prime}}33^{{}^{\prime}}2^{{}^{\prime}}22^{{}^{\prime}}2},D_{2^{{}^{\prime}}22^{{}^{\prime}}233^{{}^{\prime}}33^{{}^{\prime}}}],$
(24)

and

$[D_{33^{{}^{\prime}}2^{{}^{\prime}}233^{{}^{\prime}}2^{{}^{\prime}}2},D_{2^{{}^{\prime}}233^{{}^{\prime}}2^{{}^{\prime}}233^{{}^{\prime}}}].$
(25)

A systematic approach based on the above speculations was first proposed in
[29]. It consists of two crucial building blocks. First, one identifies a
specific type of commutator corresponding to vanishing residual laser noise.
Second, these commutators can be mapped to the TDI solution characterized by
the corresponding coefficients $q_{i(i^{\prime})}$. The first point will be
addressed shortly, and Sec. III.2 will be devoted to the second point.

In [31], the following relation was established

$\begin{gathered}{\left[D_{x_{1}x_{2}\ldots x_{n}},D_{y_{1}y_{2}\ldots
y_{n}}\right]\phi(t)}=\left(\sum_{k=1}^{n}L_{x_{k}}\sum_{m=1}^{n}\dot{L}_{y_{m}}-\sum_{m=1}^{n}L_{y_{m}}\sum_{k=1}^{n}\dot{L}_{x_{k}}\right)\dot{\phi}\left(t-\sum_{k=1}^{n}L_{x_{k}}-\sum_{m=1}^{n}L_{y_{m}}\right).\end{gathered}$
(26)

For the lower order cases, it can be readily verified using Eqs. (15)-(17). A
general proof of Eq. (26) is relegated to Appendix A.

Apparently, the first factor on the r.h.s. of Eq. (26) vanishes as long as

$y_{i}=x_{\pi(i)}.$ (27)

where $\pi\in\mathcal{S}_{n}$ is an arbitrary element of the permutation group
of degree $n$. In other words, Eq. (26) vanishes for a specific type of
commutator characterized by a given number of indices and specific
permutations. Moreover, if one can show that, under certain circumstances,
such commutator can be recognized as, for instance, the l.h.s. of Eq. (7), it
furnishes a valid TDI solution. The latter proposition will be explored in the
following subsection. For the remainder of the present subsection, however, we
elaborate further on the converse proposition by revisiting the Michelson
combinations Eq. (21) discussed above.

As a matter of fact, this class of solutions has been extensively explored in
[29], referred to as one-arm dysfunctional ones. Without loss of generality,
one assumes that the communications through the armlength connecting SC2 and
SC3 are interrupted. In other words, we have
$\eta_{2}=\eta_{{3}^{{}^{\prime}}}=0$, or equivalently,

$q_{2}=q_{3^{{}^{\prime}}}=0.$ (28)

Substituting Eq. (28) into Eq. (6), we have

$\displaystyle
q_{1}+q_{1^{\prime}}-q_{2^{\prime}}{D}_{3^{\prime}}-q_{3}{D}_{2}=0,$ (29a)
$\displaystyle q_{2^{\prime}}-q_{1}{D}_{3}=0,$ (29b) $\displaystyle
q_{3}-q_{1^{\prime}}{D}_{2^{\prime}}=0.$ (29c)

Eliminating Eqs. (29b) and (29c) by substituting the forms of $q_{2^{\prime}}$
and $q_{3}$ into Eq. (29a), one finds the desired equation

$q_{1}(1-D_{33^{{}^{\prime}}})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}2})=0.$
(30)

We note that the above elimination process algebraically ensures that the
residual laser noise is entirely governed by $p_{1}$. If one had chosen to
eliminate two other equations, the residual would have been determined by
other laser noise. The difference, however, is irrelevant from a modified
second-generation TDI perspective. Also, by adding distinct solutions, one may
construct an optimal solution that minimizes a specific target quantity.

By observing the specific form of Eq. (30), it is also apparent that
$D_{33^{{}^{\prime}}}$ and $D_{2^{{}^{\prime}}2}$ are the elementary units to
furnish any solution. For convenience, we denote $a=D_{33^{{}^{\prime}}}$ and
$b=D_{2^{{}^{\prime}}2}$, and Eq. (30) simplifies to read

$q_{1}(1-a)+q_{1^{{}^{\prime}}}(1-b)=0.$ (31)

Also, the solution Eq. (21) corresponds to the coefficients $q_{1}$ and
$q_{1^{{}^{\prime}}}$

$\displaystyle q_{1}$ $\displaystyle=1-b-ba+ab^{2},$ (32) $\displaystyle
q_{1^{{}^{\prime}}}$ $\displaystyle=-(1-a-ab+ba^{2}),$

and the residual Eq. (22) reads

$[ba,ab]p_{1}(t).$ (33)

Following [29], one can interpret the above residual as that of a cancelation
taking place between the two terms on the l.h.s. of Eq. (31) in an order-by-
order fashion. In other words, this motivated a procedure regarding how each
term of the two expressions given by Eqs. (32) is written down. As all the
lower order terms are canceled out identically, the procedure continues until
the last two remaining terms of the highest order coincidently give rise to a
commutator, which possesses the form of the l.h.s. of Eq. (26) while
satisfying Eq. (27). Subsequently, the commutator vanishes, reassuring that
Eqs. (21) or (32) is indeed a valid TDI solution.

Dhurandhar et al. introduced an interchange operator $\mathcal{I}$ between the
symbols $a$ and $b$ [29]. To be specific, for a given monomial $s$ composed of
$a$ and $b$, the string $t=\mathcal{I}(s)$ is obtained by replacing $a$ with
$b$, and meanwhile substituting original $b$ by $a$, in $s$. For instance,
$\mathcal{I}(abba)=baab$. The authors pointed out that for the one-arm
dysfunctional case, namely, the Michelson combinations, the solutions possess
the form $[s,\mathcal{I}(s)]$. Moreover, in accordance with the l.h.s. of Eq.
(26), if one has $n$ instances of $a$ in the string $s$, the count of $b$’s
must be the same. Therefore, the length of the string $s$ is $2n$, and the
commutator consists of strings of length $4n$.

###  III.2 TDI solutions derived from the commutators

The discussions elaborated by the end of the last subsection indicate that one
may derive TDI solutions by exhaustively enumerating all possible commutators.
To be specific, for $n=1$, the only commutator is $[ab,ba]$, which corresponds
to Eq. (33). For $n=2$, one has three options: $[a^{2}b^{2},b^{2}a^{2}]$,
$[abab,baba]$, and $[ab^{2}a,ba^{2}b]$. For an arbitrary integer number $n$,
there are $\frac{(2n)!}{2n!n!}$ relevant combinations of the form

$\Delta=[s_{2n},\mathcal{I}(s_{2n})]=s_{2n}\mathcal{I}(s_{2n})-\mathcal{I}(s_{2n})s_{2n},$
(34)

where $s_{2n}$ is an monomial comprised of $n$ instances of $a$ and $b$. Also,
$q_{1^{\prime}}=-\mathcal{I}(q_{1})$, which implies that we only need the
explicit form of $q_{1}$ for the solution of Eq. (31). The remaining
coefficients $q_{3},q_{2^{{}^{\prime}}}$ are subsequently obtained by Eqs.
(29b) and (29c).

Based on the above arguments, an algorithm was proposed [29] to derive TDI
solutions for the one-arm dysfunctional case systematically. The idea of the
algorithm is first to enumerate all possible commutators in question, then
deduce the polynomial coefficients $q_{i(i^{\prime})}$ of the TDI combinations
from a given commutator. The latter process is reiterated below, illustrated
by the control-flow diagram Fig. 2.

  1. 1.

Consider the first term in Eq. (34), $t_{4n}\equiv s_{2n}\mathcal{I}(s_{2n})$,
it is a monomial of degree $4n$. It is noted that $t_{4n}$ ends in either $a$
or $b$, that is, $t_{4n}=t_{4n-1}a$ or $t_{4n}=t_{4n-1}b$.

     * •

If $t_{4n}=t_{4n-1}a$, let $t_{4n-1}=t_{4n-1}$;

     * •

If $t_{4n}=t_{4n-1}b$, let $t_{4n-1}=-\mathcal{I}(t_{4n-1})$.

  2. 2.

Repeat the above procedure for $t_{4n-1}$ until the degree of the monomial
vanishes, namely, $t_{0}$. We note that $t_{0}=\pm 1$.

  3. 3.

The TDI coefficients are obtained by summing up $4n$ monomials:
$q_{1}=\sum_{k=0}^{4n-1}t_{k}$ and $q_{1^{{}^{\prime}}}=-\mathcal{I}(q_{1})$.

As an example, for the case of $[ab,ba]$ which $n=1$. The first term is
$t_{4}=ab^{2}a$, then we have $t_{3}=ab^{2}$, $t_{2}=-\mathcal{I}(ab)=-ba$,
$t_{1}=-b$, and $t_{0}=1$. It is readily verified that Eq. (32) is retrieved
by summing up the above $4$ monomials.

Figure 2:  The control-flow diagram of the algorithm proposed in [29] for
Michelson TDI combinations.

##  IV The extended combinatorial algebraic algorithm

In the last section, we revisited an existing algorithm dedicated to deriving
the modified second-generation Michelson TDI combinations using a specific
type of commutator. The present section extends the above approach to other
classes of TDI solutions. The generalization is essentially based on the
following considerations:

  * •

The algorithm proposed in [29] does not depend on the specific forms of $a$
and $b$, and it is viable as long as one encounters an equation of the form
Eq. (31).

  * •

Algebraically, to derive something similar to Eq. (31), one only needs to
introduce two additional constraints (e.g. Eq. (28)) into Eq. (6).

  * •

The introduction of the interchange operation $\mathcal{I}$ is not an
obligation. One only needs the condition Eq. (27) for commutator Eq. (26) to
vanish.

  * •

The number of variables of the non-commutative ring $\mathscr{R}$ can be
expanded to include time-advance operators, which leads to a generalized
commutation relation.

These points give rise to an extended version of the algorithm, which can be
used to derive TDI solutions of other classes, such as Beacon, Relay, Monitor,
Sagnac, and full symmetric Sagnac ones. Moreover, one encounters novel Sagnac-
inspired solutions that cannot be straightforwardly derived using the
geometric TDI.

In the remainder of the present section, we will elaborate on the general
algorithm. In particular, the generalized communtation relation will be
discussed in Sec. IV.1. Explicit examples will be explored in Sec. V.

###  IV.1 The generalized commutator

By observing other classes of TDI combinations well-known in the literature,
it is noted that the polynomial coefficients often contain time-advance
operators $D_{\bar{i}(\bar{i^{\prime}})}$. They are the inverse of the
corresponding time-delay operators, satisfying

$D_{\bar{i}}D_{i}=D_{i}D_{\bar{i}}=\mathscr{I},$ (35)

where $\mathscr{I}$ is the identity operator. When a time-advance operator is
applied to a time-dependent variable $\phi(t)$, Eq. (35) implies, to the first
order,

$D_{\bar{i}}\phi(t)=\phi(t+L_{i}(t+L_{i})),$ (36)

which gives

$\displaystyle{D}_{\overline{i}}\phi(t)$
$\displaystyle\simeq\phi\left({t}+{L}_{{i}}\right)+\dot{\phi}\left(t+L_{i}\right)\left(t+L_{i}\right)\dot{L}_{i}$
(37)
$\displaystyle=\phi\left({t}+{L}_{{i}}\right)+\dot{\phi}\left(t+L_{i}\right)t\dot{L}_{i}+\dot{\phi}\left(t+L_{i}\right)L_{i}\dot{L}_{i}.$

By comparing Eq. (37) against Eq. (15), one observes that in the argument of
the first and second terms, there is an extra $(-1)$ factor for $L_{i}$ and
$\dot{L}_{i}$. Besides, the last term on the second line of Eq. (37) gives
rise to a novel contribution.

Now, we proceed to show that Eq. (26) continues to be valid when the time-
advance operator is included. This can be done by appropriately modifying the
proof given in Appendix A, as follows. We assume that some operators in the
l.h.s. of Eq. (26) are time-advance ones and introduce modifications to the
existing derivations. Without loss of generality, let us assume there are $l$
instances of time-advance operators in $D_{x_{1}x_{2}\dots x_{n}}$ and $m$
instances in $D_{y_{1}y_{2}\dots y_{n}}$. To be specific, we denote the
subscripts corresponding to these operators by $\lambda_{i}$ with
($i=1,\cdots,l$) and $\gamma_{j}$ with ($j=1,\cdots,m$). Also, the additional
$(-1)$ factors in Eq. (37) can be accounted for by the factor

$\displaystyle\delta x_{k}$
$\displaystyle=\left\\{\begin{matrix}-1&&\mathrm{if}\ k=\lambda_{i}\
\mathrm{for\ any}\ i=1,\cdots,l\\\ +1&&\mathrm{otherwise}\end{matrix}\right.,$
(38) $\displaystyle\delta y_{k^{\prime}}$
$\displaystyle=\left\\{\begin{matrix}-1&&\mathrm{if}\ k^{\prime}=\gamma_{j}\
\mathrm{for\ any}\ j=1,\cdots,m\\\ +1&&\mathrm{otherwise}\end{matrix}\right..$

By substituting Eq. (37) into $\left[D_{x_{1}x_{2}\ldots
x_{n}},D_{y_{1}y_{2}\ldots y_{n}}\right]$, the novel contributions owing to
the third term in Eq. (37) read

$\displaystyle\left(\sum_{i=1}^{l}\dot{L}_{x_{\lambda_{i}}}L_{x_{\lambda_{i}}}+\sum_{j=1}^{m}\dot{L}_{y_{\gamma_{j}}}L_{y_{\gamma_{j}}}-\sum_{i=1}^{l}\dot{L}_{x_{\lambda_{i}}}L_{x_{\lambda_{i}}}-\sum_{j=1}^{m}\dot{L}_{y_{\gamma_{j}}}L_{y_{\gamma_{j}}}\right)\dot{\phi}\left(t-\sum_{k=1}^{n}\delta_{x_{k}}L_{x_{k}}-\sum_{k^{\prime}=1}^{n}\delta_{y_{k^{\prime}}}L_{y_{k^{\prime}}}\right)=0,$
(39)

which manifestly vanishes. Therefore, by putting all the pieces together, we
arrive at the following generalized form of Eq. (26)

$\begin{gathered}{\left[D_{x_{1}x_{2}\ldots x_{n}},D_{y_{1}y_{2}\ldots
y_{n}}\right]\phi(t)}=\left(\sum_{k=1}^{n}\delta_{x_{k}}L_{x_{k}}\sum_{k^{{}^{\prime}}=1}^{n}\delta_{y_{k^{{}^{\prime}}}}\dot{L}_{y_{k^{{}^{\prime}}}}-\sum_{k^{{}^{\prime}}=1}^{n}\delta_{y_{k^{{}^{\prime}}}}L_{y_{k^{{}^{\prime}}}}\sum_{k=1}^{n}\delta_{x_{k}}\dot{L}_{x_{k}}\right)\dot{\phi}\left(t-\sum_{k=1}^{n}\delta_{x_{k}}L_{x_{k}}-\sum_{k^{\prime}=1}^{n}\delta_{y_{k^{\prime}}}L_{y_{k^{\prime}}}\right).\end{gathered}$
(40)

It is now apparent that the vanishing condition Eq. (27) remains unchanged
when the time-advance operators are considered. By using a broadened choice of
variables provided by the generalized commutation relation Eq. (40), in the
following subsection, we elaborate on an extended combinatorial algebraic
approach for the modified second-generation TDI.

###  IV.2 The extended combinatorial algorithm

We start by discussing an explicit example not embraced by the original
algorithm. To be specific, we consider alternative Michelson combination [16,
15] given by

$\displaystyle{X}_{2}(t)$
$\displaystyle=\left[\eta_{1}+D_{3}\eta_{2^{\prime}}+D_{33^{\prime}}\eta_{1^{\prime}}+D_{33^{\prime}2^{\prime}}\eta_{3}+D_{33^{\prime}2^{\prime}2}\eta_{1}\right.$
(41)
$\displaystyle\left.+D_{33^{\prime}2^{\prime}23}\eta_{2^{\prime}}-D_{33^{\prime}2^{\prime}233^{\prime}\overline{2}}\eta_{3}-D_{33^{\prime}2^{\prime}233^{\prime}\bar{2}\bar{2}^{\prime}}\eta_{1^{\prime}}\right]$
$\displaystyle-\left[\eta_{1^{\prime}}+D_{2^{\prime}}\eta_{3}+D_{2^{\prime}2}\eta_{1}+D_{2^{\prime}23}\eta_{2^{\prime}}-D_{2^{\prime}233^{\prime}\overline{2}}\eta_{3}\right.$
$\displaystyle\left.-D_{2^{\prime}233^{\prime}\overline{2}\overline{2}^{\prime}}\eta_{1^{\prime}}+D_{2^{\prime}233^{\prime}\overline{2}\overline{2}^{\prime}}\eta_{1}+D_{2^{\prime}233^{\prime}\overline{2}\overline{2}^{\prime}3}\eta_{2^{\prime}}\right].$

As pointed out by some authors, the above combination occupies a shorter space
in the time domain, which can effectively avoid the influence of instrument
gaps and glitches. Moreover, in the high-frequency region, its gravitational
wave response and noise power spectral density also display better
performance, as discussed in [16].

The coefficients $q_{1}$ and $q_{1^{{}^{\prime}}}$ of the above solution can
be extracted and found to be

$\displaystyle q_{1}$ $\displaystyle=1-b+ab-ba\bar{b},$ (42) $\displaystyle
q_{1^{{}^{\prime}}}$ $\displaystyle=-(1-a-ba\bar{b}+aba\bar{b}).$

By substituting the solution into Eq. (31), the residual is found to be
$\Delta=[ba\bar{b},a]$.

At first glance, it seems that the form of the residual does not satisfy the
condition Eq. (27), as the two terms in the commutator do not even have the
same length. They are, nonetheless, indeed related by a permutation by making
use of Eq. (35) and, therefore

$\Delta=[ba\bar{b},a]=[ba\bar{b},a\bar{b}b].$ (43)

Moreover, following the train of thought of the combinatorial approach
discussed in the previous section, one can explicitly show how the residual
Eq. (43) is related to the equation Eq. (31). This is achieved by successively
subtracting out terms proportional to the factor $(1-a)$ or $(1-b)$ until the
last trailing “$1$”, which gives

$ba\bar{b}a=-ba\bar{b}(1-a)+ba\bar{b}(1-b)-b(1-a)-(1-b)+1,$ (44)

and

$aba\bar{b}=aba\bar{b}(1-b)-ab(1-a)-a(1-b)-(1-a)+1.$ (45)

Subsequently, one has

$\Delta=(1-b+ab-ba\bar{b})(1-a)-(1-a-ba\bar{b}+aba\bar{b})(1-b),$ (46)

which essentially indicates that the coefficients of Eq. (31) are given by Eq.
(42). It is not difficult to observe that the above derivations are generally
applicable to solving the TDI equation of the following form

$\alpha(1-a)+\beta(1-b)=0,$ (47)

where the forms of $a$ and $b$ are not specified. Its valid solution is
associated with a commutator of a rather arbitrary form which is composed of
two monomials of time-delay and time-advance operators

$\Delta=\left[s_{n},\bar{s}_{n}\right]\equiv t_{2n}-\bar{t}_{2n},$ (48)

once it satisfies the condition Eq. (27). Also, as demonstrated by the above
example, the monomials in question do not necessarily have the same counts of
$a$s and $b$s. Naturally, the two terms in the commutator are not obliged to
be related by the interchange operation $\mathcal{I}$.

For a given commutator, the extended combinatorial algorithm is summarized as
follows, which is also illustrated by the control-flow diagram Fig. 3.

  1. 1.

Initiate $\alpha=0$ and $\beta=0$

  2. 2.

Consider the first term on the r.h.s. of Eq. (48), $t_{2n}$, it is a monomial
of degree $2n$. It is noted that $t_{2n}$ ends in either $a$, $\bar{a}$, $b$,
or $\bar{b}$, namely, $t_{2n}=t_{2n-1}a$, $t_{2n}=t_{2n-1}\bar{a}$,
$t_{2n}=t_{2n-1}b$, or $t_{4n}=t_{2n-1}\bar{b}$.

     * •

If $t_{2n}=t_{2n-1}a$, let $\alpha=\alpha-t_{2n-1}$;

     * •

If $t_{2n}=t_{2n-1}\bar{a}$, let $\alpha=\alpha+t_{2n-1}\bar{a}$;

     * •

If $t_{2n}=t_{2n-1}b$, let $\beta=\beta-t_{2n-1}$;

     * •

If $t_{2n}=t_{2n-1}\bar{b}$, let $\beta=\beta+t_{2n-1}\bar{b}$;

  3. 3.

Repeat the above procedure 2 for $t_{2n-1}$ until the degree of the monomial
vanishes, namely, $t_{0}$. We note that $t_{0}=1$.

  4. 4.

Perform the steps similar to 2 and 3 for $\bar{t}_{2n}$,

     * •

If $\bar{t}_{2n}=\bar{t}_{2n-1}a$, let $\alpha=\alpha+\bar{t}_{2n-1}$;

     * •

If $\bar{t}_{2n}=\bar{t}_{2n-1}\bar{a}$, let
$\alpha=\alpha-\bar{t}_{2n-1}\bar{a}$;

     * •

If $\bar{t}_{2n}=\bar{t}_{2n-1}b$, let $\beta=\beta+\bar{t}_{2n-1}$;

     * •

If $\bar{t}_{2n}=\bar{t}_{2n-1}\bar{b}$, let
$\beta=\beta-\bar{t}_{2n-1}\bar{b}$;

until the degree of the monomial vanishes. It is noted that $\bar{t}_{0}=1$.

  5. 5.

Both $t_{2n}-t_{0}$ and $\bar{t}_{2n}-\bar{t}_{0}$ can be rewritten as a
summation of multipliers of either $(1-a)$ or $(1-b)$. Therefore, the TDI
coefficients of Eq. (47) are the resulting $\alpha$ and $\beta$.

As an example, for the case of Eq. (43), we have $t_{4}=ba\bar{b}a$ and
$\bar{t}_{4}=aba\bar{b}$. Subsequently, we have $t_{3}=ba\bar{b}$, $t_{2}=ba$,
$t_{1}=b$, $t_{0}=1$; $\bar{t}_{3}=aba$, $\bar{t}_{2}=ab$, $\bar{t}_{1}=a$,
and $\bar{t}_{0}=1$. So that
$\alpha=-t_{3}-t_{1}+\bar{t}_{2}+\bar{t}_{0}=-ba\bar{b}+ab-b+1$ and
$\beta=t_{2}\bar{b}-t_{0}-\bar{t}_{3}\bar{b}+\bar{t}_{1}=-aba\bar{b}+ba\bar{b}+a-1$,
consistent with Eq. (46).

Figure 3:  The control-flow diagram of the generalized algorithm proposed in
the present study.

##  V Applications to TDI combinations

In this section, we explore the applications of the extended combinatorial
approach developed in the last section. We elaborate on the Monitor-type TDI
solutions in the main text and relegate the remaining results on other TDI
solutions to Appendix B. In particular, we present a novel class of Sagnac-
inspired combinations, which cannot be straightforwardly obtained using the
geometric TDI. Moreover, in Appendix C, we give explicit forms of the relevant
lower-order commutators in an exhaustive fashion that furnish the TDI solution
investigated in the present study. Appendix D is devoted to discussing how to
construct specific higher-order TDI solutions based on existing lower-order
ones.

###  V.1 The Monitor-type combinations

From the specific form of the modified first-generation and modified second-
generation Monitor-E combinations [14], namely,

$\displaystyle E_{1}(t)$
$\displaystyle=[(1-D_{311^{{}^{\prime}}\bar{3}})\eta_{1}+D_{3}\eta_{2}+D_{31}\eta_{{3}^{{}^{\prime}}}]$
(49)
$\displaystyle-[(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}})\eta_{1^{{}^{\prime}}}+D_{2^{{}^{\prime}}}\eta_{3^{{}^{\prime}}}+D_{2^{{}^{\prime}}1^{{}^{\prime}}}\eta_{2}],$

and

$\displaystyle E_{2}(t)$
$\displaystyle=(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}})[(1-D_{311^{{}^{\prime}}\bar{3}})\eta_{1}+D_{3}\eta_{2}+D_{31}\eta_{{3}^{{}^{\prime}}}]$
(50)
$\displaystyle-(1-D_{311^{{}^{\prime}}\bar{3}})[(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}})\eta_{1^{{}^{\prime}}}+D_{2^{{}^{\prime}}}\eta_{3^{{}^{\prime}}}+D_{2^{{}^{\prime}}1^{{}^{\prime}}}\eta_{2}],$

one observes the following constraint equations

$q_{3}=0,q_{2^{{}^{\prime}}}=0.$ (51)

Substituting Eq. (51) into Eq. (6), one finds

$\displaystyle q_{1}+q_{1^{{}^{\prime}}}=0,$ (52a) $\displaystyle
q_{2}-q_{3^{{}^{\prime}}}D_{1^{{}^{\prime}}}-q_{1}D_{3}=0,$ (52b)
$\displaystyle
q_{3^{{}^{\prime}}}-q_{1^{{}^{\prime}}}D_{2^{{}^{\prime}}}-q_{2}D_{1}=0.$
(52c)

Eliminating Eqs. (52b) and (52c) by substituting the forms of $q_{1}$ and
$q_{1^{\prime}}$ into Eq. (52a), one finds the desired equation

$q_{2}(D_{\bar{3}}-D_{1\bar{2}^{{}^{\prime}}})+q_{3^{{}^{\prime}}}(D_{\bar{2}^{{}^{\prime}}}-D_{1^{{}^{\prime}}\bar{3}})=0.$
(53)

The above equation is essentially Eq. (47) by recognizing
$\alpha=q_{2}D_{\bar{3}},\beta=q_{3^{{}^{\prime}}}D_{\bar{2}^{{}^{\prime}}}$,
$a=D_{31\bar{2}^{{}^{\prime}}}$, and
$b=D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}$. To proceed, the algorithm
proposed in the Sec. IV.2 can be applied to extract the TDI coefficients of
the Monitor-type combination for a given valid commutator consisting of $a$,
$b$, $\bar{a}$, $\bar{b}$.

For instance, for the commutator
$[ba,ab]=[D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}},D_{311^{{}^{\prime}}\bar{3}}]$,
the corresponding TDI solution expressed in terms of the coefficients
$q_{2},q_{3^{{}^{\prime}}}$ reads

$\displaystyle q_{2}$
$\displaystyle=(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}-D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}}+D_{311^{{}^{\prime}}\bar{3}2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}})D_{3},$
(54) $\displaystyle q_{3^{{}^{\prime}}}$
$\displaystyle=-(1-D_{31\bar{2}^{{}^{\prime}}}-D_{311^{{}^{\prime}}\bar{3}}+D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}31\bar{2}^{{}^{\prime}}})D_{2^{{}^{\prime}}},$

which is the modified second-generation standard Monitor-E combination.

As a second example, consider $[ba\bar{b},a]\bar{a}$, the corresponding TDI
solution is found to be

$\displaystyle q_{2}$ $\displaystyle=\alpha
D_{3}=(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}+D_{311^{{}^{\prime}}\bar{3}}-D_{311^{\prime}1\bar{2}^{\prime}3\bar{1}^{\prime}\bar{1}\bar{3}})D_{3},$
(55) $\displaystyle q_{3^{{}^{\prime}}}$ $\displaystyle=\beta
D_{2^{{}^{\prime}}}=-(1-D_{31\bar{2}^{{}^{\prime}}}-D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}+D_{311^{{}^{\prime}}1\bar{2}^{{}^{\prime}}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}})D_{2^{{}^{\prime}}}.$

Substitute Eqs. (55) into Eq. (52a) and Eq. (52c) for the remaining
coefficients $q_{1}$ and $q_{1^{{}^{\prime}}}$, one finds

$\displaystyle{E}({t})$
$\displaystyle=\left(1-{D}_{2^{\prime}1^{\prime}1\overline{2}^{\prime}}+{D}_{311^{\prime}1\overline{2}^{\prime}}-{D}_{{2}^{\prime}1^{\prime}1\overline{2}^{\prime}3\overline{1}^{\prime}\overline{2}^{\prime}}\right)\eta_{1}$
(56)
$\displaystyle+\left(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}+D_{311^{{}^{\prime}}\bar{3}}-D_{311^{\prime}1\bar{2}^{\prime}3\bar{1}^{\prime}\bar{1}\bar{3}}\right)D_{3}\eta_{2}$
$\displaystyle-\left(1-{D}_{2^{\prime}1^{\prime}1\overline{2}^{\prime}}+{D}_{311^{\prime}1\overline{2}^{\prime}}-{D}_{2^{\prime}1^{\prime}1\overline{2}^{\prime}3\overline{1}^{\prime}\overline{2}^{\prime}}\right)\eta_{1^{\prime}}$
$\displaystyle-\left(1-D_{31\bar{2}^{{}^{\prime}}}-D_{2^{{}^{\prime}}1^{{}^{\prime}}1\bar{2}^{{}^{\prime}}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}+D_{311^{{}^{\prime}}1\bar{2}^{{}^{\prime}}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}\right)D_{2^{\prime}}\eta_{3^{\prime}},$

which is recognized as the alternative Monitor-E combination (c.f. Fig. 9(b)
of Ref. [15]).

The same procedure can be carried out for other commutator such as
$[ab^{2},bab]$, $[a\bar{b},\bar{b}a]$, and $[aab,aba]$, which subsequently
gives rise to a class of (higher-order, from the geometric TDI perspective)
Monitor-E combinations. Moreover, by introducing cyclic permutations between
the indices: $3\rightarrow 2,2^{{}^{\prime}}\rightarrow 1^{{}^{\prime}}$ and
$3\rightarrow 1,2^{{}^{\prime}}\rightarrow 3^{{}^{\prime}}$, one obtains
Monitor-F-type and Monitor-G combinations. We refer to the above solutions
obtained by considering the constraints Eq. (51) as Monitor-type solutions.

We point out that this strategy can be utilized to derive other types of TDI
combinations, such as the Beacon, Relay, and Sagnac ones. We relegate the
derivations to Appendix B of the paper. In Tab. 1, we summarize the relevant
types of TDI solutions that can be dealt with using the present approach and
the associated constraint equations. Moreover, in Appendix C, we enumerate the
explicit forms of the lower-order commutators, which furnish the TDI
solutions.

Table 1: The TDI combinations and the associated constraints that are adopted systematically by the proposed algorithm. TDI combination | Constraint equation | TDI equation
---|---|---
Sagnac-$\alpha$ combination |  $q_{2}=q_{1}D_{3}$, $q_{3}=q_{1}D_{31}$ | $q_{1}(1-D_{312})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}3^{{}^{\prime}}})=0$
Fully symmetric Sagnac combination |  $q_{3}=-q_{3^{{}^{\prime}}}$, $q_{2}=-q_{2^{{}^{\prime}}}$ | $q_{1}(1-D_{3\bar{1}^{{}^{\prime}}2})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}})=0$
Michelson-X combination |  $q_{2}=0$, $q_{3^{{}^{\prime}}}=0$ | $q_{1}(1-D_{33^{{}^{\prime}}})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}2})=0$
Relay-U combination |  $q_{2}=0$, $q_{3}=0$ | $q_{2^{{}^{\prime}}}(D_{\bar{3}}-D_{3^{{}^{\prime}}})+q_{3^{{}^{\prime}}}(D_{\bar{2}^{{}^{\prime}}}-D_{1^{{}^{\prime}}\bar{3}})=0$
Beacon-P combination |  $q_{3}=0$, $q_{3^{{}^{\prime}}}=0$ | $q_{1}(1-D_{33^{{}^{\prime}}})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}})=0$
Monitor-E combination |  $q_{3}=0$, $q_{2^{{}^{\prime}}}=0$ | $q_{2}(D_{\bar{3}}-D_{1\bar{2}^{{}^{\prime}}})+q_{3^{{}^{\prime}}}(D_{\bar{2}^{{}^{\prime}}}-D_{1^{{}^{\prime}}\bar{3}})=0$

Before closing this subsection, we compared the derived TDI solutions with
those obtained using the geometric TDI approach. In Ref. [15], a total of nine
sixteen-link modified second-generation TDI combinations were reported. All
these solutions can be retrieved using the present method, and the associated
commutator and link trajectory are enumerated in Tab. 2.

Table 2: The relevant commutators of the nine sixteen-link modified second-generation TDI combinations. The superscript of a TDI combination indicates the number of links, and the subscript represents the index of the combination for a specific solution family [15]. TDI combination | Commutator | Laser link trajectory
---|---|---
$[X]^{16}_{1}$ | $[ba,ab]$ | $1\leftarrow 2\leftarrow 1\leftarrow 3\leftarrow 1\leftarrow 3\leftarrow 1\leftarrow 2\leftarrow 1\to 3\to 1\to 2\to 1\to 2\to 1\to 3\to 1$
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}2}$
$[X]^{16}_{2}$ | $[ba\bar{b},a]$ | $1\leftarrow 2\leftarrow 1\leftarrow 3\leftarrow 1\leftarrow 2\leftarrow 1\to 3\to 1\to 2\to 1\leftarrow 3\leftarrow 1\to 2\to 1\to 3\to 1$
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}2}$
$[U]^{16}_{1}$ | $[ba,ab]$ | $1\leftarrow 2\leftarrow 1\leftarrow 3\leftarrow 2\to 1\leftarrow 3\leftarrow 2\leftarrow 1\leftarrow 2\to 3\to 1\to 2\to 1\to 2\to 3\to 1$
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}$
$[U]^{16}_{2}$ | $[ba\bar{b},a]$ | $1\leftarrow 2\leftarrow 1\leftarrow 3\leftarrow 2\leftarrow 1\leftarrow 2\to 3\to 1\to 2\to 1\leftarrow 3\leftarrow 2\to 1\to 2\to 3\to 1$
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}$
$[U]^{16}_{3}$ | $[ab\bar{a},b]\bar{b}$ | $1\leftarrow 2\leftarrow 1\leftarrow 3\leftarrow 2\to 1\to 2\to 1\leftarrow 3\leftarrow 2\leftarrow 1\leftarrow 2\to 3\to 1\to 2\to 3\to 1$
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}$
$[E]^{16}_{1}$ | $[ba,ab]$ | $1\leftarrow 2\leftarrow 3\leftarrow 2\to 1\leftarrow 3\leftarrow 2\leftarrow 3\to 1\leftarrow 2\to 3\to 2\to 1\leftarrow 3\to 2\to 3\to 1$
$a=D_{31\bar{2}^{{}^{\prime}}},b=D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}$
$[E]^{16}_{2}$ | $[ba\bar{b},a]\bar{a}$ | $1\leftarrow 2\leftarrow 3\leftarrow 2\leftarrow 3\to 1\leftarrow 2\to 3\to 2\to 1\leftarrow 3\leftarrow 2\to 1\leftarrow 3\to 2\to 3\to 1$
$a=D_{31\bar{2}^{{}^{\prime}}},b=D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}$
$[P]^{16}_{1}$ | $[ba\bar{b},a]$ | $1\leftarrow 2\leftarrow 1\leftarrow 3\to 2\leftarrow 1\leftarrow 2\leftarrow 3\to 1\to 2\to 1\leftarrow 3\to 2\to 1\to 2\leftarrow 3\to 1$
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}$
$[P]^{16}_{2}$ | $[b,ab\bar{a}]$ | $1\leftarrow 2\leftarrow 1\leftarrow 3\to 2\to 1\leftarrow 3\to 2\leftarrow 1\leftarrow 2\leftarrow 3\to 1\to 2\to 1\to 2\leftarrow 3\to 1$
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}$

###  V.2 The fully symmetric Sagnac-type combinations

Unlike other, the fully symmetric Sagnac combinations constitute a unique
family of solutions. Their distinctive nature partly resides in the fact that
these solutions do not often possess a straightforward geometric TDI
interpretation. In the last two subsections, we focus on these peculiar
solutions. The present subsection will be devoted to retrieving the existing
fully symmetric TDI combinations already explored in the literature. In the
following subsection, we elaborate on some novel Sagnac-inspired solutions
that have not been reported.

Again, we first discuss the constraint equations through which the appropriate
equation Eq. (47) can be established. The available TDI solutions of fully
symmetric Sagnac-type are then shown to be consistent with the derived
equation. Subsequently, the algorithm is employed straightforwardly to solve
for modified second-generation solutions.

The first-generation fully symmetric Sagnac solution is given by

$\zeta_{1}({t})={D}_{1}\eta_{1}+D_{2}\eta_{2}+D_{3}\eta_{3}-D_{1^{\prime}}\eta_{1^{\prime}}-D_{2^{\prime}}\eta_{2^{\prime}}-D_{3^{\prime}}\eta_{3^{\prime}},$
(57)

for which the residual laser frequency noise reads

$\Sigma_{i}(D_{i^{{}^{\prime}}}-D_{i}+D_{(i-1)(i+1)}-D_{(i+1)^{{}^{\prime}}(i-1)^{{}^{\prime}}})p_{i}(t).$
(58)

The modified first-generation fully symmetric Sagnac solution possesses the
form [14]

$\displaystyle\zeta_{2}(t)$
$\displaystyle=(D_{2^{{}^{\prime}}3^{{}^{\prime}}}-D_{1})(D_{3}\eta_{3}-D_{3}\eta_{3^{\prime}}+D_{1^{{}^{\prime}}}\eta_{1})$
(59)
$\displaystyle-(D_{32}-D_{1^{{}^{\prime}}})(D_{1}\eta_{1^{{}^{\prime}}}-D_{2^{{}^{\prime}}}\eta_{2}+D_{2^{{}^{\prime}}}\eta_{2^{{}^{\prime}}}),$

where the residual laser frequency noise is found to be

$[D_{2^{{}^{\prime}}3^{{}^{\prime}}}-D_{1},D_{32}-D_{1^{{}^{\prime}}}]p_{1}(t).$
(60)

It is noted that Eq. (60) is in accordance with the assertion that an
arbitrary commutator between different delay operators vanishes, namely,

$[D_{i(i^{{}^{\prime}})},D_{j(j^{{}^{\prime}})}]=0.$ (61)

It was pointed out in [14] that although the residual Eq. (60) does not vanish
regarding the terms $\dot{L}$, the first-order derivative in time, the noise
has been suppressed below the desired level. In other words, for a second- or
higher-generation TDI solution, one would demand the more stringent condition
imposed by Eq. (27), which ensures an explicit cancelation of the terms
$\dot{L}$ in the residual. In what follows, we will first retrieve Eq. (59)
using the combinatorial algebraic approach, then explore further by aiming at
the modified second-generation results.

For the first-generation TDI, besides Eq. (61), one has

$D_{i^{{}^{\prime}}}=D_{i}.$ (62)

Subsequently, we consider the following constraints
$q_{3}+q_{3^{{}^{\prime}}}=0,q_{2}+q_{2^{{}^{\prime}}}=0$. Note that it is
also viable to use the constraint involving $q_{1}$ and $q_{1^{{}^{\prime}}}$
by permutating the indices in a cyclic order. As pointed out in [14], the
Sagnac effect from the rotation of the array breaks the uniqueness of the
“$\zeta$-like” combinations, which incorporate with the constraints
$q_{1}+q_{1^{{}^{\prime}}}=0$. By substituting

$q_{3}=-q_{3^{{}^{\prime}}},q_{2}=-q_{2^{{}^{\prime}}}$ (63)

into Eq. (6), we have

$q_{1}(1-D_{3\bar{1}^{{}^{\prime}}2})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}})=0,$
(64)

which is equivalent to Eq. (47) by identifying
$a=D_{3\bar{1}^{{}^{\prime}}2},b=D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}$.
From Eq. (57), one has $q_{1}=D_{1}$ and $q_{1^{\prime}}=D_{1^{{}^{\prime}}}$,
which manifestly satisfies Eq. (64) while taking into account Eq. (62).

For the case of modified first-generation TDI, we replace
$D_{\bar{1}^{{}^{\prime}}}$ and $D_{2^{{}^{\prime}}}$ with $D_{3}$ and
$D_{\bar{1}}$ in the above expression to obtain

$q_{1}(1-D_{\bar{1}^{{}^{\prime}}32})+q_{1^{{}^{\prime}}}(1-D_{\bar{1}2^{{}^{\prime}}3^{{}^{\prime}}})=0.$
(65)

Again, it is straightforward to show that
$q_{1}=(D_{2^{{}^{\prime}}3^{{}^{\prime}}}-D_{1})D_{1^{{}^{\prime}}}$ and
$q_{1^{\prime}}=-(D_{32}-D_{1^{{}^{\prime}}})D_{1}$ read off from Eq. (59)
satisfies the above equation with Eq. (61).

We now use the algorithm proposed in Sec. IV.2 to solve for the modified
second-generation fully symmetric Sagnac combinations. To start with, consider
the commutator
$[ba,ab]=[D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}3\bar{1}^{{}^{\prime}}2},D_{3\bar{1}^{{}^{\prime}}22^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}]$,
the TDI coefficients are found to be

$\displaystyle q_{1}$
$\displaystyle=1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}3\bar{1}^{{}^{\prime}}2}+D_{3\bar{1}^{{}^{\prime}}22^{{}^{\prime}}\bar{1}3^{{}^{\prime}}2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}},$
(66) $\displaystyle q_{1^{{}^{\prime}}}$
$\displaystyle=-(1-D_{3\bar{1}^{{}^{\prime}}2}-D_{3\bar{1}^{{}^{\prime}}22^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}+D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}3\bar{1}^{{}^{\prime}}23\bar{1}^{{}^{\prime}}2}).$

The corresponding TDI solution reads

$\displaystyle\zeta(t)$
$\displaystyle=\left(1-D_{2^{\prime}\overline{1}3^{\prime}}-D_{2^{\prime}\overline{1}3^{\prime}3\overline{1}^{\prime}2}+D_{3\overline{1}^{\prime}22^{\prime}\overline{1}3^{\prime}2^{\prime}\overline{1}3^{\prime}}\right)\eta_{1}$
(67)
$\displaystyle+\left(1-D_{3\overline{1}^{\prime}2}-D_{3\overline{1}^{\prime}22^{\prime}\overline{1}3^{\prime}}+D_{2^{\prime}\overline{1}3^{\prime}3\overline{1}^{\prime}23\overline{1}^{\prime}2}\right)D_{2^{\prime}\overline{1}}\eta_{2}$
$\displaystyle+\left(1-D_{2^{\prime}\overline{1}3^{\prime}}-D_{2^{\prime}\overline{1}3^{\prime}3\overline{1}^{\prime}2}+D_{3\overline{1}^{\prime}22^{\prime}\overline{1}3^{\prime}2^{\prime}\overline{1}3^{\prime}}\right)D_{3\overline{1}^{\prime}}\eta_{3}$
$\displaystyle-\left(1-D_{3\bar{1}^{{}^{\prime}}2}-D_{3\bar{1}^{{}^{\prime}}22^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}+D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}3\bar{1}^{{}^{\prime}}23\bar{1}^{{}^{\prime}}2}\right)\eta_{1^{\prime}}$
$\displaystyle-\left(1-D_{3\overline{1}^{\prime}2}-D_{3\overline{1}^{\prime}22^{\prime}\overline{1}3^{\prime}}+D_{2^{\prime}\overline{1}3^{\prime}3\overline{1}^{\prime}23\overline{1}^{\prime}2}\right)D_{2^{\prime}\overline{1}}\eta_{2^{\prime}}$
$\displaystyle-\left(1-D_{2^{\prime}\overline{1}3^{\prime}}-D_{2^{\prime}\overline{1}3^{\prime}3\overline{1}^{\prime}2}+D_{3\overline{1}^{\prime}22^{\prime}\overline{1}3^{\prime}2^{\prime}\overline{1}3^{\prime}}\right)D_{3\overline{1}^{\prime}}\eta_{3^{\prime}}.$

The above procedure can be similarly applied to other commutators to give rise
to other combinations. We refer to these combinations associated with the
constraints Eq. (63) as the fully symmetric Sagnac-type solution.

It is rather interesting to point out that some fully symmetric Sagnac
combinations possess a geometric TDI interpretation. As shown in Fig. 4, Eq.
(67) corresponds to the following space-time diagram regarding the geometric
TDI.

Figure 4:  The space-time diagram for the modified second-generation fully
symmetric Sagnac combination given by Eq. (67). The notation is consistent
with Ref. [15]. The red solid and blue dashed lines represent the two
synthetic paths, and the black square and dot indicate the start and end
points of the light path, respectively.

###  V.3 Novel Sagnac-inspired combinations

In this subsection, we elaborate on a novel class of Sagnac-inspired solutions
that do not associate with any geometric TDI trajectory. The distinction is
based on the fact that any valid geometric TDI solution demands successive
transmissions of laser signal between the spacecraft. In other words, only
specific time-delay or time-advance operators can be applied to a signal
emitted from a given optical bench. For instance, the science data stream
$\eta_{1}$ cannot be applied by the time-delay operation associated with the
opposite armlength, $D_{1}$. On the other hand, it is apparent that such a
physical constraint is not necessary from an algebraic perspective.

To this end, we consider the following two constraints
$q_{1}=q_{2}D_{1},q_{2}=q_{3}D_{2}$. One observes the similarity between the
above constraints and those of the Sagnac-type combinations. Therefore, the
resulting solutions will be referred to as Sagnac-inspired. Subsequently, Eq.
(6) gives

$\displaystyle
q_{3}D_{21}+q_{1^{\prime}}-q_{2^{\prime}}{D}_{3^{\prime}}-q_{3}{D}_{2}$
$\displaystyle=0,$ (68a) $\displaystyle
q_{3}D_{2}+q_{2^{\prime}}-q_{3^{\prime}}{D}_{1^{\prime}}-q_{3}D_{213}$
$\displaystyle=0,$ (68b) $\displaystyle
q_{3}+q_{3^{\prime}}-q_{1^{\prime}}{D}_{2^{\prime}}-q_{3}D_{21}$
$\displaystyle=0.$ (68c)

Eliminating Eqs. (68a) and (68b) by substituting the forms of $q_{1^{\prime}}$
and $q_{2^{\prime}}$ into Eq. (68c), one finds the desired equation

$q_{3}\left(1-A\right)+q_{3^{\prime}}\left(1-b\right)=0,$ (69)

where
$A=D_{2133^{\prime}2^{\prime}}-D_{23^{\prime}2^{\prime}}+D_{22^{\prime}}-D_{212^{\prime}}+D_{21}=\sum_{i}a_{i}$
and $b={D}_{1^{\prime}3^{\prime}2^{\prime}}$. Since $A$ is not a monomial but
a polynomial, we note that the condition Eq. (27) is not satisfied. However,
one may argue that the vanishing condition of the associated commutator can be
generalized to include polynomial, such as $[bA,Ab]$, by noticing

$[bA,Ab]=\left[b\left(\sum_{i}a_{i}\right),\left(\sum_{j}a_{j}\right)b\right]=\sum_{ij}[ba_{i},a_{j}b]\sim\sum_{i}[ba_{i},a_{i}b],$
(70)

where “$\sim$” means identical as an operator when applied to an arbitrary
function of time. This is because the cross terms vanish

$[ba_{i},a_{j}b]+[ba_{j},a_{i}b]\sim 0,$ (71)

for $i\neq j$. To be specific, one has

$\left\\{[D_{b}D_{a_{i}},D_{a_{j}}D_{b}]+[D_{b}D_{a_{j}},D_{a_{i}}D_{b}]\right\\}\phi(t)=0,$
(72)

whose validity will be shown shortly. The r.h.s. of Eq. (70) subsequently
vanishes as it satisfies the condition Eq. (27).

The above result is a special case of a generalized relation of Eq. (40),
which applies to polynomials $A=\sum_{i}^{n}a_{i}$ and $B=\sum_{j}^{m}b_{j}$:

$\left[AB,BA\right]\phi(t)=\left[\sum_{i=1}^{n}a_{i}\sum_{j=1}^{m}b_{j},\sum_{k=1}^{m}b_{k}\sum_{l=1}^{n}a_{l}\right]\phi(t)=\sum_{ij}\left[a_{i}b_{j},b_{j}a_{i}\right]\phi(t),$
(73)

to the first-order time-derivative terms $\dot{L}$. To confirm that all the
cross terms vanish, we show that they cancel out in pairs. To be specific,
there are $mn(mn-1)/2$ pairs of the form

$\Big{(}[a_{i}b_{j},b_{k}a_{l}]+[a_{l}b_{k},b_{j}a_{i}]\Big{)}\phi(t),$ (74)

where either $i\neq l$ or $j\neq k$. Using Eq. (40), we have

$\displaystyle{\left[a_{i}b_{j},b_{k}a_{l}\right]\phi(}t)$ (75)
$\displaystyle=\left[\left(L_{a_{i}}+L_{b_{j}}\right)\left(\dot{L}_{b_{k}}+\dot{L}_{a_{l}}\right)\right.$
$\displaystyle\left.-\left(L_{b_{k}}+L_{a_{l}}\right)\left(\dot{L}_{a_{i}}+\dot{L}_{b_{j}}\right)\right]$
$\displaystyle\dot{\phi}\left(t-\left(L_{a_{i}}+L_{b_{j}}\right)\right.\left.-\left(L_{b_{k}}+L_{a_{l}}\right)\right),$

and similarly,

$\displaystyle{\left[a_{l}b_{k},b_{j}a_{i}\right]\phi(}t)$ (76)
$\displaystyle=\left[\left(L_{a_{l}}+L_{b_{k}}\right)\left(\dot{L}_{b_{j}}+\dot{L}_{a_{i}}\right)\right.$
$\displaystyle\left.-\left(L_{b_{j}}+L_{a_{i}}\right)\left(\dot{L}_{a_{l}}+\dot{L}_{b_{k}}\right)\right]$
$\displaystyle\dot{\phi}\left(t-\left(L_{a_{l}}+L_{b_{k}}\right)\right.\left.-\left(L_{b_{j}}+L_{a_{i}}\right)\right),$

where $L_{a_{i}},L_{b_{j}},\dot{L}_{a_{i}},\dot{L}_{b_{j}}$ represent the
armlengths and their rates of change regarding the associated time-translation
operations. The sum of the two terms manifestly vanishes.

Based on the above arguments, one applies the commutator and finds

$\displaystyle q_{3}$ $\displaystyle=1-b-bA+Ab^{2},$ (77) $\displaystyle
q_{3^{{}^{\prime}}}$ $\displaystyle=-(1-A-Ab+bA^{2}).$

The remaining coefficients are found to be

$\displaystyle q_{2^{\prime}}$
$\displaystyle=-\left(1-A-Ab+bA^{2}\right){D}_{1^{\prime}}$ (78)
$\displaystyle+\left(1-b-bA+Ab^{2}\right)D_{2}\left(D_{13}-1\right),$
$\displaystyle q_{1^{\prime}}$
$\displaystyle=(-(1-A-Ab+bA^{2}){D}_{1^{\prime}}$
$\displaystyle+(1-b-bA+Ab^{2})D_{2}(D_{13}-1)){D}_{3^{\prime}}$
$\displaystyle+(1\left.-b-bA+Ab^{2}\right){D}_{2}\left(1-D_{1}\right),$
$\displaystyle{q}_{2}$ $\displaystyle=\left(1-b-bA+Ab^{2}\right)D_{2},$
$\displaystyle{q}_{1}$ $\displaystyle=\left(1-b-bA+Ab^{2}\right)D_{21}.$

Due to the terms such as $D_{21}\eta_{1}$ in $q_{1}\eta_{1}$, $D_{2}\eta_{2}$
in $q_{2}\eta_{2}$, the above solution cannot be straightforwardly addressed
by the geometric TDI approach.

To analyze the performance of the combination obtained by the end of the last
section, we evaluate its response function and sensitivity curve. They are
shown in Fig. 5 and 6 in comparison with those of the conventional Sagnac
combinations derived in [32, 33]. Also, we adopt typical parameters in LISA
mission, by assuming that armlength $L=2.5\times 10^{6}\mathrm{km}$ and the
corresponding ASDs of the test mass and shot noise:
$s^{\mathrm{LISA}}_{a}=3\times
10^{-15}\mathrm{m}\cdot\mathrm{s^{-2}}/\sqrt{\mathrm{Hz}}$ and
$s^{\mathrm{LISA}}_{x}=15\times 10^{-12}\mathrm{m}/\sqrt{\mathrm{Hz}}$ [34].
The specific forms of the averaged response functions of gravitational waves
and noise power spectral densities are given in Appendix E. For the averaged
response function, the Sagnac-inspired combination shows some advantage in
higher frequencies. On the other hand, the difference in the resultant
sensitivity curves between the two combinations is not significant.

It is worth noting, besides the two constraints considered in this section,
other choices of constraint conditions such as $q_{1}=q_{2}D_{1}$ and
$q_{1^{{}^{\prime}}}=q_{2^{{}^{\prime}}}D_{1^{{}^{\prime}}}$ also leads to
solutions without any geometric TDI interpretation. Therefore, the extended
algebraic, combinatorial approach proposed in this study indicates a much
broader solution space beyond the scope of the geometric TDI.

Figure 5: The gravitational waves averaged response functions of the Sagnac
and Sagnac-inspired combinations. Figure 6: Sensitivity curves of the Sagnac
and Sagnac-inspired combinations.

##  VI Concluding remarks

In this paper, an extended algorithm was proposed to solve for the modified
second-generation TDI combinations. The approach is based on a combinatorial
algebraic scheme first introduced by Dhurandhar et al. applied to the
Michelson-type second-generation TDI solutions. By introducing the time-
advance operator and adopting different constraint conditions, we showed that
various second-generation TDI combinations, such as the Monitor, Beacon, and
Relay combinations, can be retrieved. We also reproduced the fully symmetric
Sagnac combinations whose laser-link trajectory cannot be understood in the
framework of the geometric TDI algorithm. Inspired by such constraint
conditions, we deduced a novel class of Sagnac-inspired TDI solutions. On the
mathematical side, we generalized the commutation relation and its vanishing
condition to a more general context by considering the time-advance operators
and their polynomials. Moreover, we demonstrated that the present scheme is
rather general, and the solution space is broader than what has already been
explored in the literature. Subsequently, the present study constitutes an
attempt to enrich our understanding of the TDI solutions from an algebraic
perspective.

The proposed enumerating scheme is more efficient than geometric TDI, as the
latter is potentially time-consuming at higher orders. Nonetheless, the
exhaustive nature of the present algorithm is not entirely clear. The
constraint conditions explicitly delimit the solution. As a result, it only
addresses a specific subspace of the entire solution space. Also, the
commutator relation is only valid up to the first-order time-derivative terms.
Therefore further generalization to higher-order scenarios is not
straightforward. We plan to address these topics in further studies.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China
(Grant No.11925503), the Postdoctoral Science Foundation of China (Grant
No.2022M711259), Guangdong Major project of Basic and Applied Basic Research
(Grant No.2019B030302001), Key Laboratory of TianQin Project(Sun Yat-sen
University), Ministry of Education, and the Fundamental Research Funds for the
Central Universities, HUST: 2172019kfyRCPY029. We also gratefully acknowledge
the financial support from Brazilian agencies Fundação de Amparo à Pesquisa do
Estado de São Paulo (FAPESP), Fundação de Amparo à Pesquisa do Estado do Rio
de Janeiro (FAPERJ), Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), and Coordenação de Aperfeiçoamento de Pessoal de Nível
Superior (CAPES).

##  Appendix A A proof of the commutation relation

In this Appendix, we give proof of Eq. (26), which furnishes the basis for its
generalization in Sec. IV.

Using mathematical induction, we first show the following proposition for all
nature number $n$: to the first order time-derivative, one has

$\begin{split}&P(n):\\\
&D_{x_{n}}...D_{x_{1}}\phi(t)=\phi\left(t-\sum^{n}_{i=1}L_{x_{i}}\right)+\dot{\phi}\left(t-\sum^{n}_{i=1}L_{x_{i}}\right)\left(\sum^{n-1}_{j=1}\dot{L}_{x_{j}}\left(\sum^{n}_{k=j+1}L_{x_{k}}\right)\right)-\dot{\phi}\left(t-\sum^{n}_{i=1}L_{x_{i}}\right)t\left(\sum^{n}_{i=1}\dot{L}_{x_{i}}\right).\end{split}$
(79)

Proof:

It is straightforward to show that $P(1)$ is true

$\begin{split}&P(1):\\\
&D_{x_{1}}\phi(t)=\phi(t-L_{x_{1}})-\dot{\phi}(t-L_{x_{1}})t\dot{L}_{x_{1}}\end{split}$
(80)

Induction step: if $P(n)$ holds, we have,

$\displaystyle P(n+1):$ (81) $\displaystyle
D_{x_{n+1}}D_{x_{n}}...D_{x_{1}}\phi(t)$
$\displaystyle=D_{x_{n+1}}\left(\phi\left(t-\sum^{n}_{i=1}L_{x_{i}}\right)+\dot{\phi}\left(t-\sum^{n}_{i=1}L_{x_{i}}\right)\left(\sum^{n-1}_{j=1}\dot{L}_{x_{j}}\left(\sum^{n}_{k=j+1}L_{x_{k}}\right)\right)-\dot{\phi}\left(t-\sum^{n}_{i=1}L_{x_{i}}\right)t\left(\sum^{n}_{i=1}\dot{L}_{x_{i}}\right)\right)$
$\displaystyle=\phi\left(t-\sum^{n+1}_{i=1}L_{x_{i}}\right)+\dot{\phi}\left(t-\sum^{n+1}_{i=1}L_{x_{i}}\right)\left(\sum^{n}_{j=1}\dot{L}_{x_{j}}\left(\sum^{n+1}_{k=j+1}L_{x_{k}}\right)\right)-\dot{\phi}\left(t-\sum^{n+1}_{i=1}L_{x_{i}}\right)t\left(\sum^{n+1}_{i=1}\dot{L}_{x_{i}}\right),$

so that $P(n+1)$ also holds true. Q.E.D.

It is readily shown that Eq. (79) implies

$\displaystyle D_{y_{n}}...D_{y_{1}}D_{x_{n}}...D_{x_{1}}\phi(t)$
$\displaystyle=\phi\left(t-\sum^{n}_{i=1}L_{x_{i}}-\sum^{n}_{i=1}L_{y_{i}}\right)-\dot{\phi}\left(t-\sum^{n}_{i=1}L_{x_{i}}-\sum^{n}_{i=1}L_{y_{i}}\right)t\left(\sum^{n}_{i=1}\dot{L}_{x_{i}}+\sum^{n}_{i=1}\dot{L}_{y_{i}}\right)$
(82)
$\displaystyle+\dot{\phi}\left(t-\sum^{n}_{i=1}L_{x_{i}}-\sum^{n}_{i=1}L_{y_{i}}\right)\left(\sum^{n-1}_{j=1}\dot{L}_{y_{j}}\left(\sum^{n}_{k=j+1}L_{y_{k}}\right)+\sum^{n}_{j=1}\dot{L}_{x_{j}}\left(\sum^{n}_{k=j+1}L_{x_{k}}+\sum^{n}_{k=1}L_{y_{k}}\right)\right),$

and

$\displaystyle D_{x_{n}}...D_{x_{1}}D_{y_{n}}...D_{y_{1}}\phi(t)$
$\displaystyle=\phi\left(t-\sum^{n}_{i=1}L_{y_{i}}-\sum^{n}_{i=1}L_{x_{i}}\right)-\dot{\phi}\left(t-\sum^{n}_{i=1}L_{y_{i}}-\sum^{n}_{i=1}L_{x_{i}}\right)t\left(\sum^{n}_{i=1}\dot{L}_{y_{i}}+\sum^{n}_{i=1}\dot{L}_{x_{i}}\right)$
(83)
$\displaystyle+\dot{\phi}\left(t-\sum^{n}_{i=1}L_{y_{i}}-\sum^{n}_{i=1}L_{x_{i}}\right)\left(\sum^{n-1}_{j=1}\dot{L}_{x_{j}}\left(\sum^{n}_{k=j+1}L_{x_{k}}\right)+\sum^{n}_{j=1}\dot{L}_{y_{j}}\left(\sum^{n}_{k=j+1}L_{y_{k}}+\sum^{n}_{k=1}L_{x_{k}}\right)\right).$

By subtracting Eq. (82) from Eq. (83), one finds Eq. (26) given in the main
text.

##  Appendix B The derivations of other TDI combinations

Here, we present the derivations of other TDI combinations using the
combinatorial algorithm proposed in this paper. As the procedure is mainly
reminiscent of what is discussed in Sec. V, only the essential arguments are
given below.

###  B.1 Sagnac-type combinations

From the specific forms of the first-generation Sagnac-$\alpha$ combination

$\alpha_{1}(t)=\eta_{1}-\eta_{1^{\prime}}+{D}_{3}\eta_{2}-{D}_{2^{\prime}1}\eta_{2^{\prime}}+{D}_{31}\eta_{3}-{D}_{2^{\prime}}\eta_{3^{\prime}},$
(84)

and the modified second-generation standard Sagnac-$\alpha$ combination

$\displaystyle\alpha_{2}(t)$
$\displaystyle=\left(D_{2^{\prime}1^{\prime}3^{\prime}}-1\right)\left(\eta_{1}+D_{3}\eta_{2}+D_{31}\eta_{3}\right)$
(85)
$\displaystyle-\left(D_{312}-1\right)\left(\eta_{1^{\prime}}+D_{2^{\prime}}\eta_{3^{\prime}}+D_{2^{\prime}1^{\prime}}\eta_{2^{\prime}}\right),$

it is observed that the coefficients satisfy the following relations

${q}_{2}={q}_{1}D_{3},{q}_{3}=q_{1}D_{31},{q}_{3^{\prime}}=q_{1^{\prime}}D_{2^{\prime}},{q}_{2^{\prime}}=q_{1^{\prime}}D_{2^{\prime}1^{\prime}}.$
(86)

The above relations introduce four constraints. However, it can be shown that
these four constraints are not entirely independent from Eq. (6). In
particular, the second and third lines of Eq. (6) are manifestly linear
combinations of the above constraints, and therefore, Eq. (86) only provides
two independent conditions. Subsequently, the relevant equation is obtained
from the first line of Eq. (6) by substituing the above conditions and
eliminating $q_{2},q_{3},q_{2^{\prime}}$, and $q_{3^{\prime}}$. One finds

$q_{1}(1-D_{312})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}1^{{}^{\prime}}3^{{}^{\prime}}})=0.$
(87)

Again, the above equation is nothing but Eq. (47) by identifying $a=D_{312}$
and $b=D_{2^{{}^{\prime}}1^{{}^{\prime}}3^{{}^{\prime}}}$. Subsequently, the
algorithm may proceed as described in Sec. III.2, as long as no time-advance
operator is involved. In this case, one may essentially follow [29] to
systematically enumerate the relevant commutators related to the
Sagnac-$\alpha$ type combinations. The simplest choice is
$\Delta=[ba,ab]=[D_{2^{{}^{\prime}}1^{{}^{\prime}}3^{{}^{\prime}}312},D_{3122^{{}^{\prime}}1^{{}^{\prime}}3^{{}^{\prime}}}]$,
whose the coefficients $q_{1},q_{1^{{}^{\prime}}}$,

$\displaystyle
q_{1}=1-D_{2^{\prime}1^{\prime}3^{\prime}}-D_{2^{\prime}1^{\prime}3^{\prime}312}+D_{3122^{\prime}1^{\prime}3^{\prime}2^{\prime}1^{\prime}3^{\prime}},$
(88) $\displaystyle
q_{1^{\prime}}=-\left(1-D_{312}-D_{3122^{\prime}1^{\prime}3^{\prime}}+D_{2^{\prime}1^{\prime}3^{\prime}312312}\right),$

are recognized as those of the modified second-generation Sagnac-$\alpha$
combination given above.

On the other hand, one may consider commutator that contains the time-advance
operator. As an example, for the case of $\Delta=[ba\bar{b},a]$, we arrive at
an alternative Sagnac combination

$\displaystyle\alpha({t})=\left(1-D_{2^{\prime}1^{\prime}3^{\prime}}+{D}_{3122^{\prime}1^{\prime}3^{\prime}}-D_{2^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}\right)\eta_{1}$
(89)
$\displaystyle+\left(1-D_{2^{\prime}1^{\prime}3^{\prime}}+{D}_{3122^{\prime}1^{\prime}3^{\prime}}-D_{2^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}\right)D_{3}\eta_{2}$
$\displaystyle+\left(1-D_{2^{\prime}1^{\prime}3^{\prime}}+{D}_{3122^{\prime}1^{\prime}3^{\prime}}-D_{2^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}\right)D_{31}\eta_{3}$
$\displaystyle-\left(1-{D}_{312}-D_{2^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}+{D}_{3122^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}\right)\eta_{1^{\prime}}$
$\displaystyle-\left(1-{D}_{312}-D_{2^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}+{D}_{3122^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}\right)D_{2^{\prime}}\eta_{2^{\prime}}$
$\displaystyle-\left(1-{D}_{312}-D_{2^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{\prime}\overline{2}^{\prime}}+{D}_{3122^{\prime}1^{\prime}3^{\prime}312\overline{3}^{\prime}\overline{1}^{1}\overline{2}^{\prime}}\right)D_{2^{\prime}1^{\prime}}\eta_{3^{\prime}}.$

It is readily verified that the commutators $[ab^{2},bab]$,
$[a\bar{b},\bar{b}a]$, $[aab,aba]$, among others, give rise to a class of
higher-order (from the perspective of the geometric TDI) Sagnac-$\alpha$
combinations. Similarly, Sagnac-$\beta$ and Sagnac-$\gamma$ combinations can
be obtained, respectively, by rotating the indices of the constraints
$2\rightarrow 3,3\rightarrow 1$ and $2\rightarrow 1,3\rightarrow 2$.

###  B.2 Beacon-type combinations

From the specific forms of the modified first-generation Beacon-P combination

$\displaystyle{P}_{1}(t)$
$\displaystyle=D_{\bar{2}^{{}^{\prime}}}\eta_{1}+D_{\bar{2}^{{}^{\prime}}3}\eta_{2^{\prime}}+D_{\bar{2}^{{}^{\prime}}33^{\prime}}\eta_{1^{\prime}}+D_{\bar{1}}\eta_{2}$
(90)
$\displaystyle-\left(D_{\bar{1}}\eta_{2^{\prime}}+D_{\bar{1}3^{\prime}}\eta_{1}+D_{\bar{1}3^{\prime}3}\eta_{2}+D_{\bar{2}^{{}^{\prime}}}\eta_{1^{{}^{\prime}}}\right),$

and the modified second-generation standard Beacon-P combination

$\displaystyle P_{2}(t)$
$\displaystyle=\left(1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}}+D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}\right)\eta_{1}$
(91)
$\displaystyle+\left(-1+D_{33^{{}^{\prime}}}+D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}}-D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}}\right)\eta_{1^{{}^{\prime}}}$
$\displaystyle+\left(D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}3}-D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}}-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}3}+D_{2^{{}^{\prime}}\bar{1}}\right)\eta_{2}$
$\displaystyle+\left(D_{3}+D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}}-D_{2^{{}^{\prime}}\bar{1}}-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}3}\right)\eta_{2^{{}^{\prime}}},$

it is observed that the coefficients satisfy the following relations

$q_{3}=0,q_{3^{{}^{\prime}}}=0.$ (92)

By substituting Eq. (92) into Eq. (6), one finds

$q_{1}(1-D_{33^{{}^{\prime}}})+q_{1^{{}^{\prime}}}(1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}})=0.$
(93)

Again, the above equation is nothing but Eq. (47) by identifying
$a=D_{33^{{}^{\prime}}},b=D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}$. Since
there is a time-advance operator $D_{\bar{1}}$ in $b$, one applies the process
described in Sec. IV.2 to systematically enumerate the relevant commutators
related to the Beacon-P type combinations. One possible choice is
$\Delta=[ba\bar{b},a]=[D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}},D_{33^{{}^{\prime}}}]$,
whose the coefficients $q_{1},q_{1^{{}^{\prime}}}$,

$\displaystyle q_{1}$
$\displaystyle=(1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}}+D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}),$
(94) $\displaystyle q_{1^{{}^{\prime}}}$
$\displaystyle=-(1-D_{3{3}^{{}^{\prime}}}-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}}+D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}31\bar{2}^{{}^{\prime}}}),$

are recognized as those of the modified second-generation standard Beacon-P
combination given above.

On the other hand, for the case of $\Delta=[b,ab\bar{a}]$, we arrive at an
alternative Beacon combination

$\displaystyle{P}({t})=\left(1-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}}-D_{33^{\prime}2^{{}^{\prime}}\bar{1}\bar{3}}+D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}\bar{3}}\right)\eta_{1}$
(95)
$\displaystyle+\left(-1+D_{33^{{}^{\prime}}}-D_{{2}^{\prime}\bar{1}3^{\prime}33^{{}^{\prime}}}+D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}\bar{3}}\right)\eta_{1^{{}^{\prime}}}$
$\displaystyle+\left({D}_{{2}^{\prime}\bar{1}}-D_{33^{\prime}{2}^{\prime}\bar{1}}+D_{{2}^{\prime}\bar{1}3^{{}^{\prime}}33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}}-D_{33^{\prime}2^{{}^{\prime}}\bar{1}\bar{3}2^{{}^{\prime}}\bar{1}}\right)\eta_{2}$
$\displaystyle+\left(D_{3}-D_{2^{{}^{\prime}}\bar{1}}-D_{2^{{}^{\prime}}\bar{1}3^{{}^{\prime}}3}+D_{33^{{}^{\prime}}2^{{}^{\prime}}\bar{1}\bar{3}2^{{}^{\prime}}\bar{1}}\right)\eta_{2^{\prime}}.$

It is readily verified that the commutators $[ab^{2},bab]$,
$[a\bar{b},\bar{b}a]$, $[aab,aba]$, among others, give rise to a class of
higher-order (from the perspective of the geometric TDI) Beacon-P
combinations. Similarly, Beacon-Q and Beacon-R combinations can be obtained,
respectively, by rotating the indices of the constraints $3\rightarrow
2,3^{{}^{\prime}}\rightarrow 2^{{}^{\prime}}$ and $3\rightarrow
1,3^{{}^{\prime}}\rightarrow 1{{}^{\prime}}$.

###  B.3 Relay-type combinations

From the specific forms of the modified first-generation Relay-U combination

$\displaystyle{U}_{1}(t)$
$\displaystyle=\eta_{1}+D_{3}\eta_{2^{\prime}}+D_{33^{\prime}}\eta_{1^{\prime}}+D_{33^{\prime}2^{\prime}}\eta_{3^{\prime}}$
(96)
$\displaystyle-\left(\eta_{1^{\prime}}+D_{2^{\prime}}\eta_{3^{\prime}}+D_{2^{\prime}1^{\prime}}\eta_{2^{\prime}}+D_{2^{\prime}1^{\prime}3^{\prime}}\eta_{1}\right),$

and the modified second-generation standard Relay-U combination

$\displaystyle{U}_{2}(t)$
$\displaystyle=\left(D_{33^{\prime}2^{\prime}1^{\prime}\overline{3}}-1\right)\left(\eta_{1^{\prime}}+D_{2^{\prime}}\eta_{3^{\prime}}+D_{2^{\prime}1^{\prime}}\eta_{2^{\prime}}+D_{2^{{}^{\prime}}1^{{}^{\prime}}3^{{}^{\prime}}}\eta_{1}\right)$
(97)
$\displaystyle-\left(D_{2^{\prime}1^{\prime}3^{\prime}}-1\right)D_{3}\left(\eta_{2^{{}^{\prime}}}+D_{3^{{}^{\prime}}}\eta_{1^{\prime}}+D_{3^{\prime}2^{{}^{\prime}}}\eta_{3^{\prime}}\right)$
$\displaystyle-\left(D_{33^{\prime}2^{\prime}1^{\prime}\overline{3}}-1\right)\eta_{1},$

it is observed that the coefficients satisfy the following relations

$q_{2}=0,q_{3}=0.$ (98)

By substituting Eq. (98) into Eq. (6), we have

$q_{2^{{}^{\prime}}}(D_{\bar{3}}-D_{3^{{}^{\prime}}})+q_{3^{{}^{\prime}}}(D_{\bar{2}^{{}^{\prime}}}-D_{1^{{}^{\prime}}\bar{3}})=0.$
(99)

The above equation is essentially Eq. (47) by recognizing
$\alpha=q_{2^{\prime}}D_{\bar{3}},\beta=q_{3^{{}^{\prime}}}D_{\bar{2}^{{}^{\prime}}}$,
$a=D_{33^{{}^{\prime}}}$, and $b=D_{2^{{}^{\prime}}1^{{}^{\prime}}\bar{3}}$.
Since there is a time-advance operator $D_{\bar{3}}$ in $b$, one applies the
algorithm described in Sec. IV.2 to systematically enumerate the relevant
commutators for the Relay-U type combinations. The simplest choice is
$\Delta=[ba,ab]=[D_{\bar{1}3^{{}^{\prime}}31},D_{\bar{2}^{{}^{\prime}}33^{{}^{\prime}}2^{{}^{\prime}}}]$,
whose the coefficients $q_{2^{{}^{\prime}}},q_{3^{{}^{\prime}}}$,

$\displaystyle{q}_{2^{\prime}}=\left(1-D_{2^{\prime}1^{\prime}\overline{3}}-D_{2^{\prime}1^{\prime}3^{\prime}}+D_{33^{\prime}2^{\prime}1^{\prime}\overline{3}2^{\prime}1^{\prime}\overline{3}}\right)D_{3},$
(100)
$\displaystyle{q}_{3^{\prime}}=-\left(1-D_{33^{\prime}}-D_{33^{\prime}2^{\prime}1^{\prime}\overline{3}}+D_{2^{\prime}1^{\prime}3^{\prime}33^{\prime}}\right)D_{2^{\prime}},$

are recognized as those of the modified second-generation Relay-U combination
given above.

On the other hand, for the case of $\Delta=[ba\bar{b},a]$, we arrive at an
alternative Relay combination

$\displaystyle{U}({t})$
$\displaystyle=\left(1-{D}_{2^{\prime}1^{\prime}3^{\prime}}-{D}_{2^{\prime}1^{\prime}3^{\prime}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}+D_{33^{\prime}2^{\prime}1^{\prime}3^{\prime}}\right)\eta_{1}$
(101)
$\displaystyle-\left(1-D_{33^{\prime}}-D_{2^{\prime}1^{\prime}3^{\prime}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}+D_{33^{\prime}2^{\prime}1^{\prime}3^{\prime}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}\right)\eta_{1^{\prime}}$
$\displaystyle+\left(1-D_{2^{\prime}1^{\prime}\overline{3}}+D_{33^{\prime}2^{\prime}1^{\prime}\overline{3}}-D_{2^{\prime}1^{\prime}3^{\prime}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}\right)D_{3}\eta_{2^{\prime}}$
$\displaystyle-\left(1-D_{33^{\prime}}-D_{2^{\prime}1^{\prime}3^{\prime}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}+D_{33^{\prime}2^{\prime}1^{\prime}3^{\prime}3\bar{1}^{{}^{\prime}}\bar{2}^{{}^{\prime}}}\right)D_{2^{\prime}}\eta_{3^{\prime}}.$

It is readily verified that the commutators $[ab^{2},bab]$,
$[a\bar{b},\bar{b}a]$, $[aab,aba]$, among others, give rise to a class of
higher-order (from the perspective of the geometric TDI) Relay-U combinations.
Similarly, Relay-V, Relay-W combinations can be obtained, respectively, by
rotating the indices of the constraints $2\rightarrow 3,3\rightarrow 1$ and
$2\rightarrow 1,3\rightarrow 2$.

##  Appendix C The specific forms of the lowest-order commutators

In this Appendix, we enumerate the forms of the lowest-order commutators for
our proposed algorithm. Using the notations in Sec. IV.2, we denote the length
of the commutator by $2n$. Due to the presence of the time-advance operators,
it is noted that the two terms of the commutator do not necessarily have the
same length. For $n=2$, the total count of $a$’s, $\bar{a}$’s, $b$’s, and
$\bar{b}$’s is $2n=4$. This implies two possibilities, $4=1+3$ or $4=2+2$,
such as $[ab,ba]$ and $[a,ba\bar{b}]$. By taking $n=2$ and $n=3$ as examples,
we enumerate all possible forms of the relevant commutators that satisfy Eq.
(27) for the proposed algorithm in Tab. 3. It is noted that even at the lowest
order, the number of possible commutators is much more significant compared to
the original algorithm presented in Sec. III.

Table 3: The specific forms of the commutators with $n=2$ and $n=3$ Length |  | All possible relevant commutators
---|---|---
$2n=4$ | 1+3 |  $[a,ba\bar{b}]$ $[a,\bar{b}ab]$ $[ab\bar{a},b]$ $[a\bar{b}\bar{a},\bar{b}]$
$[b,\bar{a}ba]$ $[b\bar{a}\bar{b},\bar{a}]$ $[\bar{a},\bar{b}\bar{a}b]$
$[\bar{a}\bar{b}a,\bar{b}]$
2+2 |  $[ab,ba]$ $[a\bar{b},\bar{b}a]$ $[b\bar{a},\bar{a}b]$ $[\bar{a}\bar{b},\bar{b}\bar{a}]$
$2n=6$ | 1+5 |  $[a,aab\bar{a}\bar{b}]$ $[a,aa\bar{b}\bar{a}b]$ $[a,aba\bar{b}\bar{a}]$ $[a,ab\bar{a}\bar{b}a]$ $[a,a\bar{b}ab\bar{a}]$ $[a,a\bar{b}\bar{a}ba]$ $[a,baa\bar{b}\bar{a}]$ $[a,bba\bar{b}\bar{b}]$
$[a,b\bar{a}\bar{b}aa]$ $[a,\bar{a}baa\bar{b}]$ $[a,\bar{a}ba\bar{b}a]$
$[a,\bar{a}\bar{b}aab]$ $[a,\bar{a}\bar{b}aba]$ $[a,\bar{b}aab\bar{a}]$
$[a,\bar{b}\bar{a}baa]$ $[a,\bar{b}\bar{b}abb]$
$[aab\bar{a}\bar{a},b]$ $[aa\bar{b}\bar{a}\bar{a},\bar{b}]$
$[abb\bar{a}\bar{b},b]$ $[ab\bar{a}\bar{a}\bar{b},\bar{a}]$
$[ab\bar{a}\bar{b}\bar{a},\bar{a}]$ $[ab\bar{a}\bar{b}\bar{b},\bar{b}]$
$[a\bar{b}\bar{a}bb,b]$ $[a\bar{b}\bar{a}b\bar{a},\bar{a}]$
$[a\bar{b}\bar{a}\bar{a}b,\bar{a}]$ $[a\bar{b}\bar{b}\bar{a}b,\bar{b}]$
$[b,bab\bar{a}\bar{b}]$ $[b,ba\bar{b}\bar{a}b]$ $[b,bba\bar{b}\bar{a}]$
$[b,bb\bar{a}\bar{b}a]$ $[b,b\bar{a}ba\bar{b}]$ $[b,b\bar{a}\bar{b}ab]$
$[b,\bar{a}bba\bar{b}]$ $[b,\bar{a}\bar{a}baa]$ $[b,\bar{a}\bar{b}abb]$
$[b,\bar{b}abb\bar{a}]$ $[b,\bar{b}ab\bar{a}b]$ $[b,\bar{b}\bar{a}bab]$
$[b,\bar{b}\bar{a}bba]$ $[ba\bar{b}\bar{a}\bar{a},\bar{a}]$
$[ba\bar{b}\bar{a}\bar{b},\bar{b}]$ $[ba\bar{b}\bar{b}\bar{a},\bar{b}]$
$[bb\bar{a}\bar{b}\bar{b},\bar{a}]$ $[b\bar{a}\bar{a}\bar{b}a,\bar{a}]$
$[b\bar{a}\bar{b}a\bar{b},\bar{b}]$ $[b\bar{a}\bar{b}\bar{b}a,\bar{b}]$
$[\bar{a},\bar{a}ba\bar{b}\bar{a}]$ $[\bar{a},\bar{a}b\bar{a}\bar{b}a]$
$[\bar{a},\bar{a}\bar{a}ba\bar{b}]$ $[\bar{a},\bar{a}\bar{a}\bar{b}ab]$
$[\bar{a},\bar{a}\bar{b}ab\bar{a}]$ $[\bar{a},\bar{a}\bar{b}\bar{a}ba]$
$[\bar{a},\bar{b}ab\bar{a}\bar{a}]$ $[\bar{a},\bar{b}\bar{a}\bar{a}ba]$
$[\bar{a},\bar{b}\bar{b}\bar{a}bb]$ $[\bar{a}ba\bar{b}\bar{b},\bar{b}]$
$[\bar{a}\bar{a}\bar{b}aa,\bar{b}]$ $[\bar{a}\bar{b}\bar{b}ab,\bar{b}]$
$[\bar{b},\bar{b}ab\bar{a}\bar{b}]$ $[\bar{b},\bar{b}a\bar{b}\bar{a}b]$
$[\bar{b},\bar{b}\bar{a}ba\bar{b}]$ $[\bar{b},\bar{b}\bar{a}\bar{b}ab]$
$[\bar{b},\bar{b}\bar{b}ab\bar{a}]$ $[\bar{b},\bar{b}\bar{b}\bar{a}ba]$
2+4 |  $[aa,aba\bar{b}]$ $[aa,a\bar{b}ab]$ $[aa,baa\bar{b}]$ $[aa,ba\bar{b}a]$ $[aa,\bar{b}aab]$ $[aa,\bar{b}aba]$ $[aab\bar{a},ab]$ $[aab\bar{a},ba]$
$[aa\bar{b}\bar{a},a\bar{b}]$ $[aa\bar{b}\bar{a},\bar{b}a]$ $[ab,bba\bar{b}]$
$[ab,\bar{a}baa]$ $[ab,\bar{b}abb]$ $[abb\bar{a},bb]$ $[ab\bar{a}b,bb]$
$[ab\bar{a}\bar{a},b\bar{a}]$
$[ab\bar{a}\bar{a},\bar{a}b]$ $[a\bar{b},ba\bar{b}\bar{b}]$
$[a\bar{b},\bar{a}\bar{b}aa]$ $[a\bar{b},\bar{b}\bar{b}ab]$
$[a\bar{b}\bar{a}\bar{a},\bar{a}\bar{b}]$
$[a\bar{b}\bar{a}\bar{a},\bar{b}\bar{a}]$
$[a\bar{b}\bar{a}\bar{b},\bar{b}\bar{b}]$
$[a\bar{b}\bar{b}\bar{a},\bar{b}\bar{b}]$
$[ba,bba\bar{b}]$ $[ba,\bar{a}baa]$ $[ba,\bar{b}abb]$ $[bab\bar{a},bb]$
$[ba\bar{b}\bar{b},\bar{b}a]$ $[bb,b\bar{a}ba]$ $[bb,\bar{a}bab]$
$[bb,\bar{a}bba]$
$[bb\bar{a}\bar{b},b\bar{a}]$ $[bb\bar{a}\bar{b},\bar{a}b]$
$[b\bar{a},\bar{a}\bar{a}ba]$ $[b\bar{a},\bar{b}\bar{a}bb]$
$[b\bar{a}\bar{a}\bar{b},\bar{a}\bar{a}]$
$[b\bar{a}\bar{b}\bar{a},\bar{a}\bar{a}]$
$[b\bar{a}\bar{b}\bar{b},\bar{a}\bar{b}]$
$[b\bar{a}\bar{b}\bar{b},\bar{b}\bar{a}]$
$[\bar{a}b,\bar{a}\bar{a}ba]$ $[\bar{a}b,\bar{b}\bar{a}bb]$
$[\bar{a}b\bar{a}\bar{b},\bar{a}\bar{a}]$
$[\bar{a}\bar{a},\bar{a}\bar{b}\bar{a}b]$
$[\bar{a}\bar{a},\bar{b}\bar{a}b\bar{a}]$
$[\bar{a}\bar{a},\bar{b}\bar{a}\bar{a}b]$
$[\bar{a}\bar{a}\bar{b}a,\bar{a}\bar{b}]$
$[\bar{a}\bar{a}\bar{b}a,\bar{b}\bar{a}]$
$[\bar{a}\bar{b},\bar{b}\bar{b}\bar{a}b]$ $[\bar{a}\bar{b}aa,\bar{b}a]$
$[\bar{a}\bar{b}a\bar{b},\bar{b}\bar{b}]$
$[\bar{a}\bar{b}\bar{b}a,\bar{b}\bar{b}]$ $[\bar{b}a,\bar{b}\bar{b}ab]$
$[\bar{b}a\bar{b}\bar{a},\bar{b}\bar{b}]$
$[\bar{b}\bar{a},\bar{b}\bar{b}\bar{a}b]$
$[\bar{b}\bar{a}\bar{b}a,\bar{b}\bar{b}]$
3+3 |  $[aab,aba]$ $[aab,baa]$ $[aa\bar{b},a\bar{b}a]$ $[aa\bar{b},\bar{b}aa]$ $[aba,baa]$ $[abb,bab]$ $[abb,bba]$ $[ab\bar{a},\bar{a}ba]$
$[a\bar{b}a,\bar{b}aa]$ $[a\bar{b}\bar{a},\bar{a}\bar{b}a]$
$[a\bar{b}\bar{b},\bar{b}a\bar{b}]$ $[a\bar{b}\bar{b},\bar{b}\bar{b}a]$
$[bab,bba]$ $[ba\bar{b},\bar{b}ab]$ $[bb\bar{a},b\bar{a}b]$
$[bb\bar{a},\bar{a}bb]$
$[b\bar{a}b,\bar{a}bb]$ $[b\bar{a}\bar{a},\bar{a}b\bar{a}]$
$[b\bar{a}\bar{a},\bar{a}\bar{a}b]$ $[b\bar{a}\bar{b},\bar{b}\bar{a}b]$
$[\bar{a}b\bar{a},\bar{a}\bar{a}b]$
$[\bar{a}\bar{a}\bar{b},\bar{a}\bar{b}\bar{a}]$
$[\bar{a}\bar{a}\bar{b},\bar{b}\bar{a}\bar{a}]$
$[\bar{a}\bar{b}\bar{a},\bar{b}\bar{a}\bar{a}]$
$[\bar{a}\bar{b}\bar{b},\bar{b}\bar{a}\bar{b}]$
$[\bar{a}\bar{b}\bar{b},\bar{b}\bar{b}\bar{a}]$
$[\bar{b}a\bar{b},\bar{b}\bar{b}a]$
$[\bar{b}\bar{a}\bar{b},\bar{b}\bar{b}\bar{a}]$

##  Appendix D Specific higher-order TDI combinations obtained using the
lower-order commutators

This section elaborates on constructing specific higher-order TDI solutions
based on the lower-order commutators. We will be focused on two possible
schemes. The first feasible scheme is multiplying an arbitrary monomial to the
left of a relevant commutator. For instance, $a[ba,ab]$ is a valid commutator
satisfying Eq. (27), which is obtained by multiplying $a$ to the left of
$[ba,ab]$. Similarly, one may derive a higher-order commutator by multiplying
an additional monomial to the right of a valid commutator. However, as will be
shown shortly, the second choice implies some subtlety in deriving the
corresponding TDI coefficients.

For the first scheme, it is not difficult to derive the corresponding TDI
coefficients defined in Eq. (31). Taking $a[ba,ab]$ as an example, one only
needs to multiply to the left of the existing TDI solution, Eq. (32), by the
monomial in question, $a$, namely

$\displaystyle{\tilde{q}}_{1}$ $\displaystyle=a{q}_{1}=a(1-b-ba+ab^{2}),$
(102) $\displaystyle{\tilde{q}}_{1^{\prime}}$
$\displaystyle=a{q}_{1^{\prime}}=-a\left(1-a-ab+ba^{2}\right).$

The validity and generality of the above scheme can be verified
straightforwardly by either substituting Eq. (102) back into Eq. (31) or
closely observing the process discussed in Sec. III.2.

For the second scheme, although the validity of the commutator is apparent,
the corresponding TDI coefficients do not possess a simple relation with the
lower-order counterpart. We first consider a simple example, $[ba,ab]b$. It is
not difficult to show that the TDI coefficients of Eq. (31) for this example
is

$\displaystyle{\tilde{q}}_{1}$ $\displaystyle=1-b-ba+ab^{2},$ (103)
$\displaystyle{\tilde{q}}_{1^{\prime}}$
$\displaystyle=-\left(1-a-ab+ba^{2}+[ba,ab]\right),$

which does not have a simple relation with the TDI solution of the original
commutator $[ba,ab]$, Eq. (32).

However, the above result motivates us to assume an ansatz for the TDI
solution and then solve for its specific form. To this end, we will illustrate
the approach by a more complicated example, $[ab^{2},bab]b\bar{a}$. We utilize
the following ansatz

$\displaystyle{\tilde{q}}_{1}$
$\displaystyle={q}_{1}+\left[ab^{2},bab\right]\alpha,$ (104)
$\displaystyle{\tilde{q}}_{1^{\prime}}$
$\displaystyle=q_{1^{\prime}}+\left[ab^{2},bab\right]\beta,$

where $\alpha$ and $\beta$ are operator polynomials to be determined, $q_{1}$
and $q_{1^{{}^{\prime}}}$ are the TDI solution of Eq. (31) associated with the
commutator $[ab^{2},bab]$. Substitute Eq. (104) back into Eq. (31), one finds
an equation

$\left[ab^{2},bab\right](1+\alpha(1-a)+\beta(1-b)),$ (105)

which should be nothing but $[ab^{2},bab]\bar{b}a$. This implies

$\alpha(1-a)+\beta(1-b)=\bar{b}a-1.$ (106)

Solving Eq. (106) for $\alpha$ and $\beta$ will provide us with the desired
TDI coefficients. Employing a similar trick in the derivation of Eq. (44), we
have

$\bar{b}a=-\bar{b}(1-a)+\bar{b}=-\bar{b}(1-a)+\bar{b}(1-b)+1,$ (107)

in other words,

$\bar{b}a-1=(-\bar{b})(1-a)+\bar{b}(1-b).$ (108)

The simplest choice (that is, the lowest-order solution) is therefore
$-\alpha=\beta=\bar{b}$, and subsequently, we have

$\displaystyle{\tilde{q}}_{1}$
$\displaystyle=q_{1}-\left[ab^{2},bab\right]\bar{b},$ (109)
$\displaystyle{\tilde{q}}_{1^{\prime}}$
$\displaystyle=q_{1^{\prime}}+\left[ab^{2},bab\right]\bar{b}.$

We observe that the above procedure is, by and large, general. It provides an
approach to construct two specific classes of higher-order (from the geometric
TDI perspective) TDI solutions from the lower-order ones.

##  Appendix E Sensitivity funtions for Sagnac and novel Sagnac-inspired
combinations

This Appendix presents the expressions for the averaged response functions of
gravitational waves and noise power spectral density for Sagnac-type
combinations elaborated in the main text. Here, we consider that the residual
noise primarily comprises the shot noise and test mass vibrations, whereas the
other ones, such as the clock jitter noise, are suppressed below the noise
floor.

For the Sagnac combination, the noise power spectral density $N_{\alpha}(u)$
can be expressed as

$N_{\alpha}(u)=\left[16\frac{s_{a}^{2}L^{2}}{c^{4}u^{2}}\Big{(}3-2\mathrm{cos}u-2\mathrm{cos}3u\Big{)}+24\frac{s_{x}^{2}u^{2}}{L^{2}}\right]\mathrm{sin}^{2}\frac{3u}{2},$
(110)

and the gravitational waves averaged response function $R_{\alpha}(u)$ reads

$\displaystyle R_{\alpha}(u)=\frac{1}{6}\mathrm{sin}^{2}\frac{3u}{2}$
$\displaystyle\left[28-24\mathrm{cos}u-4\mathrm{cos}3u+48\mathrm{log}2\left(1+2\mathrm{cos}u\right)+72\left(2\mathrm{log}2-\mathrm{log}3\right)\left(2+\mathrm{cos}3u\right)\right.$
(111)
$\displaystyle\left.+\frac{-204\mathrm{sin}u+147\mathrm{sin}2u-30\mathrm{sin}3u}{u}+\frac{-81+144\mathrm{cos}u-81\mathrm{cos}2u+18\mathrm{cos}3u}{u^{2}}\right.$
$\displaystyle\left.+\frac{-108\mathrm{sin}u+81\mathrm{sin}2u-18\mathrm{sin}3u}{u^{3}}+48\left(4\mathrm{Ci}u-7\mathrm{Ci}2u+3\mathrm{Ci}3u\right)+96\mathrm{cos}u\left(\mathrm{Ci}u-\mathrm{Ci}2u\right)\right.$
$\displaystyle\left.+72\mathrm{cos}3u\left(\mathrm{Ci}u-2\mathrm{Ci}2u+\mathrm{Ci}3u\right)+72\mathrm{sin}3u\left(\mathrm{Si}u-2\mathrm{Si}2u+\mathrm{Si}3u\right)\right],$

where $u=2\pi fc/L$ and $f$ is observed frequency, SinIntergral
$\mathrm{Si}\left(z\right)=\int_{0}^{z}\mathrm{sin}t/tdt$ and CosIntegral
$\mathrm{Ci}\left(z\right)=-\int_{z}^{\infty}\mathrm{cos}t/tdt$.

For the Sagnac-inspired combination, the noise power spectral density
$N_{s}(u)$ can be expressed as

$\displaystyle N_{s}(u)=$ $\displaystyle
64\left(4+3\mathrm{cos}u+2\mathrm{cos}2u+3\mathrm{cos}3u+2\mathrm{cos}4u+\mathrm{cos}5u+2\mathrm{cos}6u+\mathrm{cos}7u\right)\mathrm{sin}^{4}\frac{u}{2}\times$
(112)
$\displaystyle\left(\frac{2s_{a}^{2}L^{2}}{c^{4}u^{2}}\left(13+\mathrm{cos}u-8\mathrm{cos}2u-4\mathrm{cos}3u-2\mathrm{cos}4u\right)+\frac{s_{x}^{2}u^{2}}{L^{2}}\left(16+10\mathrm{cos}u-2\mathrm{cos}2u+2\mathrm{cos}3u+\mathrm{cos}4u\right)\right),$

and the averaged response function of gravitational waves $R_{s}(u)$ is found
to be

$\displaystyle R_{s}(u)=$
$\displaystyle\frac{2}{3}\left(4+3\mathrm{cos}u+2\mathrm{cos}2u+3\mathrm{cos}3u+2\mathrm{cos}4u+\mathrm{cos}5u+2\mathrm{cos}6u+\mathrm{cos}7u\right)\mathrm{sin}^{4}\frac{u}{2}\times$
(113)
$\displaystyle\left[96\left(41+31\mathrm{cos}u+4\mathrm{cos}2u+22\mathrm{cos}3u+10\mathrm{cos}4u\right)\mathrm{log}2-144\left(11+6\mathrm{cos}u+7\mathrm{cos}3u+3\mathrm{cos}4u\right)\mathrm{log}3\right.$
$\displaystyle\left.+944\mathrm{sin}^{2}\frac{u}{2}+64\mathrm{sin}^{2}\frac{u}{2}\left(15\mathrm{cos}u+3\mathrm{cos}2u+\mathrm{cos}3u\right)-\frac{3\left(684\mathrm{sin}u-273\mathrm{sin}2u-146\mathrm{sin}3u+86\mathrm{sin}4u-10\mathrm{sin}5u+\mathrm{sin}6u\right)}{u}\right.$
$\displaystyle\left.+\frac{12\mathrm{sin}^{2}\frac{u}{2}\left(-16+78\mathrm{cos}u-29\mathrm{cos}3u+16\mathrm{cos}4u+5\mathrm{cos}5u\right)}{u^{2}}-\frac{24\mathrm{sin}^{3}\frac{u}{2}\left(96\mathrm{cos}\frac{u}{2}-50\mathrm{cos}\frac{3u}{2}-18\mathrm{cos}\frac{5u}{2}+21\mathrm{cos}\frac{7u}{2}+5\mathrm{\frac{9u}{2}}\right)}{u^{3}}\right.$
$\displaystyle\left.+48\left(49\mathrm{Ci}u-82\mathrm{Ci}2u+33\mathrm{Ci}3u\right)+96\mathrm{cos}u\left(22\mathrm{Ci}u-31\mathrm{Ci}2u+9\mathrm{Ci}3u\right)+384\mathrm{cos}2u\left(\mathrm{Ci}u-\mathrm{Ci}2u\right)\right.$
$\displaystyle\left.+48\mathrm{cos}3u\left(23\mathrm{Ci}u-44\mathrm{cos}2u+21\mathrm{Ci}3u\right)+48\mathrm{cos}4u\left(11\mathrm{Ci}u-20\mathrm{Ci}2u+9\mathrm{Ci}3u\right)\right.$
$\displaystyle\left.+432\left(-\mathrm{sin}u+2\mathrm{sin}3u+\mathrm{sin}4u\right)\left(\mathrm{Si}u-2\mathrm{Si}2u+\mathrm{Si}3u\right)\right].$

The resultant sensitivity function is

$S(u)=\frac{\sqrt{N(u)}}{\sqrt{\frac{2}{5}}\sqrt{R(u)}}.$ (114)

Fig. 5 and Fig. 6 are evaluated using Eqs. (110)$-$(114).

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