A Table of Handcuff Graphswith up to Seven Crossings

Hiromasa MORIUCHI

Abstract

We enumerate all prime handcuff graphs with up to seven crossings byextending Conway’s method. These handcuff graphs are mutually distin-guished by the Yamada polynomial.

1 Introduction

A spatial graph is a graph embedded inS3, and two spatial graphsG, G′ areequivalentif there is a homeomorphismh : (S3, G) → (S3, G′). There existearlier studies on tabulation of spatial graphs as follows: In [1], J. H. Conwayenumerated prime knots with up to eleven crossings, and links with up to tencrossings. In [16], J. Simon enumeratedθ-curves with up to five crossings, andK4 graphs with up to four crossings. In [9], R. Litherland announced a table ofprimeθ-curves with up to seven crossings. However, there is no published proofof the completeness of his table. In [10], the author enumerated all the primeθ-curves with up to seven crossings by extending Conway’s method. Theseθ-curvesare the exactly same ones as in Litherland’s.

In this paper we enumerate all prime handcuff graphs with up to seven cross-ings by extending Conway’s method. The following are our main results.

Theorem 1.1.Table1 lists all prime handcuff graphs with up to seven crossings.

Theorem 1.2. There exist only three achiral handcuff graphs in Table1 : 21, 61,and717.

The handcuff graphs in Table 1 are listed in order of increasing crossing num-ber and their constituent links.

1

Table 1: Prime handcuff graphs with up to seven crossings.

2

This paper is organized as follows: In Section2, we give some definitionsfor a handcuff graph. In Section3, we introduce the concept of a prime basicθ-polyhedron. In Section4, we classify the handcuff graphs in Table 1 by theYamada polynomials and consider the chirality of these handcuff graphs.

2 Handcuff graphs

A handcuff graphΦ is a graph embedded inS3 consisting of two vertices(v1, v2)and three edges(e1, e2, e3), wheree3 has distinct endpointsv1 andv2, ande1 ande2 are loops based atv1 andv2, respectively. Aconstituent linkΦ12 is a subgraphof Φ that consists of two vertices(v1, v2) and two edges(e1, e2). A handcuff graphis said to betrivial if it can be embedded in a2-sphere inS3.

Let Φ be a handcuff graph andΣ a 2-sphere which decomposesS3 into 3-ballsB1, B2. If Σ ∩ Φ consists of a single pointw, and neither(Φ − w) ∩ B1 nor(Φ − w) ∩ B2 is empty, thenΣ is called anadmissible sphere of typeI for Φ. IfΣ∩Φ consists of two points, and the annulusA = Σ \ IntN(Φ; S3) is essential inS3 \ IntN(Φ; S3), thenΣ is called anadmissible sphere of typeII for Φ; cf. [17].If Σ ∩ Φ consists of three points, and neitherΦ ∩ B1 nor Φ ∩ B2 is an unknottedbouquet (Fig. 1), thenΣ is called anadmissible sphere of typeIII for Φ; cf. [2].By anadmissible sphere, we mean either an admissible sphere of type I, II, or III.

Fig. 1: An unknotted bouquet.

Definition 2.1. A handcuff graphΦ is said to beprime if Φ is nontrivial and doesnot have an admissible sphere.

For example, handcuff graphs in Fig. 2 (a),(b),(c) have admissible spheres oftypes I, II, III, respectively. Therefore, these handcuff graphs are nonprime.

3

Fig. 2: Nonprime handcuff graphs.

3 Prime basicθ-polyhedra

In this section, we introduce the concept of a prime basicθ-polyhedron in orderto enumerate prime handcuff graphs.

Let PΘ be a connected planar graph. ThenPΘ is called aθ-polyhedronif itstwo vertices are 3-valent and the other vertices are 4-valent. Aθ-polyhedronPΘ

is said to bebasicif it contains no loop and no bigon.

Remark .If PΘ contains a loop, thenPΘ produces a handcuff graph diagram whichhas an admissible sphere of type I or II (Fig. 3).

Fig. 3: Admissible spheres.

Remark .If PΘ contains a bigon, then a handcuff graph diagram obtained fromPΘ

is also obtained from another polyhedronP ′Θ with fewer 4-valent vertices thanPΘ.

In fact, adding two algebraic tangles, we obtain another algebraic tangle (Fig. 4).

4

Fig. 4: The sum of algebraic tangles.

Let C be a circle which decomposesS2 into 2-disksD1, D2. If C meetsPΘ inless than or equal to three points, and bothPΘ ∩D1 andPΘ ∩D2 contain4-valentvertices, thenC is called acutting circlefor PΘ (Fig. 5).

Fig. 5: Nonprimeθ-polyhedra.

Definition 3.1. A θ-polyhedronPΘ is said to beprime if PΘ does not have acutting circle.

Remark .A nonprimePΘ produces a nonprime handcuff graph. However, a primebasicPΘ may produce a nonprime handcuff graph; see Table 3.

In [10], we constructed a prime basicθ-polyhedron. We consider two typesof PΘ: type-× and type-∗. Here, a type-× PΘ has adjacent 3-valent vertices,

5

and a type-∗ PΘ does not have adjacent 3-valent vertices. From now on, 4-valentvertices will be denoted by or in the figures.

We obtained the following theorems ([10]):

Theorem 3.2.There exist seven type-× prime basicθ-polyhedra with up to seven4-valent vertices as in Fig.6.

Fig. 6:(a)11

×a (b) 51×a.b.c.d.e (c) 61

×a.b.c.d.e.f (d) 71×a.b.c.d.e.f.g

(e)72×a.b.c.d.e.f.g (f) 73

×a.b.c.d.e.f.g (g) 74×a.b.c.d.e.f.g

Theorem 3.3. There exist seventeen type-∗ prime basicθ-polyhedra with up toseven4-valent vertices as in Fig.7.

6

Fig. 7:(a)31

∗a.b.c (b) 41∗a.b.c.d (c) 51

∗a.b.c.d.e (d) 61∗a.b.c.d.e.f

(e)62∗a.b.c.d.e.f (f) 63

∗a.b.c.d.e.f (g) 64∗a.b.c.d.e.f

(h) 71∗a.b.c.d.e.f.g (i) 72

∗a.b.c.d.e.f.g (j) 73∗a.b.c.d.e.f.g (k) 74

∗a.b.c.d.e.f.g(l) 75

∗a.b.c.d.e.f.g (m) 76∗a.b.c.d.e.f.g (n) 77

∗a.b.c.d.e.f.g(o) 78

∗a.b.c.d.e.f.g (p) 79∗a.b.c.d.e.f.g (q) 710

∗ a.b.c.d.e.f.g

7

4 Conclusion

From Theorems 3.2 and 3.3, we can obtain all prime handcuff graph diagramsfrom prime basicPΘ by substitutingalgebraic tangles([1]) for their 4-valent ver-tices. We note that the vertex corresponds to a tangleT , and corresponds tothe mirror image ofT .

We introduce the following invariant due to Yamada ([20]) in order to classifythese handcuff graphs up to ambient isotopy.

Definition 4.1. Let g be a spatial graph diagram. ThenR(g) ∈ Z[x±1] is definedby the following recursive formulas:

(1) R(∅) = 1, where∅ denotes the empty graph,

(2) R( )

= xR( )

+ x−1R( )

+ R( )

,

(3) R( )

= R( )

+ R( )

, wheree is a nonloop edge,

(4) R(g1⊔g2) = R(g1)R(g2), whereg1⊔g2 denotes the disjoint union of spatialgraph diagramsg1 andg2,

(5) R(Bn) = −(−x − 1 − x−1)n, whereBn is then-leafed bouquet.

In particular,R(•) = R(B0) = −1, R( )

= R(B1) = x + 1 + x−1.

Theorem 4.2. If g is a diagram of a spatial graph whose maximum degree is lessthan 4, thenR(g) is ambient isotopy invariant ofg up to multiplying(−x)n forsome integern.

We call R(g) the Yamada polynomial. We can show that handcuff graphsin Table 1 are mutually distinct by investigating their constituent links and theYamada polynomial; see Table 2.

We can also compute the Yamada polynomial of a handcuff graph by the fol-lowing way. In [21], Yamada introduced a polynomial invariant for a 3-regularspatial graphΓ with some good weightω, which is the linear combination ofthe bracket polynomial. First, we mention theTemperley-Lieb algebra. ThemthTemperley-Lieb algebraVm is an algebra over the field of complex numbersC

8

generated by the elements1m, ε1, . . . , εm−1 and the following relations:

(TL1) εiεj = εjεi if |i − j| ≥ 2,

(TL2) ε2i = δεi where δ = −q − q−1 andq ∈ C,

(TL3) εiεi±1εi = εi.

The elements1m andεi diagrammatically appear as in Fig. 8.

Fig. 8: The elements1m andεi.

For any non-negative integerm, we set

[m] =qm − q−m

q − q−1.

Then we define elementsgn andfn of Vm by

gn =1

[n]

n∑j=1

[j]εj . . . εn−1,

fn = gngn−1 . . . g1.

The elementsgn andfn hold the following recursive formulas,

gn+1 =[n]

[n + 1]gnεn + 1,

fn+1 = gn+1fn.

By induction, it follows easily thatgn − 1 and fn − 1 are elements in theproper subalgebraA(ε1, . . . , εn−1) generated byε1, . . . , εn−1. So that,gn andfn

are commute withεn+1, . . . , εm−1.The following lemma is given by Wenzl ([18]), Lickorish ([8]), and Yamada

([21]).

9

Lemma 4.3.fn is the unique element inVm which is generated by1, ε1, ε2, . . . , εn−1

and holds the following:

(E1) εifn = fnεi = 0, for all i ≤ n − 1,

(E2) f2n = fn.

We callfn themagic knitting of degreen.Let Γ be a3-regular graph andω : E(Γ) → Z+ be a positive integer valued

map. We sayω is agood weightonΓ if the following conditions are satisfied. Foreach vertex ofΓ, if e1, e2, ande3 are the edges incident with the vertex, then

(V1) ω(e1) + ω(e2) + ω(e3) ∈ Z2,

(V2) |ω(e1) − ω(e2)| ≤ ω(e3) ≤ ω(e1) + ω(e2).

From now on, we consider the caseω(e) = 2 for each edgee. Letω be a goodweight on a spatial 3-regular graphΓ andγ be a diagram ofΓ. Thenγω is definedas the linear combination of link diagrams derived fromγ:

(W1) Parallelize each edgee by the weightω(e) and immerse the magic knittingof degreeω(e) as in Fig. 9(a).

(W2) Connect those parallelized edges at each vertex as in Fig. 9(b).

(a)

(b)

Fig. 9: How to makeγω.

The bracket polynomial⟨l⟩ ∈ Z[A±1] of a non-oriented link diagraml is

10

defined by the following formulas (originally introduced in [6]):

(B1) ⟨⃝⟩ = −A2 − A−2,

(B2) ⟨l ⊔⃝⟩ = (−A2 − A−2)⟨l⟩,

(B3)⟨ ⟩

= A⟨ ⟩

+ A−1⟨ ⟩

.

Yamada ([21]) defined⟨γω⟩ by the linear extension of the bracket polynomial,and gave a relation between⟨γω⟩ andR(γ).

Lemma 4.4. Let γ be a diagram of a spatial3-regular graphΓ with the goodweight definedω(e) = 2 for each edgee. Then

⟨γω⟩(A) = (−δ)χ(Γ)R(γ)(A4),

whereδ = −A2 − A−2 andχ(Γ) the Euler characteristic ofΓ.

We assumeΓ is a handcuff graphΦ. Then we can describe⟨γω⟩ concretely.Here, the magic knitting of degree2 is decomposed as follows:⟨ ⟩

=⟨ ⟩

− δ−1⟨ ⟩

. (1)

Sinceχ(Φ) = −1, we obtain the following proposition.

Proposition 4.5.For a handcuff graph diagramϕ and its constituent link diagramϕ12, let ϕ(2) be the link diagram obtained fromϕ by parallelizing each edge, andϕ

(2)12 be the link diagram obtained fromϕ12 by (2, 0)-cabling. Then

R(ϕ)(x) = −δ ⟨ϕ(2)⟩(x1/4) + ⟨ϕ(2)12 ⟩(x1/4),

whereδ = −x1/2 − x−1/2.

Proof. From Lemma 4.4, we obtain the following equation:

R(θ)(x) = (−δ)⟨ϕω⟩(x1/4), (2)

11

whereδ = −x1/2 − x−1/2. Assumeϕ12 = ϕ1 ∪ ϕ2, then we can describe⟨ϕω⟩concretely by using the equation (1):

⟨ϕω⟩ = ⟨ϕ(2)⟩ − δ−1(⟨ϕ(2)

12 ⟩ + ⟨ϕ(2)1 ⟩ + ⟨ϕ(2)

2 ⟩)

+δ−2(⟨ϕ(2)

1 ⊔⃝⟩ + ⟨ϕ(2)2 ⊔⃝⟩ + ⟨⃝⟩

)− δ−3⟨⃝ ⊔⃝⟩

= ⟨ϕ(2)⟩ − δ−1⟨ϕ(2)12 ⟩.

Then we replace⟨ϕω⟩ in the equation (2) with the above.

By using Proposition 4.5 and KNOT program ([7]), we compute the Yamadapolynomial of all the prime handcuff graphs with up to seven crossings immedi-ately; Table 2. For example, the entry for21 appears as follows:

21 {-4} (1 1 1 1 0 0 -1 -1 -1 -1),

which means thatR(21) = x−4 + x−3 + x−2 + x−1 − x2 − x3 − x4 − x5. FromTheorem 4.2, we note that there is ambiguity for degrees and coefficients of theYamada polynomial.

Table 2: The Yamada polynomial of handcuff graphs with up to seven crossings.

Φ R(ϕ)

21 {-4} (1 1 1 1 0 0 -1 -1 -1 -1)41 {-6} (1 1 1 1 1 0 0 0 -1 0 -1 0 -1 -1 0 -1)51 {-8} (1 0 -1 1 0 -1 1 0 -1 0 -1 0 -1 0 2 -1 1 1 -1)61 {-10} (1 -1 -3 2 0 -3 3 1 -1 1 0 1 -1 1 3 -3 2 -3 -1 1)62 {-9} (1 1 0 0 0 -1 -2 0 -1 -1 0 -1 0 -1 1 1 0 2 1)63 {-11} (1 -1 0 3 -2 0 1 -2 -1 -2 0 -1 -1 2 0 -1 2 1 -1 1 1)64 {-10} (1 0 -1 1 1 -1 2 2 0 1 -1 0 -2 -1 1 -2 0 1 -1 -1 )65 {-8} (1 1 1 1 1 0 0 0 0 0 0 0 -1 0 -1 0 -1 0 0 -1 0 -1)66 {-10} (2 0 1 1 -1 0 -1 1 0 1 2 1 0 0 -1 -2 -1 -1 -1 -1)67 {-10} (1 0 0 2 1 -1 2 0 -1 1 -1 1 -1 1 1 -2 0 0 -2 -1 0 -1)68 {-10} (1 0 0 3 0 -1 3 -1 -1 1 -1 0 -2 0 0 -3 1 1 -2 1 1 -1)69 {-9} (-2 -1 1 -3 -2 2 -2 0 3 1 2 0 2 0 -3 2 0 -2 2 1 -1)71 {-12} (1 -2 -1 6 -4 -4 7 -4 -4 3 0 0 -1 4 2 -5 5 3 -7 2 3 -4 -1 1)72 {-10} (1 2 -2 -1 3 -3 -1 2 -1 0 -1 1 -1 -2 3 0 -2 2 1 -2 0 1)73 {-11} (1 0 -2 2 2 -3 1 3 -3 0 2 -1 0 -1 2 -2 -3 4 -2 -2 3 0 -1)74 {-12} (1 -2 -2 5 -2 -3 7 -1 -2 3 0 0 -2 2 1 -6 2 2 -6 1 3 -2 0 1)

12

Table 2: The Yamada polynomial of handcuff graphs with up to seven crossings(continued).

Φ R(ϕ)

75 {-11} (1 0 -1 1 1 -1 0 2 -2 -1 0 -2 -1 -2 1 -1 -1 3 0 0 2 1)76 {-13} (1 -1 1 2 -4 2 2 -2 1 0 0 -3 -2 0 -3 -1 3 0 -1 3 1 -1 1 1)77 {-12} (1 -1 -2 3 -1 -4 3 -1 -3 1 0 1 -1 2 3 -3 2 3 -3 0 2 -1 -1)78 {-12} (1 -1 -1 5 -2 -4 6 -2 -3 3 0 0 -2 2 1 -5 3 3 -5 1 3 -3 -1 1)79 {-9} (1 1 -1 0 0 -1 0 0 1 0 0 1 -1 0 1 0 0 0 0 -1 -1)710 {-10} (1 1 0 0 1 0 -1 1 1 -1 1 1 -1 0 -1 0 -2 -1 1 -2 0 1)711 {-12} (1 -1 0 4 -2 -2 5 -2 -2 3 0 1 -1 2 0 -5 2 0 -5 1 2 -2 0 1)712 {-11} (1 0 -3 2 3 -4 1 5 -3 0 3 -1 0 -1 3 -2 -4 5 -2 -4 3 0 -2)713 {-11} (1 1 -2 -1 2 -2 -3 3 -1 -2 1 -1 0 -2 2 1 -3 3 2 -2 1 2)714 {-11} (1 0 -2 0 1 -3 -1 3 -2 0 2 0 1 0 3 0 -2 3 -1 -3 1 0 -1)715 {-10} (1 1 -1 0 2 -1 -2 2 0 -3 1 0 -2 0 0 1 -2 1 3 -3 1 2 -1)716 {-11} (1 -1 -3 2 2 -4 2 5 -4 0 2 -2 -1 -1 3 -2 -2 6 -2 -3 4 0 -2)717 {-9} (-2 -1 1 -1 -2 2 -1 -2 1 -1 1 -1 2 1 -2 2 1 -1 1 2)718 {-10} (1 0 -1 1 0 -2 0 0 -2 0 1 -1 0 1 0 0 -1 2 0 -1 3 -1 0 1 -1)719 {-10} (1 0 -1 1 1 -1 1 2 -1 0 1 -1 -1 0 -1 -1 -2 1 0 -2 3 0 0 1 -1)720 {-12} (-1 2 0 -3 3 -1 -2 2 1 1 0 2 1 -3 0 1 -3 0 2 -1 -1 1 0 -1)721 {-11} (2 -2 -2 4 -3 -2 2 -1 -1 -1 2 0 -2 3 1 -3 2 2 -2 0 2 0 -1)722 {-12} (1 -1 -1 3 -2 -2 4 -3 -2 2 -2 0 -1 2 1 -3 3 1 -3 2 2 -2 1 1 -1)723 {-11} (1 -1 -2 2 0 -3 3 2 -3 3 1 -1 1 0 2 -3 -1 3 -4 -1 3 -2 0 1 -1)724 {-12} (1 -1 0 3 -3 -1 5 -2 4 -1 0 -2 0 -2 -5 3 0 -3 3 2 -2 1 1 -1)725 {-11} (-1 2 1 -5 2 3 -6 1 3 -3 0 0 2 -2 -1 6 -3 -2 5 -1 -3 2 1 -1)726 {-12} (1 0 0 2 -2 -1 3 -3 -1 2 -2 0 -1 1 0 -2 3 0 -2 2 1 -2 1 1 -1)727 {-11} (-1 1 1 -3 2 3 -4 1 2 -3 0 0 2 -1 -1 4 -3 -2 3 -1 -2 2 1 -1)728 {-10} (1 0 -1 2 1 -2 1 1 -3 -1 1 -2 -1 1 0 0 -1 3 -2 4 -1 -1 1 -1)729 {-12} (-1 2 -1 -3 5 -1 -2 3 0 -1 -2 1 0 -3 2 2 -3 1 3 -2 -1 2 0 -1)730 {-11} (1 -1 -2 3 -1 -4 4 1 -4 4 2 -1 2 1 3 -3 0 4 -6 -1 3 -4 -1 1 -1)731 {-10} (-2 0 4 -3 -2 6 -1 -3 6 1 -2 2 0 0 -5 2 3 -7 3 3 -5 0 1 -1)732 {-11} (-2 1 2 -6 0 4 -7 0 4 -2 1 1 4 -1 -2 6 -2 -4 5 0 -4 2 1 -1)733 {-11} (1 -2 -2 5 -2 -5 7 0 -6 5 1 -3 1 1 2 -5 1 6 -8 1 6 -5 0 2 -1)734 {-11} (-2 3 4 -8 2 6 -10 0 5 -3 0 2 5 -2 -3 8 -4 -7 7 -1 -6 3 2 -1)735 {-11} (-1 -1 -1 -1 -1 -1 0 0 0 1 1 1 2 1 1 0 0 0 -1)736 {-12} (-1 0 1 0 0 1 1 -1 1 0 -1 0 -1 0 -2 0 1 -1 1 1)

13

Moreover, we consider the chirality of handcuff graphs. A spatial graphΓ is saidto beachiral if Γ is isotopic to its mirror image, andΓ is said to bechiral if Γ isnot achiral. In [20], S. Yamada also gave the following proposition:

Proposition 4.6. Let γ be the mirror image of a spatial graph diagramγ. ThenR(γ)(x) = R(γ)(x−1).

From Theorem 4.2 and Proposition 4.6, we can conclude almost all handcuffgraphs in Table 1 are chiral.

For example,R(41) = x−6 +x−5 +x−4 +x−3 +x−2−x2−x4−x6−x7−x9.SoR(41) = −x−9 − x−7 − x−6 − x−4 − x−2 + x2 + x3 + x4 + x5 + x6. Then, forarbitrary integern, R(41) ̸= (−x)nR(41). Thus the handcuff graph41 is chiral.

On the other hand, we prove that the handcuff graphs21, 61, and717 are achiralby deformations as in Fig. 10.

Finally, we give some open problems.

Problem 4.7. Are all of the handcuff graphs in Table 1 prime?

We hope so, but we do not know how to conclude the primeness of a handcuffgraph. When making Table 1, we omit handcuff graphs which can be decomposedobviously. More generally, we give the following.

Problem 4.8. Develop the way of decision whether a handcuff graph is prime ornot.

14

Fig. 10: The handcuff graphs21, 61, and717 are achiral.

15

Appendix

We give an enumeration of handcuff graphs with up to seven crossings by us-ing our notation. Links in the second column correspond to Rolfsen’s knot table([15]), and handcuff graphs in the last column correspond to Table 1. A linkLand a handcuff graphΦ denote mirror images ofL andΦ, respectively. Moreover,#n (n = 2, 3) denotesan ordern vertex connected sum([19]), and3θ

1 (resp.4θ1)

denotes theθ-curve31 (resp.41) ([11]).

Table 3: Handcuff graphs with up to seven crossings.

notation constituent link handcuff graph

11× 2 22

1 21

11× 4 42

1 41

11× 2 1 2 52

1 51

11× 6 62

1 65

11× 3 3 62

2 67

11× 2 2 2 62

3 68

11× 3, 3 62

1 66

11× 2 1, 2 1 62

3 69

31∗ 2 0.2̄.1 42

1 41

31∗ 2 0.2.1̄ 02

1 21#321

31∗ 2̄ 0.2̄ 0.2 22

1 62

31∗ 2 1 0.2 0.1̄ 42

1 64

41∗ 3.1.1.1 22

1 62

41∗ 3 0.1.1.1 22

1 63

41∗ 3̄ 0.1.1.1 42

1 64

41∗ 2.2.1.1 42

1 64

41∗ 2 0.2.1.1 22

1 63

41∗ 2̄ 0.2.1.1 22

1 62

41∗ 2 0.1.1.2 0 02

1 61

41∗ 2 0.1.1.2̄ 0 42

1 64

61∗ 1.1.1.1.1.1 02

1 61

63∗ 1.1.1.1.1.1 52

1 51

63∗ 1.1.1.1.1̄.1̄ 22

1 21#33θ1

63∗ 1.1.1̄.1.1.1 22

1 21#33θ1

16

Table 3: Handcuff graphs with up to seven crossings (continued).

notation constituent link handcuff graph

11× 4 1 2 72

1 718

11× 2 1 4 72

1 719

11× 2 3 2 72

3 726

11× 3 1 1 2 72

2 722

11× 2 1 1 3 72

2 723

11× 2 1 1, 3 72

1 720

11× 2 2, 2 1 72

3 721

11× 3, 2 1+ 72

2 724

11× (3, 2) 2 72

4 728

11× (3, 2) 2̄ 72

7 735

11× (3, 2̄) 2̄ 72

8 736

11× (2 1, 2) 2 72

5 730

11× (2 1, 2) 2̄ 72

8 736

31∗ 2 2.2.1̄ 52

1 714

31∗ 2 1.2.2 22

1 63

31∗ 2 1.2.2̄ 22

1 77

31∗ 2 1.2̄.2 22

1 77

31∗ 2 1.2̄.2̄ 22

1 73

31∗ 3.2̄ 0.2 42

1 64

31∗ 3 0.2 0.2̄ 42

1 710

31∗ 3 0.2̄ 0.2 42

1 64

31∗ 2 1 0.2 0.2̄ 22

1 75

41∗ 2 1 1.1.1.1 22

1 73

41∗ 2 1 1 0.1.1.1 22

1 74

41∗ 2̄ 1̄ 1̄ 0.1.1.1 52

1 714

41∗ 3.2 0.1.1 42

1 710

41∗ 3 0.2 0.1.1 42

1 711

41∗ 3̄ 0.2 0.1.1 22

1 77

41∗ 2 1.2.1.1 22

1 77

41∗ 2 1 0.2.1.1 42

1 711

41∗ 2̄ 1̄ 0.2.1.1 42

1 710

41∗ 3.2.1.1 22

1 75

17

Table 3: Handcuff graphs with up to seven crossings (continued).

notation constituent link handcuff graph

41∗ 3 0.2.1.1 22

1 76

41∗ 3̄ 0.2.1.1 22

1 75

41∗ 2 1.1.1.2 0 42

1 712

41∗ 2 1.1.1.2̄ 0 52

1 714

41∗ 2.2 1 0.1.1 52

1 714

41∗ 2 0.2 1 0.1.1 22

1 74

41∗ 2̄ 0.2 1 0.1.1 22

1 73

41∗ 2 0.2.2.1 42

1 711

41∗ 2̄ 0.2.2.1 02

1 72

41∗ 2 0.2.2 0.1 22

1 76

41∗ 2̄ 0.2.2 0.1 22

1 79

41∗ 2.2 0.1.2 0 52

1 715

41∗ 2.2 0.1.2̄ 0 42

1 710

41∗ 2 0.2.1.2 0 22

1 78

41∗ 2 0.2.1.2̄ 0 22

1 77

41∗ 2̄ 0.2.1.2 0 22

1 75

41∗ 2̄ 0.2.1.2̄ 0 22

1 21#33θ1

51× 2.2.1.1.1 72

5 731

51× 2.2̄.1.1.1 72

8 736

51× 2̄.2̄.1.1.1 72

7 735

51× 2.1.2.1.1 72

5 732

51× 2.1.2̄.1.1 72

8 736

51× 2̄.1.2.1.1 62

3 69

51× 2̄.1.2̄.1.1 52

1 51

51× 2.1.1.2 0.1 72

2 725

51× 2̄.1.1.2 0.1 62

2 717

51× 2 0.1.2.1.1 72

4 729

51× 2 0.1.2̄.1.1 72

7 735

51× 2 0.2 0.1.1.1 72

1 721

51× 2̄ 0.2̄ 0.1.1.1 62

2 717

51∗ 2 1 0.1.1.1.1 22

1 77

51∗ 2̄ 1̄ 0.1.1.1.1 22

1 77

51∗ 2 0.2 0.1.1.1 02

1 72

51∗ 2̄ 0.2 0.1.1.1 42

1 713

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Table 3: Handcuff graphs with up to seven crossings (continued).

notation constituent link handcuff graph

51∗ 2 0.1.2 0.1.1 22

1 77

51∗ 2̄ 0.1.2 0.1.1 22

1 73

51∗ 2 0.1.2̄ 0.1.1 22

1 21#33θ1

51∗ 2̄ 0.1.2̄ 0.1.1 22

1 62

51∗ 2 0.1.1.2 0.1 22

1 73

51∗ 2̄ 0.1.1.2 0.1 22

1 77

51∗ 2 0.1.1.2̄ 0.1 22

1 62

51∗ 2̄ 0.1.1.2̄ 0.1 22

1 21#33θ1

51∗ 2 0.1.1.1.2 0 42

1 713

51∗ 2̄ 0.1.1.1.2 0 02

1 72

51∗ 1̄.2 0.2 0.1.1 62

3 69

51∗ 1̄.2̄ 0.2 0.1.1 42

1 64

51∗ 1̄.2 0.2̄ 0.1.1 52

1 51

51∗ 1̄.2̄ 0.2̄ 0.1.1 52

1 51

51∗ 1.1.1.2 0.2 0 62

3 69

51∗ 1.1.1.2̄ 0.2 0 52

1 51

51∗ 1.1.1.2 0.2̄ 0 42

1 64

51∗ 1.1.1.2̄ 0.2̄ 0 52

1 51

51∗ 1.1.2.1.2 0 62

2 717

51∗ 1̄.2 0.1.2.1 62

2 717

61× 2.1.1.1.1.1 72

6 733

61× 2.1.1.1.1.1̄ 22

1 21#33θ1

61× 2̄.1.1.1.1.1 72

8 736

61× 2̄.1.1.1.1.1̄ 22

1#231 21#33θ1

61∗ 2.1.1.1.1.1 02

1 71

61∗ 2̄.1.1.1.1.1 02

1 72

61∗ 1.1.2.1.1.1 22

1 78

61∗ 2 0.1.1.1.1.1 52

1 716

61∗ 1.1.2 0.1.1.1 22

1 78

61∗ 1.1.2̄ 0.1.1.1 22

1 62

62∗ 1.2.1.1.1.1 22

1 77

62∗ 1.1.2.1.1.1 02

1 72

62∗ 1.1.1.2 0.1.1 22

1 73

62∗ 1.1.1.2̄ 0.1.1 22

1 62

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Table 3: Handcuff graphs with up to seven crossings (continued).

notation constituent link handcuff graph

63∗ 1.1.1.1.1.2 02

1 72

63∗ 1.1.1.1.1̄.2̄ 22

1 21#34θ1

63∗ 1.1.1̄.1.1.2 02

1 21#341

63∗ 1.1.1̄.1.1̄.2̄ 22

1 21#33θ1

63∗ 1.1.1̄.2 0.1.1 22

1 21#34θ1

63∗ 1.1.1.2 0.1̄.1̄ 22

1 21#34θ1

63∗ 1.1.1.1.1.2̄ 0 02

1 72

63∗ 1.1.1.1.1̄.2̄ 0 02

1 21#341

63∗ 1.1.1̄.1.1.2 0 22

1 21#34θ1

63∗ 1.1.1̄.1.1̄.2 0 22

1 21#33θ1

64∗ 2.1.1.1.1.1 22

1 63

64∗ 2̄.1.1.1.1.1 22

1 21

64∗ 2.1.1̄.1.1.1 22

1 21#33θ1

64∗ 2.1̄.1.1.1.1 42

1 41

64∗ 2̄.1.1̄.1.1.1 22

1 21#33θ1

64∗ 2̄.1̄.1.1.1.1 02

1 trivial64∗ 2.1̄.1̄.1.1.1 42

1 64

64∗ 2̄.1̄.1̄.1.1.1 52

1 51

64∗ 1.2.1.1.1.1 02

1 61

64∗ 1̄.2.1̄.1.1.1 42

1 64

64∗ 1.2̄.1.1.1.1 42

1 62

64∗ 1̄.2̄.1̄.1.1.1 52

1 51

73× 1.1.1.1.1.1.1 72

6 734

73× 1.1.1.1.1̄.1̄.1̄ 72

7 735

73× 1.1.1̄.1̄.1.1̄.1 72

7 735

73× 1.1.1̄.1̄.1̄.1̄.1̄ 72

8 736

72∗ 1.1.1.1.1.1.1 02

1 71

72∗ 1̄.1̄.1.1.1.1.1 22

1 21#33θ1

72∗ 1̄.1̄.1̄.1.1.1.1 22

1 79

72∗ 1̄.1̄.1̄.1.1.1̄.1 02

1 21#321#321

77∗ 1.1.1.1.1.1̄.1̄ 52

1 51

77∗ 1.1.1.1.1̄.1.1 42

1 64

20

Acknowledgement

The author would like to thank Professor Taizo Kanenobu and Professor AkioKawauchi for their valuable advice and encouragement.

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Osaka City University, Sugimoto, Sumiyoshi-ku, Osaka, Japanmoriuchi@sci.osaka-cu.ac.jp

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