# Rational Link Fertility

Andrew Ducharme

###### Abstract.

A knot $K$ is the resultant of a knot $H$ if there exists a minimal crossing
diagram $D$ of $K$ such that some crossings of $D$ can be altered to produce
$H$. $K$ is fertile if every prime knot $H$ with crossing number less than
$c(K)$ is a resultant of $K$. $K$ is $n$-fertile if every prime knot $H$ with
crossing number less than $n$ is a resultant of $K$. We classify the fertility
and fertility number of all rational links. This requires the introduction of
the analogous concept of link fertility.

##  1\. Introduction

Consider a minimal diagram of a knot $K$. Would the knot type be changed by
switching the parity of a crossing from over to under or vice versa? How many
other knots can be formed from a single knot? Can we form all smaller knots?
The concept of knot fertility, introduced in [MR3844207], formalizes these
questions.

######  Definition 1.1.

A knot $K$ is _fertile_ if every prime knot $H$ with $c(H)<c(K)$ can be formed
from the shadows of the minimal crossing diagrams of $K$.

######  Definition 1.2.

A knot $K$ has _fertility number_ $F(K)=m$, where all knots $H$ with $c(H)\leq
m$ can be formed from the shadows of minimal crossing diagrams of $K$. We say
$K$ is $p$-_fertile_ for all crossing numbers $p\leq m$.

Instead of working with true knot diagrams, we start with shadows, or
projections, of the diagrams, where all over/undercrossing information is
forgotten. We then examine the knots formed by particular choices of
over/under parities for each crossing. The knot formed by a complete set of
choices for each crossing is a resultant. In Figure 1, for example, 1(a) is a
diagram of the trefoil knot, 1(b) is the shadow of that diagram, and 1(c) is a
particular resultant of that shadow.

(a)(b)(c) Figure 1. (a) The trefoil knot, (b) the shadow of a trefoil, and (c)
a resultant isotopic to the unknot.

The initial study of knot fertility calculated the fertility numbers of all
knots with 10 or fewer crossings, discovering $3_{1}$, $4_{1}$, $5_{2}$,
$6_{2}$, $6_{3}$, and $7_{6}$ are the only such fertile knots [MR3844207].
Recent work showed there are no fertile alternating knots with greater than
seven crossings [MR4193872] while another duology [MR4323911, kf2] determined
any other extant fertile knot $K$ to have an odd crossing number $11\leq
c(K)\leq 21$, be genus 2, and have braid index 4.

In this paper, we completely determine the fertility and fertility number of
all rational knots and links. This partially answers Question 2 posed in
[MR3844207]. In doing so, we must (trivially) extend the definition of
fertility to links.

######  Definition 1.3.

A link $L$ has $\mu(L)$ components. This is its _component number_.

######  Definition 1.4.

A link $L$ is _fertile_ if every prime link $M$ with $c(M)<c(L)$ and
$\mu(M)=\mu(L)$ can be formed from the shadows of the minimal crossing
diagrams of $L$. $L$ has _fertility number_ $F(L)=n$, where all M with
$c(M)\leq n$ and $\mu(M)=\mu(L)$ are resultants of $L$.

######  Theorem 1.5.

(Corollary 2.13) The only fertile rational two-component links are
$2^{2}_{1}$, $4^{2}_{1}$, $5^{2}_{1}$, $6^{2}_{2}$, $6^{2}_{3}$, and
$7^{2}_{2}$.

Like with knots, we see the number of fertile links dropping rapidly as the
number of link types abound with crossing number. Showing that these are the
sole fertile rational links is actually very simple, a direct consequence of
the maximum allowed number of unique resultants for each link. As for
fertility number specifically, we explicitly determine every rational links’
fertility number, finding a limiting value for both component lengths.

######  Theorem 1.6.

(Corollary 3.29) Rational two-component links
$L=N[a_{1}\;a_{2}\;...\;a_{n}]$ have $F(L)=6$ for $n>3$.

######  Theorem 1.7.

(Theorem 4.18) Rational knots $K=N[a_{1}\;a_{2}\;...\;a_{n}]$ have $F(K)=7$
for $n>7$.

And while we do not touch on this idea in this paper, we suggest future study
on the rational resultants of a link could be promising.

######  Definition 1.8.

A knot $K$ has _rational fertility number_ $F_{R}(K)=n$, where all rational
knots $H$ with $c(H)\leq n$ can be formed from the shadows of any minimal
crossing diagram of $K$.

######  Question 1.9.

What is $F_{R}(L)$ for all rational links?

Since all knots with seven or fewer crossings are rational, $F_{R}(K)=F(K)\leq
7$. The smallest crossing number rational knots with $F_{R}(K)=8$ are
$11a77=N[2\;2\;1\;1\;1\;1\;1\;2]$, $11a91=N[2\;1\;1\;2\;1\;1\;1\;2]$,
$11a96=N[2\;2\;2\;1\;1\;1\;2]$, and $11a159=N[2\;2\;2\;2\;1\;2]$. We believe
the techniques used in this paper could be extended to completely classify
$F_{R}$ over the rational links.

The paper is outlined as follows: Section 2 answers the binary question of
fertility, Section 3 considers the fertility number of rational 2-component
links, and Section 4 considers the fertility number of rational knots. Whereas
the rest of the paper considers the existence of various resultants, Section 5
counts the number of each for some rational links. Finally, in the Appendix,
we tabulate the fertility number of every rational link.

##  2\. Rational Link Fertility

######  Theorem 2.1.

The only fertile rational knots are $3_{1}$, $4_{1}$, $5_{2}$, $6_{2}$,
$6_{3}$, and $7_{6}$. The only fertile rational links are $2^{2}_{1}$,
$4^{2}_{1}$, $5^{2}_{1}$, $6^{2}_{2}$, $6^{2}_{3}$, and $7^{2}_{2}$.

In this section, we’ll define rational tangles and links, establish our
procedure for determining their resultants, then prove the above statement
about the binary invariant fertility.

######  Definition 2.2.

A _rational tangle_ $T$ is a multiplication of integer tangles, defined
pictorially below for two (left) and three or more (right) tangles, and is
denoted by the list of such composing tangles $T=a_{1}\;a_{2}\;...\;a_{n}$,
$a_{i}\in\mathbb{Z}$. $T$ has length $d(T)=n$. The integer tangle $a_{i}$ has
$|a_{i}|$ crossings.

$a_{1}$

$a_{2}$$a_{1}$

$a_{2}$

$a_{3}$+…

We have two motivations for studying rational tangles: they are completely
classified by their continued fraction, and low crossing number links are
predominantly closures of rational tangles.

######  Definition 2.3.

A _continued fraction_ $(a_{1}\;a_{2}\;...\;a_{n})$, $a_{i}\in\mathbb{R}$ is a
real number

$a_{1}+\frac{1}{a_{2}+\frac{1}{...+\frac{1}{a_{n-1}+\frac{1}{a_{n}}}}}$

A continued fraction is in _canonical form_ if $|a_{1}|>1$, $a_{i}\not=0$,
$1<i\leq n$, $sgn(a_{1})=sgn(a_{j})$, $j\leq n$, and $a_{i}\in\mathbb{Z}$.
Canonical forms are unique to a continued fraction.

######  Theorem 2.4.

[MR0258014] Two rational tangles are isotopic to each other if their continued
fraction is equivalent.

Consequently, we have unique canonical forms of rational tangles. Note $0$,
$\infty$, and $\pm 1$ are the canonical tangles of their isotopy class that do
not fit with the above definition. Unless otherwise noted, in all our later
results, a given rational link is in canonical form.

######  Definition 2.5.

A tangle $T$ has two closures which maintain the diagram’s crossing number:
the _numerator_ $N[T]$ and _denominator_ $D[T]$.

$T$$N[T]$$T$$D[T]$

Note $N[T\;0]\simeq D[T]\simeq N[0\;T]$ and $N[T\;\infty]\simeq N[T]$. We can
concern ourself solely with numerator closures because
$D[a_{1}\;a_{2}\;...\;a_{n}]\simeq N[a_{1}\;a_{2}\;...\;a_{n-1}]$.

######  Theorem 2.6.

[MR82104, MR1953344] Suppose there exists two rational tangles with continued
fractions $\frac{p}{q}$ and $\frac{p_{0}}{q_{0}}$, where p, q and
$p_{0},q_{0}$ are relatively prime. If N$\left[\frac{p}{q}\right]$ and
N$\left[\frac{p_{0}}{q_{0}}\right]$ are the links formed by the closure of
these tangles, then the links are equivalent (up to isotopy) iff:

  * •

$p=p_{0}$

  * •

either q $\equiv$ $q_{0}$ mod p or $qq_{0}$ $\equiv$ 1 mod p.

######  Corollary 2.7.

[MR1953344] $N[a_{1}\;a_{2}\;...\;a_{n-1}\;a_{n}]\simeq
N[a_{n}\;a_{n-1}\;...\;a_{2}\;a_{1}]$.

This forces $a_{n}>1$ in reduced rational links, since the continued fraction
must be in canonical form when written forwards and backwards.

######  Notation 2.8.

The shadow of an integer tangle $a_{i}$, or a _shadow tangle_ , is denoted
$a_{i}^{\prime}$.

The 5’ tangle

######  Theorem 2.9.

Let $T^{\prime}$ be the shadow of an integer tangle with $t$ crossings. If $t$
is odd, $T^{\prime}$ has resultants $t$, $t-2$, $t-4$, …, $+1$, $-1$, $-3$, …,
$-t$. If $t$ is even, $T^{\prime}$ has resultants $t$, $t-2$, $t-4$, …, $+2$,
$0$, $-2$, …, $-t$.

###### Proof.

We can label crossings as either positive or negative by the sign of the
overstrand’s slope. Given a shadow tangle, we choose some to be positive and
the rest negative. Neighboring positive and negative crossings are removed via
RII, decreasing the crossing number by two, until only crossings of one type
remain. ∎

Because having a link $L$ as a resultant implies the mirror image of $L$ is
also a resultant, we can consider opposite sign tangles $+t$ and $-t$ as one
case $\pm t$ for all $t\in\mathbb{N}$. For multiple integer tangles, we need
to have all the same parity $\pm$ or $\mp$ to ensure the overall tangle is in
canonical form. For example, $\pm 3\;\pm 4\;\pm 2$ is in canonical form and
completely simplified.

The next result, fundamental to this work, determines how to handle tangles
with different parities like $\pm 3\;\mp 4\;\pm 2$.

######  Lemma 2.10.

(The Untangling Lemma) Let $\pm a_{i}\;\mp a_{i+1}$, $a_{i}\geq 1$, be an
assignment of two adjacent integer tangles inside a rational link shadow
$L^{\prime}=N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]$, $1<i<n$.
Then

$\pm a_{i}\;\mp a_{i+1}\approx\begin{cases}\pm(a_{i}-1)\;\pm
1\;\pm(a_{i+1}-1),&a_{i+1}\geq 2\\\ \pm(a_{i}-1)\;a_{i+2}^{\prime}\pm
1,&a_{i+1}=1\\\ \pm a_{i}+a_{i+2}^{\prime},&a_{i+1}=0,\end{cases}.$

For $a_{i+1}>0$, this move reduces the diagram’s crossing number by one.

###### Proof.

Assign the first $j$ tangles $a_{i}^{\prime}\rightarrow\pm c_{i}$ and
$a_{j+1}\rightarrow\mp c_{j+1}$ so $c_{j}\;c_{j+1}$ is the leftmost case of
disagreeing parities and $2\leq c_{j}\leq a_{j}$. Then

$L^{\prime}\rightarrow N[\pm c_{1}\;\pm c_{2}\;...\;\pm c_{j}\;\mp
c_{j+1}\;a_{j+2}^{\prime}\;...\;a_{n}^{\prime}].$

Let $A\in\mathbb{R}$ be the continued fraction of the rational tangle composed
of the $j+2$th through $n$th integer tangles once assigned. Then the rational
tangle composed of the $j$th through $n$th integer tangles has a continued
fraction

$c_{j}+\frac{1}{-c_{j+1}+\frac{1}{A}}=c_{j}-1+\frac{-(c_{j+1}-1)+\frac{1}{A}}{-c_{j+1}+\frac{1}{A}}=(c_{j}-1)+\frac{1}{1+\frac{1}{(c_{j}-1)-\frac{1}{A}}}.$

For a continued fraction $x=(a,b,...,n)$, $-x=(-a,-b,...,-n)$ so the negation
of $A$ flips the sign of all its components. Translating the above result into
a partial shadow of $L^{\prime}$, we get

$N[\pm c_{1}\;\pm c_{2}\;...\;\pm(c_{j}-1)\;\pm
1\;\pm(c_{j+1}-1)\;-a_{j+2}^{\prime}\;...\;-a_{n}^{\prime}]$

However, a shadow tangle does not change upon negation—it already contains the
positive and negative assignments.

The second and third cases are consequences of $k\;0\;\ell\simeq(k+\ell)$ for
generic integer tangles $k$, $\ell$. ∎

The key behind this lemma is $N[\pm c_{1}\;...\;\pm
c_{j}\;a_{j+1}^{\prime}\;...\;a_{n}^{\prime}]$ as a notational shorthand for
the set of rational links formed from all assignments of the remaining shadow
tangles. The rational link $N[\pm c_{1}\;\mp c_{2}\;\pm c_{3}\;...\;\pm
c_{n}]$ is isotopic to $N[\pm(c_{1}-1)\;\pm 1\;\pm(c_{2}-1)\;\mp
c_{3}\;...\;\mp c_{n}]$ but the set $N[\pm c_{1}\;\mp
c_{2}\;a_{3}^{\prime}\;...\;a_{n}^{\prime}]$ is identical to the set
$N[\pm(c_{1}-1)\;\pm 1\;\pm(c_{2}-1)\;a_{3}^{\prime}\;...\;a_{n}^{\prime}]$.
We can save ourselves a lot of accounting for negative signs by “absorbing”
the negative sign into the shadow tangles and treating the remaining tangles
as untouched.

Technically, on an individual link level, we’re choosing a particular
assignment for the first part of a link, applying the isotopy of the
Untangling Lemma, then going to the set of remaining possible assignments and
trading in for another link entirely. As such, the link we start with is, in
general, not isotopic to the link we end with. However, this trade is
bijective; the entire resolution set is unchanged.

$N[\pm a_{1}\;\mp a_{2}\;\pm a_{3}]$$N[\pm(a_{1}-1)\;\pm 1\;\pm(a_{2}-1)\;\pm
a_{3}]$$N[\pm(a_{1}-1)\;\pm 1\;\pm(a_{2}-1)\;\mp
a_{3}]$$\approx$$\simeq$$\\{N[\pm a_{1}\;\mp a_{2}\;a_{3}^{\prime}]\\}$

To remind ourselves that something more complicated is going on under the
hood, we use the symbol $\approx$ for equivalence instead of the standard
isotopy symbol $\simeq$.

######  Theorem 2.11.

A rational link $L=N[\pm a_{1}\;\pm a_{2}\;...\;\pm a_{n}]$ has only one
resultant $R$ with $c(R)=c(L)$ and at most $n-1$ unique resultants with
$c(R)=c(L)-1$. The same crossing number resultant is $L$.

###### Proof.

Changing a single crossing inside an integer tangle will decrease that
tangle’s crossing number by two. Changing more crossings will decrease the
crossing number by two until half the crossings have been changed, at which
point the crossing number will start to increase by two, until which every
crossing has been changed. Therefore, the only resultant diagrams $D$ with
$c(D)=c(L)$ are those where no crossings are changed, or all the crossings in
some tangles $a_{i}$ have been changed.

The former diagram is $L$. The latter diagrams are not in canonical form, but
can isotoped so by the above lemma. For each integer tangle with all crossings
changed, the resultant crossing number decreases by 1, so the only route to
resultants with $c(R)=c(L)-1$ is to alter a single tangle. There are
${n\choose 1}=n$ such diagrams, however, we already account for the mirroring
of the first tangle by the $\pm$ notation as $N[\pm
a_{1}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]$ has the both the initial
assignment ($+$) and its mirror ($-$). ∎

This result alone(!) explains why knots $5_{1}=N[5]$, $6_{1}=N[2\;4]$,
$7_{1}=N[7]$, $7_{3}=N[3\;4]$, $7_{4}=N[3\;1\;3]$, and $7_{5}=N[3\;2\;2]$, and
two-component links $6^{2}_{1}=N[6]$, $7^{2}_{1}=N[4\;1\;3]$, and
$7^{2}_{3}=N[2\;3\;2]$ are infertile. They are all too short to produce the
variety of links with one fewer crossing!

$c(K)$ | # $K$ | $c(L^{2})$ | # $L^{2}$
---|---|---|---
0 | 1 | 0 | 1
3 | 1 | 2 | 1
4 | 1 | 4 | 1
5 | 2 | 5 | 1
6 | 3 | 6 | 3
7 | 7 | 7 | 8
8 | 21 | 8 | 16
9 | 49 | 9 | 61
Table 1. The number of one- and two-component links $K$ and $L^{2}$ for
particular crossing numbers.

######  Theorem 2.12.

$d(L)\leq c(L)-2$.

###### Proof.

We can construct the longest possible link with $n$ crossings by multiplying
$n$ single crossing tangles. Two neighboring $\pm 1$ tangles on the interior
of a rational link only have one direct connection and remain distinct.
However, there are two direct $a_{1}$-$a_{2}$ and $a_{n-1}$-$a_{n}$
connections in the numerator closure, so the $a_{1}$ and $a_{2}$ tangles, and
the $a_{n-1}$ and $a_{n}$ tangles each cohere into integer tangles with two
crossings, decreasing link length by two. ∎

######  Corollary 2.13.

A rational link $L$ is infertile if $c(L)\geq 8$. Specifically, $F(L)\leq
c(L)-2$.

###### Proof.

$L$ has less than or equal to $c(L)-2$ resultants $R$ with $c(R)=c(L)-1$.
There are 7 knots and 8 two component links with seven crossings, but eight
crossing links can form at most 5 resultants with that crossing number. The
number of rational links grows much faster than crossing number. ∎

With this, we’ve proven all rational links are infertile except the fertile
knots $3_{1}$, $4_{1}$, $5_{2}$, $6_{2}$, $6_{3}$, and $7_{6}$, the fertile
links $2^{2}_{1}$, $4^{2}_{1}$, $5^{2}_{1}$, $6^{2}_{2}$, $6^{2}_{3}$, and
$7^{2}_{2}$, and the pesky $7_{7}=N[2\;1\;1\;1\;2]$. This length based
approach puts a rigorous spin on the idea that low crossing number knots are
fertile because their resultants are pidgeonholed into the same knots. At
small crossing numbers, a reduced resultant can only be one of very few
options. But as crossing number increases, unlocking more diverse knots,
resultants of a particular crossing number are no longer so constrained.

For completeness, we’ll give a lack-of-resultants proof here for why $7_{7}$
is infertile, but we’ll have more to say about knots like $N[2\;1\;1\;1\;2]$
later.

######  Theorem 2.14.

If a rational link $L$’s list $\\{a_{i}\\}$ is symmetric
$N[a_{1}\;a_{2}\;...\;a_{n-1}\;a_{n}]=N[a_{1}\;a_{2}\;...\;a_{2}\;a_{1}]$, it
has $\lfloor\frac{n}{2}\rfloor$ resultants $R$ with $c(R)=c(L)-1$.

###### Proof.

Let $d(L)=n$ so $k=\lceil\frac{n}{2}\rceil$ and $a_{k}$ is the fulcrum of
$L$’s reversal symmetry. As we’re only changing one tangle, the number of
unique resultants is the number of unique pairs
$(1,2),(2,3),(3,4),...,(k-1,k)$. If $n$ is odd, this is $\frac{n-1}{2}$. If
$n$ is even, we have the additional pair $(k,k)$ and $\frac{n}{2}$ unique
pairs, hence the floor function. ∎

######  Corollary 2.15.

$7_{7}=N[2\;1\;1\;1\;2]$ is infertile.

###### Proof.

$\lfloor\frac{d(7_{7})}{2}\rfloor=2$. There are three prime knots with six
crossings. ∎

##  3\. Rational Link Fertility Number

######  Theorem 3.1.

A rational knot $K$ has $F(K)\leq 7$ and a two component rational link $L$ has
$F(L)\leq 6$.

###### Proof.

A rational link’s resultants must be rational. Both bounds are the largest
crossing number with only rational links for both component numbers. ∎

######  Definition 3.2.

A rational link $L$ is _locally fertile_ if it is a knot and $F(L)=7$, or it
is a two-component link and $F(L)=6$.

We’d expect once rational links are large enough, they should be locally
fertile. But what do we mean by “large enough?” It’s reasonable to anticipate
fertility number growing roughly with crossing number (the first locally
fertile knot is $9_{27}$ and the next smallest are $10_{23},10_{25},10_{26}$
and $10_{39}$–$10_{45}$, barring $10_{43}$) but as we’ve seen so far, it is
described more accurately by length. Our goal in this and the next section is
to build and use tools to determine the minimal rational link length for local
fertility.

######  Theorem 3.3.

$F(N[a_{1}\;...\;a_{i-1}\;(a_{i}+2)\;a_{i+1}\;...\;a_{n}])\geq$
$F(N[a_{1}\;...\;a_{i-1}\;a_{i}\;a_{i+1}\;...\;a_{n}])$, $1\leq i\leq n$.

###### Proof.

Call $N[a_{1}\;...\;a_{i-1}\;a_{i}\;a_{i+1}\;...\;a_{n}]$ $L_{1}$ and
$N[a_{1}\;...\;a_{i-1}\;(a_{i}+2)\;a_{i+1}\;...\;a_{n}]$ $L_{2}$. The shadow
of $L_{2}$ can be resolved into three mutually distinct sets:
$N[a_{1}^{\prime}\;...\;a_{i-1}^{\prime}\;a_{i}^{\prime}\;a_{i+1}^{\prime}\;...\;a_{n}^{\prime}]$,
$N[a_{1}^{\prime}\;...\;a_{i-1}^{\prime}\;\pm(a_{i}+2)\;a_{i+1}^{\prime}\;...\;a_{n}^{\prime}]$,
and
$N[a_{1}^{\prime}\;...\;a_{i-1}^{\prime}\;\mp(a_{i}+2)\;a_{i+1}^{\prime}\;...\;a_{n}^{\prime}]$.
The first set is simply all the resultants of $L_{1}$, thus $F(L_{2})\geq
F(L_{1})$. ∎

We can use this heredity to build the least complicated rational links at
every length and keep adding crossings to increase our lower bounds on
fertility number. We will call this set of least complicated links the trunk.
A more technical definition is:

######  Definition 3.4.

The _trunk_ $\mathcal{K}_{\ell}$ is the set of rational links
$L=N[a_{1}\;a_{2}\;...\;a_{\ell}]=N[\\{a_{i}\\}]$ where
$(a_{1},a_{\ell})=(2,2),(3,2)$, or $(3,3)$, and $\\{a_{j}\\}$, $1<j<\ell$ are
$(\ell-2)$-tuples with elements $\\{1,2\\}$.

######  Example 3.5.

$\mathcal{K}_{1}=\\{N[2],N[3]\\}$,
$\mathcal{K}_{2}=\\{N[2\;2],N[2\;3],N[3\;3]\\}$, and
$\mathcal{K}_{3}=\\{N[2\;1\;2],N[2\;2\;2],N[3\;1\;2],N[3\;2\;2],N[3\;1\;3],N[3\;2\;3]\\}$

For any link $L\in\mathcal{K}_{\ell}$, $\ell+2\leq c(L)\leq 2\ell+2$.
$|\mathcal{K}_{\ell}|\leq 3\times 2^{\ell-1}$, but the bound will not be
achieved for $\ell>3$. If we included a link for each $(\ell-2)$-tuple,
nonsymmetric tuples would duplicate its reverse if $a_{1}=a_{n}$, e.g. the
2-tuple $(1,2)$ is included in six links $N[2\;1\;2\;2]=N[2\;2\;1\;2]$,
$N[3\;1\;2\;2]$, $N[3\;2\;1\;2]$, and $N[3\;1\;2\;3]=N[3\;2\;1\;3]$.

######  Definition 3.6.

Every rational link $L_{2}$ of length $\ell$ can be written as
$N[(a_{1}+2m_{1})\;(a_{2}+2m_{2})\;...\;(a_{\ell}+2m_{\ell})]$, where
$L_{1}=N[a_{1}\;a_{2}\;...\;a_{\ell}]\in\mathcal{K}_{\ell}$. We call $L_{2}$ a
_branch_ of $L_{1}$.

######  Definition 3.7.

$g(l,m)$ is the smallest fertility number of any m-component link in a
particular trunk $\mathcal{K}_{l}$.

######  Corollary 3.8.

Let $L$ be a rational link. Then $F(L)\geq g(d(L),\mu(L))$.

This is why trunks are meaningful: they are not just the least complicated,
but the least fertile rational links for a given length.

######  Theorem 3.9.

Let $L=N[a_{1}\;a_{2}\;...\;a_{n}]$. Then

$F(L)\geq\begin{cases}F(N[a_{1}\;a_{2}\;...\;a_{n-2}]),&a_{n}\;even\\\
F(N[a_{1}\;a_{2}\;...\;(a_{n-1}+1)]),&a_{n}\;odd\\\ \end{cases}.$

###### Proof.

If $a_{n}$ even, $a_{n}\rightarrow 0\implies L\rightarrow
N[a_{1}\;a_{2}\;...\;a_{n-2}]$, so all the resultants of the shorter link are
the resultants of $L$. If $a_{n}$ odd, assignments $L\rightarrow
N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n-1}^{\prime}\;\pm 1]$ and
$N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n-1}^{\prime}\;\mp 1)]$ form all
the resultants of
$N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;(a_{n-1})^{\prime}\;1^{\prime}]$
$\approx N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;(a_{n-1}+1)^{\prime}]$. ∎

######  Notation 3.10.

$N[x\;1^{\ell}\;y]\equiv N[x\underbrace{1\;1\;...\;1}_{\ell\text{ unit
tangles}}y]$

######  Theorem 3.11.

If $N[2\;1^{n-2}\;2]$ and all rational links $L_{B}$ of $\ell$ components in
$\mathcal{K}_{n-1}$ and $\mathcal{K}_{n-2}$ are $k$-fertile, all rational
links $L$ of length $d(L)\geq n$ are $k$-fertile.

###### Proof.

Let $L=N[a_{1}\;a_{2}\;...\;a_{n}]$ be a rational link of $\ell$ components.
If $a_{1}$ or $a_{n}$ are odd, $F(L)\geq
F(N[a_{1}\;a_{2}\;...\;(a_{n-1}+1)])\geq g(n-1,\ell)$. Similarly, if
$a_{1},a_{n}$, and some tangles $a_{j}$, $1<j<n$, are even, $F(L)\geq
g(n-2,\ell)$. This leaves one pathological counterexample: if all $a_{j}$,
$1<j<n$, are odd, $a_{1}$ and $a_{n}$ are even, and $a_{3}=a_{n-2}=1$,
assigning an end tangle to zero forms $M=N[a_{1}\;...\;a_{n-3}\;1]\approx
N[a_{1}\;...\;(a_{n-3}+1)]$, so $d(M)=n-3$. Hence, $M$ is a branch of
$N[2\;1^{n-5}\;2]\in\mathcal{K}_{n-3}$.

By definition, $L$ in this case is a branch of $N[2\;1^{n-2}\;2]$, which by
having more crossings and being of longer length, is likely to have a larger
fertility number than $N[2\;1^{n-5}\;2]$. So while it is sufficient to require
all links in $\mathcal{K}_{n-1}$, $\mathcal{K}_{n-2}$, and $\mathcal{K}_{n-3}$
to be $k$-fertile, simply using $\mathcal{K}_{n-1}$, $\mathcal{K}_{n-2}$, and
$N[2\;1^{n-2}\;2]\in\mathcal{K}_{n}$ is a much stronger condition. ∎

Our next step is to find $g(\ell,1)$ and $g(\ell,2)$ for increasingly large
$\ell$ to determine the minimum fertility number at every length. Once these
minima reach the local fertility number, we will have squeezed out the length
where all longer links are locally fertile.

###  3.1. Length 1 and 2 Knots and Links

######  Theorem 3.12.

$N[a_{1}]$ links, or the $T(p,2)$ torus links, have resultants $N[\pm k]$,
where $a_{1}$ and $k$ have the same parity.

###### Proof.

This follows directly from the resolution of integer tangles. ∎

######  Corollary 3.13.

$F(T(p,2))=3$, $p>1$, odd. The trefoil is the only fertile $T(p,2)$ knot.

###### Proof.

These knots cannot have the figure-eight knot $N[2\;2]$ as a resultant. ∎

This has been repeatedly discovered [MR3844207, MR3084750, MR1001742,
MR4190429, MR3356086, MR2514545].

######  Corollary 3.14.

The Hopf link $2^{2}_{1}=N[2]$ and $4^{2}_{1}=N[4]$ are fertile ($F(N[2])=2$,
$F(N[4])=4$). Otherwise, $N[a_{1}]$ links, $a_{1}>4$, have fertility number
$4$.

###### Proof.

These links cannot form $5^{2}_{1}=N[2\;1\;2]$. ∎

######  Theorem 3.15.

All resultants of $N[2^{\prime}\;a_{2}^{\prime}]$ shadows are $N[2\;\ell]$
knots, $0\leq\ell\leq a_{2}$.

This too is well-trodden [MR3844207, MR3084750, MR4190429, MR3356086].

######  Theorem 3.16.

Let $a_{1},a_{2},k,\ell$ be even, positive integers, $2\leq k\leq a_{1}$,
$2\leq\ell\leq a_{2}$. Then the $N[a_{1}^{\prime}\;a_{2}^{\prime}]$ knot
shadow has the $N[k\;\ell]$ and $N[(k-1)\;1\;(\ell-1)]$ knots and the unknot
as resultants.

######  Note 3.17.

$N[\pm(2-1)\;\pm 1\;\pm(2-1)]\approx N[\pm 3]$.

###### Proof.

We can automatically untangle the knot shadow for assignments $a_{1}^{\prime}$
or $a_{2}^{\prime}\rightarrow 0$. We now must identify $N[m\;n]$ diagrams,
where $m$, $n\not=0$. For specific positive indices $k$, $\ell$, we have knot
diagrams $N[\pm k\;\pm\ell]$ and $N[\pm k\;\mp\ell]\simeq N[\pm(k-1)\;\pm
1\;\pm(\ell-1)]$. Diagrams with matching signs are alternating, thus fully
reduced. ∎

######  Corollary 3.18.

The nontrivial knots $N[a_{1}\;a_{2}]$, $a_{1}$, $a_{2}$ even, have
$F(N[a_{1}\;a_{2}])=4$. The only such fertile knot is the figure-eight knot
$4_{1}=N[2\;2]$.

###### Proof.

The choice $a_{1}^{\prime}\rightarrow\pm 2$ and $a_{2}^{\prime}\rightarrow\mp
2$ produces the trefoil. Any other nontrivial resultant $R$ has $d(R)\geq 2$,
thus $5_{1}=N[5]$ cannot be a resultant. ∎

######  Theorem 3.19.

Let $a_{1},k$ be even integers and $a_{2},\ell$ be odd integers, $2\leq k\leq
a_{1}$, $3\leq\ell\leq a_{2}$. Then the $N[a_{1}^{\prime}\;a_{2}^{\prime}]$
knot shadow has resultants of the forms $N[k+1]$, $N[k\;\ell]$, and
$N[(k-1)\;1\;(\ell-1)]$, in addition to the unknot.

###### Proof.

Without loss of generality, choose $a_{1}$ to be even. Like above, we can take
$a_{1}\rightarrow 0$ to produce unknots, and can otherwise form $N[\pm
k\;\pm\ell]$ and $N[\pm k\;\mp\ell]\approx N[\pm(k-1)\;\pm 1\;\pm(\ell-1)]$
diagrams.

This leaves $N[\pm k\;\pm 1]\simeq N[\pm(k+1)]$ and $N[\pm k\;\mp 1]\simeq
N[\pm(k-1)]$. $N[\pm 2\;\mp 1]\simeq N[\pm 1\;\pm 1\;0]\simeq D[\pm 2],$ or
the unknot. ∎

Theorem 3.15 follows from $k=2\implies
N[(k-1)\;1\;(\ell-1)]=N[1\;1\;(\ell-1)]\simeq N[2\;(\ell-1)]$.

######  Corollary 3.20.

The only fertile knots of the form $N[2\;a_{2}]$, $a_{2}\in\mathbb{N}$, are
$N[2\;3]=5_{2}$, $N[2\;2]=4_{1}$, $N[2\;1]=3_{1}$, and $N[2\;0]=0_{1}$. If
$a_{2}\geq 2$, $F(N[2\;a_{2}])=4$.

###### Proof.

$5_{1}=N[5]$ cannot be written as a $N[2\;\ell]$ knot:
$\ell+\frac{1}{2}\not=5$ for any $\ell\in\mathbb{Z}$. ∎

######  Corollary 3.21.

Nontrivial knots $N[a_{1}\;a_{2}]$, $a_{1}$ even, $a_{2}$ odd, $4\leq a_{1}$,
$2<a_{2}$ are infertile with $F(N[a_{1}\geq 4\;\;a_{2}])=5$.

###### Proof.

$a_{2}$ odd allows the formation of $N[5]=5_{1}$, however, all resultants $R$
have $d(R)\leq 3$, precluding formation of $6_{3}=N[2\;1\;1\;2]$. ∎

For two-component rational links $L$, we already know if $d(L)>1$, $F(L)\geq
5$. For further information, we just need to determine if a rational link can
form $6^{2}_{1}=N[6]$, $6^{2}_{2}=N[3\;3]$, and $6^{2}_{3}=N[2\;2\;2]$.

######  Theorem 3.22.

Let $L=N[a_{1}\;a_{2}]$ be a rational two-component link. Then $F(L)=5$.

###### Proof.

The longest resultants of any $L$ are of the form $N[\pm c_{1}\;\pm 1\;\pm
c_{3}]\implies 6^{2}_{3}=N[2\;2\;2]$ cannot be a resultant. ∎

###  3.2. Length 3 and Higher Links

######  Theorem 3.23.

Let $L$ be a nontrivial prime two-component link. If $L=2_{1}^{2}=T(2,2)$,
$F(L)=2$. If $L=T(p,2)$, $p$ even, $p\geq 4$, $F(L)=4$. Otherwise, $F(L)\geq
5$.

###### Proof.

In [MR1030506], Taniyama proved any prime two-component link of five or more
crossings majorizes the links $2_{1}^{2}$, $4_{1}^{2}$, and $5_{1}^{2}$.
[MR3844207] showed a link that majorizes some link $L$ has $L$ as a resultant.
∎

######  Theorem 3.24.

Let $L=N[a_{1}\;a_{2}\;a_{3}]$ be a nontrivial two-component link. If
$a_{2}=1$ or $a_{1}=a_{3}=2$, $F(L)=5$. Otherwise $F(L)=6$.

The rational link $N[a_{1}\;a_{2}\;a_{3}]$ has two components when
$a_{1},a_{3}$ even, $a_{2}$ odd, or $a_{1},a_{3}$ odd, $a_{2}$ even, or
$a_{1},a_{2},a_{3}$ even.

######  Lemma 3.25.

Let $L=N[a_{1}\;a_{2}\;a_{3}]$ be a nontrivial two-component link. $6^{2}_{3}$
is a resultant iff $a_{2}>1$.

###### Proof.

Suppose $a_{2}=1$. Flipping the sign of one or more integer tangles will
produce resultants $R$ of the forms $N[\pm c_{1}\;\pm 1\;\mp c_{3}]$ or $N[\pm
c_{1}\;\mp 1\;\pm c_{3}]\implies d(R)\leq 2$. Therefore, the only length three
resultants will be $N[\pm c_{1}\;\pm 1\;\pm c_{3}]$, thus $6^{2}_{3}$ is not a
resultant.

We now prove if $a_{2}>1$, $6^{2}_{3}$ is a resultant of $L$. If $a_{2}$ is
even, a direct assignment $N[\pm 2\;\pm 2\;\pm 2]=6^{2}_{3}$ is possible. If
$a_{2}>1$ is odd, the assignment $N[\pm 2\;\mp 3\;\pm 2]$ is possible. In the
remaining case, $a_{1},a_{3}$ odd and $a_{2}$ even, the assignment $N[\pm
3\;\mp 2\;\mp 3]$ is possible. ∎

######  Lemma 3.26.

Let $L=N[a_{1}\;a_{2}\;a_{3}]$ be a nontrivial two-component link. $6^{2}_{1}$
is a resultant iff one of $a_{1},a_{3}\not=2$.

###### Proof.

Let $a_{1}=a_{3}=2$. $a_{1}^{\prime}$ or $a_{3}^{\prime}\rightarrow 0$ produce
only the un- or Hopf links, so we must consider $N[\pm 2\;a_{2}^{\prime}\;\pm
2]$ and $N[\pm 2\;a_{2}^{\prime}\;\mp 2]$. The only length one resultant of
$N[\pm 2\;a_{2}^{\prime}\;\pm 2]$ for either parity of $a_{2}$ is $4^{2}_{1}$.

For $N[\pm 2\;a_{2}^{\prime}\;\mp 2]$, $a_{2}^{\prime}\rightarrow 0$ forms the
unlink. Otherwise, we have $N[\pm 2\;\pm a_{2}\;\mp 2]$ and $N[\pm 2\;\mp
a_{2}\;\mp 2]\approx N[\pm 2\;\pm(a_{2}-1)\;\mp 2]$. Since $a_{2}$ can be
either parity, we are essentially considering the same case $R=N[\pm
2\;\pm(c_{2}-2)\;\pm 2]$. If $c_{2}<3$, $c(R)<6$. If $c_{2}\geq 3$, $d(R)=3$.
Therefore, $6^{2}_{1}$ cannot be a resultant.

We now show if one of $a_{1},a_{3}>2$, then we can form $6^{2}_{1}$. If
$a_{2}$ odd, we can form $N[\pm 4\;\pm 1\;\mp 2]\approx N[\pm 6]$. If $a_{2}$
even, we can form $N[\pm 4\;0\;\pm 2]$ or $N[\pm 3\;0\;\pm 3]$. ∎

We now prove Theorem 3.24, completing the classification of length 3 rational
links.

###### Proof.

By the above lemmas, if $a_{2}=1$ or $a_{1}=a_{3}=2$, $F(L)=5$. We now check
if larger links can form $6^{2}_{2}$.

Case 1: $a_{1},a_{3}$ even, $a_{2}$ odd If one of $a_{1},a_{3}>2$, then we can
form $N[\pm 4\;\mp 1\;\pm 2]\approx N[\pm 3\;\pm 3]\implies F(L)\geq 6$.

Case 2: $a_{1},a_{3}$ odd, $a_{2}$ even The first such link is
$8^{2}_{4}=N[3\;2\;3]$. $N[\pm 1\;\pm 2\;\pm 3]=6^{2}_{2}\implies F(L)\geq 6$.

Case 3: $a_{1},a_{2},a_{3}$ even If one of $a_{1},a_{3}>2$, we can form $N[\pm
4\;\mp 2\;\mp 2]=6^{2}_{2}\implies F(L)=6$.

The upper bound $F(H)\leq 6$ on rational two-component links $H$ proves
$F(L)=6$. ∎

######  Theorem 3.27.

All length 4 rational two-component links are locally fertile.

###### Proof.

The only two-component links in $\mathcal{K}_{4}$ are
$7^{2}_{2}=N[3\;1\;1\;2]$,
$8^{2}_{5}=N[3\;1\;2\;2]$, and $N[3\;2\;2\;3]$. All have the three six
crossing two-component links as resultants.

For $7^{2}_{2}$, $N[\pm 3\;\pm 1\;\mp 1\;\pm 2]\approx 6^{2}_{1}$, $N[\pm
3\;\pm 1\;\pm 1\;\mp 2]\approx 6^{2}_{2}$, and $N[\pm 3\;\mp 1\;\pm 1\;\pm
2]\approx 6^{2}_{3}$.

For $8^{2}_{5}$, $N[\pm 3\;\pm 1\;\mp 2\;\mp 2]\approx 6^{2}_{1}$, $N[\pm
3\;\pm 1\;0\;\pm 2]\approx 6^{2}_{2}$, and $N[\pm 1\;\pm 1\;\pm 2\;\pm
2]\approx 6^{2}_{3}$.

For $N[3\;2\;2\;3]$, $N[\pm 3\;0\;\pm 2\;\pm 1]\approx 6^{2}_{1}$, $N[\pm
3\;0\;0\;\pm 3]\approx 6^{2}_{2}$, and $N[\pm 3\;\mp 2\;\mp 2\;\pm 1]\approx
6^{2}_{3}$. ∎

######  Theorem 3.28.

All length 5 rational two-component links are locally fertile.

###### Proof.

There are eight two-component links in $\mathcal{K}_{5}$. First consider
$N[3\;1\;1\;1\;3]$, $N[3\;1\;2\;1\;3]$, $N[3\;2\;2\;2\;3]$,
$N[3\;2\;1\;1\;2]$, and $N[3\;2\;1\;2\;2]$. By Theorem 3.9, since each link
has odd $a_{1}$ or $a_{n}$ and all length 4 rational links are locally
fertile, these links are too.

Now consider $N[2\;2\;2\;2\;2]$. Choose $a_{2}\rightarrow 0$ to form
$N[4\;2\;2]$. This is a length three rational link with $a_{2}>1$ and one of
$a_{1},a_{3}>2$.

The remaining two-component links in $\mathcal{K}_{5}$ are
$8^{2}_{7}=[2\;1\;2\;1\;2]$ and $9^{2}_{11}=N[2\;2\;2\;1\;2]$.

For $8^{2}_{7}$, choose $a_{3}\rightarrow 0$ to form $N[2\;2\;2]$. Also,
$N[\pm 2\;\mp 1\;\pm 2\;\pm 1\;\mp 2]\approx 6^{2}_{1}$ and $N[\pm 2\;\pm
1\;\mp 2\;\pm 1\;\mp 2]\approx 6^{2}_{2}$.

For $9^{2}_{11}$, choose $a_{2}\rightarrow 0$ to form $N[4\;1\;2]$ and have
$6^{2}_{1}$ and $6^{2}_{2}$ as resultants. Also, $N[2\;2\;2\;1\;0]\simeq
6^{2}_{3}$. ∎

From Theorem 3.11, the last two results nearly complete our classification of
rational two-component links.

######  Theorem 3.29.

All rational two component links $L$ with $d(L)\geq 4$ are locally fertile.

###### Proof.

It remains to show $N[2\;1\;1\;1\;1\;2]$ is 6-fertile.
$N[\pm 2\;\mp 1\;\pm 1\;\pm 1\;\mp 1\;\pm 2]\approx 6^{2}_{1}$, $N[\pm 2\;\mp
1\;\pm 1\;\pm 1\;\pm 1\;\mp 2]\approx 6^{2}_{2}$, and $N[\pm 2\;\mp 1\;\pm
1\;\mp 1\;\pm 1\;\pm 2]\approx 6^{2}_{3}$. ∎

######  Corollary 3.30.

The only non-locally fertile two-component rational links are the torus links
$T(p,2)$, $p$ even, and length three links $N[2\;a_{2}\;2]$ and
$N[a_{1}\;1\;a_{3}]$.

##  4\. Rational Knot Fertility Number

There are far more rational knots than links in trunks: $\mathcal{K}_{6}$ is
composed of 12 links and 24 knots. Moreover, rational knots can have fertility
numbers between 4 and 7 instead of the binary of 5 or 6 for rational links $L$
with $c(L)\geq 5$. To balance clarity with brevity, we will explicitly discuss
length 3 and 4 rational knots to demonstrate the general approach, then
provide the fertility number for all other higher length rational knots
without explanation.

To tabulate $F(L)$, we wrote a program in Mathematica (available on Github
[gitRepository]) which identifies rational link resultants by their continued
fraction. Its results matched those in Table 2 of [MR3844207], which were
calculated for all rational knots through 10 crossings, and our results from
[MR4190429], which were calculated by computing the Jones polynomial of all
resultant diagrams. See Appendix 1 for all compiled results.

###  4.1. Length 3 Knots

######  Theorem 4.1.

[MR3844207] Let $K$ be a nontrivial prime knot. If $K=T(p,2)$, $p$ odd,
$F(K)=3$. If $K=4_{1}$, $F(K)=4$. Otherwise, $F(K)\geq 4$ and $K$ has $5_{2}$
as a resultant.

######  Theorem 4.2.

A rational knot $K$, $d(K)\geq 3$, has $F(K)\geq 5$ unless $d(K)=3$,
$a_{2}=1$, and $a_{1},a_{3}$ are both odd.

###### Proof.

[MR3844207] similarly show nontrivial prime knots that are not pretzel knots
$P(p_{1},p_{2},p_{3})$ with $p_{i}$ all odd, positive, integers have $5_{1}$
as a resultant. Of the rational knots $K=N[a_{1}\;a_{2}\;...\;a_{n}]$ with
$d(K)\geq 3$, only rationals with $d(K)=3$ and $a_{2}=1$ can be written as
pretzels knots described by exactly three positive $p_{i}$. Further
requirements on $a_{i}$ follow from the requirements from [MR3844207]. ∎

We point the reader unfamiliar with pretzel links to [pretzel].

######  Lemma 4.3.

Length three rational knots have $F(K)\leq 6$.

###### Proof.

$7_{7}=N[\pm 2\;\pm 1\;\pm 1\;\pm 1\;\pm 2]$ can only be formed from an
assignment $N[\pm a_{1}\;\mp a_{2}\;\mp a_{3}]\approx N[\pm(a_{1}-1)\;\pm
1\;\pm(a_{2}-2)\;\pm 1\;\pm(a_{3}-1)]$. Therefore $7_{7}$ is a resultant iff
all $a_{i}$ are odd.

When all $a_{i}$ are odd, if $c_{i}\geq 3$, $c(R)\geq 7\implies 6_{3}$ is a
resultant iff at least one of $a_{i}^{\prime}\rightarrow 1^{\prime}$. When
$a_{2}=1$, $F(K)=4$. The only remaining possibilities for six crossing
resultants are $N[1^{\prime}\;a_{2}^{\prime}\;a_{3}^{\prime}]\approx
N[(a_{2}+1)^{\prime}\;a_{3}^{\prime}]$ and similarly
$N[a_{1}^{\prime}\;(a_{2}+1)^{\prime}]$, which have resultants of length three
or less, so $6_{3}$ cannot be a resultant. ∎

######  Corollary 4.4.

Let $K=N[a_{1}\;a_{2}\;a_{3}]$ be a nontrivial rational knot where
$a_{1},a_{2},a_{3}$ are odd. If $a_{2}=1$, then $F(K)=4$. Otherwise,
$F(K)=5$.

######  Theorem 4.5.

Let $K=N[a_{1}\;a_{2}\;a_{3}]$ be a nontrivial rational knot where
$a_{1},a_{2}$ are odd and $a_{3}$ is even. If $a_{2}=1$, then $F(K)=5$.
Otherwise $F(K)=6$.

###### Proof.

$F(K)\geq 5$. If $a_{2}=1$, all resultants are of length three or below, so
$6_{3}$ is not a resultant. Otherwise the following assignments are always
possible: $N[\pm 1\;\pm 3\;\pm 2]\approx 6_{1}$, $N[\pm 3\;\pm 1\;\pm
2]\approx 6_{2}$, and $N[\pm 3\;\mp 3\;\mp 2]\approx 6_{3}\implies F(K)\geq
6$. ∎

######  Theorem 4.6.

Let $K=N[a_{1}\;a_{2}\;a_{3}]$ be a rational knot where $a_{1}$ is odd and
$a_{2},a_{3}$ are even. If $a_{2}=2$, then $F(K)=5$. Otherwise $F(K)=6$.

###### Proof.

Trunk knot $7_{5}=N[3\;2\;2]$ has at most two six crossing resultants, so
$F(7_{5})=5$. It can form $N[\pm 3\;\pm 2\;\mp 2]\approx 6_{2}$ and $N[\pm
3\;\mp 2\;\pm 2]\approx 6_{3}$, but not $6_{1}=N[4\;2]$.

When $a_{2}\geq 4$, we have $N[\pm 1\;\pm 4\;\mp 2]\approx 6_{1}\implies
F(K)=6$. However, branches with $a_{2}=2$ cannot form $6_{1}$.
$2^{\prime}\rightarrow 0$ produces length one knots, leaving
$N[a_{1}^{\prime}\;\pm 2\;a_{3}^{\prime}]$ and $N[a_{1}^{\prime}\;\mp
2\;a_{3}^{\prime}]$. $a_{1}^{\prime}\rightarrow c_{1}\geq 3$ forces either
$N[\pm c_{1}\;\pm 2\;a_{3}^{\prime}]$ (further resultants all length three or
greater, i.e. not $6_{1}$) or $R_{1}=N[\pm(c_{1}-1)\;\pm 1\;\pm
1\;a_{3}^{\prime}]$.

  * •

$a_{3}^{\prime}\rightarrow\pm c_{1}\geq 2\implies d(R_{1})=4$,

  * •

$a_{3}^{\prime}\rightarrow 0\implies d(R_{1})=1$,

  * •

$a_{3}^{\prime}\rightarrow\mp 2\implies d(R_{1})=2$, and

  * •

$a_{3}^{\prime}\rightarrow\mp c_{1}\leq 4\implies d(R_{1})=3$.

The only viable route to $6_{1}$ here is $N[\pm(c_{1}-1)\;\pm 1\;\pm 1\;\mp
2]\approx N[\pm(c_{1}-1)\;\pm 1\;0\;\pm 1\;\pm 1]\approx N[\pm(c_{1}-1)\;\pm
3]\not\approx 6_{1}$.

The last unexamined cases are $N[\pm 1\;\pm 2\;a_{3}^{\prime}]\approx N[\pm
3\;a_{3}^{\prime}]$ and $N[\pm 1\;\mp 2\;a_{3}^{\prime}]\approx N[\mp
1\;a_{3}^{\prime}]$. The former case will have at least one tangle with an odd
number of crossings since $a_{3}$ is even. The latter is length one for all
$a_{3}$. ∎

###  4.2. Length 4 Knots

There are 7 knots in $\mathcal{K}_{4}$. We’ve already seen
$6_{3}=N[2\;1\;1\;2]$ is 5-fertile and $7_{6}=N[2\;2\;1\;2]$ is 6-fertile.

######  Theorem 4.7.

$8_{11}=N[3\;2\;1\;2]$ and $8_{9}=N[3\;1\;1\;3]$ are 5-fertile.

######  Theorem 4.8.

$8_{12}=N[2\;2\;2\;2]$, $9_{13}=N[3\;2\;1\;3]$, and $9_{18}=N[3\;2\;2\;2]$ are
6-fertile.

###### Proof.

See Appendix 1. ∎

######  Theorem 4.9.

Knots $K=N[a_{1}\;1\;1\;a_{4}]$ have $F(K)=5$.

###### Proof.

$a_{1},a_{4}$ are either both even or both odd. The eight forms of resultants
are

  * •

$N[\pm c_{1}\;\pm 1\;\pm 1\;\pm c_{4}]$

  * •

$N[\pm c_{1}\;\mp 1\;\pm 1\;\pm c_{4}]\approx N[\pm(c_{1}-1)\;\pm 2\;\pm
c_{4}]$

  * •

$N[\pm c_{1}\;\pm 1\;\mp 1\;\pm c_{4}]\approx N[\pm(c_{1}+c_{4}+1)]$

  * •

$N[\pm c_{1}\;\pm 1\;\pm 1\;\mp c_{4}]\approx N[\pm c_{1}\;\pm
2\;\pm(c_{4}-1)]$

  * •

$N[\pm c_{1}\;\mp 1\;\mp 1\;\pm c_{4}]\approx N[\pm(c_{1}+c_{4}-1)]$

  * •

$N[\pm c_{1}\;\mp 1\;\pm 1\;\mp c_{4}]\approx N[\pm(c_{1}-1)\;\pm 1\;\pm
1\;\pm(c_{4}-1)]$

  * •

$N[\pm c_{1}\;\pm 1\;\mp 1\;\mp c_{4}]\approx N[\pm(c_{1}-c_{4}+1)]$

  * •

$N[\pm c_{1}\;\mp 1\;\mp 1\;\mp c_{4}]\approx N[\pm(c_{1}-c_{4}-1)]$

where $c_{i}=a_{i}-2m_{i}$, $m_{i}\in\mathbb{N}$. Obviously, for either
parity, $6_{3}$ is a resultant. Also, $N[\pm 4\;\mp 1\;\pm 1\;\mp 2]$ and
$N[\pm 3\;\pm 1\;\pm 1\;\pm 1]\approx 6_{2}$. Can we get $6_{1}$? In the
length one cases and the first length four case, the answer is obviously no.

The remaining cases are $N[\pm(c_{1}-1)\;\pm 2\;\pm c_{4}]$ and
$N[\pm(c_{1}-1)\;\pm 1\;\pm 1\;\pm(c_{4}-1)]$. Let $T=\pm(c_{i}-1)$. If
$c_{i}\geq 3$, $T$, and by extension $K$, is in canonical form. If $c_{i}=2$,
$T=\pm 1$ is absorbed, forming $N[\pm 3\;\pm c_{4}]$ or $N[\pm 2\;\pm
1\;\pm(c_{4}-1)]$, where $c_{4}$ must be even. The only $N[2\;n]$ resultants
possible with this choice are $N[2\;2]$ and $N[2\;3]$. If $c_{1}=1$, $T=0$ and
each form collapses to a length one knot. Thus $6_{1}$ is not a resultant and
these knots have $F(N[a_{1}\;1\;1\;a_{4}])=5$. ∎

######  Corollary 4.10.

The only length 4 knots $K$ with $F(K)=5$ are branches of $N[3\;2\;1\;2]$
where $a_{3}=1$ and $N[a_{1}\;1\;1\;a_{4}]$.

###### Proof.

Note $F(N[3\;2\;3\;2])=F(N[2\;3\;1\;2])=F(N[3\;3\;1\;3])=6$. ∎

###  4.3. Length 5 and Longer Knots

First, length 5 knots. There are 12 knots in $\mathcal{K}_{5}$.

######  Theorem 4.11.

$8_{14}=N[2\;2\;1\;1\;2]$, $9_{23}=N[2\;2\;1\;2\;2]$,
$8_{13}=N[3\;1\;1\;1\;2]$, $9_{20}=N[3\;1\;2\;1\;2]$,
$9_{21}=N[3\;1\;1\;2\;2]$, $10_{29}=N[3\;1\;2\;2\;2]$,
$11a337=N[3\;2\;2\;1\;3]$, and $11a357=N[3\;2\;1\;2\;3]$ are 6-fertile.

######  Theorem 4.12.

$10_{25}=N[3\;2\;2\;1\;2]$, $10_{26}=N[3\;2\;1\;1\;3]$, and
$11a236=N[3\;2\;2\;2\;2]$ are 7-fertile.

###### Proof.

See Appendix 1. ∎

######  Corollary 4.13.

$7_{7}=N[2\;1\;1\;1\;2]$ is the only 5-fertile knot in $\mathcal{K}_{5}$.

######  Corollary 4.14.

Knots $K=N[a_{1}\;1\;a_{3}\;1\;a_{5}]$, $a_{1},a_{5}$ even and $a_{3}$ odd are
the only length 5 knots with $F(K)=5$.

###### Proof.

By the same methods as above. Note $F(N[2\;3\;1\;1\;2])=6$. ∎

######  Theorem 4.15.

All knots $K\in\mathcal{K}_{6}$ have $F(K)>5$.

######  Theorem 4.16.

The sole knots in $\mathcal{K}_{6}$ that are only 6-fertile are
$9_{26}=N[3\;1\;1\;1\;1\;2]$, $10_{27}=N[3\;2\;1\;1\;1\;2]$,
$10_{30}=N[3\;1\;2\;1\;1\;2]$, $10_{32}=N[3\;1\;1\;1\;2\;2]$,
$10_{33}=N[3\;1\;1\;1\;1\;3]$, $10_{43}=N[2\;1\;2\;2\;1\;2]$,
$11a192=N[3\;1\;2\;1\;2\;2]$, and $12a1039=N[3\;1\;2\;2\;1\;3]$.

######  Theorem 4.17.

The sole knots in $\mathcal{K}_{7}$ that are only 6-fertile are
$9_{31}=N[2\;1^{5}\;2]$, $11a306=N[3\;1^{5}\;3]$, and $N[3\;1\;2\;1\;2\;1\;3]$

######  Theorem 4.18.

All knots of length 8 or higher are locally fertile.

###### Proof.

All knots in $\mathcal{K}_{8}$ and $\mathcal{K}_{9}$ are 7-fertile, as is
$N[2\;1^{8}\;2]$. ∎

We have now completely classified the fertility number of all rational knots.
Interestingly, the stragglers at each length were the knots
$N[2\;1^{\ell-2}\;2]$. Where other rational knots of the same length were
approaching local fertility, these knots needed more component $1$ tangles to
gain rational resultants. One could say they are the least rational rational
knots, just as the golden ratio
$\varphi=\frac{1+\sqrt{5}}{2}=(1\;1\;1\;1\;1\;...)$ has a superlatively slow
rate of convergence in its approximation via continued fractions. Lo and
behold, $N[2\;1^{\ell-2}\;2]\simeq N[1^{\ell}]$ with continued fraction
$(2\;1^{\ell-1})$.

##  5\. Knot Resultant Combinatorics

######  Note 5.1.

We may have a binomial coefficient ${n\choose k}$ where it is possible, in
application, for $k>n$. In these cases, the binomial coefficient should be
understood as equal to zero.

######  Theorem 5.2.

A rational link $L=N[\pm a_{1}\;\pm a_{2}\;...\;\pm a_{n}]$ has at most
${n\choose 1}+{n-1\choose 2}$ unique resultants $R$ with $c(R)=c(L)-2$ and at
most ${n\choose 1}{n-1\choose 1}+{n-1\choose 3}$ unique resultants with
$c(R)=c(L)-3$.

######  Corollary 5.3.

$L$ has at most $\frac{1}{2}(n^{2}-n+2)$ unique resultants $R$ with
$c(R)=c(L)-2$ and $\frac{1}{6}(n^{3}+5n-6)$ unique resultants with
$c(R)=c(L)-3$.

######  Theorem 5.4.

A rational shadow $L=N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]$
has at most $\lceil\frac{a_{1}+1}{2}\rceil\prod_{i=2}^{n}(a_{i}+1)$ unique
resultants.

###### Proof.

The integer tangle $x^{\prime}$ has resolutions with $x-2j$ crossings, $0\leq
j\leq x\implies$ there are $x+1$ resolutions of each integer tangle. Our
possible resolutions are nearly halved for the first tangle, as it sets the
sign convention for the link, i.e. $N[\pm
c_{1}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]$ is indistinguishable from $N[\mp
c_{1}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]$ as we can freely redefine
crossing parity here before assigning any other crossings. ∎

For any link shadow $L$, all we know it has at most $2^{c(L)-1}$ unique
resultants when we count a knot and its mirror image the same. This theorem
gives a significantly sharper upper bound for rational links, yet there is a
long way to go. The knot $8_{12}=N[2\;2\;2\;2]$ has 11 unique resultants, far
fewer than $2\times 3\times 3\times 3=54$, yet still much sharper than
$2^{7}=128$. Theorem 2.7 is likely useful for improvements.

We will now explicitly enumerate the number of each link type $L^{\prime}$
produced by rational links $L$ of lengths 1 and 2. This can be converted to a
probability distribution by dividing by the total number of resultants
$2^{c(L)}$.

######  Theorem 5.5.

$D[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]\rightarrow
2^{a_{n}}N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n-1}^{\prime}]$, $n\geq 2$.
$D[a_{1}^{\prime}]\rightarrow 2^{a_{1}}$ unknots.

This notation means we get $2^{a_{n}}$ diagrams described by
$N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n-1}^{\prime}]$.

###### Proof.

A shadow RI move is an invariant on knot shadows by creating two copies (one
for each resolution) of the shadow prior to its application to a straight
strand. Applying the denominator to a rational tangle connects the right ends
of the $a_{n}$ tangle, allowing it to be untwisted $a_{n}$ times. For $n\geq
2$, what remains is the rotated version of the numerator closure. For $n=1$,
the complicating twists must have been done on an unknot. ∎

######  Corollary 5.6.

$N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}\;0]\rightarrow
2^{a_{n}}N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n-1}^{\prime}]$, $n\geq 2$.
$N[a_{1}^{\prime}\;0]\rightarrow 2^{a_{1}}$ unknots.

######  Theorem 5.7.

An $a_{1}^{\prime}$ $(N[a_{1}^{\prime}])$ shadow has
${{a_{1}}\choose{\frac{a_{1}-k}{2}}}$ $+k$ $(N[+k])$ and $-k$ $(N[-k])$
resultants, where $a_{1}$ and $k$ have the same parity.

Interestingly, links lack many unlink resultants. Unknots are the most likely
resultant from $N[a_{1}^{\prime}]$ knot shadows. Trivial resultants are fairly
dominant in general, especially at low crossing numbers. [MR3084750] found
experimentally all knot shadows with twelve or fewer crossings had a higher
unknot probability than any nontrivial resultant probability (they did show
the 14-crossing torus knot T(7,3) was a counterexample). Furthermore, we are
unaware of any knot shadow where the resultant probability of any nontrivial
knot is greater than or equal to 50%.

$N[2^{\prime}]$, the first nontrivial link shadow, has four total resultants:
two 2-component unlinks and two Hopf links. The $N[a_{1}^{\prime}]$ link has
more Hopf links than unlinks for all $a_{1}\geq 4$. In fact, with a long
enough initial link shadow $N[a_{1}^{\prime}]$, there will be more particular
nontrivial link resultants $N[c_{1}]$ than unlinks.

######  Lemma 5.8.

For all $k$, there is some $n$ such that

$2{{n}\choose{\frac{n-k}{2}}}\geq{{n}\choose{\frac{n}{2}}}.$

$n,k$ are even integers, $n\geq k$.

###### Proof.

Converting to the factorial form of the binomial coefficient, the above
inequality holds iff

$\displaystyle 2\left[\left(\frac{n}{2}\right)!\right]^{2}$
$\displaystyle\geq\left(\frac{n}{2}-\frac{k}{2}\right)!\left(\frac{n}{2}+\frac{k}{2}\right)!\iff$
$\displaystyle
2\left(\frac{n}{2}\right)\left(\frac{n}{2}-1\right)...\left(\frac{n}{2}-\frac{k}{2}+1\right)$
$\displaystyle\geq\left(\frac{n}{2}+\frac{k}{2}\right)\left(\frac{n}{2}+\frac{k}{2}-1\right)...\left(\frac{n}{2}+1\right)$

The left polynomial has a leading coefficient two times larger than that of
the right and will eventually surpass the right polynomial for large $n$. ∎

######  Corollary 5.9.

For any link $N[c_{1}]$, there is some $A$ such that $N[a_{1}^{\prime}]$,
$a_{1}>A$, has more $N[c_{1}]$ than unlink resultants.

For $k=2$, or Hopf link resultants, the above inequality is simply
$n\geq\frac{n}{2}+1$, which is true for all integers greater than 1. For
$k=4$, or $4^{2}_{1}$ resultants, the inequality reduces to
$n(\frac{n}{2}-1)\geq(\frac{n}{2}+1)(\frac{n}{2}+2)$, or $n\geq
5+\sqrt{33}\approx 10.745$. Indeed, we see $N[10^{\prime}]$ has 252 unlink
resultants and 240 $4^{2}_{1}$ resultants, while $N[12^{\prime}]$ has 924
unlink and 990 $4^{2}_{1}$ resultants.

######  Lemma 5.10.

The shadow of a rational tangle

(5.1)
$N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]\rightarrow\sum^{a_{1}}_{m=-a_{1}}{{a_{1}}\choose{\frac{a_{1}-|m|}{2}}}N[m\;a_{2}^{\prime}\;...\;a_{n}^{\prime}],$

where $m$ is restricted to the parity of $a_{n}$.

###### Proof.

This follows directly from the resolution of integer tangles. ∎

######  Theorem 5.11.

Let $a_{1},a_{2},k,\ell$ be even, positive integers, $2\leq k\leq a_{1}$,
$2\leq\ell\leq a_{2}$. Then the $N[a_{1}^{\prime}\;a_{2}^{\prime}]$ knot
shadow has

${{a_{1}}\choose{\frac{a_{1}-k}{2}}}{{a_{2}}\choose{\frac{a_{2}-\ell}{2}}}+{{a_{1}}\choose{\frac{a_{1}-\ell}{2}}}{{a_{2}}\choose{\frac{a_{2}-k}{2}}}$

$N[k\;\ell]$ and $N[(k-1)\;1\;(\ell-1)]$ resultants and

$2^{a_{2}}{{a_{1}}\choose{\frac{a_{1}}{2}}}+\left[2^{a_{1}}-{{a_{1}}\choose{\frac{a_{1}}{2}}}\right]{{a_{2}}\choose{\frac{a_{2}}{2}}}$

unknot resultants.

###### Proof.

Using Lemma 5.10, we start with ${{a_{1}}\choose{\frac{a_{1}}{2}}}$
$N[0\;a_{2}^{\prime}]$ diagrams, which are all unknots. We can also
automatically untangle the knot shadow if we take $a_{2}\rightarrow\ell=0$,
but we’ve already counted the cases where $k=0$. New unknots come from
assignments where $k$ is non-trivial. There are exactly
$2^{a_{1}}-{{a_{1}}\choose{\frac{a_{1}}{2}}}$ of these.

What remains are
$\sum_{m}{{a_{1}}\choose{\frac{a_{1}-|m|}{2}}}\sum_{n}{{a_{2}}\choose{\frac{a_{2}-|n|}{2}}}N[m\;n]$
diagrams, where $m$, $n\not=0$. By symmetry, for specific positive indices
$k$, $\ell$, we have the proposed number of knot diagrams $N[\pm k\;\pm\ell]$
and $N[\pm k\;\mp\ell]\approx N[\pm(k-1)\;\pm 1\;\pm(\ell-1)]$. ∎

The combinatorics of the odd-odd length two rational links is a natural
extension.

######  Theorem 5.12.

Let $a_{1},k$ be even integers and $a_{2},\ell$ be odd integers, $2\leq k\leq
a_{1}$, $2\leq\ell\leq a_{2}$. Then the $N[a_{1}^{\prime}\;a_{2}^{\prime}]$
knot shadow has

$2{{a_{1}}\choose{\frac{a_{1}-k}{2}}}{{a_{2}}\choose{\frac{a_{2}-\ell}{2}}}$

$N[k\;\ell]$ and $N[(k-1)\;1\;(\ell-1)]$ resultants,

$2{{a_{2}}\choose{\frac{a_{2}-1}{2}}}\left[{{a_{1}}\choose{\frac{a_{1}-k}{2}}}+{{a_{1}}\choose{\frac{a_{1}-(k+2)}{2}}}\right]$

$N[k+1]$ resultants, and

$2^{a_{2}}{{a_{1}}\choose{\frac{a_{1}}{2}}}+2{{a_{1}}\choose{\frac{a_{1}-2}{2}}}{{a_{2}}\choose{\frac{a_{2}-1}{2}}}$

unknot resultants.

###### Proof.

Without loss of generality, choose $a_{1}$ to be even. Like above, we can take
$a_{1}\rightarrow 0$, making $2^{a_{2}}{{a_{1}}\choose{\frac{a_{1}}{2}}}$
unknot diagrams.

By Lemma 5.10, if $a_{1}\rightarrow\pm k$, $k\geq 0$, then for all $\ell\leq
a_{2}$, there are
${{a_{1}}\choose{\frac{a_{1}-k}{2}}}{{a_{2}}\choose{\frac{a_{2}-\ell}{2}}}$
$N[\pm k\;\pm\ell]$ diagrams and
${{a_{1}}\choose{\frac{a_{1}-k}{2}}}{{a_{2}}\choose{\frac{a_{2}-\ell}{2}}}$
$N[\pm k\;\mp\ell]\approx N[\mp(k-1)\;\mp 1\;\mp(\ell-1)]$ diagrams. For
$k,\ell\geq 3$, these diagrams are in canonical form, hence reduced, and
unique from each other.

This leaves $N[\pm k\;\pm 1]\approx N[\pm(k+1)]$ and $N[\pm k\;\mp 1]\approx
N[\pm(k-1)]$. $N[\pm 2\;\mp 1]\approx N[\pm 1\;\pm 1\;0]\approx D[\pm 2],$ or
the unknot. ∎

######  Theorem 5.13.

[MR3084750] The $N[a_{1}^{\prime}\;a_{2}^{\prime}\;...\;a_{n}^{\prime}]$
shadow has $\prod^{n}_{i=1}{{a_{i}}\choose{\frac{a_{i}-c_{i}}{2}}}$ resultants
of the form $N[c_{1}\;c_{2}\;...\;c_{n}]$.

This implicitly details all resultants of a rational link, and guarantees that
they will all be rational. It would be nice to be more specific about what
particular rational links occur, but this rapidly becomes a grueling
accounting exercise. For example, using Lemma 5.10, one can show the
$N[a_{1}^{\prime}\;a_{2}^{\prime}\;a_{3}^{\prime}]$ shadow produces

$\displaystyle
2^{a_{2}+1}{{a_{1}}\choose{\frac{a_{1}}{2}}}{{a_{3}}\choose{\frac{a_{3}-1}{2}}}$
$\displaystyle+2{{a_{1}}\choose{\frac{a_{1}-2}{2}}}{{a_{2}}\choose{\frac{a_{2}-2}{2}}}{{a_{3}}\choose{\frac{a_{3}-1}{2}}}$
$\displaystyle+2{{a_{2}}\choose{\frac{a_{2}}{2}}}\sum_{c_{1}>0}{{a_{1}}\choose{\frac{a_{1}-c_{1}}{2}}}\left[{{a_{3}}\choose{\frac{a_{3}-c_{1}+1}{2}}}+{{a_{3}}\choose{\frac{a_{3}-c_{1}-1}{2}}}\right]$

unknots if $a_{3}$ is odd but $a_{1}$ and $a_{2}$ and even. However, if
$a_{2}$ and $a_{3}$ are odd and only $a_{1}$ is even, there are instead

$\displaystyle
2^{a_{2}+1}{{a_{1}}\choose{\frac{a_{1}}{2}}}{{a_{3}}\choose{\frac{a_{3}-1}{2}}}$
$\displaystyle+2{{a_{1}}\choose{\frac{a_{1}-2}{2}}}{{a_{2}}\choose{\frac{a_{2}-1}{2}}}{{a_{3}}\choose{\frac{a_{3}-3}{2}}}$
$\displaystyle+2{{a_{2}}\choose{\frac{a_{2}-1}{2}}}{{a_{3}}\choose{\frac{a_{3}-1}{2}}}\left[2^{a_{1}}-{{a_{1}}\choose{\frac{a_{1}}{2}}}\right]$

unknots. For length three rational links alone, there still remains the
nontrivial resultants for these cases, the remaining case where all three
$a_{i}$ are odd, and the three cases where the rational tangle is a true link.

##  6\. Acknowledgements

Thank you to Dr. Emily Peters for her helpful suggestions on improving this
text.

## References

##  7\. Appendix I: Rational Link Fertility Numbers

Table 2. Fertility numbers of rational knots through 10 crossings. K | CF | F | K | CF | F | K | CF | F
---|---|---|---|---|---|---|---|---
$3_{1}$ | 3 | 3 | $9_{8}$ | 2 4 1 2 | 6 | $10_{17}$ | 4 1 1 4 | 5
$4_{1}$ | 2 2 | 4 | $9_{9}$ | 4 2 3 | 5 | $10_{18}$ | 4 1 1 2 2 | 6
$5_{1}$ | 5 | 3 | $9_{10}$ | 3 3 3 | 5 | $10_{19}$ | 4 1 1 1 3 | 6
$5_{2}$ | 3 2 | 4 | $9_{11}$ | 4 1 2 2 | 6 | $10_{20}$ | 3 5 2 | 6
$6_{1}$ | 4 2 | 4 | $9_{12}$ | 4 2 1 2 | 6 | $10_{21}$ | 3 4 1 2 | 5
$6_{2}$ | 3 1 2 | 5 | $9_{13}$ | 3 2 1 3 | 6 | $10_{22}$ | 3 3 1 3 | 6
$6_{3}$ | 2 1 1 2 | 5 | $9_{14}$ | 4 1 1 1 2 | 5 | $10_{23}$ | 3 3 1 1 2 | 7
$7_{1}$ | 7 | 3 | $9_{15}$ | 2 3 2 2 | 6 | $10_{24}$ | 3 2 3 2 | 6
$7_{2}$ | 5 2 | 4 | $9_{17}$ | 2 1 3 1 2 | 5 | $10_{25}$ | 3 2 2 1 2 | 7
$7_{3}$ | 4 3 | 5 | $9_{18}$ | 3 2 2 2 | 6 | $10_{26}$ | 3 2 1 1 3 | 7
$7_{4}$ | 3 1 3 | 4 | $9_{19}$ | 2 3 1 1 2 | 6 | $10_{27}$ | 3 2 1 1 1 2 | 6
$7_{5}$ | 3 2 2 | 5 | $9_{20}$ | 3 1 2 1 2 | 6 | $10_{28}$ | 3 1 3 1 2 | 6
$7_{6}$ | 2 2 1 2 | 6 | $9_{21}$ | 3 1 1 2 2 | 6 | $10_{29}$ | 3 1 2 2 2 | 6
$7_{7}$ | 2 1 1 1 2 | 5 | $9_{23}$ | 2 2 1 2 2 | 6 | $10_{30}$ | 3 1 2 1 1 2 | 6
$8_{1}$ | 6 2 | 4 | $9_{26}$ | 3 1 1 1 1 2 | 6 | $10_{31}$ | 3 1 1 3 2 | 6
$8_{2}$ | 5 1 2 | 5 | $9_{27}$ | 2 1 2 1 1 2 | 7 | $10_{32}$ | 3 1 1 1 2 2 | 6
$8_{3}$ | 4 4 | 5 | $9_{31}$ | 2 1 1 1 1 1 2 | 6 | $10_{33}$ | 3 1 1 1 1 3 | 6
$8_{4}$ | 4 1 3 | 5 | $10_{1}$ | 8 2 | 4 | $10_{34}$ | 2 5 1 2 | 6
$8_{6}$ | 3 3 2 | 6 | $10_{2}$ | 7 1 2 | 5 | $10_{35}$ | 2 4 2 2 | 6
$8_{7}$ | 4 1 1 2 | 5 | $10_{3}$ | 6 4 | 4 | $10_{36}$ | 2 4 1 1 2 | 6
$8_{8}$ | 2 3 1 2 | 6 | $10_{4}$ | 6 1 3 | 5 | $10_{37}$ | 2 3 3 2 | 6
$8_{9}$ | 3 1 13 | 5 | $10_{5}$ | 6 1 1 2 | 5 | $10_{38}$ | 2 3 1 2 2 | 6
$8_{11}$ | 3 2 1 2 | 5 | $10_{6}$ | 5 3 2 | 6 | $10_{39}$ | 2 2 3 1 2 | 7
$8_{12}$ | 2 2 2 2 | 6 | $10_{7}$ | 5 2 1 2 | 5 | $10_{40}$ | 2 2 2 1 1 2 | 7
$8_{13}$ | 3 1 1 1 2 | 6 | $10_{8}$ | 5 1 4. | 5 | $10_{41}$ | 2 2 1 2 1 2 | 7
$8_{14}$ | 2 2 1 1 2 | 6 | $10_{9}$ | 5 1 1 3 | 5 | $10_{42}$ | 2 2 1 1 1 1 2 | 7
$9_{1}$ | 9 | 3 | $10_{10}$ | 5 1 1 1 2 | 6 | $10_{43}$ | 2 1 2 2 1 2 | 6
$9_{2}$ | 7 2 | 4 | $10_{11}$ | 4 3 3 | 6 | $10_{44}$ | 2 1 2 1 1 1 2 | 7
$9_{3}$ | 6 3 | 5 | $10_{12}$ | 4 3 1 2 | 6 | $10_{45}$ | 2 1 1 1 1 1 1 2 | 7
$9_{4}$ | 5 4 | 4 | $10_{13}$ | 4 2 2 2 | 6 |  |  |
$9_{5}$ | 5 1 3 | 4 | $10_{14}$ | 4 2 1 1 2 | 6 |  |  |
$9_{6}$ | 5 2 2 | 5 | $10_{15}$ | 4 1 3 2 | 6 |  |  |
$9_{7}$ | 3 4 2 | 6 | $10_{16}$ | 4 1 2 3 | 5 |  |  |
Table 3. Fertility number of rational 2-component links through 9 crossings. K | CF | F | K | CF | F
---|---|---|---|---|---
$2^{2}_{1}$ | 2 | 2 | $8^{2}_{7}$ | 2 1 2 1 2 | 6
$4^{2}_{1}$ | 4 | 4 | $8^{2}_{8}$ | 2 1 1 1 1 2 | 6
$5^{2}_{1}$ | 2 1 2 | 3 | $9^{2}_{1}$ | 6 1 2 | 5
$6^{2}_{1}$ | 6 | 4 | $9^{2}_{2}$ | 5 1 1 2 | 6
$6^{2}_{2}$ | 3 3 | 5 | $9^{2}_{3}$ | 4 3 2 | 6
$6^{2}_{3}$ | 2 2 2 | 5 | $9^{2}_{4}$ | 4 1 4 | 5
$7^{2}_{1}$ | 4 1 2 | 5 | $9^{2}_{5}$ | 4 1 1 3 | 6
$7^{2}_{2}$ | 3 1 1 2 | 6 | $9^{2}_{6}$ | 3 3 1 2 | 6
$7^{2}_{3}$ | 2 3 2 | 5 | $9^{2}_{7}$ | 3 2 2 1 2 | 6
$8^{2}_{1}$ | 8 | 4 | $9^{2}_{8}$ | 3 1 3 2 | 6
$8^{2}_{2}$ | 5 3 | 5 | $9^{2}_{9}$ | 3 1 1 1 3 | 6
$8^{2}_{3}$ | 4 2 2 | 6 | $9^{2}_{10}$ | 2 5 2 | 5
$8^{2}_{4}$ | 3 2 3 | 6 | $9^{2}_{11}$ | 2 2 2 1 2 | 6
$8^{2}_{5}$ | 3 1 2 2 | 6 | $9^{2}_{12}$ | 2 2 1 1 1 2 | 6
$8^{2}_{6}$ | 2 4 2 | 5 |  |  |
Table 4. Fertility numbers of rational 2-component links with 10 crossings. Ordering follows [MR0258014]. CF | F | CF | F
---|---|---|---
10 | 2 | 3 2 1 2 2 | 6
7 3 | 5 | 3 1 4 2 | 6
6 2 2 | 6 | 3 1 2 1 3 | 6
5 5 | 5 | 3 1 1 2 1 2 | 6
5 2 3 | 6 | 3 1 1 1 1 1 2 | 6
5 1 2 2 | 6 | 2 6 2 | 5
4 4 2 | 6 | 2 3 2 1 2 | 6
4 2 4 | 6 | 2 3 1 1 1 2 | 6
4 2 1 3 | 6 | 2 2 2 2 2 | 6
4 1 2 1 2 | 6 | 2 2 1 1 2 2 | 6
4 1 1 1 1 2 | 6 | 2 1 4 1 2 | 6
3 4 3 | 6 | 2 1 3 1 1 2 | 6
3 3 2 2 | 6 | 2 1 1 2 1 1 2 | 6
3 2 2 3 | 6 |  |

In the following tables, a number $m$ is fixed, but a number $n^{*}$ can take
any value $n+2k$, $k\in\mathbb{N}$ and the knot will retain its fertility
number.

F(L) | Length 1 | Length 2 | Length 3
---|---|---|---
2 | 2 |  |
4 | $4^{*}$ |  |
5 |  |  $3^{*}$ $3^{*}$ |  |  $2$ $3^{*}$ $2$
---
$4^{*}$ $1$ $2^{*}$
6 |  |  |  $4^{*}$ $3^{*}$ $2^{*}$
Table 5. Length 1, 2, and 3 rational link fertility number. Links of longer length have fertility number 6. F(L) | Length 1 | Length 2
---|---|---
3 | $3^{*}$ |
4 |  |  |  $2^{*}$ $2^{*}$
---
$2$ $3^{*}$
5 |  |  $4^{*}$ $3^{*}$
Table 6. Length 1 and 2 rational knot fertility number. F(K) | 3 1 2 | 3 1 3 | 3 2 2
---|---|---|---
4 |  |  $3^{*}$ 1 $3^{*}$ |
5 |  $3^{*}$ 1 $2^{*}$ |  $3^{*}$ $3^{*}$ $3^{*}$ |  $3^{*}$ 2 $2^{*}$
6 |  $3^{*}$ $3^{*}$ $2^{*}$ |  |  $3^{*}$ $2^{*}$ $2^{*}$
Table 7. Length 3 rational knot fertility number. F(K) | 2 1 1 2 | 2 2 1 2 | 3 1 1 3 | 3 2 1 2 | 2 2 2 2 | 3 2 1 2 | 3 2 2 2
---|---|---|---|---|---|---|---
5 |  $2^{*}$ 1 1 $2^{*}$ |  |  $3^{*}$ 1 1 $3^{*}$ |  $3^{*}$ $2^{*}$ 1 $2^{*}$ |  |  |
6 |  |  $2^{*}$ $3^{*}$ 1 $2^{*}$
---
$2^{*}$ $3^{*}$ $3^{*}$ $2^{*}$
|  $2^{*}$ $2^{*}$ $1^{*}$ 2
---
$2^{*}$ $2^{*}$ 1 $2^{*}$
2 $2^{*}$ $1^{*}$ $2^{*}$
|  $3^{*}$ $3^{*}$ 1 $3^{*}$
---
$3^{*}$ $3^{*}$ $3^{*}$ $3^{*}$
$3^{*}$ 2 $3^{*}$ $2^{*}$ |  $2^{*}$ $2^{*}$ $2^{*}$ $2^{*}$ |  $3^{*}$ $2^{*}$ 1 $2^{*}$ |  $3^{*}$ 2 $2^{*}$ 2
7 |  |  $4^{*}$ $2^{*}$ $3^{*}$ $4^{*}$ |  |  $3^{*}$ $4^{*}$ $3^{*}$ $2^{*}$ |  |  $3^{*}$ $2^{*}$ $3^{*}$ $2^{*}$ |  |  $3^{*}$ $4^{*}$ $2^{*}$ 2
---
$3^{*}$ 2 $2^{*}$ $4^{*}$
$3^{*}$ $4^{*}$ $2^{*}$ $4^{*}$
Table 8. Fertility number for non-locally-fertile length 4 rational knots. F(K) | 2 1 1 1 2 | 3 1 1 1 2 | 2 2 1 1 2 | 2 2 1 2 2 | 3 1 2 1 2 | 3 1 1 2 2
---|---|---|---|---|---|---
5 |  $2^{*}$ 1 $1^{*}$ 1 $2^{*}$ |  |  |  |  |
6 |  $2^{*}$ $3^{*}$ 1 $1^{*}$ $2^{*}$ |  |  $3^{*}$ 1 $1^{*}$ $1^{*}$ 2
---
$3^{*}$ 1 $1^{*}$ 1 $2^{*}$
| 2 $2^{*}$ 1 $1^{*}$ $2^{*}$
---
$2^{*}$ $2^{*}$ 1 $1^{*}$ 2
$2^{*}$ $2^{*}$ 1 $2^{*}$ $2^{*}$ |  |  $3^{*}$ $1^{*}$ $2^{*}$ 1 $2^{*}$
---
$3^{*}$ 1 $2^{*}$ $1^{*}$ $2^{*}$
$3^{*}$ 1 $1^{*}$ $2^{*}$ $2^{*}$
7 |  $2^{*}$ $3^{*}$ $3^{*}$ $1^{*}$ $2^{*}$ |  |  $3^{*}$ 1 1 $3^{*}$ $4^{*}$
---
$3^{*}$ $3^{*}$ $1^{*}$ $1^{*}$ $2^{*}$
|  $4^{*}$ 2 $1^{*}$ $1^{*}$ $4^{*}$
---
$2^{*}$ $2^{*}$ $3^{*}$ $1^{*}$ $2^{*}$
$2^{*}$ $2^{*}$ $3^{*}$ $2^{*}$ $2^{*}$ |  $3^{*}$ $3^{*}$ $2^{*}$ $3^{*}$ $2^{*}$ |  $3^{*}$ $3^{*}$ $1^{*}$ $2^{*}$ $2^{*}$
Table 9. Fertility number for non-locally-fertile length 5 rational knots with 9 or fewer crossings. F(K) | 3 1 2 2 2 | 3 2 2 1 3 | 3 2 1 2 3
---|---|---|---
6 |  $3^{*}$ 1 $2^{*}$ $2^{*}$ 2 |  $3^{*}$ 2 $2^{*}$ 1 $3^{*}$ |  $3^{*}$ $2^{*}$ 1 $2^{*}$ $3^{*}$
7 |  |  $3^{*}$ 1 $2^{*}$ $2^{*}$ $4^{*}$
---
$3^{*}$ $3^{*}$ $2^{*}$ $2^{*}$ $2^{*}$
|  $3^{*}$ $4^{*}$ $2^{*}$ $1^{*}$ $3^{*}$
---
$3^{*}$ $2^{*}$ $2^{*}$ $3^{*}$ $3^{*}$
$3^{*}$ $2^{*}$ $3^{*}$ $2^{*}$ $3^{*}$
Table 10. Fertility number for non-locally fertile length 5 rational knots with 10 or more crossings. F(K) | 3 1 1 1 1 2 | 3 2 1 1 1 2 | 3 1 2 1 1 2 | 3 1 1 1 2 2
---|---|---|---|---
6 |  $3^{*}$ 1 $1^{*}$ 1 $1^{*}$ 2 |  $3^{*}$ $2^{*}$ 1 $1^{*}$ 1 $2^{*}$ |  $3^{*}$ 1 $2^{*}$ 1 $1^{*}$ $2^{*}$ |  $3^{*}$ 1 $3^{*}$ 1 $2^{*}$ $2^{*}$
7 |  |  $3^{*}$ $3^{*}$ $1^{*}$ 1 $1^{*}$ 2
---
$3^{*}$ 1 $1^{*}$ $3^{*}$ $1^{*}$ 2
$3^{*}$ 1 $1^{*}$ 1 $1^{*}$ $4^{*}$
$3^{*}$ $3^{*}$ $1^{*}$ $3^{*}$ $1^{*}$ $4^{*}$
|  $3^{*}$ $2^{*}$ $3^{*}$ $1^{*}$ 1 $2^{*}$
---
$3^{*}$ $2^{*}$ 1 $1^{*}$ $3^{*}$ $2^{*}$
$3^{*}$ $2^{*}$ $3^{*}$ $1^{*}$ $3^{*}$ $4^{*}$
|  $3^{*}$ $3^{*}$ $2^{*}$ 1 $1^{*}$ $2^{*}$
---
$3^{*}$ 1 $2^{*}$ $3^{*}$ $1^{*}$ $2^{*}$
$3^{*}$ $3^{*}$ $2^{*}$ $3^{*}$ $1^{*}$ $2^{*}$
|  $3^{*}$ $3^{*}$ $3^{*}$ 1 $2^{*}$ $2^{*}$
---
$3^{*}$ 1 $3^{*}$ $3^{*}$ $2^{*}$ $2^{*}$
$3^{*}$ $3^{*}$ $3^{*}$ $3^{*}$ $2^{*}$ $2^{*}$
Table 11. Fertility number for some non-locally-fertile length 6 rational knots. F(K) | 3 1 1 1 1 3 | 2 1 2 2 1 2 | 3 1 2 1 2 2 | 3 1 2 2 1 3
---|---|---|---|---
6 |  $3^{*}$ 1 $1^{*}$ $1^{*}$ 1 $3^{*}$ |  $2^{*}$ 1 $2^{*}$ $2^{*}$ 1 $2^{*}$ |  $3^{*}$ 1 $2^{*}$ 1 $2^{*}$ 2 |  $3^{*}$ 1 $2^{*}$ $2^{*}$ 1 $3^{*}$
7 |  |  $3^{*}$ $3^{*}$ $1^{*}$ $1^{*}$ 1 $3^{*}$
---
$3^{*}$ $3^{*}$ $1^{*}$ $1^{*}$ $3^{*}$ $3^{*}$
|  $2^{*}$ $3^{*}$ $2^{*}$ $2^{*}$ 1 $2^{*}$
---
$2^{*}$ $3^{*}$ $2^{*}$ $2^{*}$ $3^{*}$ $2^{*}$
|  $3^{*}$ $3^{*}$ $2^{*}$ 1 $2^{*}$ 2
---
$3^{*}$ 1 $2^{*}$ $3^{*}$ $2^{*}$ 2
$3^{*}$ 1 $2^{*}$ 1 $2^{*}$ $4^{*}$
$3^{*}$ $3^{*}$ $2^{*}$ $3^{*}$ $2^{*}$ $4^{*}$
|  $3^{*}$ $3^{*}$ $2^{*}$ $2^{*}$ 1 $3^{*}$
---
$3^{*}$ $3^{*}$ $2^{*}$ $2^{*}$ $3^{*}$ $3^{*}$
Table 12. Fertility number for the remaining non-locally-fertile length 6 rational knots. F(K) | 2 1 1 1 1 1 2 | 3 1 1 1 1 1 3 | 3 1 2 1 2 1 3
---|---|---|---
6 |  $2^{*}$ 1 $1^{*}$ 1 $1^{*}$ 1 $2^{*}$ |  $3^{*}$ 1 $1^{*}$ $3^{*}$ $1^{*}$ 1 $3^{*}$ |  $3^{*}$ 1 $2^{*}$ 1 $2^{*}$ 1 $3^{*}$
7 |  |  $2^{*}$ $1^{*}$ $1^{*}$ $3^{*}$ $1^{*}$ $1^{*}$ $2^{*}$
---
$2^{*}$ $3^{*}$ $1^{*}$ $1^{*}$ $1^{*}$ $1^{*}$ $2^{*}$
|  $3^{*}$ $1^{*}$ $1^{*}$ $3^{*}$ $1^{*}$ $1^{*}$ $3^{*}$
---
$3^{*}$ $3^{*}$ $1^{*}$ $1^{*}$ $1^{*}$ $1^{*}$ $3^{*}$
|  $3^{*}$ $1^{*}$ $2^{*}$ $3^{*}$ $2^{*}$ $1^{*}$ $3^{*}$
---
$3^{*}$ $3^{*}$ $2^{*}$ $1^{*}$ $2^{*}$ $1^{*}$ $3^{*}$
Table 13. Fertility number for non-locally-fertile length 7 rational knots.


PASA 2024

# First Search for Low-Frequency CH with a Square Kilometre Array Precursor
Telescope

Tremblay  C.D.1 Green  J.A. 1 Mader  S.L.2 Phillips  C.J.3 and Whiting  M.3
1CSIRO Astronomy and Space Science, PO Box 1130, Bentley WA 6102, Australia
2CSIRO Astronomy and Space Science, Parkes Radio Telescope, NSW, Australia
3CSIRO Astronomy and Space Science, PO Box 76, Epping NSW 1710, Australia

###### Abstract

The diatomic free radical methylidyne (CH) is an important tracer of the
interstellar medium and the study of it was critical to our earliest
understanding of star formation. Although it is detectable across the
electromagnetic spectrum, observations at radio frequencies allow for a study
of the kinematics of the diffuse and dense gas in regions of new star
formation. There is only two published (single-dish) detections of the low-
frequency hyperfine transitions between 700 and 725 MHz, despite the precise
frequencies being known. These low-frequency transitions are of particular
interest as they are shown in laboratory experiments to be more sensitive to
magnetic fields than their high-frequency counterparts (with more pronounced
Zeeman splitting). In this work we take advantage of the radio quiet
environment and increased resolution of the Australian Square Kilometre Array
Pathfinder (ASKAP) over previous searches to make a pilot interferometric
search for CH at 724.7883 MHz (the strongest of the hyperfine transitions) in
RCW 38. We found the band is clean of radio frequency interference, but we did
not detect the signal from this transition to a five sigma sensitivity limit
of 0.09 Jy, which corresponds to a total column density upper limit of
1.9$\times$1018 cm-2 for emission and 1.3$\times$1014 cm-2 for absorption with
an optical depth limit of 0.95. Achieved within 5 hrs of integration, this
column density sensitivity should have been adequate to detect the emission or
absorption in RCW 38, if it had similar properties to the only previous
reported detections in W51.

###### doi:

10.1017/pas.2024.xxx

###### keywords:

astrochemistry $-$ molecular data $-$ radio lines: stars: surveys $-$
ISM:Hii Regions $-$ ISM:molecules

##  1 INTRODUCTION

Free radicals are molecules that are uncharged, highly reactive and short
lived. In the early 1920s emission bands due to CH, OH, and CN radicals were
first discovered in the laboratory, then in 1937 the diatomic free radical of
methylidyne, CH, was discovered (Swings & Rosenfeld, 1937; Swings, 1942) as
one of the first gas phase molecules. In 1941, a group of Belgian astronomers
worked with laboratory chemists to identify CH in the optical spectra of
comets (Cunningham 1940c). This sparked an interest by laboratories and
astronomers alike in CH (Herzberg, 1971) and since the 1940’s, scientists have
suggested that there are two forms of CH found around interstellar objects;
the molecules that form on grains and are observed in ultra-violet and
infrared through the vibrational bands (although never detected); and the
molecules that form in the gas phase and are observed in the ultra-violet to
the radio parts of the electromagnetic spectrum.

The production of CH is expected to form in the gas phase through a radiative
association reaction of C+ \+ H $\rightarrow$ CH + $h\nu$ (Federman & Willson,
1982). Danks et al. (1984) suggested a more thorough description of the
formation of diffuse interstellar CH by having H2 playing a major role: by
following the same radiative association reaction with a ion-molecule reaction
CH${}_{2}^{+}$ \+ H2 $\rightarrow$ CH${}_{3}^{+}$ \+ H with a 67 per cent
efficiency at creating a CH molecule. It is now well accepted that CH
originates from the cold neutral medium of interstellar gas via the slow
radiative association of C+ with H2 forming CH${}^{+}_{2}$, which converts to
CH via dissociative recombination (Godard et al., 2014; Gerin et al., 2016).
The rate constants for the radio frequencies are not well understood
(Dagdigian, 2018). The chemical models of this reaction can produce the column
densities of CH as observed in the gas phase (Crawford, 1989; Gerin et al.,
2016) and found to be highly dependant on the presence of free carbon
(Suutarinen et al., 2011). In work published by Dagdigian (2018), they found
that when investigating collision rates with atomic and molecular hydrogen the
column densities from historical observations do not fit the model if local
thermodynamic equilibrium (LTE) conditions are assumed, except in regions
where the gas is dense. They also found that the calculations using only
excitation by H2 do not reproduce the excitation pattern of the lamda doubling
transitions near 3.3 GHz.

There have been a number of publications regarding CH; its formation routes
(e.g. Millar, 2015, references therein), its role as a proxy for H2 (e.g.
Sheffer et al. 2008; Wiesemeyer et al. 2018; Dailey et al. 2020), and its
ability to trace the dynamics of diffuse and dense gas (e.g. Federman &
Willson 1982; Gerin et al. 2016). Since CH is found in Hi regions, Hii
regions, and dust clouds, there is reason to believe the molecule is
widespread within our Galaxy (Rydbeck et al., 1973) and has been widely
observed in diffuse interstellar gas clouds at 3 GHz (Chastain et al., 2010).

A recent resurgence of the study of CH has started with laboratory work by
Truppe et al. (2013), completing a focused study on the 700 and 3300 MHz radio
transitions and Wiesemeyer et al. (2018) completing a study of the N$=$2
ground state line in the infrared. Truppe et al. (2013) combined laboratory
experiments with past astronomical observations to determine the precise
frequencies of emission, the interaction with the Zeeman effect and the use of
CH to determine fundamental universal constants. The interaction with the
Zeeman Effect (the splitting of transitions due to the presence of a magnetic
field) is of particular interest as it has been found that these transitions
have a splitting factor comparable to that of mainline OH, 0.93 Hz $\mu$G-1,
and thus has the potential to allow for this molecule to trace the impact of
magnetic fields on the chemical evolution of the interstellar medium (see
Crutcher & Kemball 2019 for a full review on the importance of Zeeman Effect
in star formation). Wiesemeyer et al. (2018) used observations at 2 THz to
compare CH with traditional proxies of molecular hydrogen in cold gas and
determined that CH provided more information over traditional models,
including CO.

Based on its expected abundance and appearance in diffuse and dense gas,
analysis of CH at low radio frequencies is an ideal target for the Square
Kilometre Array (SKA) to study the role magnetic fields play in the early
stages of star formation. The Australian Square Kilometre Array Pathfinder
(ASKAP), with operational frequencies of 700–1800 MHz and stationed in a
protected radio quiet environment, makes for an ideal precursor instrument to
determine the feasibility of these low frequency studies. This paper presents
the first interferometric search using early science observations with ASKAP.

##  2 Previous Radio Observations of CH

Most of the detections and study of the CH radical are in the ultraviolet
(e.g. Herzberg 1971), optical (e.g. Danks et al. 1984) and infrared (e.g.
Wiesemeyer et al. 2018) parts of the electromagnetic spectrum. The first radio
detection of interstellar CH (in the 3.3 GHz transitions, once observational
wavelengths were known) was toward seventeen Hi and Hii regions (almost all
with previous OH detections) by Rydbeck et al. (1973) and Federman (1982).
Rydbeck et al. (1973) found all detections were in emission and toward OH
maser sites, with the CH transition showing evidence of weak masing toward
some sources suggesting CH can be detected where the lower rotational lines of
CO could not.

A search for CH (3.3GHz) along the Galactic plane by Johansson (1979) found
the molecular distribution more extended than CO(1-0) surveys of the time.
Additional studies of the CH 3 GHz distribution towards small dark molecular
(Federman, 1982; Mattila, 1986) and high latitude translucent (Magnani &
Onello, 1995; Chastain et al., 2010; Cotten & Magnani, 2013) clouds have shown
the CH molecule to be an excellent tracer for H2 in low-density (n $\leq$ 103
cm3; AV $\leq$ 5 mag.) gas, with a linear relationship existing between N(CH)
and N(H2) for these regions. Dailey et al. (2020) studied the possibility of
using the 3 GHz transitions as a proxy of the molecular hydrogen gas and found
that the excitation temperatures vary significantly depending on the line-of-
sight.

Whilst the detection of the 3335 MHz transition of CH is readily obtained
across a wide range of molecular environments, there are very few published
detection of CH at the lower transitions of $\sim$ 700 MHz. Ziurys & Turner
(1985) observed the F$=$2–2 and F$=$1–1, N$=$1, J$=$3/2 transitions between
700 and 725 MHz with the Arecibo 300m telescope and NRAO Green Bank 300 foot
(91m) telescope followed by Turner (1988). Ziurys & Turner (1985) detected two
lines in absorption toward W51 and the F$=$2–2, N$=$1, J$=$3/2 transition was
reported as detected toward W3, W43, and Ori B in absorption with Arecibo. The
lines detected toward W51A showed a matching profile to the 3.3 GHz
observations by Genzel et al. (1979). Turner (1988) detected all four
hyperfine transitions toward W51A. Ziurys & Turner (1985) found large
fractional abundances of CH compared to expectations from chemical modelling
of the W51 region. Given the lack of observations at the lower frequencies,
more data is required to ascertain the extent of the W51 results across other
regions.

##  3 New Radio Observations of CH

The CH molecule has four hyperfine transitions in the range $\sim$700–725 MHz
representing the N=1 J=3/2 transition. The 724.788 MHz transition (F$=$1–1) is
shown in Figure 1 along with the higher frequency N=1 J=1/2 transition at 3349
MHz.

Figure 1: Relevant energy levels in CH from Figure 1 of Truppe et al. (2013).
Approximate frequencies of separation are given in MHz, with the 724.788315
MHz transition highlighted in green. The sensitivity coefficients (K) for the
two $\lambda$-doublet transitions are shown.

The transition at 701.6777 MHz was found to be the strongest by Ziurys &
Turner (1985) followed by the transition at 724.7883 MHz. The intervening
transitions at 722.4876 and 703.9783 MHz were weakly detected by Turner
(1988). These low-frequency hyperfine transitions have been found to be both
weaker (1/25th the sensitivity) and less prevalent (5% of molecules) than the
more well studied $\sim$ 3.3 GHz J=1/2 transitions (Truppe et al., 2013).
Observations with the Effelsberg 100 m telescope of the J=1/2 transitions by
Genzel et al. (1979) detected the lines in Hii regions with 2–3 hours of
integrated time, thus implying a significant integration time is required for
the J=3/2 transitions (consistent with the observations by Ziurys & Turner
1985 where they specify observing across several days). New studies of these
transitions therefore need to exploit the most sensitive new telescopes, such
as the Australian Square Kilometre Array Pathfinder.

###  3.1 Australian Square Kilometre Array Pathfinder (ASKAP)

Table 1: ASKAP Observing Parameters

Parameter | Value
---|---
Central frequency | 724 MHz
Total bandwidth | 18 MHz
Channel Resolution | 1 kHz
Synthesised beam FWHM |  30′′
Primary beam FWHM⋆ |  30 deg2
Phase centre of image (J2000) | 09h59m05s
| –47d30m39s
Time on source | 5 hours
Reference Frame | Local Standard

$\star$The primary beam size represents the size of a full processed field
with 36 PAF beams. A single PAF beam has a primary beam of 1 deg2.

The Australian Square Kilometre Array Pathfinder (ASKAP; Johnston et al. 2008,
Hotan et al. submitted) is located at the Murchison Radio-astronomy
Observatory (MRO) in Western Australia. Observations were taken on the 23
February 2019 with a central frequency of 724 MHz and 18 MHz of total
bandwidth, with a frequency resolution of 1 kHz. This provides a 0.25 km s-1
velocity resolution. Each of the antennas are equipped with a Phased Array
Feed (PAF; Schinckel & Bock 2016) giving it a field of view, over 36 beams, of
$\sim$ 30 deg2.

The observations were taken as part of the commissioning of the low-band of
ASKAP (McConnell et al., 2016) and 35111Antenna 11 was offline for these
observations. antennas were online with a minimum baseline of 22 m and maximum
baseline of 6 km. The observations were taken in full polarisation and
observations of PKS B1934–638 were performed immediately adjacent in time to
the target field for the purposes of bandpass calibration. Each calibration
observation contained one calibrator scan of duration five minutes at the
centre of each PAF beam before mosiacing to the full instrument field-of-view
(FOV).

All data were processed using ASKAPsoft (Guzman et al., 2019) for calibration
and imaging. This included flagging, bandpass calibration, phase self-
calibration, imaging, continuum subtraction, spectral cube imaging and
correction of the data to the local standard of rest reference frame. During
processing antennas 1, 4 and 10, from the inner core, were flagged222
Information on the antenna layout and configuration is found at
https://www.atnf.csiro.au/projects/askap/config.html.. This combined with
antenna 11 being offline, reduced our surface brightness sensitivity by
$\sim$40 per cent and thus our sensitivity to diffuse emission. Nine PAF beams
were processed individually and then mosaiced together to form an image of the
region used in this paper. However, the Hii region of interest fit within a
single central beam.

###  3.2 RCW 38: Hii Region

We completed a targeted survey with ASKAP toward RCW 38 to look for the low-
frequency transition of CH at 724.788 MHz with five hours of integration time
and a pointing centre of RA 09:59:05 Dec –47:30:39 (J2000). The molecule CH
has been detected in a number of different astrophysical environments but the
only detection of a low-frequency transition published was toward the OH maser
site of W51A (Ziurys & Turner, 1985). As RCW 38 is also an Hii region thought
to be fuelled by an old supernova explosion and colliding clouds (e.g.Wolk et
al. 2006; Schneider et al. 2010; Sano et al. 2018), we completed our first
attempt to search for the molecule with ASKAP toward this region.

RCW 38, an Hii region located in the Vela Molecular Cloud Complex, is
1.7$\pm$0.9 kpc away (Wolk et al., 2006), and is nearly as dense as the Orion
Kleinmann-Low Nebula with young stellar objects (Schneider et al., 2010). See
Wolk et al. (2008) for a review of the region and the historical observations.
Torii et al. (2019)recently completed a high-resolution survey of CO and its
isotopologues with the Atacama Large Millimeter/submillimeter Array. They
detected CO in the lobes of RCW 38, often referred to as IRS1 and IRS2 (from
Wynn-Williams et al., 1972), and the bridge of gas that expands between the
two with two different radial velocities of $\sim$ 2 and $\sim$ 12 km s-1.
Bourke et al. (2001) detected the 1665 and 1667 OH lines in absorption with
detectable Zeeman Splitting (with a corresponding magnetic field strength of
38$\pm$3 $\mu$G). The source is often characterised as being two clouds
colliding which fuels the significant star formation with over 1000 objects
detected by X-ray and infrared (e.g. Fukui et al. 2016; Sano et al. 2018). In
Figure 2, the ASKAP radio continuum is shown in contours over an infrared
image from the Wide-field Infrared Survey Explorer (WISE) Survey. The contours
reveal new structure to the IRS1 cloud but is found to be similar to the
structure detected by Bourke et al. (2004) at 1.6 GHz with the Australia
Telescope Compact Array. Greater detail regarding this new structure will be
presented in a future paper.

Figure 2: Left: An image of the eastern lobe (IRS1) of RCW 38, where the cyan
contours from the ASKAP radio continuum emission are overlaid onto an infrared
image of the region from the Wide-field Infrared Survey Explorer (WISE)
survey. The contours show greater detail of the lobes typically denoted as
“IRS1” in the literature (Wynn-Williams et al., 1972). The contours are set at
levels of 3–8 $\sigma$ (0.003 to 0.09 Jy beam-1) and the bottom right-hand
corner shows the size of the synthesised beam. Right: The spectrum averaged
over the region marked with a red box on the image on the left. The rest
frequency of the CH transition (724.7883 MHz) is denoted by a green vertical
line, the grey shading represents noise (one sigma) and the five sigma
sensitivity limit is taken as 0.09 Jy.

##  4 Results and limits for detection

As shown in the spectrum on the right-hand side of Figure 2, we did not detect
the CH transition towards RCW 38 with our ASKAP observations to a five sigma
limit of 0.09 Jy after averaging the signal over the region marked by the red
box in the left-hand image of the eastern lobe of RCW38.

We calculated the upper limit in emission (at five sigma) of the total column
from Equation A1 of Tremblay et al. (2017) at a Tex of 50 K to be
1.9$\times$1018 cm-2. Ziurys & Turner (1985) determined the column density
derived for the lower energy state to be 9.8$\times$1017 cm-2 for CH in W51A,
assuming similar physical conditions. However, Turner (1988) suggests this
calculation may be simplistic and not a good limit on the gas content.

The previous detections of the low-frequency CH lines by Ziurys & Turner
(1985) and Turner (1988) toward W51 were both in absorption. Using the
spectrum in Figure 2, we can make some estimate of the optical depth in
relation to the peak intensity in the continuum. For this we get an optical
depth limit of 0.95. If we use the modified column density Equations 1 and 2
from Turner (1988) we get a five-sigma upper limit on the column density of
1.3$\times$1014 cm-2 which is consistent with historical values in W51 and
other Hii regions (around 1$\times$1014 cm-2 Genzel et al. 1979; Rydbeck et
al. 1973). However, as noted by both Dailey et al. (2020) and Dagdigian
(2018), the values of column density are likely non-physical measurements as
the radio lines do not follow LTE conditions.

We do note that the previously reported low-frequency detections of CH was
achieved using the 300 m Arecibo telescope, which as a single dish with a beam
size of more than 6.3 arc minutes and has greater sensitivity to diffuse (low
surface brightness) emission. In comparison, a single ASKAP’s PAF equipped on
the antennas (at these frequencies) have a primary beam of $\sim$1 degree, and
the resultant imaging has a 30 arc second synthesised beam. In these
observations we lost a large percentage of the shorter baselines, thus making
us less sensitive to the diffuse gas but this will not effect sensitivity to
the dense gas. Danks et al. (1984) noted that the signal from the diffuse gas
was stronger than that from the dense gas and the two components are easily
differentiated through resolved velocity components.

As such, observations with the full ASKAP array of 36 antennas, with all of
the shorter baselines, may be required for future success. Additionally, we
note that Ziurys & Turner (1985) quote a velocity resolution 1.4 km s-1
whereas our observations have a velocity resolution of 0.25 km s-1. With the
Parkes 64m Telescope fitted with the new Ultra-wideband receiver (Hobbs et
al., 2020), we detected CH in the eastern lobe (IRS1) at 3.264 GHz in a 10
minute snapshot image. Therefore, we know the molecule to exist in the region
and more observing time should yield a detection of the low-frequency
transitions.

##  5 Implications for future studies

Based on the current presented observations, the best chance for detection of
the J=3/2 transition at 724.788 MHz is to obtain observations for at least 8
hours of integration with ASKAP (using all 36 antennas). However, this is the
first published search for CH with an interferometer. In the previous single-
dish searches the beam filling factor is assumed to be 1, so interferometry
will help determine if this is true (Wiesemeyer et al., 2018). It will also
help identify if the excitation temperature is indeed negative. This may
impact the length of observation required to detect CH with an interferometer
such as ASKAP.

It is also important to further consider the choice of target: although we now
have a wealth of information on the astrophysical properties of regions from
Galactic plane surveys, which can form the basis of targeted observations, all
of the previously published CH detections are towards OH maser sites. The
combination of OH and CH (with their respective broad Zeeman splitting
factors) can provide a useful diagnostic of magnetic fields in differing
stages of the interstellar medium. Surveys of OH that are planned with ASKAP
(Dickey et al., 2013) along the Galactic plane and long-term surveys like the
The Southern Parkes Large-Area Survey in Hydroxyl (SPLASH; Dawson et al. 2014)
may provide the best targets to be used for understanding CH and its role in
the interstellar medium.

Both of the components (mid- and low-frequency arrays) of the future Square
Kilometre Array (SKA) telescope will be in the southern hemisphere, whereas
most of the searches for CH at radio wavelengths have been towards targets
accessible by the northern hemisphere. As such the precursors like ASKAP can
provide us with useful regions to target later with the SKA.

The CSIRO Parkes 64m Telescope, an SKA technology pathfinder, has a new ultra-
wide band receiver (Hobbs et al., 2020), which covers the frequency range of
704–4000 MHz, allowing simultaneous studies of the 724.788 MHz and $\sim$3.3
GHz CH transitions. Rydbeck et al. (1973) observed CH ground state transitions
at 3.264 GHz in seventeen HII regions. They found a resemblance of CH to the
OH excited state of 2$\Pi$1/2, J=1/2 transitions, which can also be observed
with Parkes.

Searches for other low-frequency molecular transitions associated with CH may
also prove to be beneficial to understanding the interstellar medium. Rydbeck
et al. (1973) found that CH velocities matched those of formaldehyde, an
important tracer of core collapse in early high-mass stars. Both formaldehyde
and deuterated formaldehyde have transitions detectable by the Murchison
Widefield Array (MWA; also a SKA precursor) between 70–300 MHz and deuterated
formaldehyde was found in the Orion complex with the MWA by Tremblay et al.
(2018).

A further consideration for future studies, as alluded to earlier, is that
Truppe et al. (2013) found that the 722–724 MHz transitions of CH to be
particularly sensitive to magnetic fields and the line shows polarisation,
which ASKAP is capable of detecting. As CH is detectable in diffuse and dense
gas, this provides a new way of studying the role magnetic fields play in the
early stages of star formation in comparison to OH which is only detectable in
dense gas.

##  6 Conclusion

Using the Australian Square Kilometre Array Pathfinder telescope, an SKA
precursor in Western Australia, we have completed the first interferometric
search of the low frequency transition of the CH molecule at 724.7883 MHz
towards RCW 38. We established a five sigma upper limit for non-detection of
0.09 Jy, which corresponds to a total column density upper limit of
1.9$\times$1018 cm-2 for emission and 1.3$\times$1014 cm-2 for absorption with
an optical depth limit of 0.95. Based on the column density limit of previous
detection of CH, this should have been adequate for detection. These limits,
combined with a spectrum free from radio frequency interference, show that
detection of the 724 MHz transition should be possible with ASKAP with an
integration time of $\sim$8 hours, but would benefit from having the full 36
antenna array to increase the sensitivity to the diffuse gas.

###### Acknowledgements.

We thank the anonymous referee for their comments and assistance on improving
this manuscript. This scientific work makes use of the Murchison Radio-
astronomy Observatory, operated by CSIRO. We acknowledge the Wajarri Yamaji
people as the traditional owners of the Observatory site. Operation of ASKAP
is funded by the Australian Government with support from the National
Collaborative Research Infrastructure Strategy. Establishment of ASKAP, the
Murchison Radio-astronomy Observatory and the Pawsey Supercomputing Centre are
initiatives of the Australian Government, with support from the Government of
Western Australia and the Science and Industry Endowment Fund. This research
has made use of NASA’s Astrophysics Data System Bibliographic Services. We
acknowledge the Pawsey Supercomputing Centre which is supported by the Western
Australian and Australian Governments. Access to Pawsey Data Storage Services
is governed by a Data Storage and Management Policy (DSMP). This publication
makes use of data products from the Wide-field Infrared Survey Explorer, which
is a joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, funded by the
National Aeronautics and Space Administration.

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