# Families of not perfectly straight knots

Nicholas Owad  Topology and Geometry of Manifolds Unit
Okinawa Institute of Science and Technology Graduate University
Okinawa, Japan 904-0495<EMAIL_ADDRESS>
###### Abstract.

We present two families of knots which have straight number higher than
crossing number. In the case of the second family, we have computed the
straight number explicitly. We also give a general theorem about alternating
knots that states adding an even number of crossings to a twist region will
not change whether the knots are perfectly straight or not perfectly straight.

2016 Mathematics Subject Classification. 57M25, 57M27

##  1\. Introduction

Knot diagrams are most commonly drawn with the minimum number of crossings.
This is how they appear in the knot table in Rolfsen [12] which is often
referred to as the standard knot table. Other common ways of presenting knots
are with braids closures, in bridge position, thin position, übercrossing and
petal diagrams and numerous others. From most of these presentations of
diagrams, invariants are created which are interesting in their own respect.
Jablan and Radović defined the Meander number and OGC number, see [8]. The
author answered questions of theirs in [11] and defined the invariant, the
straight number of a knot. In these two papers, Jablan and Radović and the
author only succeed in calculating the straight number for the standard table
of knots and a few simple families with straight number equal to the crossing
number. Here, we present the first known infinite families of knots with
straight number strictly larger than crossing number.

Adams, Shinjo, and Tanka in [1] have the following result which we make use of
here.

######  Theorem 1.1.

[1, Theorem 1.2] Every knot has a projection that can be decomposed into two
sub-arcs such that each sub-arc never crosses itself.

From this result, via a planar isotopy, one can produce a diagram with a
single straight strand that contains all of the crossings, and we say the
diagram is in straight position. By convention, we will draw this straight arc
horizontally. The number of crossings might need to increase to draw a diagram
in straight position, and so we say the minimum number of crossings over all
straight diagrams for a knot $K$ is the straight number, $\mathtt{str}(K)$.

In [11], we calculated the straight number of all the knots in the standard
Rolfsen table [12]. We say that a knot $K$ is perfectly straight if
$\mathtt{str}(K)=c(K)$. Also in [11], we proved a few basic families of knots
are perfectly straight, including torus knots $T_{2,q}$, pretzel knots, and
2-bridge knots with a continued fraction decomposition length less than 6. But
these results relied on finding an arc which meets every crossing before
meeting itself a second time for some nice diagrams. It is a harder problem to
prove that a family of knots is not perfectly straight, as noted in [11] by
the following question.

######  Question 1.2.

[11] Can we find families of knots which are not perfectly straight?

In this paper, we produce two new families of knots which are not perfectly
straight. The second family comes from a more general theorem about twist
regions in knots and straight number, which can be used on any alternating
knot to create a new family, for either perfectly straight or not perfectly
straight.

A spiral knot $S(n,m,\epsilon)$, first defined in [3], see Definition 3.1, is
a generalization of torus knots by changing the crossings in the standard
braid diagram. Champanerkar, Kofman, and Purcell, [4], defined weaving knots,
$W(n,m)$, which are alternating spiral knots.

Theorem 3.3. Let $n\geq 3,m\geq n+1$ and $\gcd(n,m)=1$. Every weaving knot
$W(n,m)$, is not perfectly straight, i.e. $\mathtt{str}(W(n,m))>c(W(n,m)).$

The next theorem is much more general and lets us create new infinite families
under an operation we call increasing the number of full twists, see
Definition 4.2. Loosely, increasing the number of full twists means adding in
an even number of crossings to a twist region.

Theorem 4.6. Let $K$ be an alternating knot. Given any minimal diagram $D$ of
$K$, let $K^{\prime}$ be the knot obtained by increasing the number of full
twists in any twist region of $D$. Then $K$ is perfectly straight if and only
if $K^{\prime}$ is perfectly straight.

By applying this theorem, we can generalize Theorem 3.3.

Corollary 4.7. Let $n\geq 3,m\geq n+1$ and $\gcd(n,m)=1$. Let
$w_{i}=(\sigma_{1}^{\epsilon_{i_{1}}}\sigma_{2}^{\epsilon_{i_{2}}}\cdots\sigma_{n-1}^{\epsilon_{i_{n-1}}})$
where each $\epsilon_{i_{j}}$ is an odd integers. Then let $w=w_{1}w_{2}\cdots
w_{m}$ and let $K$ be the closure of $w$. If $K$ is alternating, then $K$ is
not perfectly straight.

We also use Theorem 4.6 on the knot $9_{32}$, which has the property
$\mathtt{str}(9_{32})=c(9_{32})+1=10$, to create the first known family of not
perfectly straight knots with known straight number.

Theorem 4.9. Let $t=(t_{1}t_{2},\ldots,t_{6})$ be positive integers such that
$t_{1},t_{2},t_{5},$ and $t_{6}$ are odd and $t_{3}$ and $t_{4}$ are even and
let $s$ be the sum of the $t_{i}$’s. Let $K_{t}$ be the alternating knot
obtained from the template in Figure 6 with $t_{i}$ crossings in the
corresponding twist region. Then $\mathtt{str}(K_{t})=c(K_{t})+1=s+2$.

In the next section, we give some basic definitions related to straight number
and braids. In Section 3, we define and prove the family of weaving knots is
not perfectly straight. And in Section 4 we investigate how increasing the
number of full twists affects the straight number. Here we give the second
family of not perfectly straight knots, and find their straight number. We
also discuss generalizing this idea to make any number of infinite families of
not perfectly straight knots.

##  2\. Definitions and Background

We assume the reader is familiar with braids and knot theory. See Birman and
Brendles [2] and Rolfsen [12], respectively, for more information. For more
information on straight knots and their properties, see [11]. We give the
definitions that are relevant for this paper here.

A link is an ambient isotopy class of $n$ embedded circles in 3-space, i.e.
$\bigsqcup^{n}S^{1}\hookrightarrow S^{3}$. A knot is a link with a single
component, $n=1$. By Theorem 1.1, we know that every knot can be drawn with
two arcs where all crossings occur between the these two arcs. By planar
isotopy, we can make one of these arcs straight, and we say the diagram is in
straight position.

######  Definition 2.1.

Given a knot $K$, the straight number of $K$, $\mathtt{str}(K)$, is the
minimum number of crossings over all diagrams of $K$ that are in straight
position.

If a knot $K$ has $\mathtt{str}(K)=c(K)$, where $c(K)$ is the crossing number,
then we say $K$ is perfectly straight. The horizontal arc through the middle
of the diagram in straight position is called the straight strand and every
crossing occurs on this strand.

The braid group $B_{n}$ on $n$ strands has $n-1$ generators
$\sigma_{1},\sigma_{2},\dots,\sigma_{n-1}$, where $\sigma_{i}$ represents $n$
vertical strands with the $i$-th strand passing over the $i+1$-st strand,
traveling from top to bottom. An element $w$, of the braid group $B_{n}$, is
turned into a link by taking the closure, $cl(w)$. That link is a knot, or
single component link, if the permutation obtained by mapping $w$ into the
symmetric group is an $n$-cycle.

We also need to analyze behavior of knots under flypes, see [10] for more
details. For convenience, we include some definitions which will be used in
future proofs.

######  Definition 2.2.

A flype is move in a diagram described by Figure 1. The single crossing that
switches position to the other side of $F$ in a flype will be called the
flyper.

$F$$T$

$F$

$T$ Figure 1. Two diagrams, $D_{0}$ on the left and $D_{1}$ on the right,
related by a flype.

######  Definition 2.3.

[7] Each crossing $x$ that is a flyper for some flype generates a unique flype
cycle as shown in Figure 2. This cycle is minimal in the sense that each
tangle $F_{i}$ cannot be broken into to two nontrivial tangles that can each
be part of a flype.

$F_{1}$$F_{2}$$F_{3}$$F_{n}$ Figure 2. The flype cycle of length $n$ of a
crossing that is a flyper.

##  3\. Weaving knots

In [3], Brothers et al. define Spiral knots. In this section, we prove
alternating spiral knots, or weaving knots as in [4], are not perfectly
straight. The idea for spiral knots comes from taking the standard diagram of
a torus knot and changing the crossing information.

######  Definition 3.1.

[3] Let $n\geq 2$ and $m\geq 1$, and let $K$ be the closure of the braid word
$(\sigma_{1}^{\epsilon_{1}}\sigma_{2}^{\epsilon_{2}}\cdots\sigma_{n-1}^{\epsilon_{n-1}})^{m}$,
where each $\epsilon_{i}=\pm 1$. If
$\epsilon=(\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{n})$, call $K$ the
spiral link $S(n,m,\epsilon)$.

Let
$w=(\sigma_{1}^{\epsilon_{1}}\sigma_{2}^{\epsilon_{2}}\cdots\sigma_{n-1}^{\epsilon_{n-1}})^{m}$
be the braid word of an spiral link. Note that if the $\epsilon=(\pm 1,\mp
1,\pm 1,\mp 1,\ldots,\pm 1)$, then the spiral link is alternating and is
called a weaving link, [4], and call it $W(n,m)$. See Figure 3. For
convenience, we name
$w_{0}=(\sigma_{1}^{\epsilon_{1}}\sigma_{2}^{\epsilon_{2}}\cdots\sigma_{n-1}^{\epsilon_{n-1}})$
and thus, $w=w_{0}^{m}$. From [3], when $\gcd(n,m)=1$, the spiral link is a
knot.

Figure 3. An example of an alternating braid word
$w_{0}=\sigma_{1}\sigma_{2}^{-1}\sigma_{3}\sigma_{4}^{-1}\in B_{5}$.

######  Lemma 3.2.

Let $n\geq 3$ and $m\geq n+1$. An weaving link $W(n,m)$ has exactly one
reduced alternating diagram up to planar isotopy.

We omit the technical proof here and include a quick sketch. Menasco and
Thistlethwaite proved the Tait flyping conjecture [10] where, any two prime,
oriented, reduced, alternating diagrams are connected by a sequences of flypes
and planar isotopy. By Menasco’s result “an alternating knot is prime if and
only if it looks prime,” [9], we know that these $W(n,m)$ are all prime for
$n\geq 3$ and $m\geq n+1$. Finally, by analyzing the circles in the diagram
which intersect the knot 4 times, we can see that there only such
possibilities surround a single crossing, and these cannot be flyped to obtain
a new diagram. See Figure 4 for the different types of candidates circles.
Hence, there is no flype that yields a different diagram for a knot $W(n,m)$.

$w_{1}$$w_{2}$$w_{3}$ Figure 4. All of the circles, in gray, intersect the
knot more than 4 times.

Now we prove the main theorem of this section.

######  Theorem 3.3.

Let $n\geq 3,m\geq n+1$ and $\gcd(n,m)=1$. Every weaving knot $W(n,m)$, is not
perfectly straight, i.e. $\mathtt{str}(W(n,m))>c(W(n,m)).$

###### Proof.

We will show that there is no way to traverse the knot and visit every
crossing before visiting a crossing a second time. By Lemma 3.2 we know that
this diagram of $K$ is the only diagram with the minimum number of crossing.
Thus we have $m(n-1)$ crossings in the diagram of $K$.

Let $m=bn+r$, where $b\geq 1$ and $1\leq r\leq n-1$. Recall $w=w_{0}^{m}$ and
label the $w_{0}$ in order $w_{1}$ through $w_{m}$. We count the number of
crossings we can reach before we come to a crossing the second time. To
maximize the number of crossings we meet in a single path traveling once down
through the braid, we begin at strand one. Notice that anytime we travel
through a single $w_{i}$ from strand one, we meet all $n-1$ crossings of that
$w_{i}$. Any other strand will only meet one crossing. So when we start at
strand one in $w_{1}$ and travel through $w_{n}$, we will have met $2(n-1)$
crossings. So traveling all the way through to $w_{bn}$ means we meet $b\cdot
2(n-1)$ crossings. Next, we will be at strand one in $w_{bn+1}$, which gives
us another $n-1$ crossings. The remaining of $w_{i}$’s to $w_{m}$ will have a
single crossing each, giving us $r-1$ more crossings. This means we have
$2b(n-1)+(n-1)+(r-1)$ total crossings by traveling once down through $w$,
starting at strand one.

We cannot obtain more crossings by following this path further because we will
come back to the top of $w$ through the closure, and we have already meet all
the crossings in $w_{1}$. But we can travel backwards from strand one in
$w_{1}$, where we began. This will allow us to meet $r-1$ more crossings
before we come to $w_{bn+1}$, where we were last in strand one and met every
crossing. Thus, we have met the maximum number of crossings,

$2b(n-1)+(n-1)+2(r-1)=2m+n-2(b+1).$

By taking the symmetry of the braid into account, there is no other starting
position which will increase the number of crossings we can meet.

Now, we claim that the maximum number of crossings in the diagram we can meet,
$2m+n-2(b+1)$, is less than the total number of crossings, $m(n-1)$, for any
$n\geq 3$ and $m\geq n+1$. When $n\geq 4$, we see that

$2m+n-2(b+1)<2m+n<3m\leq m(n-1),$

leaving us with when $n=3$. So, assuming that $n=3$, we have
$2m+n-2(b+1)=2m+3-2(b+1)$, and since $b\geq 1$, we notice that

$2m+3-2(b+1)<2m=m(3-1),$

completing the proof.∎

This shows that the weaving knots are not perfectly straight, but does not
give many clues as to what the straight number actually is. Thus we ask the
following question.

######  Question 3.4.

What is the straight number of weaving knots?

We observe the following example which might shed some light on this question.

######  Example 3.5.

The simplest example of a knot that this theorem applies to is $8_{18}$. Let
$n=3$, $m=4$, $w_{0}=\sigma_{1}\sigma_{2}^{-1}$, and $w=w_{0}^{4}$ and take
$K$ to be the closure $w$. By Theorem 3.3 we see that this knot is not
perfectly straight, so $\mathtt{str}(8_{18})>8.$ But from [11], by
exhaustively checking all possible configurations straight knots of the
standard table, we know that $\mathtt{str}(8_{18})=10$. We can modify this
problem slightly by letting $m=5$ and then $w=w_{0}^{5}$ to obtain the knot
$10_{123}$, which we found to have $\mathtt{str}(10_{123})=12$. Thus, for both
of these examples, we have $\mathtt{str}(K)=c(K)+2$.

Therefore the bound $\mathtt{str}(K)\geq c(K)+1$ that Theorem 3.3 gives us is
not sharp for the smallest examples. It seems unlikely that, in general, this
theorem is sharp.

##  4\. Adding full twists to diagrams

In this section, we describe a general method for generating families of knots
that are not perfectly straight by adding twists into a diagram we know is not
perfectly straight.

######  Definition 4.1.

A twist region of a diagram of a link K consists of maximal collections of
bigon regions arranged end to end. A single crossing adjacent to no bigons is
also a twist region.

For convenience, we introduce the following definition.

######  Definition 4.2.

We say we are modifying a diagram by increasing the number of full twists of a
twist region when we add an even number of alternating crossings to the twist
region in such a way that no new crossings can be removed by Reidemeister Type
2 moves.

For the main theorem of this section, we need the following useful lemma that
we will then build upon.

######  Lemma 4.3.

Given any diagram $D$, let $D^{\prime}$ be the knot diagram obtained by
increasing the number of full twists in any twist region of $D$. Then $D$ in
straight position if and only if $D^{\prime}$ is in straight position.

###### Proof.

Assume that $D$ is in straight position. Then there is an arc $A$ which meets
every crossing without meeting itself. Any twist region $t$ is then in $A$.
Increasing the number of full twists of $t$ will make a new diagram
$D^{\prime}$ with $A$ now containing the new crossings. For the other
direction, assume that we increased the number of full twists of a twist
region $t$ in a diagram $D$ to obtain the new diagram $D^{\prime}$. If
$D^{\prime}$ is in straight position, then the same arc $A$ without the new
twists will also meet every crossing in $D$.∎

Next, we prove the two pieces that make up Theorem 4.6.

######  Proposition 4.4.

Let $K$ be an alternating knot that is perfectly straight. Given any minimal
diagram $D$ of $K$, let $K^{\prime}$ be the knot obtained by increasing the
number of full twists in any twist region of $D$. Then $K^{\prime}$ is also
perfectly straight.

###### Proof.

Given an alternating, perfectly straight knot $K$ and any minimal diagram $D$,
we see that $D$ is a reduced alternating diagram. Either $D$ is in straight
position or not. If it is in straight position, then increasing the number of
full twists does not change this by Lemma 4.3. If $D$ is not in straight
position, there is some sequence $S=(f_{1},f_{2},\ldots,f_{n})$ of flypes
which will create a diagram $D_{0}$ which is in straight position for the knot
$K$. Note that if we need to move a single crossing $x$ through $S$, we need
to move every crossing of the twist region, $t$, that it belongs to. Also, let
$S^{\prime}\subset S$ be the subsequence of flypes for which the crossings of
$t$ are the flypers.

Assume there are two more crossings in $t$ and call this diagram $D^{\prime}$
of $K^{\prime}$. Let $\hat{S}$ be the sequence which we produce in the
following way. For each $f_{i}$ in $S$, add $f_{i}$ to $\hat{S}$ once if
$f_{i}\not\in S^{\prime}$ and three times if $f_{i}\in S^{\prime}$. Then by
applying the sequence $\hat{S}$ to $D^{\prime}$, we will move the two new
crossings into position next to the other crossings of $t$ and obtain a new
diagram $D_{0}^{\prime}$.

We claim that this diagram $D_{0}^{\prime}$ is in straight position. If we
were to remove two crossings from $t$, then we would have our diagram $D_{0}$
of $K$ which is in straight position. Then by applying Lemma 4.3, we know that
$D_{0}^{\prime}$ is in straight position. ∎

Note that there are alternating knots which are perfectly straight but have
minimal diagrams which are not in straight position, hence the need for this
proposition. Thus, flyping can change whether a minimal diagram is in straight
position. The knot $7_{7}$ is an example of this behavior. The diagram in
Rolfsen’s standard table is not in straight position but
$\mathtt{str}(7_{7})=7$.

######  Proposition 4.5.

Let $K$ be an alternating knot that is not perfectly straight. Given any
minimal diagram $D$ of $K$, let $K^{\prime}$ be the knot obtained by
increasing the number of full twists in any twist region of $D$. Then
$K^{\prime}$ is also not perfectly straight.

###### Proof.

Let $t$ be the twist region and $x$ a crossing on one end of $t$. Then, by
Definition 2.2, $x$ is either a flyper of some flype or not – we may assume
the flype is not the rest of $t$. First, assume $x$ is not a flyper for any
flype, and we increase the number of full twists in $t$. Then for each diagram
$D^{\prime}$ of $K^{\prime}$, there is a corresponding diagram $D$ for $K$
that only differs by exactly the new full twists, which we know is not in
straight position. Then by applying Lemma 4.3 to each of these diagrams we
know that $K^{\prime}$ is still not perfectly straight.

Next we assume that $x$ is a flyper for some flype and thus has a unique flype
cycle, see Definition 2.3. We may assume that $t$ is the single crossing $x$,
as we can make the rest of $t$ a tangle new $F_{i}$. There are three cases for
what can happen when we increase the number of full twists of $t=x$ by one and
then consider all flypes:

  1. (1)

all three crossings are in the same position between $F_{i}$ and $F_{i+1}$,

  2. (2)

two crossings are in one position and the third crossing is in another
position, or,

  3. (3)

all three crossings are in different positions.

If we are in case (1), then by Lemma 4.3 we know that this diagram is not in
straight position as this is just increasing the number of full twists and
making no flypes.

For case (2), we have flyped a single crossing to another position. This is
equivalent to adding in two crossings to a position where $x$ is not. Assume
to the contrary that this diagram is in straight position. Then the arc which
meets every crossing would have been able to also meet every crossing in $D$,
the diagram with these two crossings removed.

Finally, for case (3), we have a similar situation as case (2). Relabel the
diagram so that we have one crossing to the left of the tangle $F_{1}$, just
as in Figure 2. Then the other two crossings are at positions before tangle
$F_{i}$ and after some $F_{i+j}$, flipping all the tangles from $i$ to $i+j$
upside-down. Again, if we assume to the contrary that this diagram is in
straight position, the diagram without these two extra crossings and all the
tangles right-side up would be in straight position, contradicting our
assumption and finishing the proof. ∎

Combining these two propositions give us the following theorem.

######  Theorem 4.6.

Let $K$ be an alternating knot. Given any minimal diagram $D$ of $K$, let
$K^{\prime}$ be the knot obtained by increasing the number of full twists in
any twist region of $D$. Then $K$ is perfectly straight if and only if
$K^{\prime}$ is perfectly straight.

From this theorem, one can quickly produce entire families of knots which are
perfectly straight or not perfectly straight. Here, we generalize weaving
knots.

######  Corollary 4.7.

Let $n\geq 3,m\geq n+1$ and $\gcd(n,m)=1$. Let
$w_{i}=(\sigma_{1}^{\epsilon_{i_{1}}}\sigma_{2}^{\epsilon_{i_{2}}}\cdots\sigma_{n-1}^{\epsilon_{i_{n-1}}})$
where each $\epsilon_{i_{j}}$ is an odd integers. Then let $w=w_{1}w_{2}\cdots
w_{m}$ and let $K$ be the closure of $w$. If $K$ is alternating, then $K$ is
not perfectly straight.

###### Proof.

If each $\epsilon_{i_{j}}=\pm 1$ and $K$ is alternating, then $K$ is a weaving
knot and thus, by Theorem 3.3, $K$ is not perfectly straight. If some
$\epsilon_{i_{j}}$ are odd integers and $K$ is alternating, then by applying
Theorem 4.6, we still have that $K$ is not perfectly straight.∎

######  Question 4.8.

Is a similar statement to Theorem 4.6 true when $K$ is not alternating?

Another application of this theorem is to take any alternating knot $K$ which
has $\mathtt{str}(K)=c(K)+1$ and produce from it a family of knots which is
has $\mathtt{str}(K_{n})=c(K_{n})+1$. For example, consider the knot $9_{32}$
in Figure 5. It is alternating and $\mathtt{str}(9_{32})=10$, which we showed
in [11]. To go from the reduced diagram on the left to the diagram in straight
position, we need to deal with the single crossing not on the straight strand.
Notice that if we push the gray, under arc up under the straight strand and
the bigon we introduce two crossings on the straight strand but eliminate the
single crossing not on the straight strand. The diagram on the right of Figure
5 is now in straight position.

Theorem 4.6 tells us that we can take any twist region in the alternating
diagram on the left, increase the number of full twists, and get a knot which
is also not perfectly straight. But if we only use the twist regions
identified in the template in Figure 6, we will produce knots which are not
perfectly straight, but still have the property that $\mathtt{str}(K)=c(K)+1$.

Figure 5. The knot $9_{32}$. On the left is a reduced diagram that is not in
straight position. On the right is diagram of $9_{32}$ in straight position
with 10 crossings.  $t_{1}$$t_{2}$$t_{3}$$t_{4}$$t_{5}$$t_{6}$ Figure 6. The
template for Theorem 4.9. It is made by increasing the number of full twists
of a diagram of $9_{32}$.

This template in Figure 6 has seven twist regions, one of which is a single
crossing not on the straight strand, which we will not modify. The twist
regions $t_{1},t_{2},t_{5},$ and $t_{6}$ need to have an odd number of
crossings and $t_{3}$ and $t_{4}$ need to have an even number of crossings.
Then, make the knot alternating according to the one crossing defined in the
template.

######  Theorem 4.9.

Let $t=(t_{1}t_{2},\ldots,t_{6})$ be positive integers such that
$t_{1},t_{2},t_{5},$ and $t_{6}$ are odd and $t_{3}$ and $t_{4}$ are even and
let $s$ be the sum of the $t_{i}$’s. Let $K_{t}$ be the alternating knot
obtained from the template in Figure 6 with $t_{i}$ crossings in the
corresponding twist region. Then $\mathtt{str}(K_{t})=c(K_{t})+1=s+2$.

###### Proof.

By Theorem 4.6, we know that $K_{t}$ is not perfectly straight for any choice
of $t$. We can put the diagram in straight position by pushing the gray arc up
in the same way as in Figure 5. This gives us one more crossing, completing
the proof.∎

It should be noted that $9_{32}$ is only special in the sense that it is the
first knot to have enough crossings and $\mathtt{str}(K)=c(K)+1$. This process
can easily create an infinite number of infinite families, and we will briefly
describe one such process later.

A reasonable question is what happens when we take the minimal diagram on the
left in Figure 5 and increase the number of full twists of the one crossing
not on the straight strand. Inserting one full twist here turns $9_{32}$ into
$11_{91}$. By Theorem 4.9, we know that increasing the number of full twists
in any of the other twist regions preserves the fact that
$\mathtt{str}(K)=c(K)+1$. But by previous work done for [11] that was not
published, we know that $\mathtt{str}(11_{91})=13=c(11_{91})+2$. Hence, in
general we cannot make our statement in Theorem 4.6 stronger to include how
straight number versus crossing number behaves. It seems likely that this
process of increasing the number of full twists only increases the difference
between straight number and crossing number. This leads us to the following
conjecture.

######  Conjecture 4.10.

Let $K$ be an alternating knot with $\mathtt{str}(K)=c(K)+n$, where $n\geq 1$,
and let $K^{\prime}$ be the knot obtained by increasing the number of full
twists in some twist region. Then $\mathtt{str}(K^{\prime})\geq
c(K^{\prime})+n$.

To create an infinite number of families like the one in Theorem 4.9, we
merely modify the diagram in careful way. We sketch one possible way to do
this here.

######  Example 4.11.

Consider the twist region $t_{3}$ in Figure 6. Break $t_{3}$ into two regions
side by side and pull the strand that connects them up and follow the region
of the diagram to the section of the straight strand between $t_{5}$ and
$t_{6}$. Here introduce a new twist region $t_{x_{1}}$ with the straight
strand. Again, make the diagram alternating. One can check the flypes, which
will be the same as in the original diagram of $9_{32}$, to be sure this is
not perfectly straight. Using this diagram, make a similar template and you
have a another infinite family.

Use this trick again and break up the new twist region $t_{x_{1}}$ and pull it
back between the two pieces of $t_{3}$ creating a new twist region
$t_{x_{2}}$. Again, we have a new family and repeating this process will yield
a new infinite family each time.

######  Question 4.12.

Can we find lower bounds on straight number, from other other invariants,
which are strictly greater than crossing number?

## References

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# Galactic rotation curves of spiral galaxies and dark matter in
$f(\mathcal{R},T)$ gravity theory

Gayatri Mohan<EMAIL_ADDRESS>Umananda Dev Goswami
<EMAIL_ADDRESS>Department of Physics, Dibrugarh University, Dibrugarh
786004, Assam, India

###### Abstract

Galactic rotation curve is a powerful indicator of the state of the
gravitational field within a galaxy. Flatness of these curves implies the
linear increase of mass to have constant velocity with the radial distance. In
this paper, we focus on the possibility of explaining the flatness of observed
rotation curves of spiral galaxies without postulating the existence of dark
matter in the framework of $f(\mathcal{R},T)$ gravity where the gravitational
Lagrangian is written by an arbitrary function of $\mathcal{R}$, the Ricci
scalar and of $T$, the trace of stress-energy tensor $T_{\mu\nu}$. We derive
the gravitational field equations in this gravity theory for the static
spherically symmetric spacetime and solve the equations for metric
coefficients using a specific model that has minimal coupling between matter
and geometry. The orbital motion of a massive test particle moving in a stable
circular orbit is considered and the behaviour of tangential velocity in the
halo region with the help of the considered model is studied. The linear
variation with distance of the interaction mass generated due to matter-
geometry coupling successfully explains the galactic dynamics without the
existence of dark matter at large distances from the galactic core.

Modified gravity; Galactic rotation curves; Dark datter; Minimal coupling;
Interaction mass.

##  I Introduction

The significant mismatch between the observed and dynamically estimated masses
of different astrophysical systems such as galaxies and clusters of galaxies
raises the issue of Dark Matter (DM), which is one of the leading theoretical
difficulties in modern astrophysics. The pioneering works and observational
data show that the total mass density of the universe is not dominated by the
baryonic matter, instead by some unseen, unusual form of matter called DM,
which is needed for the explanation of both the galactic dynamics as well as
the large scale structures in the universe Mar . The presence of such
mysterious matter was first pointed out by J. H. Oort in the early 1930s J_3 .
He studied the doppler shifts of some stars of our galaxy moving near the
galactic plane and thus by computing their velocities he argued for the
presence of more mass inside the galaxy than the observed mass to hold the
stars in their respective orbits. In 1933, a Swiss astronomer Fritz Zwicky
F_5a  suggested similar hidden matter from his study on the Coma cluster of
galaxies. By employing virial theorem he estimated the virial mass (total
mass) of the cluster from the velocity dispersions of galaxies using doppler
shift and then comparing the virial mass of the cluster to the luminous mass
calculated from $M/L$ ratio, he found an inequality of these two masses which
implies a large mass discrepancy in the cluster Gar_4 ; F_5a ; F_1 ; F_5 . In
1939, Horace Babcock H_9  studied the rotation curve of M31 about 20 kpc away
from the center of the galaxy and by estimating mass distribution of it, he
proposed for a large amount of mass in the exterior part of the galaxy.
Outstanding investigations on the mass distributions in galaxies were
published during late 1960s to early 1980s through the observations of
rotational velocities of stars in some galaxies by Morton Roberts M_10 , Ken
Freeman K_12 , J. C. Jackson J_8 , Roberts and Rots M_10a , Rubin and Ford
V_11 ; V_11a  etc. They noted that the mass of a galaxy must be larger than
its detectable mass and thus they confirmed about an undetectable,
significant, additional mass density at large distances from the center. From
the analyses of rotation curves based on radio observatory data, D. H. Rogstad
and G. Shostak R_D  also suggested the necessity of low luminosity material in
the outer regions of M33, M101, NGC 6946, NGC 2403 and IC 342 galaxies.

As already clear from above, two distinguished observational findings, the
mass discrepancies in clusters of galaxies and the flat shapes of rotation
curves of spiral galaxies are two striking evidences for the existence of DM
within these structures L_2 ; Anna ; C15 ; M_5b . Moreover, another
observational discovery within General Relativity (GR) is the gravitational
lensing, in which distorted image of a distant light source is formed due to
the deflection of light by a massive gravitating system such as a cluster of
galaxies, is also a strong evidence of DM within the lensing cluster. It is
also a powerful tool to determine the amount of mass within the lensing
cluster Gar_4 ; Ca . Further, rotation curves, the plots of tangential
velocities of stars and gas clouds rotating around the center of galaxies as a
function of the distance from the center, are directly related to the overall
mass distributions of galaxies and are potent means to analyze the
gravitational field within galaxies. These curves show that the tangential
velocity of a star increases linearly with the distance from the center of the
galaxy and attains approximately a constant value $v\approx$ 200 – 300 km/s
L_2 ; THar  up to 5 – 6 times Mar  of the luminous radius of a galaxy. It
infers that a visible galaxy is embedded in an extended spherical matter
distribution called the galactic halo (DM halo), where due to a constant $v$,
the mass $M(r)$ within the radius $r$ increases linearly with $r$ according to
the relation $M(r)=v^{2}r/G$ and it causes density variation as $1/r^{2}$ for
a spherically symmetric model Tg ; V_11b ; V_11c . This variation is
exceptional from the Newtonian gravity, which yields Keplerian decrease in
velocity at a large distance from the center.

Most importantly, rotation curves hint the failure of GR at large distances,
as to fit the flat parts of the curves GR needs DM in the outer parts of the
galaxies. But till now no experimental evidence has been found or no direct
evidence of DM has been detected. Consequently, attempts have been made for
searching the alternatives of Newtonian Gravity and GR. As a result, based on
the modification of Newtonian gravity or of GR different theoretical models
have been proposed to understand observations both at cosmological and
astrophysical scales C15 ; Mil ; Ad ; RH . M. Milgrom in 1983, proposed the
Modified Newtonian Dynamics (MOND) Mil  as a possible means to describe the
hidden mass in galaxies and their clusters. Basis of this modification is that
the acceleration $a$ of a particle follows the relation $a^{2}/a_{0}\approx
MG/r^{2}$ within the limit of small acceleration $a\ll a_{0}$ at a distance
$r$ from a mass $M$, where $a_{0}$ is a constant with the dimension of
acceleration. Similarly, the theory of GR has been generalised by modifying
the geometric part of standard Einstein-Hilbert (E-H) action with the higher
order curvature invariants SC  to understand both cosmological and
astrophysical observations. Among these modified theories, termed as Modified
Theories of Gravity (MTGs), the simplest one is the $f(\mathcal{R})$ gravity,
introduced by H. A. Buchdahl in 1970 HB . He considered a modification in the
Lagrangian $\mathcal{L}$ of the E-H action and formulated the theory by
choosing a nonlinear function of the Ricci scalar $\mathcal{R}$ in place of
$\mathcal{R}$ and discussed the singularity problem of the Robertson-Walker
cosmological model. Subsequently, several $f(\mathcal{R})$ gravity models have
been developed with different $f(\mathcal{R})$ functions, which are playing a
significant role in successful explanation of cosmic as well as galactic
dynamics, large scale structures without any exotic component in the universe.
Particularly, for example, the galactic dynamic of a test particle without DM
has been studied in Refs. C15 ; Bra ; Ra6 ; Cap ; Y11 ; cf ; SSh ; MkM ; Ugu .
Moreover, many investigations have also been conducted by analysing the
scalaron’s property in $f(\mathcal{R})$ gravity theory in many literatures
(see e.g. bk ; JF ; TT ; Sca ) to explain DM as the scalar field manifestation
of spacetime curvature.

The theory of GR has also been modified or extended by amending both the
geometry and mass-energy content of the Lagrangian $\mathcal{L}$ of the E-H
action. One of such interesting extended theory of gravity is the
$f(\mathcal{R},T)$ gravity theory, proposed by T. Harko et al. TH  in 2011,
where the gravitational Lagrangian $\mathcal{L}$ is a function of both the
Ricci scalar $\mathcal{R}$ and the trace $T$ of the energy-momentum tensor.
Thus this theory is based on the idea of possible coupling between the
geometry and matter. The notable point of the theory is that
$f(\mathcal{R},T)$ gravity models depend on the source term and source term
has a general expression in terms of matter Lagrangian $\mathcal{L}_{m}$.
Therefore, there would be a particular set of field equations for every choice
of matter Lagrangian. The gravitational field equations in the theory contain
two fluid sources: one is the pure or perfect fluid source which is assumed as
the matter source and other is due to the interaction between matter and
geometry, called the exotic fluid source. Harko et al. have demonstrated that
the covariant derivative of energy-momentum tensor does not vanish in the
theory giving the non-geodesic motion of a massive test particle and also it
has been noted that an extra force remains effective because of the matter-
geometry coupling developed in the theory. Later, Moraes presented that the
extra force may vanish for $\mathcal{L}_{m}=-\,p$ in the non-relativistic dust
matter case pm . Under analogous noninteracting two fluid structures as
presented by Harko et al., S. Chakraborty has described S7  that the
conservation of energy-momentum tensor can be taken into account for some
specific forms of the $f(\mathcal{R},T)$ function. Hence for such cases the
geodesic motion of a test particle in $f(\mathcal{R},T)$ gravity can be
achieved. The $f(\mathcal{R},T)$ theory encourages investigations in different
issues of cosmology MJ ; Hs ; Eb ; Mjs  as well as of astrophysics S.K. ; Jmz
; Ms  including the DM in small AVZ ; Ps ; MPS  and large scale structures Bl
; TKO . Maurya et al. have performed a study on compact structures in
$f(\mathcal{R},T)$ gravity with anisotropic matter distribution and found it
as a justified theory in explaining such astrophysical objects S.K. . Effects
of DM on galactic scale has been suitably elaborated by R. Zaregonbadi and his
collaborators considering a $f(\mathcal{R},T)$ model of minimal matter and
geometry coupling Ra6 . These remarkable studies have motivated us to
understand DM through MTGs.

In our present work, we intend to study the galactic rotation curves in the
framework of $f(\mathcal{R},T)$ gravity theory, hence to explain the mystery
of DM as a large-scale modification of gravity. For this work we consider a
minimally matter-geometry coupled $f(\mathcal{R},T)$ gravity model in the
power law form of both geometry and matter parts with the power law form of
matter density distribution in the galactic halo. Moreover, here we employ the
geodesic equations to obtain the rotation curves of galaxies in spherically
symmetric spacetime. We organize our rest of the paper as follows. In Section
II, we derive the field equations in the $f(\mathcal{R},T)$ gravity and then
transform these equations in the form of modified Einstein equations. In
Section III, the solutions of the modified field equations for the static
spherically symmetric metric and the interaction density variation are
presented. In Section IV, we considered a minimally coupled $f(\mathcal{R},T)$
model and the metric coefficients are expressed in terms of it. The orbital
motion of a test particle is analysed and the variation of interaction mass
with the radius within the galactic halo is derived in Section V. Finally we
conclude our work in Section VI. In this work we use the unit $c=1$ and the
metric signature $(-,+,+,+)$.

##  II Modified Einstein field equations in $f(\mathcal{R},T)$ gravity

The action in the $f(\mathcal{R},T)$ gravity theory takes the form:

$S=\frac{1}{2\kappa^{2}}\int\\!\sqrt{-\,g}\,{f(\mathcal{R},T)}\,d^{4}x+S_{m},$
(1)

where $S_{m}=\int\\!\sqrt{-\,g}\,\mathcal{L}_{m}\,d^{4}x$ is the matter part
of the action with $\mathcal{L}_{m}$ as the Lagrangian density of matter
field, $g$ is determinant of the metric $g_{\mu\nu}$ and $\kappa^{2}=8\pi
G=1/M^{2}_{pl}$. $G$ and $M_{pl}$ are the gravitational constant and Planck
mass (reduced) respectively. The energy-momentum tensor of matter field can be
defined as

$T_{\mu\nu}=-\,\frac{2\,\delta(\sqrt{-\,g}\mathcal{L}_{m})}{\sqrt{-\,g}\
\delta g^{\mu\nu}}=-\,2\ \frac{\delta\mathcal{L}_{m}}{\delta
g^{\mu\nu}}+g_{\mu\nu}\,\mathcal{L}_{m}.$ (2)

From this equation we obtain the trace of $T_{\mu\nu}$ as

$T=-\,2\,g^{\mu\nu}\frac{\delta\mathcal{L}_{m}}{\delta
g^{\mu\nu}}+4\,\mathcal{L}_{m}.$ (3)

In this case we assumed that the matter Lagrangian density $\mathcal{L}_{m}$
does not depend on the derivative of the metric tensor $g_{\mu\nu}$, instead
it depends on $g_{\mu\nu}$ only. Now, the variation of action (1) with respect
to $g_{\mu\nu}$ results field equations of $f(\mathcal{R},T)$ gravity as

$F_{R}\,\frac{\delta{R}}{\delta{g^{\mu\nu}}}-\frac{1}{2}\,f(\mathcal{R},T)\,g^{\mu\nu}=\kappa^{2}T_{\mu\nu}-F_{T}\,\frac{\delta{T}}{\delta{g^{\mu\nu}}},$
(4)

where

$F_{R}=\frac{\partial f(\mathcal{R},T)}{\partial R},\;\;F_{T}=\frac{\partial
f(\mathcal{R},T)}{\partial T},$ $\frac{\delta R}{\delta
g^{\mu\nu}}=R_{\mu\nu}-\nabla_{\mu}\nabla_{\nu}+g_{\mu\nu}\square\;\;\text{and}\;\;\frac{\delta
T}{\delta g^{\mu\nu}}=T_{\mu\nu}+\theta_{\mu\nu}.$ (5)

Here, $R_{\mu\nu}$ is the Ricci tensor, $\nabla_{\mu}$ is the covariant
derivative related to Levi-Civita connection $\Gamma$,
$\square=\nabla_{\mu}\nabla^{\mu}$ represents the d’Alembertian operator and

$\theta_{\mu\nu}=g^{ab}\,\frac{\delta{T}_{ab}}{\delta{g^{\mu\nu}}}=-\,2\,T_{\mu\nu}+g_{\mu\nu}\,\mathcal{L}_{m}-2\,g^{ab}\frac{\partial^{2}{\mathcal{L}_{m}}}{\partial{g^{\mu\nu}}\partial{g^{ab}}}$
(6)

is the energy tensor. As already mentioned, the basis of $f(\mathcal{R},T)$
theory is the coupling between matter and geometry, which shows by equations
(4) with equation (6) that the field equations of the theory depend on energy
tensor $\theta_{\mu\nu}$. This also indicates that corresponding to different
forms of $\mathcal{L}_{m}$, a different set of equations would be generated M3
. $\mathcal{L}_{m}$ can be written in terms of energy density $\rho$ or in
terms of thermodynamic pressure $p$ Fra . The choice of the matter Lagrangian
$\mathcal{L}_{m}$ equal to $p$ results
$\theta_{\mu\nu}=-\,2\,T_{\mu\nu}+p\,g_{\mu\nu}.$ Hence, from equations (4) by
assuming the energy-momentum tensor $T_{\mu\nu}$ for a dustlike matter field,
we finally obtain the field equations of $f(\mathcal{R},T)$ theory as follows
R5 ; Ra6 :

$F_{R}\,R_{\mu\nu}-\frac{1}{2}\,f(\mathcal{R},T)\,g_{\mu\nu}+\left(g_{\mu\nu}\square-\nabla_{\mu}\nabla_{\nu}\right)F_{R}=\left(\kappa^{2}+F_{T}\right)T_{\mu\nu}.$
(7)

With a simple algebric manipulation, these field equations (7) can be written
as the modified Einstein equations with an effective energy-momentum tensor
$T_{\mu\nu}^{eff}$ as

$G_{\mu\nu}=8\pi\,G_{eff}\,T_{\mu\nu}^{eff},$ (8)

where $G_{eff}=G/F_{R}$ and $T_{\mu\nu}^{eff}=T_{\mu\nu}+T_{\mu\nu}^{I}.$
$T_{\mu\nu}^{I}$ represents the matter-curvature interaction term defined as
interaction energy-momentum tensor and may act as a new matter source induced
by the $f(\mathcal{R},T)$ action that behaves as an extra effective fluid of
purely geometric origin R5 ; S7 . It is given as

$T_{\mu\nu}^{I}=\
\frac{1}{\kappa^{2}}\left[F_{T}\,T_{\mu\nu}+\frac{1}{2}\left(f(\mathcal{R},T)-F_{R}R\right)g_{\mu\nu}+\left(\nabla_{\mu}\nabla_{\nu}-g_{\mu\nu}\square\right)F_{R}\right].$
(9)

It is to be noted that because of the extra terms appearing in the field
equations, the $f(\mathcal{R},T)$ theory of gravity can be regarded as a two
fluid model pm . Further, defining $T_{\mu\nu}^{I}$ for an anisotropic
geometric matter distribution, we can express it as Bra

$T_{\mu}^{\nu I}=\text{diag}\left(-\rho^{I},\ P_{r}^{I},\ P_{t}^{I},\
P_{t}^{I}\right),$ (10)

where $\rho^{I}$ represents interaction energy density, $P_{r}^{I}$ and
$P_{t}^{I}$ are the radial and tangential components of interaction pressure
respectively. Thus we obtain,

$\rho^{eff}=\rho+\rho^{I},\;\;\;P_{r}^{eff}=P_{r}^{I},\;\;\;P_{t}^{eff}=P_{t}^{I}.$
(11)

Here, $\rho^{eff}$ is effective energy density. As we have considered
$T_{\mu\nu}$ for the dustlike matter, so both radial and tangential components
of the effective pressure are equal to their respective interaction terms.

##  III Gravitational Field Equations for Static Spherically Symmetric Mass
Disribution

###  III.1 Solutions of the modified field equations

As mentioned in Section I, since our study mainly concentrates on the
explanation of behaviour of galactic rotation curves of spiral galaxies in
galactic halo regions without considering the existence of DM in it, we need
to have a suitable mass distribution for such galaxies. Simply such a galaxy
can be considered to have spherically symmetric distribution of matter having
smooth gravitational field J9 . So we shall consider a general static
spherically symmetric metric of the form:

$ds^{2}=-e^{2\mu}dt^{2}+e^{2\nu}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta\,d\varphi^{2},$
(12)

where $\mu$ and $\nu$ are metric potentials and are functions of $r$ only. In
our present case the metric coefficients $e^{2\mu}$ and $e^{2\nu}$ can be
determined by solving the modified Einstein field equations (8) as follows.
The nonzero components of Einstein tensor $G_{\mu\nu}$ for this metric (12)
can be obtained as

$\displaystyle G_{0}^{0}\,$
$\displaystyle=\,-\,\frac{2\,e^{-2\nu}}{r}\,\nu^{\prime}+\frac{e^{-2\nu}}{r^{2}}-\frac{1}{r^{2}},$
(13) $\displaystyle G_{1}^{1}\,$
$\displaystyle=\,\frac{2\,e^{-2\nu}}{r}\,\mu^{\prime}+\frac{e^{-2\nu}}{r^{2}}-\frac{1}{r^{2}},$
(14) $\displaystyle G_{2}^{2}\,=\,G_{3}^{3}\,$
$\displaystyle=\,e^{-2\nu}\left[\mu^{\prime\prime}+(\mu^{\prime})^{2}-\mu^{\prime}\nu^{\prime}\right]+\frac{e^{-2\nu}}{r}\,(\mu^{\prime}-\nu^{\prime}).$
(15)

Using these components (13), (14) and (15) together with equation (11) we
obtain the effective field equations from equations (8) as given by

$\frac{2\,e^{-2\nu}}{r}\,\nu^{\prime}-\frac{e^{-2\nu}}{r^{2}}+\frac{1}{r^{2}}=8\,\pi\,G_{eff}\left(\rho+\rho^{I}\right),$
(16)
$\frac{2\,e^{-2\nu}}{r}\,\mu^{\prime}+\frac{e^{-2\nu}}{r^{2}}-\frac{1}{r^{2}}=8\,\pi\,G_{eff}\,P_{r}^{I},$
(17) $e^{-2\nu}\left[\
\mu^{\prime\prime}+(\mu^{\prime})^{2}-\mu^{\prime}\nu^{\prime}\right]+\frac{e^{-2\nu}}{r}\,(\mu^{\prime}-\nu^{\prime})=8\,\pi\,G_{eff}\,P_{t}^{I}.$
(18)

The interaction energy density $\rho^{I}$ and components of anisotropic
interaction pressure $P_{r}^{I}$ and $P_{t}^{I}$ can be obtained from equation
(9), which can be expressed as

$\displaystyle\rho^{I}\,$
$\displaystyle=\,\frac{1}{8\,\pi\,G}\left[F_{T}\,\rho+\frac{1}{2}\left(R\,F_{R}-f(\mathcal{R},T)\right)+\left(F_{R}^{\prime\prime}-\nu^{\prime}F_{R}^{\prime}+\frac{2}{r}\,F_{R}^{\prime}\right)\,e^{-2\nu}\right],$
(19) $\displaystyle P_{r}^{I}\,$
$\displaystyle=\,\frac{1}{8\,\pi\,G}\left[\frac{1}{2}\left(f(\mathcal{R},T)-R\,F_{R}\right)+\left(\mu^{\prime}\,F_{R}^{\prime}-\frac{2}{r}\,F_{R}^{\prime}\right)e^{-2\nu}\right],$
(20) $\displaystyle P_{t}^{I}\,$
$\displaystyle=\,\frac{1}{8\,\pi\,G}\left[\frac{1}{2}\left(f(\mathcal{R},T)-R\,F_{R}\right)-\left(F_{R}^{\prime\prime}+\frac{F_{R}^{\prime}}{r}+(\mu^{\prime}-\nu^{\prime})\,F_{R}^{\prime}\right)e^{-2\nu}\right].$
(21)

To solve equations (16), (17) and (18) for the metric coefficients, we just
sum equation (16) and (17) which gives a differential equation for the
functions $\mu$ and $\nu$ in the form:

$\frac{\mu^{\prime}+\nu^{\prime}}{r\,e^{2\nu}}=4\,\pi\,G_{eff}\left[\rho+\rho^{I}+P_{r}^{I}\right].$
(22)

At this point it needs to be mentioned that in GR we have
$\mu^{\prime}+\nu^{\prime}=0$ and hence $e^{2\mu}e^{2\nu}=1$. But as we are
working with the $f(\mathcal{R},T)$ modified theory of gravity, which is an
extension of GR, so have obtained equation (22). Thus in our case, we should
have $e^{2\mu}e^{2\nu}$ different from unity. It can be argued that if
$\mu^{\prime}+\nu^{\prime}$ is a well-behaved expression it should have a
solution of the following form Y11 :

$e^{2\mu}e^{2\nu}=X(r).$ (23)

Here, function $X(r)$ should be slightly different from unity to remain it in
the vicinity of GR, so that the $f(\mathcal{R},T)$ gravity behaves similarly
to GR in the low curvature regimes. To differ $X(r)$ from the unity by a small
amount we assume that Y11

$X(r)=\left(\frac{r}{h}\right)^{\delta},$ (24)

where $\delta$ is a small dimensionless parameter, $h$ is the scale length of
the system. With this form of $X(r)$ equation (23) takes the form:

$2(\mu+\nu)=\ln\left(\frac{r}{h}\right)^{\delta}.$ (25)

From which a differential equation for the metric potentials is achieved as

$\mu^{\prime}\ +\nu^{\prime}=\frac{\delta}{2\,r}.$ (26)

Also

$e^{2\mu}e^{2\nu}=e^{2\mu+2\nu}\cong 1+2(\mu+\nu).$

Hence we may write the product $e^{2\mu}e^{2\nu}$ approximately as

$e^{2\mu}e^{2\nu}=X(r)\cong 1+\ln\left(\frac{r}{h}\right)^{\delta}.$ (27)

Thus the product $e^{2\mu}e^{2\nu}$ is slightly greater than unity as
expected. Now, using equation (26) in equation (22) we can expresses the
metric coefficient $e^{2\nu}$ in the present case as

$e^{2\nu}=\delta\left[8\,\pi\,G_{eff}\,r^{2}\left(\rho+\rho^{I}+P_{r}^{I}\right)\right]^{-1}.$
(28)

And hence the coefficient of $g_{00}$ component of the metric can be written
as

$e^{2\mu}=8\,\pi\,G_{eff}\,\delta^{-1}h^{-\delta}r^{\delta+2}\left(\rho+\rho^{I}+P_{r}^{I}\right).$
(29)

Later these metric coefficients will be determined in their explicit forms for
our considered model of the $f(\mathcal{R},T)$ gravity.

###  III.2 Power law of mass density

It is observed that most of the baryonic matter of a normal spiral galaxy lies
in the flattened disk in the form of stars, dust and interstellar gas. The
mass to luminosity ratio of such galaxies increases with increasing distance
from the center up to the outer regions of the galaxies V_11c . Further
rotation curves show linear increase of mass with distance indicating higher
concentration of nonluminous matter beyond the disk. Luminosity of many such
galaxies almost follow a power law over a large range of radial distance.
Consequently, one may consider a system the mass distribution of which in far
distant galactic halo is such that the density falls off as some power of the
radial distance $r$ as J9 ; Fra

$\rho(r)=\rho_{0}\left(\frac{1}{r}\right)^{\lambda},$ (30)

where $\rho_{0}$ and $\lambda$ are positive constants, $\rho_{0}$ can be set
as unity without loss of generality. The mass profile within $r$ for this mass
distribution is

$M(r)=4\pi\int_{0}^{r}{r^{2}\rho(r)}\,dr=4\pi\rho_{0}\,\frac{r^{3-\lambda}}{3-\lambda}.$
(31)

Hence the tangential velocity from the mass profile is obtained as

$v^{2}=\frac{4\pi G\rho_{0}\,r^{2-\lambda}}{3-\lambda}.$ (32)

As the observed rotation curves dictate us about the constant tangential
velocity in far distant halo regions of galaxies, therefore, for this region
we may take $\lambda=2$ in equation (32) so that corresponding to this value
$v$ remains constant. In view of this, rewriting equation (30) with
$\lambda=2$, we obtain the following power law for matter density in the halo:

$\rho(r)\propto\frac{1}{r^{2}},$ (33)

which suggests that the mass density decreases with increase of radial
distance within the halo. Here $\lambda=1$ and $3$ are not allowed, since only
for $\lambda\neq 3$, the mass interior to $r$ is finite and for $\lambda\neq
1$ the law behaves as power law.

##  IV The Metric Coefficients from the Model of Study

One can perform a study on $f(\mathcal{R},T)$ modified theory by employing
different matter-geometry coupling models. The functional $f(\mathcal{R},T)$
in the theory may be modelled through minimally, non-minimally or purely non-
minimally coupling of the matter and geometry sectors. In the minimally
coupling case the functional form of $f(\mathcal{R},T)$ is assumed in the form
of $f(\mathcal{R},T)=q(\mathcal{R})+s(T)$, where $q(\mathcal{R})$ and $s(T)$
are arbitrary functions only of $\mathcal{R}$ and $T$ respectively. Whereas in
the non-minimal and purelly non-minimal cases $f(\mathcal{R},T)$ is assumed to
have forms: $f(\mathcal{R},T)=q(\mathcal{R})[1+s(T)]$ and
$f(\mathcal{R},T)=q(\mathcal{R})\,s(T)$ respectively H13 ; H14 .

In our work, we consider that the matter sector and geometry sector in
$f(\mathcal{R},T)$ gravity are coupled minimally and for the fulfilment of our
desired result we choose a minimal power law model of this gravity theory in
the following form:

$f(\mathcal{R},T)=a\,\mathcal{R}^{\alpha}-b\,(-T)^{\beta},$ (34)

where $a$, $b$, $\alpha$ and $\beta$ are constants. Of these constants $a$ and
$b$ are coupling constants that give the correct dimension to the action,
$\alpha$ is the slope parameter that determines the amount of curvature
effect, whereas $\beta$ determines the amount of matter effect. For the
existence of the model, parameters $a$, $b$, $\alpha$ and $\beta$ should not
be zero. It is clear that $a=1$, $\alpha=1$ and $b=0$ together reduce the
theory to GR. Further, the theory will turn down to GR for these two values of
$\alpha$ and $a$, if $T_{\mu\nu}=0$. Also $\alpha$ and $\beta$ should not take
negative values. For our model now we have,

$F_{\mathcal{R}}=a\,\alpha\,\mathcal{R}^{\alpha-1}\;\;\;\text{and}\;\;\;F_{T}=b\,\beta\,\rho^{\beta-1}.$
(35)

From equations (19) and (35) the interaction energy density for the model (34)
is obtained as

$\rho^{I}=\frac{1}{8\,\pi\,G}\left[\frac{b}{2}\,\rho^{\beta}\left\\{(2\beta+1)-\left(\frac{\alpha-1}{\alpha-2}\right)(\beta+2)\right\\}-\frac{8\,\pi\,G\,\rho}{2}\left(\frac{\alpha-1}{\alpha-2}\right)\right].$
(36)

It is immediately clear from this equation that the interaction energy density
$\rho^{I}$ is higly model dependent and for the positive as well as finite
value of $\alpha$ and $\beta$, they are disallowed to take values $2$ and
$-1/2,-2$ respectively. Moreover, the parameter $\alpha$ can not take the
value $1$, otherwise the curvature part will have no contribution to the
$\rho^{I}$. Further, it is seen that the parameter $a$ has no role to play in
the $\rho^{I}$ as this mass density is independent of the parameter $a$. We
know that galactic rotation curves obtained from observational data show the
linear increase of rotational velocities up to the disk of the galaxy and
remain constant keeping the curves flat in the halo region at large distance
from the center, where a very low amount of baryonic mass is observed J9 .
Interpretation of rotation curves at large distances is generally given by
postulating the existence of DM in this region. Here, we have the relation
(36) generated by our $f(\mathcal{R},T)$ gravity model (34) that can explain
the rotation curves without the need of DM. This would be clear from the
graphical analysis of the relation (36). For this we proceed by plotting
interaction mass density $\rho^{I}$ against the baryonic mass density $\rho$
using this relation as shown in Fig. 1. The figure indicates that beyond the
galactic disk, for the positive values of $b$, $\alpha$ and $\beta$ with
$1<\alpha<2$ (i) $\rho^{I}\geq 0$ and (ii) $\rho\ll\rho^{I}$.

Figure 1: Relations between interaction energy density and baryonic energy
density obtained from equation (36) for different values of model parameters
$b$, $\alpha$ and $\beta$.

The result (i) yields the required lower limit of parameter $\beta$ as
$\beta>0$, which is in accordance with the disallowed value of $\beta$ as
mentioned above. In particular the result (ii) tells us about the domination
of geometrically induced interaction mass density in the halo region over the
baryonic mass density. The geometrical mass associated with this interaction
mass density may be responsible for the constancy of tangential velocity in
galactic halo. Furthermore, it is seen from Fig. 1 that the interaction mass
density in the halo region has significant domination over the baryonic mass
density for larger values of $\alpha$ and $\beta$ within their allowed
range.Also it is found that dominance of interaction mass density increases
for higher values of parameter $b$ similar to $\alpha$ and $\beta$.

Again, the radial pressure in the model (34) takes the form:

$P_{r}^{I}=\frac{1}{8\,\pi\,G}\left[\frac{1}{2}a\,\mathcal{R}^{\alpha}(1-\alpha)-\frac{b}{2}(-T)^{\beta}\right].$
(37)

Using expressions (36) and (37) for $\rho^{I}$ and $P_{r}^{I}$ respectively,
we obtain,

$\rho^{I}+P_{r}^{I}=\frac{b\,\beta}{8\,\pi\,G}\,\rho^{\beta}.$ (38)

From Fig. 1 we may assume that contribution of the baryonic mass density to
the total mass density exterior to the galactic disk is very less in
comparision to the interaction mass density, consequently equation (28) can be
written using equation (38) for the halo region as

$e^{2\nu}=\frac{\delta\,F_{\mathcal{R}}}{b\,\beta\,r^{2}}\,\rho^{-\beta}.$
(39)

Substitution of $F_{\mathcal{R}}=a\alpha\mathcal{R}^{\alpha-1}$ with
$\mathcal{R}=8\,\pi G\rho$ and taking $\kappa^{2}=8\,\pi G=1$ reduce the above
equation to the form:

$e^{2\nu}=\frac{a\,\alpha\,\delta}{b\,\beta}r^{-2}\rho^{\alpha-\beta-1}.$ (40)

Accordingly the metric coefficient of the $g_{00}$ component will be

$e^{2\mu}=\frac{b\,\beta}{a\,\alpha\,\delta\,h^{\delta}}\,r^{2+\delta}\rho^{\beta-\alpha+1}.$
(41)

Equations (40) and (41) are the explicit form of the metric coefficients in
our specified $f(\mathcal{R},T)$ model (34).

To understand the behaviour of interaction energy or mass density in the halo
region with respect to distance from the galactic center we need to express it
in terms of radial distance $r$. Hence the interaction mass density with
respect to $r$ can be written from equations (30) and (36) as follows:

$\rho^{I}(r)=\frac{b}{2}\,r^{-\beta\lambda}\,\left[(2\beta+1)-\left(\frac{\alpha-1}{\alpha-2}\right)(\beta+2)\right]-\frac{r^{-\lambda}}{2}\left(\frac{\alpha-1}{\alpha-2}\right),$
(42)

where we have taken $\rho_{0}=1$ and $\kappa^{2}=1$.

Figure 2: Variation of interaction and baryonic mass densities with distance
$r$ for respective representative sets of model parameters.

Fig. 2 shows the plots of variations of interaction and baryonic mass
densities inside the halo region with respect to distance $r$ according to
equations (42) and (30) respectively. The leftmost curve is the consequence of
the second order power density law, which is substantiated to be followed by
the spiral galaxies within halo region as given by equation (30). Whereas the
rest three curves are upshots of the interaction mass density expressed by
equation (42) which are obtained from our specific $f(\mathcal{R},T)$ model.
These curves show a strong resemblance with the power law density profile
given by equation (30). Therefore, it is plausible for $\rho^{I}$ to establish
the density law similar to equation (30). Hence we can write,

$\rho^{I}\propto r^{-2}$ (43)

In the following section we will find the flatness of the rotation curve
according to variation of interaction mass with increasing $r$ at far distance
from the center of a galaxy.

##  V Galactic Rotation Curves in a Spherically Symmetric Spacetime

###  V.1 Orbital motion of a test particle in stable circular orbit

The tangential velocity of a star in an orbit of radius $r$ around the
galactic center significantly reflects the matter distribution of the galaxy
within it. Therefore, we consider a test particle (a star) moving in a static
spherically symmetric spacetime described by the metric (12), which follows a
time-like geodesic confined to the equatorial plane, $\theta=\pi/2$. The
general geodesic equation that describes the motion of such particles has the
following form M14 :

$\frac{d^{2}x^{\sigma}}{d\tau^{2}}\,+\,\Gamma_{\mu\nu}^{\sigma}\frac{dx^{\mu}}{d\tau}\,\frac{dx^{\nu}}{d\tau}=0,$
(44)

where $\tau$ is the affine parameter along the geodesic. In such time-like
geodesic it represents the proper time. The connection coefficient
$\Gamma_{\mu\nu}^{\sigma}$ is defined as

$\Gamma_{\mu\nu}^{\sigma}=\,\frac{1}{2}\,g^{\sigma\psi}\left[\frac{\partial
g_{\mu\psi}}{\partial x^{\nu}}+\frac{\partial g_{\nu\psi}}{\partial
x^{\mu}}-\frac{\partial g_{\mu\nu}}{\partial x^{\psi}}\right].$ (45)

Motion in the equatorial plane
($\theta=\pi/2\rightarrow\dot{\theta}=\ddot{\theta}=0$) immediately results in
three equations of motion for the coordinates $t,r,\varphi$ respectively
corresponding to the non-vanishing components of $\Gamma_{\mu\nu}^{\sigma}$ as
follows:

$\frac{d^{2}t}{d\tau^{2}}\,+\,2\,\Gamma_{01}^{0}\,\frac{dt}{d\tau}\,\frac{dr}{d\tau}=0,$
(46)
$\frac{d^{2}r}{d\tau^{2}}\,+\,\Gamma_{00}^{1}\left(\frac{dt}{d\tau}\right)^{2}\,+\,\Gamma_{11}^{1}\left(\frac{dr}{d\tau}\right)^{2}\,+\,\Gamma_{33}^{1}\left(\frac{d\varphi}{d\tau}\right)^{2}=0,$
(47)
$\frac{d^{2}\varphi}{d\tau^{2}}\,+\,2\,\Gamma_{13}^{3}\,\frac{dr}{d\tau}\,\frac{d\varphi}{d\tau}=0,$
(48)

where we take $x^{0}=t,\,x^{1}=r,\,x^{2}=\theta$ and $x^{3}=\varphi$. For a
stable circular orbit, after insertion of non-zero values of the connection
coefficients, equations (46) and (48) take the forms respectively as

$\frac{d}{d\tau}\left(e^{2\mu}\frac{dt}{d\tau}\right)=0\;\;\;\text{and}\;\;\;\frac{d}{d\tau}\left(r^{2}\,\frac{d\varphi}{d\tau}\right)=0.$

The integration of above two equations lead two constants of motion along the
geodesic, viz. energy, $E=e^{2\mu}\frac{dt}{d\tau}$ and angular momentum,
$l=r^{2}\,\frac{d\varphi}{d\tau}$. But at distances far from the spherically
symmetric body, the spacetime is almost flat and the field is weaker. At this
weak field limit GR agrees with the Newtonian gravity rj  and the particle
moves with a considerably slower speed relative to the speed of light. In this
low speed approximation, all the spatial components of particle’s four
velocity $u^{k}=\frac{dx^{k}}{d\tau}$, where $k=1,2,3$ are dominated by its
time component $u^{0}=\frac{dx^{0}}{d\tau}$ af . Thus, for such a sufficiently
slow moving particle, $u^{k}=\frac{dx^{k}}{d\tau}\ll\frac{dx^{0}}{d\tau}$, and
the proper time $\tau$ may be approximated to the coordinate time $t$ and the
four velocity
$\frac{dx^{\sigma}}{d\tau}(\frac{dx^{0}}{dt},\frac{dx^{1}}{dt},\frac{dx^{2}}{dt},\frac{dx^{3}}{dt})$
as $(1,0,0,0)$ wa . With this approximation none of the spatial component of
the four velocity will appear in the geodesic equation wa  and hence equation
(44) takes the form as

$\frac{d^{2}x^{\sigma}}{dt^{2}}\,+\,\Gamma_{00}^{\sigma}=0.$ (49)

Here the connection $\Gamma_{00}^{\sigma}$ can be calculated from the metric
(12) using the relation (45) as

$\Gamma_{00}^{\sigma}=\,\frac{1}{2}\,g^{\sigma\psi}\left[\frac{\partial
g_{0\psi}}{\partial x^{0}}+\frac{\partial g_{0\psi}}{\partial
x^{0}}-\frac{\partial g_{00}}{\partial
x^{\psi}}\right]=-\,\frac{1}{2}g^{\sigma\psi}\frac{\partial g_{00}}{\partial
x^{\psi}}.$

Now, the equation of motion for the radial component will be

$\frac{d^{2}r}{dt^{2}}\,+\,\Gamma_{00}^{1}=0.$ (50)

In the Newtonian limit, the radial or centripetal acceleration $a$ of the test
particle moving with the tangential (circular) velocity $v$ in an orbit of
radius $r$ is given as

$a=-\,\frac{v^{2}}{r}.$ (51)

Equations (50) and (51) together give $v^{2}=r\,\Gamma_{00}^{1}$, where after
substitution of corresponding metric elements of the metric (12) we obtain
$\Gamma_{00}^{1}$ as $\Gamma_{00}^{1}=\mu^{\prime}e^{2\mu-2\nu}$ and hence

$v^{2}=r\mu^{\prime}e^{2\mu-2\nu}.$ (52)

This equation relates the tangential velocity of a test particle to the radial
coordinate $r$, the $g_{00}$ and $g_{11}$ components of the metric (12). The
metric coefficients $e^{2\nu}$ and $e^{2\mu}$ in terms of $r$ and the model
parameters for $\lambda=2$ can be rewritten from equations (40) and (41) in
the following forms:

$e^{2\nu}=\frac{a\alpha\delta}{b\beta}\ r^{2(\beta-\alpha)}$ (53)

and

$e^{2\mu}=\left(\frac{r}{h}\right)^{\delta}\frac{b\beta}{a\alpha\delta}\
r^{2(\alpha-\beta)}.$ (54)

Now, we have to find $\mu^{\prime}$ to express the tangential velocity of the
particle by the model parameters. It may be obtained from the differentiation
of (54) with respect to $r$ as

$\mu^{\prime}=\frac{2(\alpha-\beta)+\delta}{2r}.$ (55)

Thus, insertion of relations (53), (54) and (55) in equation (52) we get the
tangential velocity in radial coordinate $r$, derived from our assumed
$f(\mathcal{R},T)$ gravity model for a test particle rotating around the
center of spiral galaxy as

$v^{2}=\left(\frac{b\beta}{a\alpha\delta}\right)^{2}\
\left(\alpha-\beta+\frac{\delta}{2}\right)\left(\frac{r}{h}\right)^{\delta}r^{4(\alpha-\beta)},$
(56)

where $(b/a)$ provides the right dimension to the velocity in the geometrized
unit. We have generated rotational curves (see Fig. 3) of a test particle
moving around the center of a galaxy for different values of model parameters
according to equation (56).

Figure 3: Rotational curves for different values of model parameters $a$, $b$,
$\alpha$, $\beta$ and $\delta$ obtained by using equation (56). Here the left
plot is for different values of ($\alpha-\beta$), middle plot is for different
values of $a$ and $b$, and the right plot is for different values of $\delta$.

The plots show that the flatness of the rotational curve increases for
decreasing difference of the parameters $\alpha$ and $\beta$ within their
permissible limit and for equal values of these two parameters the curve
becomes almost flat in the observed constant velocity range. This is in fact
clear from equation (56) that under the assumption $\delta<<1$, equal values
of $\alpha$ and $\beta$ lead to the constant velocity of the test particle.
Evidently, the value of $\delta$ that we have obtained
($\delta\simeq\mathcal{O}(10^{-6})$) from Fig. 3 agrees well with the
assumption. It has been found that $\delta\simeq\mathcal{O}(10^{-6})$ along
with the appropriate values of other model parameters can constrain the value
of rotational velocity $v\simeq 200-300$ km/s within the halo region as
indicated by observational data of a typical galaxy. This order of $\delta$
derived from the our specified model agrees with that obtained in Refs. Ra6 ;
Y11 ; rs . Also, it is seen that constants $a$ and $b$ have significant roles
in constraining $\delta$ to its probable order and $a$ supports a wide range
of values in the model (34). Furthermore, the larger as well as smaller values
of $b$ fix the velocity profile accordingly. The overall graphical view of
Fig. 3 indicates a nearly good correspondence with the observational data.

In the following section, we will express the mass distribution in the
galactic halo with the help of our specified minimally coupled model (34).

###  V.2 Mass distribution in galactic halo

For the interpretation of observed galactic curves at large distance $r$ from
the center of galaxies without introducing any exotic matter, we need to find
the interaction mass (the mass that is appeared due to matter-curvature
coupling, which is of purely geometric origin) of galaxies and its dependence
on $r$. Observations reveal that the baryonic matter distribution in the halo
region is very low and it contains only about 1% of total baryonic mass of a
galaxy in the form of low metallicity old stars J9 . That is the galactic halo
is not dominated by baryonic mass. It can be, therefore, assumed that the
contribution of baryonic mass and its density to total mass and density in
this region is negligible, and the whole matter density of the region is due
to the contribution of the interaction mass only. So, it is reasonable for
this region to assume that $\rho\approx 0$. In the modified $f(\mathcal{R},T)$
theory, under this assumption equation (16) reduces to

$\frac{2e^{-2\nu}}{r}\ \nu^{\prime}-\frac{e^{-2\nu}}{r^{2}}+\frac{1}{r^{2}}=8\
\pi\ G_{eff}\ \rho^{I}.$ (57)

Indeed, it usually describes the dark matter dominated region. Integration of
equation (60) in the limit from 0 to $r$ gives

$e^{-2\nu}=1-\frac{8\ \pi\ G_{eff}}{r}\int_{0}^{r}\rho^{I}r^{\prime
2}dr^{\prime}$ (58)

which leads

$e^{-2\nu}=1-\frac{2G_{eff}M^{I}}{r},$ (59)

where we have used the definition for $M^{I}$, the interaction mass of the
galaxy as

$M^{I}=\int_{0}^{r}4\ \pi\rho^{I}(r^{\prime})\ r^{\prime 2}dr^{\prime}.$ (60)

Now, conveniently we may define the baryonic matter boundary of the galaxy at
$r=R_{B}$ upto which a fixed amount of baryonic matter extends uniformly
showing its dominance within the boundary. Beyond this limit, i.e. from
$r>R_{B}$, the region of $\rho\approx 0$ begins and it covers to the outermost
measurable point. Within the baryonic matter boundary equation (62) gives,

$e^{-2\nu}=1-\frac{2G_{eff}M_{B}}{R_{B}},$ (61)

where the total baryonic mass of the galaxy is approximately,

$M_{B}=\int_{0}^{R_{B}}4\ \pi\rho(r^{\prime})\ r^{\prime 2}dr^{\prime}.$ (62)

Fig. 3 dictates us about the almost constant velocity region for the equality
of parameters $\alpha$ and $\beta$. From equation (53) this equality results
in $e^{2\nu}$ as constant one i.e., $e^{2\nu}=\frac{a\
\delta}{b}=\text{constant}$.

Figure 4: Behavior of metric coefficient $e^{-2\nu}$ with respect to radial
distance $r$.

On the other hand, the depiction of behavior of metric coefficient $e^{-2\nu}$
obtained from the model (34) with radial distance $r$ hints its continuity
over $r$ (see Fig 4). Therefore, across the baryonic matter boundary it is
possible to equate equations (59) and (61) with the help of above constancy
relation as

$1-\frac{2G_{eff}M^{I}}{r}=1-\frac{2G_{eff}M_{B}}{R_{B}},$

which provides a relation between $M^{I}$ and $M_{B}$ as follows:

$M^{I}=\frac{M_{B}}{R_{B}}\,r.$ (63)

It implies that

$M^{I}\propto r.$ (64)

That is the interaction mass $M^{I}$, which is not the mass for any physical
particle, increases linearly within the halo region with the increase of
distance from the galactic center. As a rotation curve reflects the mass
distribution of a system and its flat portion is usually explained by the
linear increase of the postulated DM with the radial distance, so the relation
(64) can fit the flat behavior of rotation curves quite well without the DM
within the halo. This relation also shows that the density assigned for
interaction mass follows the relation (43) and hence the interaction mass
behaves similar to the DM that is in general postulated for the explanation of
the galactic rotation curves in the region far away from the galactic center.

##  VI Conclusions

MTGs provide alternative gravitational approaches in contrast to the GR to
understand and determine the flat portion of rotation curves corresponding to
stars that are moving around the galactic nucleus at far distances away from
it. Thus MTGs can be used to give the geometric interpretation of the so-
called DM. In this present work, we have developed such an approach to address
the DM problem by considering the $f(\mathcal{R},T)$ modified theory of
gravity especially in the galactic scale and the emphasis is given to the
possibility for elucidating the rotation curves of spiral galaxies in the
constant velocity zone without the need for the exotic DM. Here, we have
employed a minimally coupled $f(\mathcal{R},T)$ gravity model of the form
$af(R)+bf(T)$, which could explain the additional matter content needed for
the observational agreement with the rotation curves of galaxies. In fact, the
extra terms appearing in the field equations of the $f(\mathcal{R},T)$ gravity
theory may be treated as an extra fluid, whose source is due to the coupling
between the matter and the geometry. The mass corresponding to the interaction
term has been assumed here as the interaction mass and has been illustrated as
an additional mass like the mass of DM.

Assuming the first fluid (the physical matter fluid) in the two fluid
structure of $f(\mathcal{R},T)$ gravity theory as a dust-like perfect fluid,
the modified Einstein equations have been solved and subsequently obtained the
metric coefficients for a static spherically symmetric spacetime in the
vicinity of GR. The DM halo on which we are interested is the least understood
component of a galaxy with approximately spherical in shape. In our work, we
emphasized to have both constant tangential velocity and power law of
interaction density within the halo so that it can explain the observed
rotation curve without DM.

We have established a relation for the variation of interaction density with
respect to distance $r$ by plotting $(\rho^{I}-r)$ graph for different values
of model parameters and then comparing it with the baryonic matter density
graph, we have determined the density profile required for the description of
mass variation within the halo. Also, we have fixed the metric coefficients in
terms of model parameters and seen that equal values of $\alpha$ and $\beta$
lead $g_{11}$ component to a constant one, which is found as specific for the
model as this equality condition makes the rotation curve almost flat within
the permissible values of other parameters. The rotation curves generated by
relation (56) are more or less comparable to the observational curves and they
constrain $\delta$ to the order of $10^{-6}$. Thus in conformity to be with
the observed data on the galactic rotation curves Ra6 ; Y11 , this result
suggests that the parameter $\delta$ should be a considerably smaller one (of
the order of $10^{-6}$) by confirming our assumption that $X(r)$ should
slightly be different from the unity.

The probability of attaining the dominance of interaction energy density in
halo region over the baryonic energy density in accordance with the working
model has been confirmed through $(\rho^{I}-\rho)$ plots for different sets of
allowed values of the model parameters. Each plot including the $\alpha=\beta$
case also shows the right agreement with our expectation. Indeed, our
specified model results in the constant tangential velocity of a test particle
needed for fitting observed rotation curves far away from the galactic center.
As flatness behavior of rotation curves suggests a considerable amount of
hidden matter, therefore, we have derived the interaction mass by fixing the
metric coefficients and found the continuous linear growth of it with distance
$r$, which nullifies the requirement of DM within the halo at all. As the
linear variation of mass is achieved only for constant velocity and our
working model has provided the constant velocity in halo, so the interaction
mass may behave as DM to fit the rotation curves in the galactic halos. In
future, one may attempt to extend this work with a non-minimally coupled
$f(\mathcal{R},T)$ model by constraining the model parameters with the
available observed data.

## Acknowledgments

UDG is thankful to the Inter-University Centre for Astronomy and Astrophysics
(IUCAA), Pune, India for the Visiting Associateship of the institute.

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