# Strongly positive amphicheiral knots with doubly symmetric diagrams

Christoph Lamm

###### Abstract.

We determine the prime strongly positive amphicheiral knots up to 16 crossings
and show that a large fraction of them admit knot diagrams with a double
symmetry (rotational symmetry for strongly positive amphicheirality and an
additional mirror symmetry for the ribbon property). The remaining knots are
presented as ‘almost doubly symmetric’ diagrams, defined as diagrams with
double symmetry where exactly one symmetric pair of crossings is switched. We
also find that the rosette knot 14a19470, which has period 7, is the first
prime strongly positive amphicheiral knot in the knot tables which is not
slice.

###### Key words and phrases:

strongly positive amphicheiral knots, symmetric unions of knots

###### 2020 Mathematics Subject Classification:

57K10

##  1\. Introduction

In the article on the tabulation of strongly invertible knots [11] we
announced a study on strongly positive amphicheiral knots. The current article
contains the results of this study. In particular, we list all prime strongly
positive amphicheiral knots up to 16 crossings and present diagrams with
additional symmetries for them.

A knot is called strongly positive amphicheiral if it has a diagram which is
mapped to its mirror image by a rotation of $\pi$, preserving the orientation.
An example is the diagram on the left in Figure 1.

Figure 1. A rotationally symmetric diagram (for $14a_{19517}$) and a doubly
symmetric diagram (for $12n_{706}$)

In many respects, strongly positive amphicheiral knots are similar to slice
knots: Their Alexander polynomial is always a square [6] and this is similar
to the $f(t)\cdot f(1/t)$ condition for the Alexander polynomial of slice
knots (compare also with Long’s result on algebraic sliceness [13]). However,
Flapan showed in 1986 that not all prime strongly positive amphicheiral knots
are slice [4].

On the other hand, the family of strongly positive amphicheiral knots is quite
smaller than that of slice knots. For instance, 2-bridge knots cannot be
strongly positive amphicheiral [6].

In the current study, we analyze diagrams of strongly positive amphicheiral
knots which are also symmetric unions, a property which implies that a knot is
slice:

In [10] we found that all prime strongly positive amphicheiral knots with
crossing number up to 12 possess diagrams with additional symmetries. This
type of diagram with additional symmetries is simultaneously a symmetric union
(with respect to a vertical axis) and has strongly invertible partial knots
(with respect to a horizontal axis of rotation in a transvergent diagram).

This combination of two simultaneous symmetries results again in a
rotationally symmetric (strongly positive amphicheiral) diagram with respect
to an axis perpendicular to the diagram plane, see Section 3 for more details.
We call these diagrams ‘doubly symmetric’.

In Figure 2 a general symmetric union with partial knot $8_{21}$ is contrasted
with a symmetric union in a doubly symmetric diagram (with the same partial
knot). The diagram on the right in Figure 1 is another example of a doubly
symmetric diagram.

Figure 2. The partial knot in a general symmetric union diagram (left) and in
a doubly symmetric diagram (right)

Knots with symmetric union diagrams are ribbon knots. Therefore, knots with
doubly symmetric diagrams combine properties of strongly positive amphicheiral
knots and ribbon knots. Our initial question was:

######  Question 1.1.

Which prime strongly positive amphicheiral knots up to 16 crossings have
doubly symmetric diagrams?

Using SnapPy [2], it was possible to determine all prime strongly positive
amphicheiral knots up to 16 crossings. There are 131 of them and they are
listed in Tables 4 and 5.

Although we found that from these at least 106 have doubly symmetric diagrams,
it is clear that not all prime strongly positive amphicheiral knots will have
this property: As mentioned, not all prime strongly positive amphicheiral
knots are ribbon.

Before continuing the summary of results, we dive into a non-mathematical
digression.

###  1.1. Doubly symmetric diagrams, knot gardens and ornaments

I encountered the term ‘knot garden’ in Andrew Ranicki’s slides [15], where on
page 6 a Tudor knot garden is mentioned. The illustration on this slide
reminded me of the symmetry of doubly symmetric knot diagrams and therefore I
include a digression on knot gardens and symmetric ornaments.

Knot gardens appeared in Europe around 1500. Richard M. Bacon (freelance
writer and horticultural consultant) wrote in 1985 in the New York Times about
them:

> The idea of weaving together lines of plants to establish this pattern came
> from monastic enclosures in medieval Europe. Herbs were most certainly used
> because they were cultivated to assure good health. The device was taken
> over later by lay gardeners and made even more decorative. For a time they
> were at the height of fashion and became the focal point for classical and
> formal gardens in this country by 1800. Then they were expanded and
> superceded by the parterre which introduced internal paths to the pattern
> and a broader selection of plant material.

The NYT article contains the illustration in Figure 3. The garden contains
four closed components and several additional elements. A knot garden in a
strict mathematical sense would contain one component only and no other
decorative additions.

Figure 3. A knot garden illustration, New York Times, May 19th, 1985

A related term, occuring in the last sentence, is the ‘parterre’. Parterres
are similar if seen from above but they miss the three-dimensional weaving of
knot gardens. As a reminiscence to horticulture, we will use the term parterre
in this article for knot shadows, i.e. knot diagrams without over- and under-
crossings. The concrete meaning of these terms for historic gardens in England
seems to be quite unclear; see the appendix on knot gardens in [5].

As an example for a knot garden in literature we found the book
‘Hypnerotomachia Poliphili’ (The Dream of Poliphilus), which appeared in 1499
and is attributed to the Italian monk Francesco Colonna. An analysis of the
knot garden theme in this work can be found in [17]. There is even an opera
with the title ‘The knot garden’ by the English composer Michael Tippett
(first performed in 1970).

We mention that the required symmetry for the parterre (gardening and doubly
symmetric diagrams) is the dihedral symmetry of a rectangle ($D_{2}$). The
additional symmetry of a square is not needed, although in many cases a garden
compartment will be a square and might have a $D_{4}$ symmetry.

Figure 4. A lino print of $16n_{872167}$ (left), and an inverted version
(right), showing the diagram $t_{4}(0,0,3\mid 1)$ in template notation

Symmetric designs also occur in celtic and islamic ornaments. As an example
which is close to a celtic knot design, we did a linoleum print of a doubly
symmetric diagram, see Figure 4. Note, though, that in celtic knot design we
almost never find non-alternating diagrams. The use of non-alternating
diagrams cannot be avoided when knot diagrams are constructed as symmetric
unions because for the arcs which traverse the vertical axis two under- or two
over-crossings are necessarily adjacent.

In Versailles we find the following parterre with two symmetry axes
($D_{2}$-symmetry):

Figure 5. Part of the ‘Parterre du Midi’, Versailles

###  1.2. Doubly symmetric diagrams with additional symmetries

Before we give a further outline of the article, we present the following
doubly symmetric knot diagrams with $D_{4}$-symmetry of their parterres. The
second diagram can be rotated by 45° (counter-clockwise) into another doubly
symmetric diagram and therefore has two sets of axes simultaneously. This is
not the case, however, for the other two diagrams shown here. (As an exercise
we invite the reader to check these statements.)

Figure 6. Three doubly symmetric knot diagrams for $14a_{19472}$,
$16n_{991381}$ and $16n_{268004}$ with additional diagonal symmetries. These
diagrams are $t_{4}(0,1,0\mid 1)$, $t_{1}(2,0\mid 2)$ and
$t_{9}(1,1,0\mid-1,1)$.

###  1.3. Plan of the article

We continue the overview of the results and the article’s structure. In the
next section we explain how, with the help of SnapPy, we determined the 131
prime strongly positive amphicheiral knots up to 16 crossings. We found
experimentally that from these at least 106 have doubly symmetric diagrams and
therefore are ribbon knots. For the remaining 25 knots we found ‘almost doubly
symmetric’ diagrams. These are diagrams with double symmetry where exactly one
symmetric pair of crossings is switched. They are no longer symmetric unions,
but still represent strongly positive amphicheiral knots.

######  Question 1.2.

Does every strongly positive amphicheiral knot have a doubly symmetric or
almost doubly symmetric diagram?

The main open problem is to find a knot invariant which enables us to show
that there is a strongly positive amphicheiral knot which is ribbon, or even a
symmetric union, but does not have a doubly symmetric diagram.

##  2\. Strongly positive amphicheiral knots up to 16 crossings

Surprisingly, prime strongly positive amphicheiral knots up to 16 crossings
have not yet been tabulated. Before we describe the SnapPy script which does
the tabulation, we explain in an example how SnapPy is used for this purpose.

Until recently it was unclear whether the knot $12a_{435}$ is strongly
positive amphicheiral (it is an amphicheiral knot and its Alexander polynomial
is a square). For the analysis of this knot we proceed as follows with SnapPy:

`S=Manifold(’12a435’).symmetry_group()`: SnapPy has stored all hyperbolic
knots up to 16 crossings and the hyperbolic structures of their complements.
It either computes the symmetry group (of isometries) on the spot or it is
also already stored. The variable $S$ now contains this symmetry group.

`S.isometries()`: Gives the list of elements of the symmetry group. In this
case there are 8 isometries and the output looks as follows (we omit the
information ‘Extends to link’ and the component mapping `0 -> 0` for each of
the isometries):

     [1 0]  [-1 0]  [1 0]  [-1 0]  [1  0]  [-1  0]  [1  0]  [-1  0]
     [0 1]  [ 0 1]  [0 1]  [ 0 1]  [0 -1]  [ 0 -1]  [0 -1]  [ 0 -1]

Explanation: The matrix refers to the meridian and longitude mapping. The
first of the eight isometries is the identity. The isometries are in this case
numbered from 0 through 7. Their effect on the orientation of $\mathbb{S}^{3}$
and on the knot is as follows:

Isometries 0 and 2 are orientation-preversing isometries preserving the knot’s
orientation.
Isometries 1 and 3 are orientation-reversing isometries preserving the knot’s
orientation.
Isometries 4 and 6 are orientation-reversing isometries reversing the knot’s
orientation.
Isometries 5 and 7 are orientation-preversing isometries reversing the knot’s
orientation.

The knot $12a_{435}$ is positive amphicheiral (isometries 1 and 3) and
negative amphicheiral (isometries 4 and 6). If one of the isometries 1 or 3 is
an involution, the knot is strongly positive amphicheiral.

Multiplication of isometry 1 with itself is done with
`S.multiply_elements(1,1)`: The result is 2, which is not the identity.
Therefore isometry 1 is not an involution. We do the same for isometry 3, and
the result is again 2. Hence the knot $12a_{435}$ is not strongly positive
amphicheiral.

The following script does this for all amphicheiral knots in a specified
range. It is called for alternating and non-alternating knots with crossing
number 10, 12, 14 and 16 (line 3 is modified for that and the range in line 2
has to be adapted as well):

     1 def print_amphicheiral():
     2     for n in range(1,888):
     3         K_id = ’12n’+str(n)
     4         K = Manifold(K_id)
     5         try:
     6             sym = K.symmetry_group()
     7         except:
     8             print(K_id, " is not hyperbolic.")
     9         if sym.is_amphicheiral():
    10             iso = sym.isometries()
    11             pa  = false  # positive amphicheiral flag
    12             spa = false  # strongly positive amphicheiral flag
    13             count = 0
    14             for i in iso:
    15                 A = i.cusp_maps()[0]
    16                 if A[0,0] == -1 and A[1,1] == 1:
    17                     pa = true
    18                     if sym.multiply_elements(count,count) == 0:
    19                         spa = true
    20                 count += 1
    21             if spa:
    22                 print(’knot ’, K_id, ’ is strongly positive amphicheiral’)
    23             elif pa:
    24                 print(’knot ’, K_id, ’ is positive amphicheiral’)
    25             else:
    26                 print(’knot ’, K_id, ’ is amphicheiral’)

To facilitate the understanding, we give more comments on individual code
lines: For non-hyperbolic knots an exception is caught in line 7 (the symmetry
group cannot be calculated by SnapPy). Line 9 filters for amphicheiral knots;
the following lines are analogous to our example and two boolean flags are
defined. Lines 14 through 20 contain the iteration through the isometries. The
matrix elements are tested in line 16 and the involution property in line 18.

The result consists of 131 prime knots; they are listed in Tables 4 and 5.
This list is complete because prime amphicheiral knots up to 16 crossings are
always hyperbolic: Torus knots are always chiral and satellite knots up to 16
crossings are satellites of the trefoil and therefore chiral (see [7], p. 43).

##  3\. Templates for doubly symmetric diagrams

In this section, we describe the templates for tabulating doubly symmetric
diagrams. In the article [10] we already used templates for symmetric unions
and complete diagrams were shown. In [11] the table for strongly invertible
knots used templates with only half of a diagram. Now, because of the double
symmetry, we may even reduce this to one quarter of a diagram!

For doubly symmetric diagrams, we combine the properties of symmetric unions
and strongly invertible (transvergent) diagrams. These are defined as follows:
Symmetric unions are the connected sum of a diagram and its mirror image with
reversed orientation (defining the partial knot of the symmetric union, which
is unique up to mirror image), with additional crossings inserted on the
symmetry axis. Transvergent diagrams for strongly invertible knots are
rotationally symmetric with respect to an axis in the diagram plane, see [11]
for details. These two constructions are therefore very similar and their
parterre diagrams are indistinguishable. Care is needed for the choice of
crossings, so that the resulting diagram with two symmetries is indeed
strongly positive amphicheiral:

######  Definition 3.1.

A knot diagram $D$ is called doubly symmetric if it possesses the two
following symmetries with respect to perpendicular axes $x$ and $y$ in the
diagram plane

  * a)

$D$ is a symmetric union with axis $y$,

  * b)

the partial knot diagrams on the left and on the right sides are (mirror
symmetric) transvergent diagrams with axis $x$,

and the inserted crossings on the axis $y$ are mirror symmetric with respect
to $x$ as shown in Figure 7. A knot is called doubly symmetric if it has a
doubly symmetric diagram.

Figure 7. The definition of doubly symmetric diagrams. The crossings on the
axis $x$ are omitted in this illustration. They are mirror symmetric with
respect to the axis $y$ because doubly symmetric diagrams are symmetric
unions.

It is quite obvious that a knot with a doubly symmetric diagram is strongly
positive amphicheiral: The combination of rotation around the $x$-axis and
reflection at the $y$-$z$-plane results in a point reflection at the origin
and the crossings on the two axes are chosen in a way that they are also point
symmetric to the origin. This is equivalent to rotating the diagram by $\pi$
into its mirror image and since the origin is not part of the diagram we have
a strongly positive amphicheiral knot.

The only subtlety is the choice of the crossings on the $y$-axis. See the
following remark.

######  Remark 3.2.

We observe that the twist numbers for the crossings on the axis $y$ are
$n_{1},\ldots,n_{k}$ and $-n_{k},\ldots,-n_{1}$. A rotationally symmetric
diagram with respect to $x$ would have twist number $n_{1},\ldots,n_{k}$ and
$n_{k},\ldots,n_{1}$. In this case, the total diagram would be a (strongly
invertible) transvergent diagram, but it would lose the rotational symmetry
with respect to the central axis perpendicular to the diagram plane which
guarantees the strongly positive amphicheirality. Therefore, although the
partial knots are strongly invertible in this construction, doubly symmetric
knots can be invertible or non-invertible.

Figure 8. Converting a doubly symmetric knot diagram into a template. The
example shows a diagram of $16n_{645918}$, transformed into template
$t_{2}(1,-1\mid 2)$.

Our tabulating approach uses templates, similar to the way we did in [10] and
[11]. Each template contains the quarter diagram in the second quadrant and
integer markings on the two axes for the twists which are inserted there.
Figure 8 shows how a template diagram is constructed from a doubly symmetric
diagram.

Figure 9. The conventions for the integer markings for twists on the two axes.
Figure 10. Converting a template into a doubly symmetric knot diagram,
illustrated by three doubly symmetric diagrams for the knot $16n_{872167}$:
$t_{4}(0,0,3\mid 1)$, $t_{6}(1,-1,0\mid 1)$ and $t_{17}(1,0,0,0\mid 0,0,1)$.

In the other direction, we proceed as in Figure 10: If a template diagram is
given, we reconstruct the other quadrants and insert the twists on the axes
given by the integer markings. The conventions for these markings are defined
in Figure 9.

The list of strongly positive amphicheiral knots up to 16 crossings in the
appendix was compiled by starting with the simplest templates made out of one
or two crossings and by testing twist variations with Knotscape. We then added
templates with three crossings, and for some knots templates with four
crossings were necessary. In the appendix, we also list variations with
different partial knots because we consider doubly symmetric knot diagrams
with different partial knots as interesting enough to be documented in detail.
For instance, Figure 10 shows three different diagrams with partial knots
$7_{6}$, $8_{20}$ and $9_{1}$ for $16n_{872167}$. If several diagrams were
found for a pair (consisting of a doubly symmetric knot and the partial knot),
we chose a diagrams with smallest crossing number for Table 4 and for the
appendix.

##  4\. Almost doubly symmetric diagrams

The total number of prime strongly positive amphicheiral knots up to 16
crossings is 131 and from these we found 106 with doubly symmetric diagrams.
Wishing to show diagrams with a visible strongly positive amphicheiral
symmetry also for the remaining knots, we can choose between a) rotationally
symmetric diagrams in braid form (see the diagram in the left of Figure 1), or
b) rotationally symmetric diagrams built from more general tangles. We found
by incidence that a small deviation from a doubly symmetric diagram yielded
additional knots, and this led to the search for the remaining 25 knots as
‘almost doubly symmetric’ diagrams. Hence, we chose b) but with additional
restrictions on the tangle form. Almost doubly symmetric diagrams are doubly
symmetric diagrams where two rotationally symmetrical crossings are modified,
see Figure 11. A knot is called almost doubly symmetric if it has an almost
doubly symmetric diagram.

Figure 11. An almost doubly symmetric diagram for $16a_{288139}$. The example
shows the diagram constructed from template $t_{24}(-2,1,-1\mid 0,-1)$. The
two switched rotationally symmetrical crossings are marked in gray.

The template notation for almost doubly symmetric diagrams consists of an
additional marking for one crossing: The quadrants in which the two
symmetrical crossings are switched are indicated in the templates as `I,III`,
or `II,IV`. Currently it is not known whether this relatively small deviation
from a double symmetry is enough to generate all strongly positive
amphicheiral knots, see Question 1.2.

We remark, that due to the vertical symmetry it is possible to use only one of
the markings, and for our templates we use `I,III`. If several rotationally
symmetric crossing pairs are modified it can be necessary to apply both
markings in one template description.

##  5\. Summary data for prime amphicheiral knots up to 16 crossings

Although the topic of this article is strongly positive amphicheiral knots, we
would like to give some data for amphicheiral knots in general. The total
number of prime knots up to 16 crossings is 1,701,936 (see the title of [7])
if for each knot we count inverse and mirrored variants only once.
Amphicheiral knots contribute only with roughly $1/900=0.1111\%$ to this
number, as there are 1892 of them (not including the trivial knot). The
distribution of amphicheiral knots into sub-classes is also interesting.

The following Table 1 splits the two symmetry classes of positive and fully
amphicheiral knots into two parts, according to having the ‘strong’ property
(indicated by ‘,s’), and summarizes the numbers of amphicheiral knots. The 131
prime strongly positive amphicheiral knots are contained in the columns
‘$+$,s’ and ‘$a$,s’, and there are 59 knots of type ‘$+$,s’ and 72 of type
‘$a$,s’.

c | $-$ | $+$ |  $+$,s | $a$ |  $a$,s | Sum
---|---|---|---|---|---|---
4 |  |  |  | 1 |  | 1
6 |  |  |  | 1 |  | 1
8 | 1 |  |  | 4 |  | 5
10 | 6 |  |  | 5 | 2 | 13
12 | 40 |  | 1 | 13 | 4 | 58
14 | 227 | 2 | 4 | 25 | 16 | 274
15 | 1 |  |  |  |  | 1
16 | 1361 | 11 | 54 | 63 | 50 | 1539
Sum | 1636 | 13 | 59 | 112 | 72 | 1892
Table 1. The numbers of prime amphicheiral knots for each minimal crossing
number $c$. The notation for the symmetry types is based on [7]: ‘$-$’ =
negative amphicheiral, ‘$+$’ = positive amphicheiral, ‘$a$’ = fully
amphicheiral, with ‘,s’ added for strong symmetry.

We give some comments on the sub-classes of amphicheiral knots: By far the
most frequent prime amphicheiral knots in this table are the negative
amphicheiral knots. The only amphicheiral knot with 15 crossings (15n139717)
is also of this type. The rarest knot family consists of positive amphicheiral
knots which are not strongly positive amphicheiral (13 knots in column ‘$+$’).

We explicitly list the knots which occur 4 times or less (i.e. with table
entries $\leq 4$): The two amphicheiral knots with 4 and 6 crossings are
$4_{1}$ and $6_{3}$. The knot with $c=8$ in column ‘$-$’ is $8_{17}$. This is
the first non-invertible knot in the knot tables. The four fully amphicheiral
knots with 8 crossings are $8_{3}$, $8_{9}$, $8_{12}$ and $8_{18}$. The two
strongly positive amphicheiral knots with 10 crossings are $10_{99}$ and
$10_{123}$ (both invertible). For $c=12$ we have the strongly positive
amphicheiral knots 12a427 (non-invertible) and 12a1019, 12a1105, 12a1202,
12n706 (invertible). For $c=14$ we have in column ‘$+$’ the knots 14a10435 and
14n14148 (non-invertible) and in column ‘$+$,s’ the knots 14a8662, 14a16309,
14a18676 and 14a18680 (also non-invertible).

###  5.1. Double symmetry and invertibility

We already mentioned that doubly symmetric knots may be invertible or non-
invertible. The following Table 2 shows that the invertibility property is
similarly distributed for doubly and almost doubly symmetric knots. Note, that
some of the knots which are recorded here as almost doubly symmetric might be
doubly symmetric.

|  $+$,s |  $a$,s | Sum
---|---|---|---
doubly symmetric | 45 | 61 | 106
almost doubly symmetric | 14 | 11 | 25
Sum | 59 | 72 | 131
Table 2. The distribution of invertible (‘$a$,s’) and non-invertible (‘$+$,s’)
knots among doubly and almost doubly symmetric knots.

###  5.2. Symmetry groups

In Table 3 we give an overview of the symmetry groups of amphicheiral knots up
to 16 crossings. This is based on Table A2A in [7] and the result of our
SnapPy script. $C_{n}$ denotes the cyclic group with $n$ elements and $D_{n}$
the dihedral group with $2n$ elements (and therefore $C_{2}=D_{1}$).

type | most frequent case | additional cases
---|---|---
$-$ | $1577\times D_{1}$ | $59\times D_{2}$
$+$ | $13\times C_{4}$ |
$+$,s | $59\times C_{2}$ |
$a$ | $106\times D_{4}$ | $4\times D_{8}\;\,$
|  | $1\times D_{16}$
$a$,s | $66\times D_{2}$ | $4\times D_{6}\;\,$
|  | $1\times D_{10}$
|  | $1\times D_{14}$
Table 3. The symmetry groups occuring for each symmetry type

The fully amphicheiral knots with dihedral symmetry $D_{2n}$, $2n\geq 6$ are:

  * -

$D_{6}$: 12a1019, 12a1202, 12n706, 16n524438

  * -

$D_{8}$: $8_{18}$, 16a357361, 16a375396, 16n995250

  * -

$D_{10}$: $10_{123}$

  * -

$D_{14}$: 14a19470

  * -

$D_{16}$: 16a379778

The last three of these knots are $n$-periodic because they are braid closures
of the form $(\sigma_{1}\sigma_{2}^{-1})^{n}$. The knots with $c=12$ with
symmetry group $D_{6}$ have period 3 [8], and $8_{18}$ has period 4.

We look at the more detailed symmetry types according to [1]. For amphicheiral
knots the following three types are possible: SNAP, RRef and SNASI. Negative
amphicheiral knots are of type SNAP. They have symmetry groups $D_{n}$ and
period $n$. Positive amphicheiral knots are of type RRef. They have symmetry
groups $C_{2n}$ and period $n$. Fully amphicheiral knots are of type SNASI,
with symmetry groups $D_{2n}$ and period $n$. In particular, 16n524438 also
has period 3, and 16a357361, 16a375396, 16n995250 have period 4. (Note, that
hyperbolic amphicheiral knots are never freely periodic.)

###  5.3. Ribbon property

The knots in Table 4 are doubly symmetric and therefore ribbon. We checked the
ribbon property of the 25 knots in Table 5 with information taken from [3] and
[14]. All knots but the following four are shown there to be ribbon knots:
14a19470, 16a288139, 16a289378, 16a309401. The ribbon property of the last
three of them is currently unknown, while 14a19470 is not slice [16].
Therefore 14a19470 is the first knot in the knot tables which is strongly
positive amphicheiral and not slice (compare [4]).

In Figure 12, we show on the left the knot 14a19470 as the closure of
$(\sigma_{1}\sigma_{2}^{-1})^{7}$. After flipping the braid inside the
rectangle in the indicated direction, we get the diagram in the middle, and
two symmetric Reidemeister 2 moves yield the almost doubly symmetric diagram
on the right (with the two exceptional crossings marked in gray).

Figure 12. A transformation of the knot $14a_{19470}$ from 7-periodic braid
form into an almost doubly symmetric diagram, given by template
$t_{24}(0,0,-1\mid 0,-1)$

###  5.4. The number of distinct partial knots

As mentioned before, Table 4 contains knot diagrams with different partial
knots if these occurred as search results. For 59 knots we found diagrams with
only one partial knot. For 33 knots there are two partial knots, for 11 knots
there are three, for 2 knots there are four, and there is one knot with five
partial knots. These results are experimental and we do not know if more
partial knot variants exist in each of these cases. The knot 12a427 has the
highest number of partial knots and these are: $7_{4}$, $8_{21}$, $9_{2}$,
$3_{1}\sharp 4_{1}$, $3_{1}\sharp 5_{1}$, all of them having determinant 15.
There are several interesting questions concerning partial knots and we note
the following explicitly:

######  Question 5.1.

Does a doubly symmetric strongly positive amphicheiral knot exist for which
the partial knot in doubly symmetric diagrams is unique but for which several
partial knots occur in general symmetric union diagrams?

###  5.5. Composite partial knots

Doubly symmetric knots with composite partial knots occur frequently and the
following templates generate them: $t_{2}$, $t_{3}$, $t_{12}$, $t_{13}$,
$t_{14}$, $t_{15}$, $t_{16}$. (Note, however, that in the templates $t_{13}$
and $t_{15}$ the knots 16n101996 and 16n102000 have the partial knot $3_{1}$
which is not composite. In this case the second factor is trivial and the
partial knot should be read as $3_{1}\sharp$ triv.)

Two more remarks: a) We did not find a doubly symmetric diagram representing a
knot up to 16 crossings with a composite partial knot consisting of more than
2 prime factors. b) Our notation for composite partial knots in the appendix
is not completely precise: A partial knot written as $3_{1}\sharp 3_{1}$ could
either be $3_{1}\sharp 3_{1}$ (with signature $\pm 4$) or $3_{1}\sharp
3_{1}^{*}$, in which case the second factor is the mirror image of the first
one and the total signature is 0. The same ambiguity occurs for other partial
knots, e.g. $3_{1}\sharp 5_{2}$.

###  5.6. Determinants and Alexander polynomials

The determinants for the 131 prime strongly positive amphicheiral knots up to
16 crossings cover all squares of odd numbers between 1 and 53. The only knot
with determinant $2809=53^{2}$ is 16a357272 and it has not yet been found to
be doubly symmetric. All other determinants between $1$ and $51^{2}$ occur for
knots with doubly symmetric diagrams. The most frequent determinant is 441
with 9 knots (all of them doubly symmetric).

As the Alexander polynomial of strongly positive amphicheiral knots is always
a square, we checked for doubly symmetric diagrams whether the square of the
Alexander polynomial of one of the partial knots equals the Alexander
polynomial of the knot. This is the case for roughly half of them (54 out of
106 knots).

Examples: a) The Alexander polynomial of 14n22073 is
$\Delta(t)=(1-t^{2}+t^{4})^{2}$ and we found doubly symmetric diagrams for
14n22073 with trivial partial knot only. The first knot in the knot table
having Alexander polynomial $\Delta(t)=1-t^{2}+t^{4}$ is 12n121 and it seems
unlikely that 14n22073 can be a symmetric union with a partial knot with
crossing number 12 or higher, as for instance 12n121.

b) In a similar example with a partial knot with crossing number 11 we find
this relationship between the Alexander polynomials, though: The doubly
symmetric knot 16n869383 has Alexander polynomial
$\Delta(t)=(2-4t+3t^{2}-4t^{3}+2t^{4})^{2}$ and we found 3 doubly symmetric
diagrams, with partial knots $7_{4}$, $8_{21}$ and 11n79. The Alexander
polynomial of 11n79 is $\Delta(t)=2-4t+3t^{2}-4t^{3}+2t^{4}$.

Remark: This relationship between the Alexander polynomials of a symmetric
union and its partial knot always occurs for even twist numbers (on the
vertical axis). Doubly symmetric diagrams with even twist numbers are quite
rare, though: Table 4 contains only the 8 cases 14n9732, 16n645918, 16n645926,
16n786382, 16n847920, 16n847983, 16n991381, 16n991505.

######  Question 5.2.

Which topological properties distinguish doubly symmetric knots with even
twist numbers from arbitrary doubly symmetric knots?

Figure 13. Doubly symmetric knot diagrams for $16a_{107998}$,
$t_{11}(0,3,-1\mid 1)$ and $16n_{845519}$, $t_{22}(0,1,-1\mid 1)$, both with
22 crossings

###  5.7. Number of crossings in doubly symmetric diagrams

We close the article with a comment on the crossing number of doubly symmetric
diagrams. As mentioned before, alternating knots cannot realize their minimal
crossing number in a doubly symmetric diagram. This is achieved, however, by
the knot 14n9732 in $t_{1}(1,0,\mid 2)$ and by 7 non-alternating knots with
$c=16$: 16n428839, 16n451788, 16n645918, 16n645926, 16n847920, 16n991381 and
16n991505. In the article [12] we will list exhaustively all doubly symmetric
diagrams up to 18 crossings which are not reducible to less complicated ones
and we will formulate this as a theorem.

The four doubly symmetric diagrams in Figures 13 and 14 additionally
illustrate the opposite case that diagrams with quite a lot of crossings were
found as minimal diagrams (22 in the first and 26 crossings in the second
figure). The maximal value of 28 crossings in a diagram in the appendix occurs
only for one knot, and this is 16n925482 in the almost doubly symmetric
template $t_{28}$.

Figure 14. Doubly symmetric knot diagrams for $16n_{225139}$,
$t_{23}(-1,1,-1,1\mid 1,0)$ and $16n_{919894}$, $t_{21}(1,-1,-2,0\mid 1,0)$,
both with 26 crossings

##  6\. Outlook

The description of the systematic enumeration of doubly symmetric diagrams
will be the topic of the article [12] (we decided not to include this in the
present article because it would become too long). There, we will prove that
no prime knots have doubly symmetric diagrams with $\leq 12$ crossings in
their symmetric diagram, and list the prime knots with doubly symmetric
diagrams with 14 crossings (we find $10_{99}$, $10_{123}$, 12n706 and 14n9732)
and with 16 and 18 crossings. We do that by enumerating all templates up to 4
crossings which are not reducible to less complicated ones; we use Knotscape
[9] to analyze the diagrams obtained by inserting crossings on the axes.

## Acknowledgments

I thank Marc Kegel and Steven Sivek for their help with my first steps with
SnapPy, Nathan Dunfield for help concerning the ribbon status of several knots
and explanations concerning SnapPy, Brendan Owens for giving me a reference
for the ribbon status of 14n19470, and Keegan Boyle for explaining the
classification of symmetries for amphicheiral knots.

$10_{99}$ | 81 | $6_{1}$ | $t_{1}$ | 16a20777 | 1521 | $3_{1}\sharp 6_{3}$ | $t_{13}$ | 16n102009 | 441 | $3_{1}\sharp 5_{2}$ | $t_{15}$
---|---|---|---|---|---|---|---|---|---|---|---
|  | $3_{1}\sharp 3_{1}$ | $t_{2}$ | 16a51808 | 1369 | $9_{14}$ | $t_{9}$ | 16n102453 | 225 | $8_{21}$ | $t_{7}$
|  | $8_{20}$ | $t_{5}$ | 16a59437 | 2401 | $9_{27}$ | $t_{9}$ |  |  | $3_{1}\sharp 5_{1}$ | $t_{14}$
|  | $9_{1}$ | $t_{17}$ | 16a66481 | 1225 | $9_{12}$ | $t_{9}$ | 16n106013 | 225 | $3_{1}\sharp 5_{1}$ | $t_{16}$
$10_{123}$ | 121 | $6_{2}$ | $t_{1}$ | 16a101002 | 1225 | $4_{1}\sharp 5_{2}$ | $t_{2}$ | 16n204537 | 49 | $5_{2}$ | $t_{7}$
|  | $7_{2}$ | $t_{9}$ |  |  | $5_{1}\sharp 5_{2}$ | $t_{12}$ |  |  | $7_{1}$ | $t_{9}$
12a427 | 225 | $7_{4}$ | $t_{7}$ | 16a107430 | 1089 | $8_{15}$ | $t_{4}$ | 16n225139 | 289 | $8_{2}$ | $t_{23}$
|  | $3_{1}\sharp 4_{1}$ | $t_{2}$ |  |  | $9_{11}$ | $t_{9}$ | 16n225152 | 289 | $7_{5}$ | $t_{9}$
|  | $8_{21}$ | $t_{4}$ | 16a107998 | 1849 | $9_{22}$ | $t_{11}$ | 16n226358 | 625 | $8_{8}$ | $t_{9}$
|  | $3_{1}\sharp 5_{1}$ | $t_{14}$ | 16a116332 | 1089 | $3_{1}\sharp 6_{2}$ | $t_{12}$ | 16n226373 | 49 | $5_{2}$ | $t_{9}$
|  | $9_{2}$ | $t_{17}$ | 16a116809 | 1521 | $3_{1}\sharp 6_{3}$ | $t_{12}$ |  |  | $7_{1}$ | $t_{10}$
12a1019 | 361 | $7_{6}$ | $t_{1}$ |  |  | $9_{17}$ | $t_{21}$ | 16n268004 | 361 | $8_{4}$ | $t_{9}$
12a1105 | 289 | $7_{5}$ | $t_{1}$ | 16a149574 | 729 | $3_{1}\sharp 6_{1}$ | $t_{2}$ |  |  | $9_{3}$ | $t_{9}$
|  | $8_{2}$ | $t_{9}$ | 16a151960 | 1369 | $9_{36}$ | $t_{11}$ | 16n268599 | 121 | $7_{2}$ | $t_{9}$
|  | $10_{1}$ | $t_{17}$ | 16a152693 | 2025 | $9_{24}$ | $t_{11}$ | 16n272647 | 121 | $6_{2}$ | $t_{17}$
12a1202 | 169 | $7_{3}$ | $t_{1}$ | 16a154813 | 1521 | $9_{17}$ | $t_{21}$ |  |  | $7_{2}$ | $t_{9}$
|  | $9_{43}$ | $t_{5}$ |  |  | $10_{144}$ | $t_{4}$ | 16n323632 | 9 | $3_{1}$ | $t_{8}$
12n706 | 49 | $5_{2}$ | $t_{1}$ | 16a168328 | 1681 | $10_{47}$ | $t_{19}$ |  |  | $8_{19}$ | $t_{7}$
|  | $7_{1}$ | $t_{9}$ | 16a193681 | 2209 | $9_{25}$ | $t_{11}$ | 16n428839 | 225 | $7_{4}$ | $t_{7}$
|  | $9_{42}$ | $t_{4}$ | 16a202161 | 625 | $4_{1}\sharp 5_{1}$ | $t_{12}$ |  |  | $8_{21}$ | $t_{4}$
14a6002 | 625 | $4_{1}\sharp 4_{1}$ | $t_{2}$ | 16a202163 | 1225 | $4_{1}\sharp 5_{2}$ | $t_{2}$ |  |  | $9_{2}$ | $t_{17}$
|  | $4_{1}\sharp 5_{1}$ | $t_{16}$ |  |  | $5_{1}\sharp 5_{2}$ | $t_{14}$ |  |  | $10_{136}$ | $t_{22}$
14a6398 | 529 | $8_{7}$ | $t_{9}$ | 16a259485 | 1089 | $8_{15}$ | $t_{7}$ | 16n451781 | 529 | $8_{7}$ | $t_{10}$
14a8662 | 441 | $7_{7}$ | $t_{4}$ |  |  | $3_{1}\sharp 6_{2}$ | $t_{14}$ | 16n451788 | 25 | $5_{1}$ | $t_{1}$
|  | $3_{1}\sharp 5_{2}$ | $t_{2}$ | 16a259789 | 1089 | $3_{1}\sharp 6_{2}$ | $t_{16}$ | 16n524438 | 49 | $9_{42}$ | $t_{4}$
14a8872 | 729 | $8_{10}$ | $t_{11}$ | 16a293262 | 1089 | $3_{1}\sharp 6_{2}$ | $t_{12}$ | 16n535891 | 225 | 11n13 | $t_{20}$
|  | $9_{6}$ | $t_{17}$ | 16a313024 | 2025 | $8_{18}$ | $t_{4}$ | 16n645918 | 81 | $3_{1}\sharp 3_{1}$ | $t_{2}$
|  | $9_{48}$ | $t_{4}$ |  |  | $10_{164}$ | $t_{4}$ | 16n645926 | 81 | $3_{1}\sharp 3_{1}$ | $t_{3}$
14a16309 | 625 | $9_{49}$ | $t_{4}$ | 16a313458 | 729 | $3_{1}\sharp 6_{1}$ | $t_{2}$ | 16n780729 | 529 | $8_{6}$ | $t_{10}$
14a16311 | 289 | $8_{3}$ | $t_{1}$ | 16a314171 | 1225 | $8_{16}$ | $t_{4}$ |  |  | $10_{2}$ | $t_{17}$
|  | $10_{130}$ | $t_{5}$ |  |  | $9_{12}$ | $t_{1}$ | 16n780731 | 1 | triv | $t_{10}$
14a16437 | 729 | $9_{6}$ | $t_{9}$ | 16a326803 | 1521 | $10_{165}$ | $t_{4}$ | 16n786382 | 1 | triv | $t_{8}$
14a17173 | 841 | $8_{12}$ | $t_{1}$ | 16a329540 | 2025 | $10_{158}$ | $t_{4}$ | 16n797553 | 361 | $8_{4}$ | $t_{9}$
14a18187 | 729 | $8_{11}$ | $t_{1}$ | 16a330218 | 1089 | $8_{15}$ | $t_{4}$ | 16n847920 | 49 | $5_{2}$ | $t_{1}$
|  | $9_{47}$ | $t_{4}$ | 16a332442 | 1369 | $9_{36}$ | $t_{11}$ | 16n847983 | 25 | $4_{1}$ | $t_{9}$
14a18362 | 361 | $8_{4}$ | $t_{9}$ | 16a332444 | 1521 | $9_{15}$ | $t_{1}$ |  |  | $5_{1}$ | $t_{8}$
14a18676 | 225 | $7_{4}$ | $t_{4}$ | 16a340727 | 961 | $10_{46}$ | $t_{19}$ | 16n847988 | 289 | $7_{5}$ | $t_{9}$
|  | $3_{1}\sharp 5_{1}$ | $t_{12}$ | 16a340770 | 1369 | $10_{6}$ | $t_{9}$ | 16n849519 | 441 | $7_{7}$ | $t_{22}$
14a18680 | 441 | $7_{7}$ | $t_{7}$ |  |  | 11a234 | $t_{17}$ | 16n855304 | 289 | $9_{44}$ | $t_{5}$
|  | $3_{1}\sharp 5_{2}$ | $t_{2}$ | 16a354511 | 1225 | $8_{16}$ | $t_{4}$ | 16n858257 | 441 | $7_{7}$ | $t_{1}$
14a18723 | 529 | $8_{6}$ | $t_{1}$ |  |  | $10_{162}$ | $t_{4}$ |  |  | $8_{5}$ | $t_{7}$
|  | $9_{5}$ | $t_{10}$ | 16a354563 | 1681 | $9_{18}$ | $t_{1}$ | 16n868471 | 169 | $7_{3}$ | $t_{10}$
|  | $10_{2}$ | $t_{17}$ | 16a356133 | 2601 | $10_{163}$ | $t_{4}$ |  |  | $8_{1}$ | $t_{9}$
14a19472 | 441 | $7_{7}$ | $t_{4}$ | 16a356811 | 961 | $9_{9}$ | $t_{9}$ | 16n869383 | 225 | $7_{4}$ | $t_{1}$
|  | $8_{5}$ | $t_{11}$ | 16a356843 | 441 | $8_{5}$ | $t_{4}$ |  |  | $8_{21}$ | $t_{4}$
|  | $9_{4}$ | $t_{18}$ |  |  | $9_{4}$ | $t_{1}$ |  |  | 11n79 | $t_{4}$
14n8213 | 169 | $6_{3}$ | $t_{1}$ |  |  | 11n64 | $t_{5}$ | 16n872167 | 81 | $8_{20}$ | $t_{6}$
|  | $7_{3}$ | $t_{10}$ | 16a358693 | 1681 | $9_{20}$ | $t_{9}$ |  |  | $9_{1}$ | $t_{17}$
|  | $8_{1}$ | $t_{9}$ | 16a368293 | 1521 | $9_{16}$ | $t_{11}$ |  |  | $9_{46}$ | $t_{4}$
14n9732 | 25 | $4_{1}$ | $t_{7}$ |  |  | $10_{9}$ | $t_{18}$ | 16n872172 | 361 | $7_{6}$ | $t_{1}$
|  | $5_{1}$ | $t_{1}$ | 16a371892 | 841 | $9_{7}$ | $t_{1}$ |  |  | $9_{3}$ | $t_{9}$
|  | $10_{132}$ | $t_{21}$ |  |  | $10_{8}$ | $t_{9}$ | 16n919894 | 441 | 11n118 | $t_{21}$
14n22073 | 1 | triv | $t_{8}$ | 16n101996 | 9 | $3_{1}$ | $t_{13}$ | 16n925976 | 9 | $8_{19}$ | $t_{7}$
14n25903 | 121 | $6_{2}$ | $t_{1}$ |  |  | $8_{19}$ | $t_{7}$ | 16n958050 | 625 | $8_{9}$ | $t_{10}$
|  | $10_{125}$ | $t_{20}$ | 16n102000 | 9 | $3_{1}$ | $t_{15}$ | 16n988939 | 1 | triv | $t_{18}$
|  | $10_{128}$ | $t_{4}$ | 16n102007 | 441 | $8_{5}$ | $t_{7}$ | 16n991381 | 81 | $6_{1}$ | $t_{1}$
16a19960 | 1521 | $3_{1}\sharp 6_{3}$ | $t_{15}$ |  |  | $3_{1}\sharp 5_{2}$ | $t_{13}$ | 16n991505 | 121 | $6_{2}$ | $t_{1}$
Table 4. A list of 171 diagrams for the 106 prime strongly positive amphicheiral knots for which doubly symmetric diagrams have been found. The second and third columns contain determinant and partial knot and the fourth column indicates the template with the diagram information. 14a19470 | 841 | $t_{24}$ | 16a288139 | 1681 | $t_{24}$ | 16a339566 | 2209 | $t_{26}$
---|---|---|---|---|---|---|---|---
14a19517 | 529 | $t_{28}$ | 16a289378 | 2209 | $t_{24}$ | 16a354193 | 2025 | $t_{28}$
16a66232 | 1849 | $t_{25}$ | 16a309401 | 2401 | $t_{28}$ | 16a357272 | 2809 | $t_{26}$
16a106900 | 1369 | $t_{27}$ | 16a312423 | 1521 | $t_{25}$ | 16a359219 | 1225 | $t_{27}$
16a150121 | 1681 | $t_{25}$ | 16a324978 | 1089 | $t_{25}$ | 16n404143 | 225 | $t_{25}$
16a155718 | 841 | $t_{27}$ | 16a326826 | 1849 | $t_{26}$ | 16n526315 | 169 | $t_{25}$
16a156092 | 961 | $t_{25}$ | 16a329518 | 1849 | $t_{25}$ | 16n918713 | 361 | $t_{25}$
16a275557 | 1369 | $t_{25}$ | 16a330763 | 2601 | $t_{26}$ | 16n925482 | 25 | $t_{28}$
16a275642 | 2025 | $t_{25}$ |  |  |  |  |  |
Table 5. A list of the 25 prime strongly positive amphicheiral knots for which
doubly symmetric diagrams have not been found. The second column contains the
determinant and the third column indicates the (almost doubly symmetric)
template with the diagram information.

##  7\. Appendix

The Appendix contains the templates generating the 196 (171 + 25) diagrams of
strongly positive amphicheiral prime knots with crossing number $\leq 16$. The
twist numbers correspond to the dotted lines in the templates read from left
to right (first part) and from top to bottom (second part). In the second
column the partial knot is indicated (notation $\partial$). The first four
pages contain the doubly symmetric diagrams and the fifth page shows the
almost doubly symmetric diagrams.

template $t_{1}$

template $t_{2}$

template $t_{3}$

$10_{99}$ , $\partial$ |  $6_{1}$ = | ( 2, |  0 $\mid$ | 1)
---|---|---|---|---
$10_{123}$ , $\partial$ |  $6_{2}$ = | ( 1, |  1 $\mid$ | 1)
$12a_{1019}$ , $\partial$ |  $7_{6}$ = | ( 2, |  1 $\mid$ | 1)
$12a_{1105}$ , $\partial$ |  $7_{5}$ = | ( 1, |  2 $\mid$ | 1)
$12a_{1202}$ , $\partial$ |  $7_{3}$ = | ( 3, |  0 $\mid$ | 1)
$12n_{706}$ , $\partial$ |  $5_{2}$ = | (-2, |  0 $\mid$ | 1)
$14a_{16311}$ , $\partial$ |  $8_{3}$ = | ( 4, |  0 $\mid$ | 1)
$14a_{17173}$ , $\partial$ |  $8_{12}$ = | ( 2, |  2 $\mid$ | 1)
$14a_{18187}$ , $\partial$ |  $8_{11}$ = | ( 3, |  1 $\mid$ | 1)
$14a_{18723}$ , $\partial$ |  $8_{6}$ = | ( 1, |  3 $\mid$ | 1)
$14n_{8213}$ , $\partial$ |  $6_{3}$ = | (-2, |  1 $\mid$ | 1)
$14n_{9732}$ , $\partial$ |  $5_{1}$ = | ( 1, |  0 $\mid$ | 2)
$14n_{25903}$ , $\partial$ |  $6_{2}$ = | (-3, |  0 $\mid$ | 1)
$16a_{314171}$ , $\partial$ |  $9_{12}$ = | ( 4, |  1 $\mid$ | 1)
$16a_{332444}$ , $\partial$ |  $9_{15}$ = | ( 2, |  3 $\mid$ | 1)
$16a_{354563}$ , $\partial$ |  $9_{18}$ = | ( 3, |  2 $\mid$ | 1)
$16a_{356843}$ , $\partial$ |  $9_{4}$ = | ( 5, |  0 $\mid$ | 1)
$16a_{371892}$ , $\partial$ |  $9_{7}$ = | ( 1, |  4 $\mid$ | 1)
$16n_{451788}$ , $\partial$ |  $5_{1}$ = | ( 1, |  0 $\mid$ | 3)
$16n_{847920}$ , $\partial$ |  $5_{2}$ = | (-2, |  0 $\mid$ | 2)
$16n_{858257}$ , $\partial$ |  $7_{7}$ = | (-3, |  1 $\mid$ | 1)
$16n_{869383}$ , $\partial$ |  $7_{4}$ = | (-4, |  0 $\mid$ | 1)
$16n_{872172}$ , $\partial$ |  $7_{6}$ = | (-2, |  2 $\mid$ | 1)
$16n_{991381}$ , $\partial$ |  $6_{1}$ = | ( 2, |  0 $\mid$ | 2)
$16n_{991505}$ , $\partial$ |  $6_{2}$ = | ( 1, |  1 $\mid$ | 2)

$10_{99}$ , $\partial$ |  $3_{1}\sharp 3_{1}$ = | ( 1, |  -1 $\mid$ | 1)
---|---|---|---|---
$12a_{427}$ , $\partial$ |  $3_{1}\sharp 4_{1}$ = | ( 1, |  -2 $\mid$ | 1)
$14a_{6002}$ , $\partial$ |  $4_{1}\sharp 4_{1}$ = | ( 2, |  -2 $\mid$ | 1)
$14a_{8662}$ , $\partial$ |  $3_{1}\sharp 5_{2}$ = | ( 1, |  -3 $\mid$ | 1)
$14a_{18680}$ , $\partial$ |  $3_{1}\sharp 5_{2}$ = | ( 3, |  -1 $\mid$ | 1)
$16a_{101002}$ , $\partial$ |  $4_{1}\sharp 5_{2}$ = | ( 2, |  -3 $\mid$ | 1)
$16a_{149574}$ , $\partial$ |  $3_{1}\sharp 6_{1}$ = | ( 1, |  -4 $\mid$ | 1)
$16a_{202163}$ , $\partial$ |  $4_{1}\sharp 5_{2}$ = | ( 3, |  -2 $\mid$ | 1)
$16a_{313458}$ , $\partial$ |  $3_{1}\sharp 6_{1}$ = | ( 4, |  -1 $\mid$ | 1)
$16n_{645918}$ , $\partial$ |  $3_{1}\sharp 3_{1}$ = | ( 1, |  -1 $\mid$ | 2)

$16n_{645926}$ , $\partial$ |  $3_{1}\sharp 3_{1}$ = | (-1, |  -1 $\mid$ | 2)
---|---|---|---|---

template $t_{4}$

$12a_{427}$ , $\partial$ |  $8_{21}$ = | (-2, | 0, |  0 $\mid$ | 1)
---|---|---|---|---|---
$12n_{706}$ , $\partial$ |  $9_{42}$ = | ( 0, | 1, |  2 $\mid$ | 1)
$14a_{8662}$ , $\partial$ |  $7_{7}$ = | ( 1, | 0, |  0 $\mid$ | 1)
$14a_{8872}$ , $\partial$ |  $9_{48}$ = | (-3, | 0, |  0 $\mid$ | 1)
$14a_{16309}$ , $\partial$ |  $9_{49}$ = | (-2, | 0, |  -1 $\mid$ | 1)
$14a_{18187}$ , $\partial$ |  $9_{47}$ = | (-2, | 1, |  0 $\mid$ | 1)
$14a_{18676}$ , $\partial$ |  $7_{4}$ = | ( 0, | 0, |  -1 $\mid$ | 1)
$14a_{19472}$ , $\partial$ |  $7_{7}$ = | ( 0, | 1, |  0 $\mid$ | 1)
$14n_{25903}$ , $\partial$ |  $10_{128}$ = | ( 0, | 2, |  2 $\mid$ | 1)

$16a_{107430}$ , $\partial$ |  $8_{15}$ = | ( 2, | 0, |  0 $\mid$ | 1)
---|---|---|---|---|---
$16a_{154813}$ , $\partial$ |  $10_{144}$ = | (-4, | 0, |  0 $\mid$ | 1)
$16a_{313024}$ , $\partial$ |  $8_{18}$ = | ( 1, | 1, |  0 $\mid$ | 1)
$16a_{313024}$ , $\partial$ |  $10_{164}$ = | (-2, | 1, |  -1 $\mid$ | 1)
$16a_{314171}$ , $\partial$ |  $8_{16}$ = | ( 0, | 1, |  -1 $\mid$ | 1)
$16a_{326803}$ , $\partial$ |  $10_{165}$ = | (-2, | 2, |  0 $\mid$ | 1)
$16a_{329540}$ , $\partial$ |  $10_{158}$ = | (-3, | 0, |  -1 $\mid$ | 1)
$16a_{330218}$ , $\partial$ |  $8_{15}$ = | ( 0, | 2, |  0 $\mid$ | 1)
$16a_{354511}$ , $\partial$ |  $8_{16}$ = | ( 1, | 0, |  -1 $\mid$ | 1)
$16a_{354511}$ , $\partial$ |  $10_{162}$ = | (-2, | 0, |  -2 $\mid$ | 1)
$16a_{356133}$ , $\partial$ |  $10_{163}$ = | (-3, | 1, |  0 $\mid$ | 1)
$16a_{356843}$ , $\partial$ |  $8_{5}$ = | ( 0, | 0, |  -2 $\mid$ | 1)
$16n_{428839}$ , $\partial$ |  $8_{21}$ = | ( 1, | 1, |  1 $\mid$ | 1)
$16n_{524438}$ , $\partial$ |  $9_{42}$ = | ( 1, | 0, |  2 $\mid$ | 1)
$16n_{869383}$ , $\partial$ |  $8_{21}$ = | ( 0, | -2, |  0 $\mid$ | 1)
$16n_{869383}$ , $\partial$ |  $11n_{79}$ = | ( 0, | 3, |  2 $\mid$ | 1)
$16n_{872167}$ , $\partial$ |  $9_{46}$ = | ( 0, | 0, |  3 $\mid$ | 1)

template $t_{5}$

template $t_{6}$

$10_{99}$ , $\partial$ |  $8_{20}$ = | (-1, | 1, |  0 $\mid$ | 1)
---|---|---|---|---|---
$12a_{1202}$ , $\partial$ |  $9_{43}$ = | (-1, | 2, |  0 $\mid$ | 1)
$14a_{16311}$ , $\partial$ |  $10_{130}$ = | (-1, | 3, |  0 $\mid$ | 1)
$16a_{356843}$ , $\partial$ |  $11n_{64}$ = | (-1, | 4, |  0 $\mid$ | 1)
$16n_{855304}$ , $\partial$ |  $9_{44}$ = | (-2, | 1, |  0 $\mid$ | 1)

$16n_{872167}$ , $\partial$ |  $8_{20}$ = | ( 1, | -1, |  0 $\mid$ | 1)
---|---|---|---|---|---

template $t_{7}$

$12a_{427}$ , $\partial$ |  $7_{4}$ = | ( 0, |  -1 $\mid$ | -1, | 1)
---|---|---|---|---|---
$14a_{18680}$ , $\partial$ |  $7_{7}$ = | ( 1, |  0 $\mid$ | -1, | 1)
$14n_{9732}$ , $\partial$ |  $4_{1}$ = | (-1, |  -1 $\mid$ | -1, | 0)
$16a_{259485}$ , $\partial$ |  $8_{15}$ = | ( 2, |  0 $\mid$ | -1, | 1)
$16n_{101996}$ , $\partial$ |  $8_{19}$ = | ( 0, |  2 $\mid$ | -1, | 1)
$16n_{102007}$ , $\partial$ |  $8_{5}$ = | ( 0, |  -2 $\mid$ | -1, | 1)
$16n_{102453}$ , $\partial$ |  $8_{21}$ = | (-2, |  0 $\mid$ | -1, | 1)
$16n_{204537}$ , $\partial$ |  $5_{2}$ = | (-1, |  -2 $\mid$ | -1, | 0)
$16n_{323632}$ , $\partial$ |  $8_{19}$ = | ( 0, |  2 $\mid$ | -1, | 0)
$16n_{428839}$ , $\partial$ |  $7_{4}$ = | ( 0, |  -1 $\mid$ | -1, | 0)
$16n_{858257}$ , $\partial$ |  $8_{5}$ = | ( 0, |  -2 $\mid$ | -1, | 2)
$16n_{925976}$ , $\partial$ |  $8_{19}$ = | ( 0, |  2 $\mid$ | -1, | 2)

template $t_{8}$

$14n_{22073}$ , $\partial$ | triv = | ( 0, |  1 $\mid$ | 1, | 0)
---|---|---|---|---|---
$16n_{323632}$ , $\partial$ |  $3_{1}$ = | ( 0, |  2 $\mid$ | 1, | 0)
$16n_{786382}$ , $\partial$ | triv = | ( 0, |  1 $\mid$ | 2, | 0)
$16n_{847983}$ , $\partial$ |  $5_{1}$ = | (-1, |  1 $\mid$ | 1, | 0)

template $t_{9}$

$10_{123}$ , $\partial$ |  $7_{2}$ = | (-2, | 0, |  0 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$12a_{1105}$ , $\partial$ |  $8_{2}$ = | ( 1, | 0, |  1 $\mid$ | 1, | 0)
$12n_{706}$ , $\partial$ |  $7_{1}$ = | ( 1, | 0, |  0 $\mid$ | 0, | 1)
$14a_{6398}$ , $\partial$ |  $8_{7}$ = | (-2, | 1, |  0 $\mid$ | 0, | 1)
$14a_{16437}$ , $\partial$ |  $9_{6}$ = | ( 1, | 0, |  2 $\mid$ | 1, | 0)
$14a_{18362}$ , $\partial$ |  $8_{4}$ = | ( 1, | 1, |  0 $\mid$ | 0, | 1)
$14n_{8213}$ , $\partial$ |  $8_{1}$ = | ( 2, | 0, |  0 $\mid$ | 0, | 1)
$16a_{51808}$ , $\partial$ |  $9_{14}$ = | (-3, | 1, |  0 $\mid$ | 0, | 1)
$16a_{59437}$ , $\partial$ |  $9_{27}$ = | (-2, | 1, |  1 $\mid$ | 0, | 1)
$16a_{66481}$ , $\partial$ |  $9_{12}$ = | (-2, | 2, |  0 $\mid$ | 0, | 1)

$16a_{107430}$ , $\partial$ |  $9_{11}$ = | ( 2, | 1, |  0 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$16a_{340770}$ , $\partial$ |  $10_{6}$ = | ( 1, | 0, |  3 $\mid$ | 1, | 0)
$16a_{356811}$ , $\partial$ |  $9_{9}$ = | ( 1, | 2, |  0 $\mid$ | 0, | 1)
$16a_{358693}$ , $\partial$ |  $9_{20}$ = | ( 1, | 1, |  1 $\mid$ | 0, | 1)
$16a_{371892}$ , $\partial$ |  $10_{8}$ = | (-5, | 0, |  0 $\mid$ | 0, | 1)
$16n_{204537}$ , $\partial$ |  $7_{1}$ = | ( 1, | 0, |  0 $\mid$ | -2, | 1)
$16n_{225152}$ , $\partial$ |  $7_{5}$ = | ( 1, | -2, |  0 $\mid$ | 0, | 1)
$16n_{226358}$ , $\partial$ |  $8_{8}$ = | (-2, | 0, |  1 $\mid$ | 0, | 1)
$16n_{226373}$ , $\partial$ |  $5_{2}$ = | ( 1, | -1, |  1 $\mid$ | 0, | 1)
$16n_{268004}$ , $\partial$ |  $8_{4}$ = | ( 1, | 1, |  0 $\mid$ | -1, | 1)
$16n_{268004}$ , $\partial$ |  $9_{3}$ = | ( 3, | 0, |  0 $\mid$ | -1, | 1)
$16n_{268599}$ , $\partial$ |  $7_{2}$ = | (-2, | 0, |  0 $\mid$ | 1, | 0)
$16n_{272647}$ , $\partial$ |  $7_{2}$ = | (-2, | 0, |  0 $\mid$ | -1, | 1)
$16n_{797553}$ , $\partial$ |  $8_{4}$ = | ( 1, | 1, |  0 $\mid$ | 1, | 0)
$16n_{847983}$ , $\partial$ |  $4_{1}$ = | ( 1, | -1, |  0 $\mid$ | 0, | 2)
$16n_{847988}$ , $\partial$ |  $7_{5}$ = | ( 1, | -2, |  0 $\mid$ | -1, | 1)
$16n_{868471}$ , $\partial$ |  $8_{1}$ = | ( 2, | 0, |  0 $\mid$ | 1, | 0)
$16n_{872172}$ , $\partial$ |  $9_{3}$ = | ( 3, | 0, |  0 $\mid$ | 0, | 1)

template $t_{10}$

$14a_{18723}$ , $\partial$ |  $9_{5}$ = | (-3, | 1, |  0 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$14n_{8213}$ , $\partial$ |  $7_{3}$ = | (-1, | -1, |  0 $\mid$ | 1, | 0)
$16n_{226373}$ , $\partial$ |  $7_{1}$ = | (-1, | -1, |  -2 $\mid$ | 1, | 1)
$16n_{451781}$ , $\partial$ |  $8_{7}$ = | (-1, | -1, |  1 $\mid$ | 1, | 0)
$16n_{780729}$ , $\partial$ |  $8_{6}$ = | (-2, | -1, |  0 $\mid$ | 1, | 0)
$16n_{780731}$ , $\partial$ | triv = | (-1, | 0, |  1 $\mid$ | 1, | 1)
$16n_{868471}$ , $\partial$ |  $7_{3}$ = | (-1, | -1, |  0 $\mid$ | 0, | 1)
$16n_{958050}$ , $\partial$ |  $8_{9}$ = | (-1, | -2, |  0 $\mid$ | 1, | 0)

template $t_{11}$

$14a_{8872}$ , $\partial$ |  $8_{10}$ = | ( 0, | 2, |  -1 $\mid$ | 1)
---|---|---|---|---|---
$14a_{19472}$ , $\partial$ |  $8_{5}$ = | ( 0, | -1, |  -1 $\mid$ | 1)
$16a_{107998}$ , $\partial$ |  $9_{22}$ = | ( 0, | 3, |  -1 $\mid$ | 1)
$16a_{151960}$ , $\partial$ |  $9_{36}$ = | ( 0, | -2, |  -1 $\mid$ | 1)
$16a_{152693}$ , $\partial$ |  $9_{24}$ = | (-1, | 2, |  -1 $\mid$ | 1)
$16a_{193681}$ , $\partial$ |  $9_{25}$ = | ( 0, | 2, |  -2 $\mid$ | 1)
$16a_{332442}$ , $\partial$ |  $9_{36}$ = | ( 0, | -1, |  -2 $\mid$ | 1)
$16a_{368293}$ , $\partial$ |  $9_{16}$ = | (-1, | -1, |  -1 $\mid$ | 1)

template $t_{12}$

$14a_{18676}$ , $\partial$ |  $3_{1}\sharp 5_{1}$ = | (-1, | 0, |  1 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$16a_{101002}$ , $\partial$ |  $5_{1}\sharp 5_{2}$ = | (-1, | 0, |  3 $\mid$ | -1, | 1)
$16a_{116332}$ , $\partial$ |  $3_{1}\sharp 6_{2}$ = | ( 3, | 0, |  1 $\mid$ | 0, | 1)
$16a_{116809}$ , $\partial$ |  $3_{1}\sharp 6_{3}$ = | ( 2, | -1, |  1 $\mid$ | 0, | 1)
$16a_{202161}$ , $\partial$ |  $4_{1}\sharp 5_{1}$ = | (-1, | 0, |  2 $\mid$ | 0, | 1)
$16a_{293262}$ , $\partial$ |  $3_{1}\sharp 6_{2}$ = | (-1, | -1, |  1 $\mid$ | 0, | 1)

template $t_{13}$

$16a_{20777}$ , $\partial$ |  $3_{1}\sharp 6_{3}$ = | ( 1, | -2, |  -1 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$16n_{101996}$ , $\partial$ |  $3_{1}$ = | ( 1, | 0, |  -1 $\mid$ | 0, | 1)
$16n_{102007}$ , $\partial$ |  $3_{1}\sharp 5_{2}$ = | ( 1, | -1, |  -1 $\mid$ | 0, | 1)

template $t_{14}$

$12a_{427}$ , $\partial$ |  $3_{1}\sharp 5_{1}$ = | ( 1, | 0, |  -1 $\mid$ | 1, | 0)
---|---|---|---|---|---|---
$16a_{202163}$ , $\partial$ |  $5_{1}\sharp 5_{2}$ = | ( 1, | 0, |  -3 $\mid$ | 1, | 0)
$16a_{259485}$ , $\partial$ |  $3_{1}\sharp 6_{2}$ = | ( 1, | 1, |  -1 $\mid$ | 0, | 1)
$16n_{102453}$ , $\partial$ |  $3_{1}\sharp 5_{1}$ = | ( 1, | 0, |  -1 $\mid$ | 0, | 1)

template $t_{15}$

$16a_{19960}$ , $\partial$ |  $3_{1}\sharp 6_{3}$ = | ( 1, | -2, |  1 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$16n_{102000}$ , $\partial$ |  $3_{1}$ = | ( 1, | 0, |  1 $\mid$ | 0, | 1)
$16n_{102009}$ , $\partial$ |  $3_{1}\sharp 5_{2}$ = | ( 1, | -1, |  1 $\mid$ | 0, | 1)

template $t_{16}$

$14a_{6002}$ , $\partial$ |  $4_{1}\sharp 5_{1}$ = | ( 1, | 0, |  2 $\mid$ | 1, | 0)
---|---|---|---|---|---|---
$16a_{259789}$ , $\partial$ |  $3_{1}\sharp 6_{2}$ = | ( 1, | 1, |  1 $\mid$ | 0, | 1)
$16n_{106013}$ , $\partial$ |  $3_{1}\sharp 5_{1}$ = | ( 1, | 0, |  1 $\mid$ | 0, | 1)

template $t_{17}$

$10_{99}$ , $\partial$ |  $9_{1}$ = | ( 1, | 0, | 0, |  0 $\mid$ | 0, | -1, | 1)
---|---|---|---|---|---|---|---|---
$12a_{1105}$ , $\partial$ |  $10_{1}$ = | ( 2, | 0, | 0, |  0 $\mid$ | 0, | -1, | 1)
$12a_{427}$ , $\partial$ |  $9_{2}$ = | (-2, | 0, | 0, |  0 $\mid$ | 0, | -1, | 1)
$14a_{8872}$ , $\partial$ |  $9_{6}$ = | ( 1, | -2, | 0, |  0 $\mid$ | 0, | 0, | 1)
$14a_{18723}$ , $\partial$ |  $10_{2}$ = | ( 1, | 0, | 0, |  1 $\mid$ | 1, | 0, | 0)
$16a_{340770}$ , $\partial$ |  $11a_{234}$ = | ( 1, | 0, | 0, |  2 $\mid$ | 1, | 0, | 0)
$16n_{272647}$ , $\partial$ |  $6_{2}$ = | ( 1, | 0, | -1, |  0 $\mid$ | 1, | 0, | 1)
$16n_{428839}$ , $\partial$ |  $9_{2}$ = | (-2, | 0, | 0, |  0 $\mid$ | 0, | 0, | 1)
$16n_{780729}$ , $\partial$ |  $10_{2}$ = | ( 1, | 0, | 0, |  1 $\mid$ | 1, | -1, | 0)
$16n_{872167}$ , $\partial$ |  $9_{1}$ = | ( 1, | 0, | 0, |  0 $\mid$ | 0, | 0, | 1)

template $t_{18}$

$14a_{19472}$ , $\partial$ |  $9_{4}$ = | ( 1, | 0, | -1, |  -1 $\mid$ | 0, | 0, | 1)
---|---|---|---|---|---|---|---|---
$16a_{368293}$ , $\partial$ |  $10_{9}$ = | ( 1, | 1, | -1, |  -1 $\mid$ | 0, | 0, | 1)
$16n_{988939}$ , $\partial$ | triv = | ( 1, | 0, | 0, |  -1 $\mid$ | 1, | 0, | 1)

template $t_{19}$

$16a_{168328}$ , $\partial$ |  $10_{47}$ = | ( 0, | 2, |  -1 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$16a_{340727}$ , $\partial$ |  $10_{46}$ = | ( 0, | -1, |  -1 $\mid$ | 0, | 1)

template $t_{20}$

$14n_{25903}$ , $\partial$ |  $10_{125}$ = | ( 0, | 1, |  -1 $\mid$ | 0, | 1)
---|---|---|---|---|---|---
$16n_{535891}$ , $\partial$ |  $11n_{13}$ = | ( 0, | 2, |  -1 $\mid$ | 0, | 1)

template $t_{21}$

$14n_{9732}$ , $\partial$ |  $10_{132}$ = | ( 1, | -1, | 0, |  -1 $\mid$ | 1, | 0)
---|---|---|---|---|---|---|---
$16a_{116809}$ , $\partial$ |  $9_{17}$ = | ( 1, | 0, | 0, |  0 $\mid$ | 1, | 0)
$16a_{154813}$ , $\partial$ |  $9_{17}$ = | ( 1, | 0, | 0, |  0 $\mid$ | 1, | -1)
$16n_{919894}$ , $\partial$ |  $11n_{118}$ = | ( 1, | -1, | -2, |  0 $\mid$ | 1, | 0)

template $t_{22}$

$16n_{428839}$ , $\partial$ |  $10_{136}$ = | (0, | -1, |  -1 $\mid$ | 1)
---|---|---|---|---|---
$16n_{849519}$ , $\partial$ |  $7_{7}$ = | (0, | 1, |  -1 $\mid$ | 1)

template $t_{23}$

$16n_{225139}$ , $\partial$ |  $8_{2}$ = | (-1, | 1, | -1, |  1 $\mid$ | 1, | 0)
---|---|---|---|---|---|---|---

template $t_{24}$

$14a_{19470}$ = | ( 0, | 0, |  -1 $\mid$ | 1, | 0)
---|---|---|---|---|---
$16a_{288139}$ = | (-2, | 1, |  -1 $\mid$ | 0, | -1)
$16a_{289378}$ = | (-1, | 1, |  -2 $\mid$ | 0, | -1)

template $t_{25}$

$16a_{66232}$ = | ( 0, | -2, |  0 $\mid$ | 0, | -1)
---|---|---|---|---|---
$16a_{150121}$ = | ( 0, | 1, |  0 $\mid$ | 0, | -1)
$16a_{156092}$ = | ( 0, | 2, |  0 $\mid$ | -1, | 0)
$16a_{275557}$ = | ( 1, | -2, |  0 $\mid$ | 1, | 0)
$16a_{275642}$ = | ( 0, | -2, |  0 $\mid$ | 1, | -1)
$16a_{312423}$ = | ( 1, | 1, |  0 $\mid$ | -1, | 0)
$16a_{324978}$ = | ( 1, | -2, |  0 $\mid$ | -1, | 0)
$16a_{329518}$ = | ( 0, | -2, |  -1 $\mid$ | 1, | 0)
$16n_{404143}$ = | ( 0, | -1, |  -1 $\mid$ | 1, | 1)
$16n_{526315}$ = | ( 1, | 0, |  1 $\mid$ | -1, | 0)
$16n_{918713}$ = | ( 0, | 0, |  1 $\mid$ | 2, | -1)

template $t_{26}$

$16a_{326826}$ = | ( 0, | 2, |  -1 $\mid$ | 0, | 1)
---|---|---|---|---|---
$16a_{330763}$ = | ( 0, | 1, |  1 $\mid$ | 1, | -1)
$16a_{339566}$ = | ( 0, | 0, |  -1 $\mid$ | 1, | 1)
$16a_{357272}$ = | (-1, | 1, |  1 $\mid$ | 1, | 0)

template $t_{27}$

$16a_{106900}$ = | ( 1, | 0, |  2 $\mid$ | 1, | 0)
---|---|---|---|---|---
$16a_{155718}$ = | ( 1, | 0, |  -1 $\mid$ | 1, | 0)
$16a_{359219}$ = | ( 0, | -2, |  -1 $\mid$ | 0, | -1)

template $t_{28}$

$14a_{19517}$ = | ( 1, | 0, | 0, |  -1 $\mid$ | 1, | -1)
---|---|---|---|---|---|---
$16a_{309401}$ = | ( 2, | 0, | 0, |  -1 $\mid$ | 1, | 0)
$16a_{354193}$ = | ( 1, | 0, | 0, |  -2 $\mid$ | 1, | 0)
$16n_{925482}$ = | ( 2, | -1, | 0, |  2 $\mid$ | 1, | 0)

## References

  * [1] K. Boyle, N. Rouse and B. Williams: A classification of symmetries of knots, preprint (2023), arxiv:math.GT/ 2306.04812
  * [2] M. Culler, N.M. Dunfield, M. Goerner and J. Weeks: SnapPy, a computer program for studying the geometry and topology of 3-manifolds
  * [3] N.M. Dunfield and S. Gong: Ribbon concordances and slice obstructions: experiments and examples, in preparation/personal communication
  * [4] E. Flapan: A prime strongly positive amphicheiral knot which is not slice, Math. Proc. Camb. Philos. Soc. 100 (1986), 533–537
  * [5] J. Francis: ’A ffitt place for any Gentleman’, Gardens, gardeners and gardening in England and Wales, c. 1560–1660, PhD thesis, U. Birmingham (2011)
  * [6] R. Hartley and A. Kawauchi: Polynomials of amphicheiral knots, Math. Ann. 243 (1979), 63–70
  * [7] J. Hoste, M. Thistlethwaite and J. Weeks: The first 1,701,936 knots, Math. Intell. 20 (1998), 33–48
  * [8] S. Jabuka and S. Naik: Periodic knots and Heegard Floer correction terms, J. Eur. Math. Soc. 18 (2016), 1651–1674
  * [9] J. Hoste and M. Thistlethwaite: Knotscape – providing convenient access to tables of knots (software), 1999
  * [10] C. Lamm: The search for nonsymmetric ribbon knots, Exp. Math. 30 (2021), 349–363
  * [11] C. Lamm: Symmetric diagrams for all strongly invertible knots up to 10 crossings, preprint (2022), arxiv: math.GT/2210.13198
  * [12] C. Lamm: The enumeration of doubly symmetric diagrams for strongly positive amphicheiral knots, in preparation
  * [13] D.D. Long: Strongly plus-amphicheiral knots are algebraically slice, Math. Proc. Camb. Philos. Soc. 95 (1984), 309–312
  * [14] B. Owens and F. Swenton: An algorithm to find ribbon disks for alternating knots, Exp. Math. (2023), online version
  * [15] A. Ranicki: On signatures of knots. For Cherry Kearton, slides for a talk at Durham (2010)
  * [16] A. Sartori: Polinomi di Alexander e nodi slice, Master thesis, U. Pisa (2010)
  * [17] A. Segre: Untangling the knot: Garden design in Francesco Colonna’s Hypnerotomachia Poliphili, Word $\&$ Image 14 (1998), 82–108

Christoph Lamm
Rückertstr. 3, 65187 Wiesbaden
Germany
e-mail<EMAIL_ADDRESS>


# Pressure-induced structural phase transition of vanadium:
A revisit from the perspective of ensemble theory

Bo-Yuan Ning  Institute of Modern Physics, Fudan University, Shanghai, 200433,
China  Applied Ion Beam Physics Laboratory, Fudan University, Shanghai,
200433, China  Department of Materials Science and Engineering, Southern
University of Science and Technology, Shenzhen 518055, China  Xi-Jing Ning
<EMAIL_ADDRESS>Institute of Modern Physics, Fudan University, Shanghai,
200433, China  Applied Ion Beam Physics Laboratory, Fudan University,
Shanghai, 200433, China

###### Abstract

For realistic crystals, the free energy strictly formulated in ensemble theory
can hardly be obtained because of the difficulty in solving the high-dimension
integral of the partition function, the dilemma of which makes it even a doubt
if the rigorous ensemble theory is applicable to phase transitions of
condensed matters. In the present work, the partition function of crystal
vanadium under compression up to $320$ GPa at room temperature is solved by an
approach developed very recently, and the derived equation of state is in a
good agreement with all the experimental measurements, especially the latest
one covering the widest pressure range up to 300 GPa. Furthermore, the derived
Gibbs free energy proves the very argument to understand most of the
experiments reported in the past decade on the pressure-induced phase
transition, and, especially, a novel phase transition sequence concerning
three different phases observed very recently and the measured angles of two
phases agree with our theoretical results excellently.

The structural phase transition of crystal vanadium (V) under high pressure at
room temperature attracts a long-time interests. Although it had been already
a consensus that the BCC phase is highly stableTakemura (2000) and early
experiments affirmed this stability up to $220$ GPaNakamoto  _et al._ (2005),
theoretical works based on calculations of phonon mode softening and trigonal
shear elastic instability predicted a likely structural transformation within
a range from $130$ GPaSuzuki and Otani (2002) to $200$ GPaLanda  _et al._
(2006a, b), which inspired an experimentDing  _et al._ (2007) in year 2007
showing that a phase transition does take place at about $63$ GPa.
Specifically, the BCC phase transits into a rhombohedral (RH) structure with
the RH angle $\alpha>\alpha_{\text{BCC}}$($109.47^{\circ}$) denoted as RH1 to
distinguish from a similar structure RH2 with $\alpha<\alpha_{\text{BCC}}$.

The above mentioned phase transition was further confirmed by later
experiments Jenei  _et al._ (2011); Yuying  _et al._ (2014); Errandonea  _et
al._ (2019); Wang  _et al._ (2021); Daniele  _et al._ (2016), and the relevant
theoretical works, based on either approximate calculations of free energyYi
X.  _et al._ (2016); Qiu and Marcus (2008) or _ab initio_ lattice dynamicsLuo
_et al._ (2007); A. K. and P. (2007); Lee  _et al._ (2007, 2008); Landa  _et
al._ (2018), reached a qualitative agreement that the transition pressure
($P_{c}$) would be $60\sim 90$ GPa for BCC$\rightarrow$RH1, and predicted that
$P_{c}$ for RH${}_{1}\rightarrow$ RH2 and RH${}_{2}\rightarrow$ BCC are around
$120$ and $250$ GPa respectively. Nevertheless, two experiments reported in
year 2021Akahama  _et al._ (2021); Stevenson  _et al._ (2021) exhibited
different results. Akahama _et al_.Akahama  _et al._ (2021) compressed foil V
up to $300$ GPa at room temperature and found that BCC lattice is a stable
phase until the RH2 and BCC phase coexisting at pressure ($P$) larger than
$242$ GPa, while the RH1 phase is a metastable phase caused by nonhydrostatic
pressure effects. Stevenson _et al_.Stevenson  _et al._ (2021), on the other
hand, observed that the BCC lattice transform into RH2 when $P>40$ GPa and
then turns back to BCC until RH1 emerges for $P>100$ GPa. It is noticeable
that the measured $\alpha$ by Stevenson _et al_. for RH1 (or RH2) is
$109.54^{\circ}$ (or $109.35^{\circ}$), which is significantly different from
the theoretical result, $110.5^{\circ}$ (or $108.5^{\circ}$) for RH1 (or RH2).

In the viewpoint of statistical mechanics, all the discrepancies stated above
as well as others mentioned in literaturesUshcats  _et al._ (2016); Martynov
(1999); Hansen and van Gunsteren (2014) should be settled down as long as the
partition function (PF) can be obtained to produce the free energy (FE).
Unfortunately, the exact solution to the PF of condensed matters is almost
impossible because of the $3N$-fold configurational integral, and various
approximationsLiu (2020), such as the one in Ref.Qiu and Marcus (2008), were
developed to calculate the FE without knowledge of the PF. As expected, those
approximated methods provided lots of interesting information on the phase
transitions of condensed matters, while may not address all the issues
substantially.

Very recently, we put forward a direct integral approach (DIA) to the PF of
condensed state systems with ultrahigh efficiency and precisionNing  _et al._
(2021); Liu  _et al._ (2019); Gong  _et al._ (2019, 2020), and has been
successfully applied to reproduce the equation of state (EOS) for solid
copperNing  _et al._ (2021), argonGong  _et al._ (2019) and $2$-D materialsLiu
_et al._ (2019) obtained from experiments or molecular dynamics simulations.
Compared with phonon model based on harmonic or quasi-harmonic approximations,
which is currently applied to produce EOS, DIA is applicable to much wider
realm with much higher precisionGong  _et al._ (2020). In the present work,
DIA is used to compute the PF of crystal V with various phases and the derived
Gibbs FE is applied to investigate the phase transitions induced by pressure
at room temperature.

For a crystal containing $N$ atoms confined within volume $\mathcal{V}$ at
temperature $T$, the atoms are regarded as $N$ point particles of the atomic
mass $m$ with Cartesian coordinate
$\mathbf{q}^{N}=\\{\mathbf{q}_{1},\mathbf{q}_{1},\ldots\mathbf{q}_{N}\\}$, and
the total potential energy, $U(\mathbf{q}^{N})$, as the function of
$\mathbf{q}^{N}$ is computed by quantum mechanics, i.e., for a given set of
$\mathbf{q}^{N}$, the total potential energy $U(\mathbf{q}^{N})$ concerned
with the motions of electrons in the field of the nucleus fixed at the lattice
sites is calculated by quantum mechanics. With knowledge of
$U(\mathbf{q}^{N})$, the PF of the system reads

$\begin{split}\mathcal{Z}&=\frac{1}{N!}\left(\frac{2\pi m}{\beta
h^{2}}\right)^{\frac{3}{2}N}\int d\textbf{q}^{N}\exp[-\beta
U(\textbf{q}^{N})]\\\ &=\frac{1}{N!}\left(\frac{2\pi m}{\beta
h^{2}}\right)^{\frac{3}{2}N}\mathcal{Q},\end{split}$ (1)

where $h$ is the Planck constant and $\beta=1/k_{B}T$ with $k_{B}$ the
Boltzmann constant. If the configurational integral $\mathcal{Q}=\int
d\textbf{q}^{N}\exp[-\beta U(\textbf{q}^{N})]$ is solved, then the pressure
$P$ and the Gibbs FE $G$ can be computed by

$\displaystyle P$ $\displaystyle=$
$\displaystyle\frac{1}{\beta}\frac{\partial\ln\mathcal{Q}}{\partial\mathcal{V}},$
(2) $\displaystyle G$ $\displaystyle=$
$\displaystyle-\frac{1}{\beta}\ln[\frac{1}{N!}\left(\frac{2\pi m}{\beta
h^{2}}\right)^{\frac{3}{2}N}]-\frac{1}{\beta}\ln\mathcal{Q}+P\mathcal{V}.$ (3)

In this way, the contributions from both the electrons and nucleus are
included in the calculations.

According to our proposed DIANing  _et al._ (2021), for a single-component
crystal with $N$ atoms placed in their lattice sites $\mathbf{Q}^{N}$ and with
the total potential energy $U_{0}(\mathbf{Q}^{N})$, we firstly introduce a
transformation,

$\mathbf{q}^{\prime N}=\mathbf{q}^{N}-\mathbf{Q}^{N},\
U^{\prime}(\mathbf{q}^{\prime N})=U(\mathbf{q}^{\prime
N})-U_{0}(\mathbf{Q}^{N}),$ (4)

where $\mathbf{q}^{\prime N}$ represents the displacements of atoms away from
their lattice positions, and then the configurational integral can be
expressed in a one-fold integral,

$\mathcal{Q}=e^{-\beta U_{0}}\left[\int e^{-\beta
U^{\prime}(q^{\prime}_{i_{x,y,z}})}dq^{\prime}_{i_{x,y,z}}\right]^{3N}=e^{-\beta
U_{0}}\mathcal{L}^{3N},$ (5)

where $q^{\prime}_{i_{x,y,z}}$ denotes the distance of the $i$th atom moving
along the $x$ (or $y$, $z$) direction relative to its lattice site while the
other two degrees of freedom of the atom and all the other atoms are kept
fixed. As shown in Fig.1, we take BCC structure of V placed in a $3\times
3\times 3$ RH supercell as an example to illustrate the implementation of the
DIA. The basis vectors of the primitive cell are set as
$\mathbf{a}_{1}=a_{\text{rh}}\cdot(1,0,0)$,
$\mathbf{a}_{2}=a_{\text{rh}}\cdot(\cos\alpha,\sin\alpha,0)$,
$\mathbf{a}_{3}=a_{\text{rh}}\cdot(\cos\alpha,\frac{\cos\alpha-\cos^{2}\alpha}{\sin\alpha},\frac{\sqrt{1-3\cos^{2}\alpha+2\cos^{3}\alpha}}{\sin\alpha})$
and the volume of the cell equals to $\displaystyle
a_{\text{rh}}^{3}\sqrt{1-3\cos^{2}\alpha+2\cos^{3}\alpha}$, where
$a_{\text{rh}}=\sqrt{3}/2a_{c}$ with $a_{c}$ for the lattice constant of a
cubic primitive cell. According to Eq.(5), an arbitrary atom is selected and
moved $0.5$ Å by a step of $0.05$ Å along the direction of $\mathbf{a}_{1}$ as
shown in Fig.1(a) with the initial and final positions of the atom colored in
blue. During the movement, the total potential energy is computed by the
density functional theory (DFT) at every steps and the spline interpolation
algorithmDierckx (1975); *spl2 is used to smooth the $U^{\prime}(x^{\prime})$
curves, which are shown in Fig.1(b).

Figure 1: (Color Online) (a) Schematic of DIA to the BCC structure in a RH
supercell and (b) the calculated $U^{\prime}(x^{\prime})$ for different
supercell volumes with $a_{c}$ changed from $0.81$ to $1.01a_{0}$
($a_{0}=3.0$Å).

The DFT calculations is performed in Vienna Ab initio Simulation PackageKresse
and Furthmüller (1996a, b) with the projector-augmented wave formalismBlöchl
(1994); Kresse and Joubert (1999), and the general gradient approximation of
the Perdew-Burke-Ernzerhf parametrizationsPerdew  _et al._ (1996) is adopted
for the exchange-correlation functional with 13 valence electrons
($3s^{2}p^{6}d^{3}4s^{2}$) consideredZhang  _et al._ (2021). A
$\Gamma$-centered $9\times 9\times 9$ uniform $k$-mesh grid is set to sample
the Brillouin zone by the Monkhorst-Pack schemeMonkhorst and Pack (1976) and
the tetrahedron method with Blöchl corrections is used to determine the
electron orbital partial occupancy, together with $342.7749$ eV set as the
cut-off energy of the plane-wave basis, $1\times 10^{-6}$ eV as the
convergence energy criterion of the electron self-consistent computations.

Figure 2: (Color Online) Isothermal $P$-$\mathcal{V}$ curve of the BCC phase
at $300$K from the PF (red dash line) and the experimental data denoted in
different colored symbols. The inset shows the detailed comparisons in the
low-pressure zone.

As shown in Fig.2, the isothermal $P$-$\mathcal{V}$ curve derived from the PF
via Eq.(2) coincides very well with the experiments, which can be seen from
the relative difference (RD) between the theoretical volume
$(V/V_{0})_{\text{PF}}$ and the experimental one $(V/V_{0})_{\text{exp}}$,
defined as
RD$=\frac{|(V/V_{0})_{\text{PF}}-(V/V_{0})_{\text{exp}}|}{(V/V_{0})_{\text{exp}}}$,
where $V_{0_{\text{PF}}}$ and $V_{0_{\text{exp}}}$ are the theoretical and
experimental atomic volumes under ambient conditions (room temperature and one
atmospheric pressure). In the low-pressure zone ($P<20$ GPa), the average RD
for all the shown experiments is about $0.25\%$, which can be seldom achieved
in common theoretical work without using empirical data or empirical EOS. In
the higher-pressure zone ($P>20$ GPa), the RD for all the experiments except
for the one of Ref.Akahama  _et al._ (2021) get a little larger with the
pressure increasing, which may be attributed to the difficulty in precise
measurements of higher pressure in considerations of the fact that the results
of different experiments diverge larger with increases of the pressure (see
the experimental points shown in the inset of Fig.2). It is worthwhile to see
that the RD for one of the latest experimentsAkahama  _et al._ (2021) (grey
diamonds shown in Fig.2), covering the widest pressure range, keeps to be
smaller than $0.7\%$ except for three pressure points at $135$, $176$ and
$240$ GPa. Such an excellent agreement enables us to theoretically determine
the lattice constant under exactly given pressure and temperature. As listed
in Table.1, the RD between the theoretical $a_{c}$ and the experimental ones
is smaller than $0.73\%$, exhibiting a good agreement.

It should be pointed out that the implementation of DIA excludes all the
artificial dependence of adjustable parameters, empirical EOS or experimental
data, and accordingly, the good agreements between the theory and the
experiments for the EOS and the lattice constant strongly indicate that the
ensemble theory is the very approach to understand the thermodynamics
properties of condensed matters.

Table 1: The lattice constant $a_{c}$ of BCC phase at ambient conditions calculated by DIA and from experiments fitting empirical EOSs, Birch-Murnaghan (B-M), Vinet and AP2. | DIA | Ref.Errandonea  _et al._ (2019) | Ref.Daniele  _et al._ (2016) | Ref.Akahama  _et al._ (2021) | Ref.Stevenson  _et al._ (2021)
---|---|---|---|---|---
$a_{c}$ (Å) | 3.008 | 3.029 | 3.03 | 3.027 | 3.023
Deviations | / | $0.693\%$ | $0.726\%$ | $0.627\%$ | $0.496\%$
EOS | / | B-M | _no report_ | Vinet | AP2

To consider the transitions of crystal V from the BCC to RH phases with the
angle $\alpha$ different from $\alpha_{\text{BCC}}$, the PF of four RH1 and
RH2 phases with the angle deviation,
$\Delta=(\alpha_{\text{RH}}-\alpha_{\text{BCC}})/\alpha_{\text{BCC}}$, $\pm
0.1\%$, $\pm 0.2\%$, $\pm 0.5\%$ and $\pm 1\%$ is solved by the DIA in the
same way as described above (see potential-energy curves of the RH phases in
supplementary material), and, the derived Gibbs FE differences between the RH
and BCC per atom, $\mathcal{G}_{\text{RH}}-\mathcal{G}_{\text{BCC}}$, are
shown in Fig.3(a). When the amplitude of the angle deviation, $|\Delta|$,
equals to $0.1\%$, the Gibbs FE of both RH1 and RH2 is close to
$\mathcal{G}_{\text{BCC}}$, and increases abruptly to be larger than
$\mathcal{G}_{\text{BCC}}$ by more than $120$ meV/atom with the $|\Delta|$
increased by only $0.1\%$, while further increasing the angle deviation leads
to a little changes of $\mathcal{G}_{\text{RH}}$. These results show clearly,
based on the Gibbs FE criterion, that the phase transition with the angle
equal to or larger than $0.2\%$ cannot take place unless the deviation being
smaller, and furthermore, Fig.3(b) indicates that the two phases, RH1 with a
deviation of $0.1\%$ and RH2 with a deviation of $-0.1\%$, would emerge when
the pressure is larger than $20$ GPa. It is interesting to note that the phase
transition was very recently examined by Stevenson et al.Stevenson  _et al._
(2021) who concluded that two kinds of phase transitions, BCC$\rightarrow$ RH1
and BCC $\rightarrow$ RH2, indeed take place, qualitatively coinciding with
the previous experiments or theories, and the determined RH angle for RH1 (or
RH2) is $109.54^{\circ}$ (or $109.35^{\circ}$), corresponding to $|\Delta|\sim
0.1\%$ as our theoretical results, which is a little smaller than the one,
$109.65^{\circ}$ for RH1, measured in previous experimentDing  _et al._
(2007); Jenei  _et al._ (2011) where RH2 was not observed, but significantly
different from the theoretically predicted $110.5^{\circ}$ (or
$108.5^{\circ}$)Luo  _et al._ (2007); Lee  _et al._ (2007); A. K. and P.
(2007); Qiu and Marcus (2008); Yi X.  _et al._ (2016), corresponding to
$|\Delta|\sim 1\%$. Since the unique difference between BCC and RH1 (or RH2)
phase is the RH angle deviation of about only $0.1\%$, it should be quite a
challenge for experimental observations of the subtle difference, which may be
the reason why early literatures reported no phase transitions for V induced
by pressure less than $220$ GPaNakamoto  _et al._ (2005).

Figure 3: (Color Online) (a) Gibbs FE of RH phases relative to BCC phase at
room temperature up to $170$ GPa and (b) the data for RH phases with $\pm
0.1\%\cdot\alpha_{\text{BCC}}$ for clarity.

Now we make a detailed comparison between our theoretical results and the
experimental observations. As shown in Fig.3(b), the Gibbs FE of the RH1 and
RH2 differ from that of BCC less than $1$ meV, which is much smaller than the
average kinetic energy, $\sim 30$ meV, of a thermal atom at room temperature,
and it seems that RH1 or RH2 is just the thermal fluctuation from the BCC
structure instead of a phase transition. Actually, the very “driving force”
accounting for the phase transition under given pressure and temperature is
not the thermal kinetic energy (or internal energy), but instead, the Gibbs FE
that determines the probability for a given phase existence. According to
ensemble theory, the relative probability for RH with respective to BCC equals
to $e^{(G_{\text{BCC}}-G_{\text{RH}})/k_{B}T}$, where
$G_{\text{BCC}}=N\mathcal{G}_{\text{BCC}}$,
$G_{\text{RH}}=N\mathcal{G}_{\text{RH}}$, and $N$ the number of the atoms in a
piece of macro bulk crystal is on an order of $10^{23}$. Thus, our calculation
results indicate clearly that the phase transition BCC $\rightarrow$ RH2 must
take place at room temperature when the pressure is larger than $18$ GPa, and
then RH1 emerges at $28$ GPa. This RH1 transition was observed by Jenei et
al.Jenei  _et al._ (2011) at $32$ GPa and by Daniele et al.Daniele  _et al._
(2016) at $35$ GPa, respectively. As the pressure increasing, our results show
that RH2 would emerge at $\sim 44$ GPa, then transforms back to BCC for the
pressure larger than $50$ GPa until RH1 emerges at $61$ GPa, displaying a
transition process, RH${}_{2}\rightarrow$ BCC $\rightarrow$ RH1, with the
pressure increased. Experimentally, Akahama et al.Akahama  _et al._ (2021)
observed a RH phase comes to existence from $45$ GPa, and the BCC
$\rightarrow$ RH1 transition was reported at $63$ GPa in Ref.Ding  _et al._
(2007), $60.5$ GPa in Ref.Yuying  _et al._ (2014) and $64$ GPa in
Ref.Errandonea  _et al._ (2019), which are all in a good agreement with our
results, in considerations of the deviations among the measured pressures
being of $\sim 2$ GPa. It is noticeable that the RH${}_{2}\rightarrow$ BCC
$\rightarrow$ RH1 transition sequence, neither observed in earlier experiments
nor proposed by previous computations, was observed in the one of the latest
experimentsStevenson  _et al._ (2021), and the measured transition pressure of
RH2, $40$ GPa, coincides with our theoretical one, $44$ GPa. When the pressure
is increased larger than $132$ GPa, our results indicate that RH1 would exist
until RH2 transition occurs at $154$ GPa and the RH2 phase keeps stable with
the pressure up to $254$ GPa, over which the BCC may coexist with RH2 until
the pressure increases up to $280$ GPa when BCC becomes the most stable for
higher pressures. It is very interesting to note that this coexistence phase
of RH2 and BCC, never reported in previous literatures, was observed by
Akahama et al.Akahama  _et al._ (2021) very recently at $242$ GPa.

The above discussions show that most results of our calculations coincide well
with the experiments, especially the EOS, the angle deviations of the RH and
some transition pressures, such as $30$, $40$, $60$, $250$ GPa for BCC
$\rightarrow$ RH1, RH2 existence, BCC $\rightarrow$ $RH_{1}$ and coexistence
of RH2 and BCC, respectively. Based on these facts, it should be reasonable to
expect that the disagreements between our results and the experimental
observations, such as the theoretical phase transition under $\sim 20$ GPa,
which was not reported in all the experiments, may be settled down by future
experiments.

In conclusion, the rigorous ensemble theory is applied, for the first time, to
investigate phase transitions of crystal V via DIA solving the PF without any
artificial tunable parameters or empirical EOS, achieving very good agreements
with experimental measurements and observations, and showing that the theory
is substantial to describe phase transitions of condensed matters. The
theoretical approach of this work may find its vast applications in the field
to predict parameter-free EOS and phase behaviors of condensed matters under
extreme conditions.

##  I acknowledgement

BYN is grateful to A. Daniele, M. McMahon and Y. Akahama for their kindly
providing the raw experimental data in Refs.Daniele  _et al._ (2016), Akahama
_et al._ (2021) and Stevenson  _et al._ (2021), respectively. Part of the
computational tasks was conducted in HPC platform supported by The Major
Science and Technology Infrastructure Project of Material Genome Big-science
Facilities Platform supported by Municipal Development and Reform Commission
of Shenzhen.

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