# Singular twisted links and singular twisted virtual braids

KOMAL NEGI, MADETI PRABHAKAR

###### Abstract

The concepts of twisted knot theory and singular knot theory inspire the
introduction of singular twisted knot theory. This study showcases similar
findings for singular twisted links, including the Alexander theorem and the
Markov theorem derived from knot theory. Moreover, in this paper we define
singular twisted virtual braids and their monoid structure. Additionally, we
provide both a monoid and a reduced monoid presentation for singular twisted
virtual braids.

MSC2020: 57K10, 57K12, 57M15

Keywords. Singular twisted knots, singular twisted virtual braids.

##  1 Introduction

S. Kamada [7] proved that every oriented virtual link can be depicted as the
closure of a virtual braid. Furthermore, it is established that two virtual
braids have isotopic closures if and only if they are connected by virtual
braid isotopy and a finite series of Markov moves, a widely recognized result.
Similar theorems have been established by Kauffman and Lambropoulou [3, 4]
through the use of $L_{v}$-equivalence. Moreover, C. Caprau et al. [1]
extended the $L_{v}$-move approach to virtual singular braids, deriving the
analogous $L_{v}$-move for virtual singular braids.

Recently, Kamada et al. [2] established Alexander and Markov’s theorems for
twisted virtual braids and twisted links [5, 6]. Furthermore, they verified
that the collection of twisted virtual braids with $n$ strands forms a group
[8]. It will be interesting to see similar kind of results for singular
twisted virtual braids.

This paper focuses on oriented singular twisted links and establishes
Alexander and Markov-type theorems for this category of links. These theorems
play a important role in understanding the structure of singular twisted knots
and links. First, we define the singular twisted virtual braid monoid through
generators and relations. This definition unveils that the singular twisted
virtual braid monoid with $n$ strands extends from the virtual singular braid
monoid with $n$ strands by a group, which is isomorphic to
$\mathbb{Z}_{2}^{n}$.

Several braiding techniques can be employed to demonstrate the extension of
the Alexander theorem to the category of singular twisted virtual braids. In
our context, we adopt the braiding algorithm outlined in [1].

This article is organized as follows, in Section 2, we provide some
preliminaries of singular twisted links and singular twisted virtual braids.
Also, we stated the Alexander theorem for singular twisted links. In Section
3, we outline $L_{v}$-moves and present the Markov-type theorems for singular
twisted links and singular twisted virtual braids. Moreover, we have
established that number of Markov moves or $L_{v}$-moves can be reduced for
singular twisted virtual braids. In Section 4, we provide a reduced
presentation of the singular twisted virtual braid monoid.

##  2 Preliminaries

In this section we define singular twisted links in topological and
combinatorial way. We also define singular twisted virtual braids. Moreover,
we give Alexander theorem for singular twisted links.

###  2.1 Singular twisted link

######  Definition 2.1.

A singular twisted link is an embedding of a 4-valent graph $G$ in
$S_{g}\times I$, where $S_{g}$ is a surface of the genus $g$ and $I$ is the
unit interval such that the thickened surface $S_{g}\times I$ is orientable.

######  Remark 2.2.

A singular twisted knot is an embedding of a one component 4-valent graph $G$
in $S_{g}\times I$.

There is also a combinatorial way to define singular twisted links. For that
we need to define singular twisted link diagrams.

######  Definition 2.3.

A singular twisted link diagram is a decorated immersion of finitely many
disjoint copies of $S^{1}$ into $\mathbb{R}^{2}$, with bars and finitely many
transverse double points such as classical crossings, virtual crossings, and
singular crossings.

Two singular twisted link diagrams are said to be equivalent(or ambient
isotopic) if they are related by a sequence of Reidemeister moves shown in
Figure 1.

Figure 1: Reidemeister moves for singular twisted link diagrams

######  Definition 2.4.

A singular twisted link is the equivalence class of a singular twisted link
diagram.

######  Example 2.5.

Consider the two singular twisted link diagrams $D_{1}$ and $D_{2}$ as shown
in Figure 2. $D_{2}$ can be obtained from $D_{1}$ by sequence of Reidemeister
moves S4, S3, and T2, respectively.

Figure 2: Singular twisted link diagrams

As we know, classical knots have a nice connection with braids. A similar
connection exists between virtual knots and virtual braids. Therefore, it is
natural to ask: Does there exist any braid-like structure in this theory? In
the next subsection, we define singular twisted virtual braids, and we
demonstrate their connection to singular twisted links.

###  2.2 Singular twisted virtual braid

######  Definition 2.6.

A singular twisted virtual braid diagram on $n$ strands (or of degree $n$) is
a union of $n$ smooth or polygonal curves, which are called strands, in
$\mathbb{R}^{2}$ connecting points $(i,1)$ with points $(q_{i},0)$
$(i=1,\dots,n)$, where $(q_{1},\ldots,q_{n})$ is a permutation of the numbers
$(1,\ldots,n)$, such that these curves are monotonic with respect to the
second coordinate and intersections of the curves are transverse double points
equipped with information as a positive/negative/virtual/singular crossing and
curves may have bars by which we mean short arcs intersecting the curves
transversely. See Figure 3, where the five crossings are negative, positive,
virtual, singular and positive from the top.

Figure 3: A singular twisted virtual braid on 3 strands

######  Definition 2.7.

Two singular twisted virtual braid diagrams $b$ and $b^{\prime}$ of degree $n$
are equivalent if they are related by classical, virtual, twisted and singular
braid moves shown in Figure 4,5,6, and 7, respectively.

Figure 4: Classical braid moves Figure 5: Virtual braid moves Figure 6:
Twisted braid moves Figure 7: Singular braid moves

######  Definition 2.8.

A singular twisted virtual braid is an equivalence class of singular twisted
virtual braid diagrams.

Similar to the case of singular braids and virtual singular braids, the set of
singular twisted virtual braids forms a monoid, where the product is defined
by the concatenation similar to the braid group. The set of isotopy classes of
singular twisted virtual braid on $n$ strands forms a monoid, which we denoted
by $STVB_{n}$. Define $STVB_{\infty}=\bigcup\limits_{n=1}^{\infty}STVB_{n}$.

Figure 8: Generators of the monoid of singular twisted virtual braids

######  Theorem 2.9.

The monoid $STVB_{n}$ is generated by standard generators, $\sigma_{i}^{\pm
1}$, $v_{i}$ $(i=1,\dots,n-1)$, $\gamma_{i}$ $(i=1,\dots,n)$, and $\tau_{i}$
$(i=1,\dots,n-1)$ satisfy the following relations, where $e$ denotes the
identity element:

$\displaystyle\sigma_{i}\sigma_{j}$ $\displaystyle=\sigma_{j}\sigma_{i}$ for
$\displaystyle|i-j|>1;$ (1) $\displaystyle\sigma_{i}\sigma_{i+1}\sigma_{i}$
$\displaystyle=\sigma_{i+1}\sigma_{i}\sigma_{i+1}$ for $\displaystyle
i=1,\ldots,n-2;$ (2) $\displaystyle v_{i}^{2}$ $\displaystyle=e$ for
$\displaystyle i=1,\ldots,n-1;$ (3) $\displaystyle v_{i}v_{j}$
$\displaystyle=v_{j}v_{i}$ for $\displaystyle|i-j|>1;$ (4) $\displaystyle
v_{i}v_{i+1}v_{i}$ $\displaystyle=v_{i+1}v_{i}v_{i+1}$ for $\displaystyle
i=1,\ldots,n-2;$ (5) $\displaystyle\sigma_{i}v_{j}$
$\displaystyle=v_{j}\sigma_{i}$ for $\displaystyle|i-j|>1;$ (6) $\displaystyle
v_{i}\sigma_{i+1}v_{i}$ $\displaystyle=v_{i+1}\sigma_{i}v_{i+1}$ for
$\displaystyle i=1,\ldots,n-2;$ (7) $\displaystyle\gamma_{i}^{2}$
$\displaystyle=e$ for $\displaystyle i=1,\ldots,n;$ (8)
$\displaystyle\gamma_{i}\gamma_{j}$ $\displaystyle=\gamma_{j}\gamma_{i}$ for
$\displaystyle i,j=1,\ldots,n;$ (9) $\displaystyle\gamma_{j}v_{i}$
$\displaystyle=v_{i}\gamma_{j}$ for $\displaystyle j\neq i,i+1;$ (10)
$\displaystyle\sigma_{i}\gamma_{j}$ $\displaystyle=\gamma_{j}\sigma_{i}$ for
$\displaystyle j\neq i,i+1;$ (11) $\displaystyle\gamma_{i+1}v_{i}$
$\displaystyle=v_{i}\gamma_{i}$ for $\displaystyle i=1,\ldots,n-1;$ (12)
$\displaystyle v_{i}\sigma_{i}v_{i}$
$\displaystyle=\gamma_{i+1}\gamma_{i}\sigma_{i}\gamma_{i}\gamma_{i+1}$ for
$\displaystyle i=1,\ldots,n-1;$ (13) $\displaystyle\sigma_{i}\sigma_{i}^{-1}$
$\displaystyle=e$ for $\displaystyle i=1,\ldots,n-1;$ (14)
$\displaystyle\tau_{i}\tau_{j}$ $\displaystyle=\tau_{j}\tau_{i}$ for
$\displaystyle|i-j|>1;$ (15) $\displaystyle\sigma_{i}\tau_{j}$
$\displaystyle=\tau_{j}\sigma_{i}$ for $\displaystyle|i-j|>1;$ (16)
$\displaystyle\sigma_{i}\tau_{i}$ $\displaystyle=\tau_{i}\sigma_{i}$ for
$\displaystyle i=1,\ldots,n;$ (17)
$\displaystyle\sigma_{i}\sigma_{i+1}\tau_{i}$
$\displaystyle=\tau_{i+1}\sigma_{i}\sigma_{i+1}$ for $\displaystyle
i=1,\ldots,n-2;$ (18) $\displaystyle\tau_{i}v_{j}$
$\displaystyle=v_{j}\tau_{i}$ for $\displaystyle|i-j|>1;$ (19) $\displaystyle
v_{i}\tau_{i+1}v_{i}$ $\displaystyle=v_{i+1}\tau_{i}v_{i+1}$ for
$\displaystyle i=1,\ldots,n-2;$ (20) $\displaystyle\tau_{i}\gamma_{j}$
$\displaystyle=\gamma_{j}\tau_{i}$ for $\displaystyle j\neq i,i+1;$ (21)
$\displaystyle v_{i}\tau_{i}v_{i}$
$\displaystyle=\gamma_{i+1}\gamma_{i}\tau_{i}\gamma_{i}\gamma_{i+1}$ for
$\displaystyle i=1,\ldots,n-1.$ (22)

######  Remark 2.10.

If we consider generators $\sigma_{i}$, $v_{i}$ $(i=1,\dots,n-1)$, and
$\gamma_{i}$ $(i=1,\dots,n)$ with the relations from (1) to (13), then this
set forms a group called the twisted virtual braid group $TVB_{n}$ [8].

As of now, it is evident that a singular twisted virtual braid can yield a
singular twisted link through the closure operation. The query arises: can we
generate a singular twisted virtual braid from a singular twisted link? The
answer is yes, there exists an algorithm for this process.

The method to convert any diagram of a singular twisted link into the form of
a singular twisted virtual braid’s closure is same as the braiding algorithm
outlined in [1], the only difference is the incorporation of bars on the
strands. The given algorithm helps to establish a theorem concerning singular
twisted links, analogous to the Alexander theorem for twisted virtual braids
and twisted links [2].

######  Theorem 2.11.

(Alexander type theorem for singular twisted links) Every oriented singular
twisted link can be represented as the closure of singular twisted virtual
braid.

######  Corollary 2.12.

Every oriented twisted link can be represented as the closure of twisted
virtual braid.

We have observed that if two singular twisted virtual braids are equivalent,
their corresponding closures are equivalent as singular twisted links.
However, the converse is not necessarily true. It is possible for two non-
equivalent singular twisted virtual braids to have equivalent closures. To
establish this converse, we can introduce a new notion of equivalence between
the singular twisted virtual braids, known as $L_{v}$-equivalence or Markov
equivalence.

##  3 $L_{v}$-moves and Markov-type theorems for singular twisted virtual
braids

In this section we show a theorem on braid presentation of singular twisted
links which is analogous to the Markov theorem for twisted links. For this, we
present the $L_{v}$-moves applicable to singular twisted virtual braids,
wheras the literature already includes $L_{v}$ moves for virtual braids and
singular virtual braids. Furthermore, we have demonstrated the capability to
decrease the number of $L_{v}$-moves required for singular twisted virtual
braids.

All types of $L_{v}$-moves for singular twisted virtual braids are stated
below:

A $L_{v}$-move of type 1, denoted by ST1-move contains two types of moves, the
virtual and real $L_{v}$-move (denoted as VST1-move and RST1-move,
respectively). These moves are illustrated in Figure 9.

Figure 9: ST1-move: Virtual or real $L_{v}$-move

A $L_{v}$-move of type 2, denoted by  ST2-move is a under threaded
$L_{v}$-move shown in Figure 10. It contains two types of moves, the left
ST2-move and the right ST2-move (denoted as LST2-move and RST2-move,
respectively).

Figure 10: ST2-move: Left and Right under threaded $L_{v}$-move

A $L_{v}$-move of type 3, denoted by  ST3-move is a $rs$-threaded
$L_{v}$-move. There are two types of them denoted as LST3 and RST3 shown in
Figure 11.

Figure 11: ST3-move: Left and Right $rs$-threaded $L_{v}$-move

(a) Singular conjugation move

(b) Real and Twist conjugation move.

Figure 12: Different types of ST4-move

A $L_{v}$-move of type 4, denoted by  ST4-move is a conjugation move. There
are three types of conjugation moves: Singular, real, and twist conjugation
(denoted as SST4, RST4 and TST4-move, respectively) as shown in Figure 12(a)
and Figure 12(b).

######  Remark 3.1.

In [4], it has been shown that virtual conjugation, basic $L_{v}$-moves, left
$rL_{v}$\- and $vL_{v}$-moves, over-threaded $L_{v}$-moves, and multi-threaded
$L_{v}$-moves can be derived from the $L_{v}$-equivalence for virtual
braids(it encompasses real conjugation, right $rL_{v}$\- and $vL_{v}$-moves,
left and right under-threaded $L_{v}$-moves, and virtual braid isotopy).
Consequently, these moves can also be derived from the ST1-ST4 $L_{v}$-moves,
thus we do not require to write them in our $L_{v}$-move Markov-type theorem
for singular twisted virtual braids.

Now we are ready to define new notion of equivalence for the singular twisted
virtual braids.

######  Definition 3.2.

Two twisted virtual braids are said to be twisted $L_{v}$-equivalent if they
differ by twisted virtual braid isotopy and a finite sequence of $L_{v}$ moves
ST1-ST2, RST4, and TST4.

######  Definition 3.3.

Two singular twisted virtual braids are said to be $L_{v}$-equivalent if they
differ by singular twisted virtual braid isotopy and a finite sequence of
$L_{v}$ moves ST1-ST4.

######  Theorem 3.4.

Two singular twisted virtual braids have isotopic closures if and only if they
are $L_{v}$-equivalent.

###### Proof.

If two singular twisted virtual braids are $L_{v}$-equivalent then they have
isotopic closures.

Figure 13: Different cases of T1 move

For the converse, we aim to demonstrate that two singular twisted virtual
braids with isotopic closures are related by $L_{v}$-equivalence. To achieve
this, our task is to validate that singular twisted link diagrams, which
differ due to Reidemeister moves, correspond to closures of singular twisted
virtual braids that are $L_{v}$-equivalent. Starting with Reidemeister moves
R1, R2, R3, V1, V2, V3, V4 moves, these moves has been checked in [4] and S1,
S2, S3 moves has been checked in [1], hence it is enough to validate T1, T2,
T3, and S4 moves from Reidemeister moves for singular twisted links. These
moves need to be considered with any given orientation of the strands. We
explore all instances of each isotopy move. If both the strands are in
downward direction, then they are related by braid isotopy and braid moves. If
at least one strand is oriented upward, then we proceed as follow:

Figure 14: Braiding of T2 move

For T1 move and T2 move, all cases are illustrated pictorially in Figure 13
and Figure 14, respectively.

For T3 move, there are four cases. Case I, when both strands are in upward
direction, we apply the braiding algorithm on the diagrams on both sides of
the move (followed by braiding isotopy), the resulting diagrams are related by
ST4-moves as shown in Figure 15. Case II, when one strand is upward and one
strand is downward as shown in Figure 16, we apply the braiding algorithm on
the diagrams on both sides of the move, the resulting diagrams are related by
ST4-move and ST1-move. Case III is a different form of Case I as shown in
Figure 17. Case IV is a different form of Case II illustrated in Figure 18. S4
move is same as T3 move.

Figure 15: Case I of T3 move Figure 16: Case II of T3 move Figure 17: Case III
of T3 move Figure 18: Case IV of T3 move

∎

######  Corollary 3.5.

Two twisted virtual braids have isotopic closures if and only if they are
twisted $L_{v}$-equivalent.

###  3.1 Algebraic Markov type theorem for singular twisted virtual braids

Consider a singular twisted virtual braid $b$ of degree $n$. Define a
monomorphism $\iota_{s}^{t}:STVB_{n}\to STVB_{n+s+t}$, where $s$ and $t$ are
non-negative integers such that $\iota_{s}^{t}(b)$ is a singular twisted
virtual braid of degree $n+s+t$ obtained from $b$ by adding $s$ trivial
strands to the left and $t$ trivial strands to the right.

######  Theorem 3.6.

Two singular twisted virtual braids have isotopic closures if and only if they
differ by a finite sequence of braid relations in $STVB_{\infty}$ together
with the following moves.

  * (i)

Real conjugation: $\sigma_{i}^{\pm 1}\alpha\sim\alpha\sigma_{i}^{\pm 1}$;
virtual conjugation: $v_{i}\alpha\sim\alpha v_{i}$;
singular conjugation: $\tau_{i}\alpha\sim\alpha\tau_{i}$; and twist
conjugation: $\gamma_{i}\alpha\sim\alpha\gamma_{i}$,

  * (ii)

Right real and right virtual stabilization:
$\iota^{1}_{0}(\alpha)\sigma_{n}^{\pm
1}\sim\alpha\sim\iota^{1}_{0}(\alpha)v_{n}$,

  * (iii)

Right algebraic under-threading:
$\alpha\sim\iota^{1}_{0}(\alpha)\sigma_{n}^{\pm 1}v_{n-1}\sigma_{n}^{\mp 1}$;
and
left algebraic under-threading:
$\alpha\sim\iota^{0}_{1}(\alpha)\sigma_{1}^{\pm 1}v_{2}\sigma_{1}^{\mp 1}$,

  * (iv)

Right algebraic rs-threading:
$\iota^{1}_{0}(\alpha)\tau_{n}v_{n-1}\sigma_{n}^{\pm
1}\sim\iota^{1}_{0}(\alpha)\sigma_{n}^{\pm 1}v_{n-1}\tau_{n}$; and
left algebraic rs-threading:
$\iota^{0}_{1}(\alpha)\tau_{1}v_{2}\sigma_{1}^{\pm
1}\sim\iota^{0}_{1}(\alpha)\sigma_{1}^{\pm 1}v_{2}\tau_{1}$.

To prove this theorem it is sufficient to show that each algebraic Markov move
can be obtained from a finite sequence of $L_{v}$-moves, which has already
been proved in [1] except for the twist conjugation. It is easy to observe
that the twist conjugation is a part of the $L_{v}$-moves.

######  Corollary 3.7.

Two twisted virtual braids have isotopic closures if and only if they differ
by a finite sequence of braid relations in $TVB_{\infty}$ together with the
following moves.

  * (i)

Real conjugation : $\sigma_{i}^{\pm 1}\alpha\sim\alpha\sigma_{i}^{\pm 1}$;
virtual conjugation: $v_{i}\alpha\sim\alpha v_{i}$; and
twist conjugation: $\gamma_{i}\alpha\sim\alpha\gamma_{i}$,

  * (ii)

Right real and right virtual stabilization:
$\iota^{1}_{0}(\alpha)\sigma_{n}^{\pm
1}\sim\alpha\sim\iota^{1}_{0}(\alpha)v_{n}$,

  * (iii)

Right algebraic under-threading:
$\alpha\sim\iota^{1}_{0}(\alpha)\sigma_{n}^{\mp 1}v_{n-1}\sigma_{n}^{\pm 1}$
and
left algebraic under-threading:
$\alpha\sim\iota^{0}_{1}(\alpha)\sigma_{1}^{\pm 1}v_{2}\sigma_{1}^{\mp 1}$.

In the following subsection, we have demonstrated the potential to minimize
the number of $L_{v}$-moves and algebraic moves in the context of singular
twisted virtual braids.

###  3.2 On threading moves of singular twisted virtual braids

It turns out that if two singular twisted virtual braids are related by a left
algebraic under-threading move (left algebraic $rs$-threading move) then they
are related by a sequence of conjugation and a right algebraic under threading
move (right algebraic $rs$-threading move). Similarly, if two singular twisted
virtual braids are related by a LST2-move (LST3-move) then they are related by
a ST4-move and a RST2-move (RST3-move). Thus we can remove left algebraic
under threading move and left algebraic $rs$-threading move from the
definition of Markov equivalence and also we can remove LST2-move and
LST3-move from the definition of $L_{v}$ equivalence. Details are provided in
the following theorems.

######  Theorem 3.8.

If two twisted virtual braids of degree $n$ are related by a left algebraic
under-threading move (left algebraic $rs$-threading move), then they are
related by a sequence of conjugation and a right algebraic under-threading
move (right algebraic $rs$-threading move).

###### Proof.

Consider a monoid isomorphism $f_{n}:STVB_{n}\to STVB_{n}$ determined by

$\displaystyle\sigma_{i}$ $\displaystyle\mapsto\sigma_{n-i},$ for
$\displaystyle i=1,\dots,n-1$ $\displaystyle v_{i}$ $\displaystyle\mapsto
v_{n-i},$ for $\displaystyle i=1,\dots,n-1$ $\displaystyle\gamma_{i}$
$\displaystyle\mapsto\gamma_{n-i+1},$ for $\displaystyle i=1,\dots,n$
$\displaystyle\tau_{i}$ $\displaystyle\mapsto\tau_{n-i},$ for $\displaystyle
i=1,\dots,n-1.$

Where, $f_{n}(b)$ a singular twisted virtual braid diagram obtained from the
diagram $b$ by applying the above correspondence to the expression in terms of
the generators.

Let $b^{*}\in STVB_{n}$ such that

$\displaystyle b^{*}=\prod_{i=1}^{n-1}(v_{i}v_{i-1}\dots
v_{1})\prod_{j=1}^{n}\gamma_{j}.$

Define $F_{n}:STVB_{n}\to STVB_{n}$ such that

$\displaystyle b$ $\displaystyle\mapsto b^{*}b{b^{*}}^{-1}$ for $\displaystyle
b\in STVB_{n}.$

It turns out to be a monoid isomorphism. It is easily seen that
${b^{*}}^{2}=e$ and $F_{n}(b)=f_{n}(b)$ for $b\in STVB_{n}$. In particular $b$
and $f_{n}(b)$ are related by conjugation.

It is observable that the map $f_{n}$, facilitating the transition from the
left diagram to the right one (or vice-versa), is indeed a conjugation map
$F_{n}$. We use this observation to support our assertion. Further elaboration
is provided below:

Let $\alpha$ and $b_{2}$ be singular twisted virtual braid diagrams related by
a left algebraic under-threading move. Suppose that

$b_{1}=\iota^{0}_{1}(\alpha)\quad\mbox{and}\quad
b_{2}=\alpha\sigma_{1}v_{2}\sigma_{1}^{-1}.$

Then

$f_{n+1}(b_{1})=\iota^{1}_{0}(f_{n}(\alpha))\quad\mbox{and}\quad
f_{n+1}(b_{2})=f_{n}(\alpha)\sigma_{n}v_{n-1}\sigma_{n}^{-1},$

and hence $f_{n}(\alpha)$ and $f_{n+1}(b_{2})$ are related by a right
algebraic under threading move. Since $b_{1}$ is conjugate to $f_{n}(b_{1})$
as elements of $STVB_{n}$ it implies $\alpha$ is conjugate to $f_{n}(\alpha)$.
Also, $b_{2}$ is conjugate to $f_{n+1}(b_{2})$. Hence, $\alpha$ and $b_{2}$
are related by a sequence of conjugation and right algebraic under-threading
move.

Now, suppose $b_{3}$ and $b_{4}$ be singular twisted virtual braid diagrams
related by a left algebraic $rs$-threading move. Suppose that

$b_{3}=\iota^{0}_{1}(\alpha)\tau_{1}v_{2}\sigma_{1}\quad\mbox{and}\quad
b_{4}=\iota^{0}_{1}(\alpha)\sigma_{1}v_{2}\tau_{1}.$

Then

$f_{n+1}(b_{3})=\iota^{1}_{0}(f_{n}(\alpha))\tau_{n}v_{n-1}\sigma_{n}\quad\mbox{and}\quad
f_{n+1}(b_{4})=\iota^{1}_{0}(f_{n}(\alpha))\sigma_{n}v_{n-1}\tau_{n},$

and hence $f_{n+1}(b_{3})$ and $f_{n+1}(b_{4})$ are related by a right
algebraic $rs$-threading move. Therefore, $b_{3}$ and $b_{4}$ are related by a
sequence of conjugation and right algebraic $rs$-threading move. ∎

######  Theorem 3.9.

Two singular twisted virtual braids have isotopic closures if and only if they
differ by a finite sequence of braid relations in $STVB_{\infty}$ together
with the following moves.

  * •

Real conjugation: $\sigma_{i}^{\pm 1}\alpha\sim\alpha\sigma_{i}^{\pm 1}$;
virtual conjugation: $v_{i}\alpha\sim\alpha v_{i}$;
singular conjugation: $\tau_{i}\alpha\sim\alpha\tau_{i}$; and twist
conjugation: $\gamma_{i}\alpha\sim\alpha\gamma_{i}$,

  * •

Right real and right virtual stabilzation: $\alpha\sigma_{n}^{\pm
1}\sim\alpha\sim\alpha v_{n}$,

  * •

Right algebraic under-threading: $\alpha\sim\alpha\sigma_{n}^{\mp
1}v_{n-1}\sigma_{n}^{\pm 1}$,

  * •

Right algebraic rs-threading: $\alpha\tau_{n}v_{n-1}\sigma_{n}^{\pm
1}\sim\alpha\sigma_{n}^{\pm 1}v_{n-1}\tau_{n}$.

###### Proof.

Use Theorem 3.6 and Theorem 3.8. ∎

######  Corollary 3.10.

Two twisted virtual braids have isotopic closures if and only if they differ
by a finite sequence of braid relations in $TVB_{\infty}$ together with the
following moves.

  * (i)

Real conjugation: $\sigma_{i}^{\pm 1}\alpha\sim\alpha\sigma_{i}^{\pm 1}$;
virtual conjugation: $v_{i}\alpha\sim\alpha v_{i}$; and twist conjugation:
$\gamma_{i}\alpha\sim\alpha\gamma_{i}$,

  * (ii)

Right real and right virtual stabilization: $\alpha\sigma_{n}^{\pm
1}\sim\alpha\sim\alpha v_{n}$,

  * (iii)

Right algebraic under-threading: $\alpha\sim\alpha\sigma_{n}^{\mp
1}v_{n-1}\sigma_{n}^{\pm 1}$.

######  Theorem 3.11.

If two singular twisted virtual braids of degree $n$ are related by a
LST2-move (LST3-move), then they are related by a sequence of ST4 and
RST2-move (RST3-move).

###### Proof.

In Figure 19, pictorially we have shown that LST2-move can be achieved by
RST2-move and ST4-move. Similarly, the LST3-move can be obtained by the
RST3-move and ST4-move as shown in Figure 20.

Figure 19: LST2-move using RST2-move and ST4-move Figure 20: LST3-move using
RST3-move and ST4-move

∎

######  Theorem 3.12.

Two singular twisted virtual braids have isotopic closures if and only if they
differ by sequence of ST1, RST2, RST3 and ST4 moves.

###### Proof.

Use Definition 3.3 and Theorem 3.11. ∎

######  Corollary 3.13.

Two twisted virtual braids have isotopic closures if and only if they differ
by sequence of ST1, RST2, RST4, and TST4 moves.

##  4 Reduced presentation for singular twisted virtual braid monoid

In this section, we show that the presentation of the singular twisted virtual
braid monoid $STVB_{n}$ given in Theorem 2.9 can be reduced to a presentation
with less generators and less relations by rewriting $\sigma_{i}$
$(i=2,\ldots,n-1)$, $\tau_{i}$ $(i=2,\ldots,n-1)$, and $\gamma_{i}$
$(i=2,\ldots,n)$ in terms of $\sigma_{1}$, $\tau_{1}$, $\gamma_{1}$ and
$v_{1},\dots,v_{n-1}$ as follows:

$\displaystyle\sigma_{i}$ $\displaystyle=(v_{i-1}\ldots v_{1})(v_{i}\ldots
v_{2})\sigma_{1}(v_{2}\ldots v_{i})(v_{1}\ldots v_{i-1})$ for $\displaystyle
i=2,\ldots,n-1,$ (23) $\displaystyle\tau_{i}$ $\displaystyle=(v_{i-1}\ldots
v_{1})(v_{i}\ldots v_{2})\tau_{1}(v_{2}\ldots v_{i})(v_{1}\ldots v_{i-1})$ for
$\displaystyle i=2,\ldots,n-1,$ (24) $\displaystyle\gamma_{i}$
$\displaystyle=(v_{i-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots v_{i-1})$ for
$\displaystyle i=2,\ldots,n.$ (25)

These can be seen geometrically from their diagrams shown in Figure 21.

Figure 21: $\sigma_{i}$, $\tau_{i}$, and $\gamma_{i}$

Many relations are already reduced in [1], [2],and [4].

######  Theorem 4.1.

The monoid $STVB_{n}$ has a presentation whose generators are $\sigma^{\pm
1}_{1},\tau_{1},\gamma_{1},v_{1},\dots,v_{n-1}$ and the defining relations are
as follows:

$\displaystyle v_{i}^{2}$ $\displaystyle=e$ for $\displaystyle
i=1,\ldots,n-1;$ (26) $\displaystyle v_{i}v_{j}$ $\displaystyle=v_{j}v_{i}$
for $\displaystyle|i-j|>1;$ (27) $\displaystyle v_{i}v_{i+1}v_{i}$
$\displaystyle=v_{i+1}v_{i}v_{i+1}$ for $\displaystyle i=1,\ldots,n-2;$ (28)
$\displaystyle\sigma_{1}(v_{2}v_{3}v_{1}v_{2}\sigma_{1}v_{2}v_{1}v_{3}v_{2})$
$\displaystyle=(v_{2}v_{3}v_{1}v_{2}\sigma_{1}v_{2}v_{1}v_{3}v_{2})\sigma_{1},$
(29)
$\displaystyle(v_{1}\sigma_{1}v_{1})(v_{2}\sigma_{1}v_{2})(v_{1}\sigma_{1}v_{1})$
$\displaystyle=(v_{2}\sigma_{1}v_{2})(v_{1}\sigma_{1}v_{1})(v_{2}\sigma_{1}v_{2}),$
(30) $\displaystyle\sigma_{1}v_{j}$ $\displaystyle=v_{j}\sigma_{1}$ for
$\displaystyle j=3,\ldots,n-1;$ (31) $\displaystyle\gamma_{1}^{2}$
$\displaystyle=e,$ (32) $\displaystyle\gamma_{1}v_{j}$
$\displaystyle=v_{j}\gamma_{1}$ for $\displaystyle j=2,\ldots,n-1;$ (33)
$\displaystyle\gamma_{1}v_{1}\gamma_{1}v_{1}$
$\displaystyle=v_{1}\gamma_{1}v_{1}\gamma_{1},$ (34)
$\displaystyle\gamma_{1}v_{1}v_{2}\sigma_{1}v_{2}v_{1}$
$\displaystyle=v_{1}v_{2}\sigma_{1}v_{2}v_{1}\gamma_{1},$ (35)
$\displaystyle\gamma_{1}v_{1}\gamma_{1}\sigma_{1}\gamma_{1}v_{1}\gamma_{1}$
$\displaystyle=\sigma_{1}.$ (36) $\displaystyle\sigma_{1}\sigma^{-1}_{1}$
$\displaystyle=e$ (37) $\displaystyle\sigma_{1}\tau_{1}$
$\displaystyle=\tau_{1}\sigma_{1}$ (38) $\displaystyle\tau_{1}v_{i}$
$\displaystyle=v_{i}\tau_{1}$ for $\displaystyle i=3,\ldots n-1;$ (39)
$\displaystyle\tau(v_{1}v_{2}\sigma_{1}v_{2}v_{1})\sigma_{1}$
$\displaystyle=(v_{1}v_{2}\sigma_{1}v_{2}v_{1})\sigma_{1}(v_{1}v_{2}\tau_{1}v_{2}v_{1})$
(40)
$\displaystyle\tau_{1}(v_{2}v_{3}v_{1}v_{2}\sigma_{1}v_{2}v_{1}v_{3}v_{2})$
$\displaystyle=(v_{2}v_{3}v_{1}v_{2}\sigma_{1}v_{2}v_{1}v_{3}v_{2})\tau_{1}$
(41) $\displaystyle\tau_{1}(v_{2}v_{3}v_{1}v_{2}\tau_{1}v_{2}v_{1}v_{3}v_{2})$
$\displaystyle=(v_{2}v_{3}v_{1}v_{2}\tau_{1}v_{2}v_{1}v_{3}v_{2})\tau_{1}$
(42) $\displaystyle\gamma_{1}v_{1}v_{2}\tau_{1}v_{2}v_{1}$
$\displaystyle=v_{1}v_{2}\tau_{1}v_{2}v_{1}\gamma_{1},$ (43)
$\displaystyle\gamma_{1}v_{1}\gamma_{1}\tau_{1}\gamma_{1}v_{1}\gamma_{1}$
$\displaystyle=\tau_{1}.$ (44)

The following lemmas are sufficient for the proof of the Theorem 4.1.

######  Lemma 4.2.

The braid relation $\tau_{i}\gamma_{j}=\gamma_{j}\tau_{i}$ follow from the
defining relations (24), and (25), the other relations, and the reduced
relations
$\gamma_{1}v_{1}v_{2}\tau_{1}v_{2}v_{1}=v_{1}v_{2}\tau_{1}v_{2}v_{1}\gamma_{1}$
and $\gamma_{1}v_{j}=v_{j}\gamma_{1},\text{ for }j\neq 1$.

###### Proof.

We can see that the relation $(\ref{2nd reduction})$ is equivalent to the
relation

$\tau_{i+1}=v_{i}v_{i+1}\tau_{i}v_{i+1}v_{i}$, $\forall 1\leq i\leq n-1$.

First we show $(\ref{1rel-height-sb})$ when $j=1$, i.e.,
$\tau_{i}\gamma_{1}=\gamma_{1}\tau_{i}$ for $i\neq 1$. We apply induction on
$i$, with initial condition $i=2$. Using the relation $(\ref{2nd reduction})$
and $(\ref{relB-height-sb2})$, the following relation follows:

$\tau_{2}\gamma_{1}=\gamma_{1}\tau_{2}$ (45)

Assuming $\tau_{i}\gamma_{1}=\gamma_{1}\tau_{i}$, we obtain
$\tau_{i+1}\gamma_{1}=\gamma_{1}\tau_{i+1}$ as follows:

$\displaystyle\tau_{i+1}\gamma_{1}$
$\displaystyle=v_{i}v_{i+1}\tau_{i}v_{i+1}\underline{v_{i}\gamma_{1}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{relB-height-bv}))}$
$\displaystyle=v_{i}v_{i+1}\tau_{i}\underline{v_{i+1}\gamma_{1}}v_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{relB-height-bv}))}$
$\displaystyle=v_{i}v_{i+1}\underline{\tau_{i}\gamma_{1}}v_{i+1}v_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{i=1}))}$
$\displaystyle=v_{i}\underline{v_{i+1}\gamma_{1}}\tau_{i}v_{i+1}v_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{relB-height-bv}))}$
$\displaystyle=\underline{v_{i}\gamma_{1}}v_{i+1}\tau_{i}v_{i+1}v_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{relB-height-bv}))}$
$\displaystyle=\gamma_{1}\underline{v_{i}v_{i+1}\tau_{i}v_{i+1}v_{i}}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{2nd reduction}))}$
$\displaystyle=\gamma_{1}\tau_{i+1}.$

Hence,

$\tau_{i}\gamma_{1}=\gamma_{1}\tau_{i}\quad\text{ for }i\neq 1.$ (46)

Now, we show relation $(\ref{1rel-height-sb})$:
$\tau_{i}\gamma_{j}=\gamma_{j}\tau_{i}$ for $j\neq i,i+1$.

Case(i) Suppose $j\leq i-1$. Then

$\displaystyle\tau_{i}\gamma_{j}$
$\displaystyle=\underline{\tau_{i}}(v_{j-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{rel-height-vv1}))}$
$\displaystyle=(v_{j-1}\ldots v_{1})\underline{\tau_{i}\gamma_{1}}(v_{1}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{4th reduction}))}$
$\displaystyle=(v_{j-1}\ldots v_{1})\gamma_{1}\underline{\tau_{i}}(v_{1}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{rel-height-vv1}))}$
$\displaystyle=\underline{(v_{j-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{j-1})}\tau_{i}\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by
(\ref{3rd reduction}))}$ $\displaystyle=\gamma_{j}\tau_{i}.$

Case(ii) Suppose $j\geq i+2$. Then

$\displaystyle\tau_{i}\gamma_{j}$
$\displaystyle=\underline{\tau_{i}}(v_{j-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{rel-height-vv1}))}$
$\displaystyle=(v_{j-1}\ldots v_{i+2})\underline{\tau_{i}}(v_{i+1}\ldots
v_{1})\gamma_{1}(v_{1}\ldots v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by
(\ref{2nd reduction}))}$ $\displaystyle=(v_{j-1}\ldots
v_{i+2})v_{i+1}v_{i}\tau_{i+1}\underline{v_{i}v_{i+1}}(\underline{v_{i+1}v_{i}}v_{i-1}\ldots
v_{1})\gamma_{1}(v_{1}\ldots v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\
\text{ (by (\ref{relB-inverse-v}))}$ $\displaystyle=(v_{j-1}\ldots
v_{i})\underline{\tau_{i+1}}(v_{i-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{rel-height-vv1}))}$
$\displaystyle=(v_{j-1}\ldots v_{i})(v_{i-1}\ldots
v_{1})\underline{\tau_{i+1}\gamma_{1}}(v_{1}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{4th reduction}))}$
$\displaystyle=(v_{j-1}\ldots
v_{1})\gamma_{1}\underline{\tau_{i+1}}(v_{1}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{rel-height-vv1}))}$
$\displaystyle=(v_{j-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{i-1})\underline{\tau_{i+1}}(v_{i}\ldots v_{j-1})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{ (by (\ref{2nd reduction}))}$
$\displaystyle=(v_{j-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{i-1})v_{i}v_{i+1}\tau_{i}\underline{v_{i+1}v_{i}}(\underline{v_{i}v_{i+1}}v_{i+2}\ldots
v_{j-1})\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{relB-inverse-v}))}$
$\displaystyle=(v_{j-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{i+1})\underline{\tau_{i}}(v_{i+2}\ldots v_{j-1})\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{ (by (\ref{2nd reduction}))}$
$\displaystyle=\underline{(v_{j-1}\ldots v_{1})\gamma_{1}(v_{1}\ldots
v_{i+1})(v_{i+2}\ldots v_{j-1})}\tau_{i}\leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \text{ (by (\ref{3rd reduction}))}$
$\displaystyle=\gamma_{j}\tau_{i}.$

∎

######  Lemma 4.3.

The braid relation
$v_{i}\sigma_{i}v_{i}=\gamma_{i+1}\gamma_{i}\tau_{i}\gamma_{i}\gamma_{i+1}$
follow from the defining relations (24) and (25), the virtual relations, and
the reduced relations
$\gamma_{1}v_{1}\gamma_{1}\tau_{1}\gamma_{1}v_{1}\gamma_{1}=\tau_{1}$ and
$\gamma_{1}v_{j}=v_{j}\gamma_{1},\text{ for }j\neq 1$.

###### Proof.

For the proof we will use the following relation:

$(v_{1}\ldots v_{i-1})v_{i}(v_{i-1}\ldots v_{1})=(v_{i}\ldots
v_{2})v_{1}(v_{2}\ldots v_{i})$ (47)
$\displaystyle\gamma_{i+1}\gamma_{i}\tau_{i}\gamma_{i}\gamma_{i+1}$
$\displaystyle=(v_{i}\ldots v_{1})\gamma_{1}(v_{1}\ldots v_{i})(v_{i-1}\ldots
v_{1})\gamma_{1}\underline{(v_{1}\ldots v_{i-1})(v_{i-1}\ldots
v_{1})}(v_{i}\ldots v_{2})$ $\displaystyle\tau_{1}(v_{2}\ldots
v_{i})\underline{(v_{1}\ldots v_{i-1})(v_{i-1}\ldots
v_{1})}\gamma_{1}(v_{1}\ldots v_{i-1})(v_{i}\ldots
v_{1})\gamma_{1}(v_{1}\ldots v_{i})\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{relB-inverse-v}))}$
$\displaystyle=(v_{i}\ldots v_{1})\gamma_{1}\underline{(v_{1}\ldots
v_{i-1}v_{i}v_{i-1}\ldots v_{1})}\gamma_{1}(v_{i}\ldots
v_{2})\tau_{1}(v_{2}\ldots v_{i})\gamma_{1}$
$\displaystyle\underline{(v_{1}\ldots v_{i-1}v_{i}v_{i-1}\ldots
v_{1})}\gamma_{1}(v_{1}\ldots v_{i})\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
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v_{2}v_{1}v_{2}\ldots v_{i})\underline{\gamma_{1}}(v_{i}\ldots
v_{2})\tau_{1}(v_{2}\ldots v_{i})\underline{\gamma_{1}}$
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(\ref{relB-height-bv}))}$ $\displaystyle=(v_{i}\ldots v_{1})(v_{i}\ldots
v_{2})\gamma_{1}v_{1}\underline{(v_{2}\ldots v_{i})(v_{i}\ldots
v_{2})}\gamma_{1}\tau_{1}\gamma_{1}\underline{(v_{2}\ldots v_{i})(v_{i}\ldots
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\leavevmode\nobreak\ \text{ (by (\ref{relB-inverse-v}))}$
$\displaystyle=(v_{i}\ldots v_{1})(v_{i}\ldots
v_{2})\underline{\gamma_{1}v_{1}\gamma_{1}\tau_{1}\gamma_{1}v_{1}\gamma_{1}}(v_{2}\ldots
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\leavevmode\nobreak\ \leavevmode\nobreak\ \text{ (by (\ref{relB-height-
sb2}))}$ $\displaystyle=v_{i}\underline{(v_{i-1}\ldots v_{1})(v_{i}\ldots
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\leavevmode\nobreak\ \text{ (by (\ref{2nd reduction}))}$
$\displaystyle=v_{i}\tau_{i}v_{i}.$

∎

##  5 Concluding remarks

In this paper, we explore singular twisted virtual braids and the associated
singular twisted virtual braid monoid. We establish theorems regarding
singular twisted links such as the Alexander theorem and the Markov theorem.
Additionally, we present a monoid presentation and a reduced monoid
presentation for the singular twisted virtual braid monoid. As a potential
direction for future research, developing invariants for both singular twisted
virtual braids and singular twisted links would be of considerable interest.

### Acknowledgements

The first author would like to thank the University Grants Commission(UGC),
India, for Research Fellowship with NTA Ref.No.191620008047. The second author
acknowledges the support given by the NBHM, Government of India under grant-
in-aid with F.No.02011/2/20223NBHM(R.P.)/R&D II/970. This work was partially
supported by the FIST program of the Department of Science and Technology,
Government of India, Reference No. SR/FST/MS-I/2018/22(C).

## References

  * [1] C. Caprau, A. Pena, S. McGahan, Virtual singular braids and links, Manuscripta Math. 151 (2016), no. 1-2, 147–175.
  * [2] K. Negi, M. Prabhakar, and S. Kamada, Twisted virtual braids and twisted links, arXiv:2302.13244v1.
  * [3] L.H. Kauffman and S. Lambropoulou, Virtual braids. Fund. Math. 184 (2004), 159–186.
  * [4] L.H. Kauffman, S. Lambropoulou, Virtual braids and the L-move, J. Knot Theory Ramif. 15(2006), no. 6, 773–811.
  * [5] M. O. Bourgoin, Twisted link theory. Algebr. Geom. Topol. 8 (2008), no. 3, 1249–1279.
  * [6] N. Kamada and S. Kamada, Virtual links which are equivalent as twisted links, Proc. Amer. Math. Soc. 148 (2020), no 5, 2273–2285.
  * [7] S. Kamada, Braid presentation of virtual knots and welded knots. Osaka J. Math. 44 (2007), no. 2, 441–458.
  * [8] V.G. Bardakov, T.A. Kozlovskaya, K. Negi, M. Prabhakar, Twisted Virtual Braid Group, arXiv:2310.04154v1.

Department of Mathematics, Indian Institute
of Technology Ropar, Punjab, India.

     Email<EMAIL_ADDRESS><EMAIL_ADDRESS>


Resonant production of dark photons from axion without a large coupling

Naoya Kitajima${}^{\,a,b}$ and Fuminobu Takahashi${}^{\,b}$

aFrontier Research Institute for Interdisciplinary Sciences, Tohoku
University, Sendai, Miyagi 980-8578, Japan
bDepartment of Physics, Tohoku University, Sendai, Miyagi 980-8578, Japan

Dark photons could be produced resonantly by the oscillating axion field in
the early universe. This resonant production mechanism has been used in
various contexts, including dark photon dark matter and primordial magnetic
field production. However, for this resonant production to work in an
expanding universe, a large axion-dark photon coupling is required, which is
not easy to realize in terms of model building and requires the introduction
of many charged fermions and/or the complex clockwork mechanism. In this
paper, we present a new scenario that efficiently produces dark photons from
the axion with a much smaller coupling. This is possible by modifying the
dynamics of axion and significantly delaying the onset of oscillations, as in
the so-called trapped misalignment mechanism. As a specific example, we
consider models in which dark photon production occurs efficiently despite the
small axion-dark photon coupling by temporally trapping an axion in a wrong
minimum and releasing it after the Hubble parameter becomes much smaller than
the axion mass. In this scenario, it is expected that the polarization
asymmetry of dark photons and gravitational waves generated from dark photons
will be significantly reduced.

##  1 Introduction

Dark photons are known as a plausible candidate for physics beyond the
Standard Model [1]. Recently they have attracted much attention as a candidate
for dark matter, which can be probed by various experiments/observations via
the kinetic mixing with the standard model photons. For reviews, see e.g.
Refs. [2, 3].

The production mechanism of dark photon dark matter has been studied in the
literature, such as the production from the inflationary fluctuations [4, 5],
the resonant production from the axion oscillation [6, 7, 8], gravitational
production during reheating [9], misalignment production of coherent
oscillation [10, 11, 12], production from cosmic strings [13, 14], and
resonant production from dark Higgs [15]. In this paper, we focus on the
resonant production of dark photons from the axion condensate.

Dark photons can be resonantly produced from the coherent oscillation of axion
field [16], and if the axion-dark photon coupling is large enough, the dark
photon production occurs efficiently until the production is saturated. Then
the axion dynamics is significantly affected by the produced dark photons
through the back reaction. In particular, when applied to the QCD axion, the
axion abundance can be reduced to be several % as a consequence of the dark
photon production [17] (see also Ref. [18]). Stochastic gravitational waves
are also sourced by the produced dark photons, which predict circular
polarization asymmetry of the gravitational wave background as a unique signal
of this scenario [19, 20, 21, 22, 23, 24, 25, 26]. In addition, if the dark
photon mass is generated by the Higgs mechanism, the dark photons produced by
the process can effectively stabilize the dark Higgs field at its origin until
late times, where the hidden U(1) symmetry is restored. Then, the dark Higgs
can lead to an additional short-term inflation phase [27] or the early dark
energy [28]. Focusing on the coupling with the standard model photons instead
of dark photons, the primordial large-scale magnetic field may also be
produced by the same mechanism (e.g. [29]).111When electrons or other charged
particles are present, they can hamper the growth of the gauge fields. This
makes it harder to generate magnetic fields using the resonant production
method.

While the resonant production of dark photons is a very interesting phenomenon
and is being investigated in a variety of contexts, it necessitates a large
axion-dark photon coupling, which, in turn, requires a non-trivial UV
completion [6]. The situation is exacerbated when the dark gauge coupling is
small, because the anomalous axion-dark photon coupling is considered to be
proportional to the gauge coupling squared. For example, one way to realize
such a large axion-dark photon coupling is to introduce many symmetry-breaking
scalar fields as in the clockwork mechanism [30] (see also Refs. [31, 32, 33,
34, 35, 36, 37, 38]). Alternatively, one may introduce a large number of
charged fermions that run in the loop diagram.

In this paper, we show a novel possibility to efficiently produce dark photons
from coherent oscillations of the axion for a much smaller coupling. This is
due to a modification of the axion dynamics. Specifically, we consider an
additional potential which traps the axion in early times. Such a potential
can significantly delay the commencement of axion oscillations compared to the
conventional scenario where the axion starts to oscillate when the Hubble
parameter becomes comparable to the axion mass. The trapping effect on the
axion has been investigated in the context of the QCD axion with an extra
Peccei-Quinn symmetry breaking term [39, 40, 41, 30, 42, 43, 44], a lighter
QCD axion [45], and the axion or axion-like particle undergoing a first-order
phase transition [46]. It was shown that the trapped misalignment mechanism
[30, 42, 44] can significantly increase or decrease the axion abundance,
depending on the minimum in which the axion is trapped, as compared to the
conventional misalignment mechanism [47, 48, 49]. Here we show that, by
temporally trapping the axion in a wrong minimum and delaying the onset of
oscillations as in the trapped misalignment mechanism, dark photons are
resonantly produced even for the axion-dark photon couplings that are
comparable to or smaller than ${\cal O}(1)$.

##  2 Model

Let us consider the system of axion ($\phi$) and dark photon ($A_{\mu}$). The
Lagrangian is given by

$\displaystyle{\cal
L}=\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m_{A}^{2}A_{\mu}A^{\mu}-\frac{\beta}{4f_{a}}\phi
F_{\mu\nu}\tilde{F}^{\mu\nu},$ (1)

where $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ is the field
strength tensor,
$\tilde{F}^{\mu\nu}=\epsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}/2\sqrt{-g}$ with
$\epsilon^{0123}=1$ is its dual, $m_{A}$ is the mass of the dark photon,
$f_{a}$ is the axion decay constant and $\beta$ is the axion-dark photon
coupling. Here we consider the following axion potential,

$\displaystyle
V(\phi)=m_{a}^{2}f_{a}^{2}\left[1-\cos\bigg{(}\frac{\phi}{f_{a}}\bigg{)}\right]+V_{\rm
trap}(\phi),$ (2)

with $V_{\rm trap}$ being the potential which traps the axion in early times.
For instance, one can consider the following trapping potential as a toy
model,

$\displaystyle V_{\rm
trap}(\phi)=\frac{1}{2}m_{*}^{2}(\phi-\phi_{*})^{2}\theta(t_{*}-t),$ (3)

where $\theta(t)$ is the step function with respect to the cosmic time $t$,
$t_{*}$ is the time when the trapping ends, and $m_{*}~{}(>m_{a})$ is the mass
stabilizing the axion around $\phi=\phi_{*}$. We also consider the model
motivated by the trapped QCD axion model [44] with the potential given by

$\displaystyle
V(\phi)=m_{a}(t)^{2}f_{a}^{2}\left[1-\cos\bigg{(}\frac{\phi}{f_{a}}\bigg{)}\right]+V_{\rm
trap}(\phi),$ (4)

where the time-dependent mass is given by

$\displaystyle
m_{a}(t)=\begin{cases}m_{a0}\bigg{(}\cfrac{t}{t_{*}}\bigg{)}^{b/2}&~{}~{}\text{for}~{}~{}t<t_{*}\\\\[8.53581pt]
m_{a0}&~{}~{}\text{otherwise}\end{cases}.$ (5)

The trapping potential is given by

$\displaystyle V_{\rm
trap}(\phi)=\Lambda_{H}^{4}\left[1-\cos\bigg{(}\frac{N_{H}\phi}{f_{a}}\bigg{)}\right],$
(6)

where $N_{H}$ is an integer. In case of the QCD axion, the time dependent mass
is expressed as $m_{a0}(\Lambda/T)^{b}$ with $\Lambda\sim O(0.1)\,{\rm GeV}$
being the QCD scale and $b\simeq 4$. In this model, first, the axion falls
into one of the minima of the trapping potential (6). Fig. 1 shows the
schematic illustration of the potential with the above two cases.

(a) Step-function trapping

(b) Trapped QCD axion

Figure 1:  Schematic illustration of the two potentials, a trapping by a step
function of time with $m_{*}=4m_{a}$ and $\phi_{*}=f_{a}$ (left) and the
trapped QCD axion with $N_{H}=3$ and $\Lambda_{H}^{2}=0.5m_{a0}f_{a}$ (right).

From the Lagrangian, one obtains the following equations of motion,

$\displaystyle\ddot{\phi}+3H\dot{\phi}-\frac{\nabla^{2}\phi}{a^{2}}+V^{\prime}(\phi)=-\frac{\beta}{4f_{a}}F_{\mu\nu}\tilde{F}^{\mu\nu},$
(7)
$\displaystyle\ddot{\bm{A}}+H\dot{\bm{A}}-\frac{\nabla^{2}\bm{A}}{a^{2}}-\frac{\beta}{f_{a}a}\left(\dot{\phi}\nabla\times\bm{A}-\nabla\phi\times(\dot{\bm{A}}-\nabla
A_{0})\right)=0,$ (8)

where $a$ is the scale factor, $H=\dot{a}/a$ is the Hubble parameter, with the
overdot denoting the derivative with respect to the cosmic time. One can
decompose the gauge field by circular polarization states,

$\displaystyle\bm{A}(t,\bm{x})=\sum_{\lambda=\pm}\int\frac{d^{3}k}{(2\pi)^{3}}\bm{e}_{\lambda}(\bm{k})e^{i\bm{k}\cdot\bm{x}}A_{\lambda}(t,\bm{k}),$
(9)

where $\bm{e}_{\pm}(\bm{k})$ are circular polarization vectors satisfying
$\bm{k}\cdot\bm{e}_{\pm}(\bm{k})=0$ and $\bm{k}\times\bm{e}_{\pm}(\bm{k})=\mp
i|\bm{k}|\bm{e}_{\pm}(\bm{k})$. Then, assuming the homogeneous background
including the axion field, the evolution equation for each mode is simplified
as

$\displaystyle\ddot{A}_{\pm}+H\dot{A}_{\pm}+\left(m_{A}^{2}+\frac{k^{2}}{a^{2}}\mp\frac{k}{a}\frac{\beta\dot{\phi_{0}}}{f_{a}}\right)A_{\pm}=0,$
(10)

where $\phi_{0}$ is the background value of the axion coherent oscillation.
Note that it shows that the dark photon experiences tachyonic instabilities
depending on the coupling $\beta$ and the sign of $\dot{\phi}_{0}$. Soon after
the beginning of the axion oscillation, the dark photon field can be produced
exponentially. The equation of motion for the axion coherent oscillation is

$\displaystyle\ddot{\phi}_{0}+3H\dot{\phi}_{0}+V^{\prime}(\phi_{0})=\frac{\beta}{f_{a}}\langle\bm{E}\cdot\bm{B}\rangle,$
(11)

where the right-hand-side represents the backreaction by the produced dark
photons and the angular bracket represents an average over volume. It can be
calculated in terms of the Fourier modes of the circular polarization states
as,

$\displaystyle\langle\bm{E}\cdot\bm{B}\rangle=-\frac{1}{a^{3}}\int\frac{d^{3}k}{(2\pi)^{3}}\frac{k}{2}\frac{d}{dt}\big{(}|A_{+}|^{2}-|A_{-}|^{2}\big{)}.$
(12)

The electric and magnetic fields can be represented respectively by
$E_{i}=-\dot{A_{i}}/a$, $B_{i}=\epsilon_{ijk}\partial_{j}A_{k}/a^{2}$ under
the temporal gauge ($A_{0}=0$). The energy density of the dark photon can be
calculated as follows

$\displaystyle\rho_{A}^{(\pm)}=\frac{1}{2}\int\frac{dk}{k}\left[{\cal
P}^{(\pm)}_{E}(k)+{\cal P}^{(\pm)}_{B}(k)\right],$ (13)

with the power spectra given by

$\displaystyle{\cal
P}^{(\pm)}_{E}(k)=\frac{k^{3}|\dot{A}_{\pm}|^{2}}{2\pi^{2}a^{2}},~{}~{}{\cal
P}^{(\pm)}_{B}(k)=\frac{k^{5}|A_{\pm}|^{2}}{2\pi^{2}a^{4}}.$ (14)

In the case with small $\beta$, and a sufficient long trapping of the axion
before the onset of oscillations, the gauge field is amplified by the narrow
resonance rather than the tachyonic instability. When it begins to oscillate,
the homogeneous axion field evolves as $\phi(t)=\theta_{i}f_{a}(a_{\rm
osc}/a)^{3/2}\cos(m_{a}t)$. Then Eq. (10) can be expressed as the following
Mathieu equation,

$\displaystyle\frac{\partial^{2}\mathcal{A}_{\pm}}{\partial
z^{2}}+\left[p-2q\cos(2z)\right]\mathcal{A}_{\pm}=0,$ (15)

where we have defined $z\equiv m_{a}t/2+\pi/4$, $\mathcal{A}_{\pm}\equiv
a^{1/2}A_{\pm}$, $p=4k^{2}/(am_{a})^{2}+4m_{A}^{2}/(m_{a})^{2}$ and $q=\mp
2k\beta\theta_{i}(a_{\rm osc}/a)^{3/2}/(am_{a})$, and we have assumed $H\ll
m_{a}$. Here, we are interested in the case of the narrow resonance, $|q|<1$,
with $\beta\theta_{i}\lesssim 1$. Then, the parametric amplification occurs in
the first resonance band satisfying $1-|q|<p<1+|q|$ [50, 51]. Neglecting the
dark photon mass for simplicity, the resonance condition reads

$\displaystyle 1-\frac{\beta\theta_{i}}{2}\bigg{(}\frac{a_{\rm
osc}m_{a}}{2k}\bigg{)}^{3/2}\lesssim\frac{am_{a}}{2k}\lesssim
1+\frac{\beta\theta_{i}}{2}\bigg{(}\frac{a_{\rm osc}m_{a}}{2k}\bigg{)}^{3/2},$
(16)

where we dropped higher order terms of $q$ since $|q|\ll 1$. Note that the
center of the instability band is at $k/a=m_{a}/2$ and the band width is
proportional to $\beta\theta_{i}$. Therefore the time interval during the
resonant amplification can be estimated as

$\displaystyle z_{\rm res}=\frac{1}{2}m_{a}(t_{\rm end}-t_{\rm
osc})\approx\frac{\beta\theta_{i}m_{a}}{H_{\rm osc}}\left(\frac{2k}{a_{\rm
osc}m_{a}}\right)^{-3/2},$ (17)

where $t_{\rm end}$ is the time at the end of the resonance. Neglecting the
backreaction to the axion, the gauge field is maximally amplified as
$\mathcal{A}_{\pm}\propto\exp(\mu z_{\rm res})$ with $\mu\simeq|q|/2$ being
the growth rate. Thus, the total growth exponent can be computed as follows

$\displaystyle\mu z_{\rm res}\simeq\frac{1}{4}|q|m_{a}(t_{\rm end}-t_{\rm
osc})\simeq\frac{1}{2\sqrt{2}}\left(\frac{k}{a_{\rm
osc}m_{a}}\right)^{-1/2}\frac{(\beta\theta_{i})^{2}m_{a}}{H_{\rm osc}}.$ (18)

Note that the wave number giving the maximum enhancement is $k\approx a_{\rm
osc}m_{a}/2$ since the momentum of produced dark photons should kinematically
satisfy $k/a_{\rm osc}\gtrsim m_{a}/2$.

In the conventional scenario with $H_{\rm osc}\sim m_{a}$, the growth exponent
is $(\beta\theta_{i})^{2}$. Then the significant dark photon production never
occurs for a small coupling ($\beta\ll 1$). In the trapped axion scenario, on
the other hand, the growth exponent can be enhanced by a factor of
$m_{a}/H_{\rm osc}$. Thus, the explosive dark photon production is possible
even for a small coupling if the onset of the axion oscillation is delayed
significantly.

In the next section, we will numerically solve the equations of motion for the
axion and dark photon fields and examine whether the trapping effect of the
axion can induce resonant production of dark photons even for small $\beta$.
Although we could perform lattice simulations in principle, our goal only
requires us to estimate the growth rate of dark photon fields. Therefore, we
will restrict ourselves to the linear analysis and ignore the feedback of the
produced dark photons on the axion dynamics.

##  3 Numerical results

We solve the equation of motion (10) for the dark photon numerically, assuming
a homogeneous background that includes the axion field. Since our interest is
in the growth of dark photon fields for various axion potentials, we neglect
the back reaction and set the r.h.s. of Eq. (11) to zero. We consider three
scenarios of axion dynamics: one without the trapping potential $V_{\rm
trap}$, one with a step function of time, and one with a trapped QCD axion
model.

In Fig. 2, we show the time evolution of the axion field value and the energy
density of each circular polarization mode of the dark photon field for the
three cases mentioned above. In the case without trapping shown in panel (a),
it can be seen that a large asymmetry in the circular polarization is
generated in the dark photons produced in the first few oscillations, due to
the cosmic expansion. Although the back reaction to the axion is not included,
the production of dark photons stops at some point because instability band
becomes smaller due to cosmic expansion. On the other hand, in the two trapped
cases shown in panels (b) and (c), the axion oscillation period is
consistently much shorter than Hubble time from right after the trapping ends,
and therefore both plus and minus modes are equally generated, resulting in no
circular polarization asymmetry. This is in contrast to the conventional case
(a) where there is a large difference between two modes of generated dark
photons. In both cases (b) and (c), we set $\beta=1$ , but it can be seen that
resonant production of dark photons occurs efficiently.

(a) $\beta=10$ without trap

(b) Step-function trapping, $\beta=1$, $m_{a}t_{*}=50$

(c) Trapped QCD axion, $\beta=1$, $m_{a}t_{*}=50$

Figure 2:  Time evolution of the energy density of the dark photon (blue), the
axion field value (red) and the Hubble parameter (maganta). Solid and dashed
blue lines corresponds respectively to the minus and plus circular
polarization modes. We have taken $m_{A}=0$ and $\beta=10$ (top-left),
$\beta=1$ and $m_{a}t_{*}=50$ (top-right and bottom), $m_{*}=4m_{a}$ (top-
right), $N_{H}=3$ and $\Lambda_{H}^{2}=0.5m_{a}f_{a}$ (bottom).

In Fig. 3, we show how much the energy density of dark photons increased
compared to the initial value in the three cases, by changing $\beta$ and the
end time of trapping $t_{*}$. In the case without trapping, it can be seen
that dark photons are generated more efficiently as $\beta$ increases. On the
other hand, in two trapped cases with $\beta=1$, it can be seen that dark
photons are generated more efficiently as the end time of trapping becomes
later. As before the circular polarization asymmetry is suppressed in the
trapped cases.

We also show the time evolution of the spectrum of dark photons for each case
in Fig. 4. Compared to when there is no trap, we can see that the spectrum
becomes sharper. This is what is expected from the previous discussion that
resonance is narrow. Since the horizontal axis is the comoving wavenumber, we
can also see that the fastest growing mode moves to a larger comoving
wavenumber as times goes.

(a) without trap

(b) Step-function trapping, $\beta=1$

(c) Trapped QCD axion, $\beta=1$

Figure 3:  Time evolution of the comoving energy density of the dark photon
normalized by the initial value. The thick and dashed line corresponds
respectively to the minus and the plus circular polarization modes. We set
$m_{A}=0$, $m_{*}=4m_{a}$ (top-right), $N_{H}=3$ and
$\Lambda_{H}^{2}=0.5m_{a}f_{a}$ (bottom) and neglected the backreaction to the
axion.

(a) $\beta=30$ without trap

(b) Step-function trapping, $\beta=1$, $m_{a}t_{*}=250$

(c) Trapped QCD axion, $\beta=1$, $m_{a}t_{*}=250$

Figure 4:  Time evolution of the spectrum of comoving number density of the
dark photon. We set $m_{A}=0$, $m_{*}=4m_{a}$ (top-right), $N_{H}=3$ and
$\Lambda_{H}^{2}=0.5m_{a}f_{a}$ (bottom) and neglected the backreaction to the
axion. The time evolves from bottom to top and the thick black line
corresponds to the final time $m_{a}t=1000$.

Fig. 5 shows the total growth factor of the energy density of dark photons as
a function of the growth exponent $\mu z_{\rm res}$ defined by Eq (18). The
growth can be well-fitted by the analytic formula (dotted lines) especially in
the case of trapped QCD axion model. The deviation from the fitting can be
understood by the effect of the cosmic expansion. In particular, when the
Hubble parameter is comparable to the mass, the growth shows a step-like
behavior due to the tachyonic instability as one can see in Fig. 3(a). In this
case, the growth is clearly delayed compared with the monochromatic
exponential growth which we assumed in the derivation of the total growth
exponent in Eq. (18). If the onset of the oscillation is delayed due to the
trapping and the cosmic expansion becomes negligible in one oscillation
period, the growth can be regarded as a monochromatic exponential function,
which explains why it can be well approximated by the fitting function.

(a) Without trap

(b) Step-function trapping, $\theta_{i}=1$

(c) Trapped QCD axion, $N_{H}=3$

(d) Trapped QCD axion, $N_{H}=6$

Figure 5:  Total growth factor of comoving energy density of dark photons as a
function of $\mu z_{\rm res}$ with $k=a_{\rm osc}m_{a}$ (see Eq. (18)). Top-
left: the case without trapping with $\theta_{i}=2\pi/3$. Dotted line is a
fitting function, $\exp(0.052\mu z_{\rm res})$. Top-Right: step-function
trapping case with $\theta_{i}=1$. Dotted line is a fitting function,
$\exp(0.75\mu z_{\rm res})$ Bottom panels: trapped QCD axion model with
$N_{H}=3$ (left), 6 (right). We have taken $\beta=1$ (red), 0.5 (blue), 0.3
(magenta) and 2 (cyan). Dotted lines are fitting formula, $\exp(0.19\mu z_{\rm
res})$ (left) and $\exp(0.21\mu z_{\rm res})$ (right).

##  4 Discussion and conclusions

We have proposed a modification to the axion dynamics in this paper to
demonstrate that resonant dark photon production can occur efficiently, even
for a small axion-dark photon coupling. This relies on a later onset of the
axion oscillations than is typical. We have studied the time evolution of dark
photons by varying the trapping duration for various coupling constants. Our
results have shown that the required axion-dark photon coupling constant
becomes small in proportion to the square root of the Hubble parameter at the
onset of axion oscillations, as expected from the analytical estimate of the
growth rate. In other words, the total growth exponent is proportional to
$\beta^{2}m_{a}/H_{\rm osc}$ (see Eq. (18)). This implies that the resonant
production of dark photons can occur efficiently by trapping axions for an
arbitrarily small coupling constant, eliminating the need for a complicated UV
completion to obtain the large axion-dark photon coupling. While the axion
dynamics are more complex, only two potentials are needed, and the efficiency
of dark photon production depends on when the trapping ends, which typically
happens when the two potential heights become comparable.

Our mechanism is applicable to a variety of dark photon production scenarios.
For example, it could be applied to scenarios where the abundance of the QCD
axion with a large decay constant is suppressed by producing dark photons. In
the conventional scenario where the axion starts to oscillate when the Hubble
parameter is equal to the axion mass, the axion abundance is reduced to about
$1/10$ [17], but we need to run lattice numerical simulations to determine how
much the axion abundance is reduced in our mechanism. In particular, because
of the relative weakness of the cosmic expansion effect, the interactions
between axions and dark photons may continue to be effective for a longer time
after the back reaction becomes important, and may significantly change the
final ratio of the axion and dark photon energy abundances. This is left for a
further study.

One may apply this mechanism to the early dark energy models using an axion
field [52, 28] where the axion-dark photon coupling is used either for an
efficient reheating to avoid the use of contrived axion potential of the
original model [53] or for generating a heavy mass for dark Higgs. The source
of the trapping potential could behave as dark radiation or hot dark matter.

Another application may be the production of the primordial magnetic field
[29]. However, it relies on the inverse cascade decay of helical magnetic
fields. On the other hand, in our mechanism using the trapped axion, the
circular polarization asymmetry of the produced photons is significantly
suppressed. Therefore, it is difficult to generate the primordial magnetic
field by our mechanism.

Similarly, for gravitational waves produced by dark photons, circular
polarization asymmetry is expected to be significantly reduced compared to the
conventional case due to the weak asymmetry of the first few oscillations of
the axion, but we need to run lattice numerical simulations to obtain a
quantitative evaluation.

## Acknowledgment

F.T. thanks Tomohiro Fujita for a useful comment on the magnetogenesis. The
present work is supported by the Graduate Program on JSPS KAKENHI Grant
Numbers 19H01894 (N.K.), 20H01894 (F.T.), 20H05851 (F.T. and N.K.), 21H01078
(N.K.), 21KK0050 (N.K.), JSPS Core-to-Core Program (grant number:
JPJSCCA20200002) (F.T.). This article is based upon work from COST Action
COSMIC WISPers CA21106, supported by COST (European Cooperation in Science and
Technology).

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