SYMMETRIC DIAGRAMS FOR ALL STRONGLY INVERTIBLE KNOTS UP TO 10 CROSSINGS CHRISTOPH LAMM

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SYMMETRIC DIAGRAMS FOR ALL STRONGLY INVERTIBLE KNOTS UP

TO 10 CROSSINGS

CHRISTOPH LAMM

ABSTRACT. We present a table of symmetric diagrams for strongly invertible knots up to

10 crossings, point out the similarity of transvergent diagrams for strongly invertible knots

with symmetric union diagrams and discuss open questions.

1. INTRODUCTION

A knot K⊂S3is said to be strongly invertible if there is an orientation preserving

smooth involution hof S3such that h(K) = Kand hreverses the orientation on K.

Sakuma’s article ‘On strongly invertible knots’ [15] which appeared in 1986 contains a

table of symmetric diagrams of strongly invertible knot up to crossing number 9. Strongly

invertible knots currently receive a lot of attention, see for instance [1], [2], [12] and [16].

Our own motivation to extend the table of symmetric diagrams to knots with crossing

number 10 stems from the related symmetry type of symmetric unions of knots and a

project to study strongly positive amphicheiral knots.

Strongly invertible knots can be depicted as transvergent diagrams (in which the axis

of rotation lies in the diagram plane) or as intravergent diagrams (where the axis is per-

pendicular to the diagram plane). The purpose of this article is to give one transvergent

diagram for each (prime) strongly invertible knot with c(K)≤10.

Deﬁnition 1.1. For a knot Kwe denote, as usual, by c(K)the minimal crossing number

of a diagram of K. If Kis strongly invertible we denote by ct(K)the minimal crossing

number of all transvergent diagrams of K.

Symmetric equivalence can be deﬁned for strongly invertible knots either using the

conjugacy class of hin the symmetry group of Kor by symmetric diagram moves (see

[12], Section 2).

Interestingly, especially when compared to the case of symmetric unions [4], for hyper-

bolic knots there are always one or two equivalent classes (two equivalent classes occurring

for knots with period 2). Whereas Sakuma lists diagrams showing both classes simultane-

ously (for knots having two of them), we contend with only one diagram for each knot in

the main part of this article. However, in Appendix B we give a list of diagrams for knots

with two equivalence classes.

We focus on the crossing number of the symmetric diagrams and try to ﬁnd minimal

(transvergent) diagrams. In many cases symmetric diagrams exist with c(K)crossings, so

that ct(K) = c(K). These symmetric diagrams with minimal crossing number are occasion-

ally also contained in knot tables (for instance in Rolfsen’s table the knots 939, 940, 941 are

shown with a vertical symmetry axis). For our list we cannot guarantee that the crossing

number of a diagram is minimal if it exceeds c(K)because we did not exhaustively check

all symmetric diagrams.

2020 Mathematics Subject Classiﬁcation. 57K10.

Key words and phrases. strongly invertible knots.

SYMMETRIC DIAGRAMS FOR ALL STRONGLY INVERTIBLE KNOTS UP TO 10 CROSSINGS 2

2. MOTIVATION

Symmetric unions of knots were introduced by Kinoshita and Terasaka in 1957 [6].

They consist of the connected sum of a knot and its mirror image with reversed orienta-

tion, with crossings inserted on the symmetry axis. The building blocks of transvergent

diagrams of strongly invertible knots are very similar: A connected sum of a knot diagram

and a rotational copy, with crossings inserted on the symmetry axis. Figure 1 shows an ex-

ample for this similarity. In a future study we will compare methods for symmetric unions

and strongly invertible knots and check if the transfer of results is possible.

FIGURE 1. The knot 946. Left: as a symmetric union (mirror symme-

try), right: as a strongly invertible diagram (rotational symmetry with

inversion of orientation).

Another aim is the construction of strongly positive amphicheiral knots as symmetric

unions with strongly invertible partial knots. The diagrams used for this construction pos-

sess two symmetry axes and a rotation around an axis perpendicular to the diagram plane

yields the strongly positive amphicheiral property, see Figure 2.

FIGURE 2. A strongly positive amphicheiral knot diagram with two axes

of symmetry (showing the knot 12a1019)

A knot is called strongly positive amphicheiral if it has a diagram which is mapped

to its mirror image by a rotation of π, preserving the orientation. Obviously, composite

knots consisting of a knot and its mirror image are strongly positive amphicheiral. Prime

strongly positive amphicheiral knots are rare, however: Until recently only three prime

strongly positive amphicheiral knots with 12 or fewer crossings were known: 1099, 10123

and 12a427. In [9] we found additional examples, so that currently the list consists of 1099,

10123, 12a427, 12a1019, 12a1105, 12a1202, and 12n706. A future study will extend this

list to strongly positive amphicheiral knots with crossing numbers up to 16 (remark: see

[10]).

SYMMETRIC DIAGRAMS FOR ALL STRONGLY INVERTIBLE KNOTS UP TO 10 CROSSINGS 3

3. TEMPLATE NOTATION AND GENERATION OF THE TABLE

All two-bridge knots are strongly invertible and because they are also 2-periodic they

have two equivalent classes of strong invertibility, see [15]. Sakuma shows how to ﬁnd

representatives for each knot and class. Therefore we do not need to list diagrams of two-

bridge knots in detail and we concentrate on the remaining knots up to crossing number

10 (which all have bridge number 3). For two-bridge knots we will discuss the question

whether symmetric diagrams exist with c(K)crossings, however.

Our tabulating approach uses templates, in a similar way as in [9]. As half of a diagram

contains already the information for the whole diagram, a template only shows the upper

half and integer markings on the axis for the twists inserted on the axis. The axis is placed

horizontally.

Figure 3 shows how a template diagram is constructed from a transvergent diagram.

-1 -1 -2

-1 -1

-2

FIGURE 3. Constructing a template representation from a transvergent

diagram (the example shows B1(−2,−1,−1) = 1076)

In the other direction we proceed as in Figure 4: If a template diagram is given we need

to reconstruct the other half and insert the crossings on the axis.

1 1

-2

FIGURE 4. Converting a template into a knot diagram (the example

shows C1(1,−2,1) = 10122)

The list of all strongly invertible 3-bridge knots in Appendix A was compiled in the

following way: We started with Sakuma’s table and converted his diagrams into templates.

Variation of the twist attributes on the axis then yielded many 10 crossing knots (using

Knotscape [8]). Other sources, as Rolfsen’s table and examples in [2] and [12] were used in

the same way. The remaining 8 cases were manually transformed into symmetric diagrams

SYMMETRIC DIAGRAMS FOR ALL STRONGLY INVERTIBLE KNOTS UP TO 10 CROSSINGS 4

with the help of the software KLO [7], see the Acknowledgments. From the collection of

all examples we chose one diagram with smallest crossing number for our table. As we

remarked, it is expected that transvergent diagrams exist with smaller crossing number for

some of them.

4. TWO-BRIDGE KNOTS

For two-bridge knots we use the Conway notation C(a1,...,an)(and a shorter version

[a1,...,an]in Figure 7). As usual, the plat diagram’s closing patterns are different for even

and odd n, see Figures 5 and 6.

-a1-a3

-1

-2

-a5

-a1-a3-a5

a4a6

FIGURE 5. Horizontal symmetry for alternating two-bridge knot dia-

grams

-1

-2

-a1

-a3

-a1

FIGURE 6. Vertical symme-

try for alternating two-bridge

knot diagrams

In these ﬁgures we give examples for n=6 (horizontal

placement and axis) and n=5 (vertical case, but still with

horizontal axis).

Sakuma’s representatives for the two equivalence

classes use ‘even’ continued fractions. Each two-bridge

knot can be written as C(a1,...,an)with even numbers ai.

In this case nis also even and half of nis the knot’s genus.

If all aiare even then the ﬂyping modiﬁcation shown in

Figure 5 is possible. However, to proceed as in the ﬁgure,

only ever second half-twist number needs be even.

In order to ﬁnd diagrams with ct(K) = c(K)we there-

fore try to transform the Conway notation of each 2-bridge

knot so that it realizes the minimal crossing number and

has even entries for ever second ai. For example, for 2-

bridge torus knots we may use T(2,2n+1) = C(2n+1) =

C(1,2n). This condition is the ﬁrst part of the following

proposition. The second part uses a symmetric diagram

which is placed vertically and is illustrated in Figure 6.

Proposition 4.1. A two-bridge knot K has a transvergent

diagram with c(K)crossings, if

•K can be written as C(a1,...,an)with even n and all ai>0where the twist-

numbers a2, a4, . . . are even, or

•K can be written as C(a1,...,an)with odd n and all ai>0and symmetric twist-

numbers a1=an, a2=an−1, . . . . In this case a n+1

2is necessarily odd.

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SYMMETRICDIAGRAMSFORALLSTRONGLYINVERTIBLEKNOTSUPTO10CROSSINGSCHRISTOPHLAMMABSTRACT.Wepresentatableofsymmetricdiagramsforstronglyinvertibleknotsupto10crossings,pointoutthesimilarityoftransvergentdiagramsforstronglyinvertibleknotswithsymmetricuniondiagramsanddiscussopenquestions.1.INTRODUCTIONAknotK⊂S...

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SYMMETRIC DIAGRAMS FOR ALL STRONGLY INVERTIBLE KNOTS UP TO 10 CROSSINGS CHRISTOPH LAMM

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