a partial order on the set of prime knots with up to 11

Oct. 6th 2009

A partial order on the set of prime knots with up

to 11 crossings

Mineko MATSUMOTO (Soka Univ.)

joint work with

Keiichi HORIE, Teruaki KITANO and Masaaki SUZUKI.

1 Plan of Talk• Introduction

• Approach

• N-data

• Alexander Polynomial and Wada Invariant

• Searching Epimorphism

• Result and Final Remarks

2 Introduction• K : a prime knot in S3

• G(K) : the knot group of K i.e. G(K) = π1(S3 − K)

2 Introduction• K : a prime knot in S3

• G(K) : the knot group of K i.e. G(K) = π1(S3 − K)

If there exists an epimorphism between two knot groups,

we can define a partial order.

The partial order is determined on the set of prime knots

with up to 10 crossings by Kitano-Suzuki.

The goal : to determine the partial order on the set of the

prime knots with up to 11 crossings.



”A partial order on the set of prime knots with up to 11

crossings”,

to appear in Journal of Knot Theory and Its Ramifications

arXiv : 0906.3943



”A partial order on the set of prime knots with up to 11

crossings”,

to appear in Journal of Knot Theory and Its Ramifications

arXiv : 0906.3943

Note :

Numbering of knots follows Rolfsen’s table and Knot Info.

Example

11a1, 11n4

Definition

K1 ≥ K2 ⇔ ∃ϕ : G(K1) −→−→ G(K2)

Fact

This relation ≥ is a partial order on the set of prime knots.

• K ≥ K

• K1 ≥ K2, K2 ≥ K1 ⇒ K1 = K2

• K1 ≥ K2, K2 ≥ K3 ⇒ K1 ≥ K3

Example 2.1 85 and 31

85 31

G(85) =

* x1, x2, x3, x7x2x−17 x−1

1 , x8x3x−18 x−1

2 , x6x4x−16 x−1

3 ,x4, x5, x6, x1x5x

−11 x−1

4 , x3x6x−13 x−1

5 , x4x7x−14 x−1

6 ,x7, x8 x2x8x

−12 x−1

7

+

.

G(31) = 〈y1, y2, y3 | y3y1y−13 y−1

2 , y1y2y−11 y−1

3 〉.

If generators are mapped to the following words:

x1 7→ y3, x2 7→ y2, x3 7→ y1, x4 7→ y3,

x5 7→ y3, x6 7→ y2, x7 7→ y1, x8 7→ y3.

The image of any relator in G(85) becomes trivial in G(31).

x7x2x−17 x−1

1 7→ y1y2y−11 y−1

3 = 1, . . .

There exists an epimorphism from G(85) onto G(31).Therefore, we can write

85 ≥ 31.

Geometric meaning of epimorphisms

• periodic knots

Example 818 ≥ 41

period 2−−−−→

There exists an epimorphism from G(818) onto G(41)• degree one maps

A degree one map induces an epimorphism.

Example 818 ≥ 31

Theorem 2.2 (Kitano-Suzuki)

The partial order on the set of prime knots with up to

10 crossings is given by

85, 810, 815, 818, 819, 820, 821, 91, 96, 916,923, 924, 928, 940, 105, 109, 1032, 1040, 1061,1062, 1063, 1064, 1065, 1066, 1076, 1077,1078, 1082, 1084, 1085, 1087, 1098, 1099,10103, 10106, 10112, 10114, 10139, 10140,10141, 10142, 10143, 10144, 10159, 10164

≥ 31

818, 937, 940, 1058, 1059, 1060,10122, 10136, 10137, 10138

}≥ 41

1074, 10120, 10122 ≥ 52

We extend this results for the knots with up to 11 crossings.

Number of knots with up to 11 crossings : 552



up to 10 crossings

• number of knots : 249• number of cases : 249P2 = 61, 752

up to 11 crossings

• number of knots : 249 + 552 = 801• number of cases : 801P2 =640,800



up to 10 crossings

• number of knots : 249• number of cases : 249P2 = 61, 752

up to 11 crossings

• number of knots : 249 + 552 = 801• number of cases : 801P2 =640,800

We had not decided the relation except for the pairs of

knots with up to 10 crossings.

640, 800 − 61, 752 = 579, 084 cases.

3 Approach

(i) To check nonexistence of epimorphism.

3 Approach


• Alexander polynomial

• Twisted Alexander polynomial [=Wada invariant]

3 Approach



• Twisted Alexander polynomial [=Wada invariant]

(ii) We search epimorphism for the rest of cases.

3 Approach



• Twisted Alexander polynomial [=Wada Invariant]

(ii) We search epimorphism for the rest of cases.

Objective Software

to get n-data Kodama’s KNOT

to check nonexistence of DoctorK

an epimorphism Mathematica

to find an epimorphisms Mathematica

• Kodama’s KNOT

Computer library developed by K. Kodama.

http://www.math.kobe-u.ac.jp/ kodama/index.html

• DoctorK

Computation library developed by K. Horie.

• Kodama’s KNOT

Computer library developed by K. Kodama.

http://www.math.kobe-u.ac.jp/ kodama/index.html

• DoctorK

Computation library developed by K. Horie.

The first problem is how to input knots into the computer.

4 N-dataN-data : one of methods to represent a regular diagram of

a knot as a code, introduced by Wada.



What is the N-data?



What is the N-data?

Example 4.1

• 31 : n1 n3 l2 r1 l2 u3 u1

• 41 : n1 n3 r2 l1 r2 l1 u2 u1

• 11a258 : n1 n3 l2 n3 r1 l2 r3 l4 r1 u3 l2 r3 r3 r1 l2 u3 u1

• 11n130 : 3n1 r2 r4 l3 r4 l5 r4 l3 r2 u1 n3 l4 r3 u2 l2 2u1

(i) Draw a regular diagram of a knot.

(ii) Modify the regular diagram so that any two points of

crossings, local maximum points or local minimum points

don’t have same height.

(iii) Slice a regular diagram along some lines that are parallel

to the horizontal line.

(iv) Correspond as follows:

local maximum point : ni, local minimum point : ui,

right twist crossing : ri, left twist crossing : li.n1

n3

r2

l１

ｒ2

ｒ2

u3

u1

5 Alexander Polynomial

∆K(t) : Alexander polynomial of K.



Proposition 5.1

∆K1(t) is not divisible by ∆K2(t) ⇒ G(K1) −→6−→ G(K2).



Proposition 5.1

∆K1(t) is not divisible by ∆K2(t) ⇒ G(K1) −→6−→ G(K2).

To prove the nonexistence of an epimorphism, Alexander

polynomial is strong.

We can reduce the number of cases

579, 048 → 3, 072.

Example 5.2

11a1 ≥ 41?

41 11a1

∆41(t) = t2 − 3t + 1∆11a1 = 2t6 − 12t5 + 30t4 − 39t3 + 30t2 − 12t + 2∆11a1(t) is not divisible by ∆41(t).Therefore ,

11a1 � 41.

Example 5.3 11a8 and 11a38

11a8 11a38

∆11a8(t)∆11a38(t)

=2t6 − 11t5 + 27t4 − 37t3 + 27t2 − 11t + 22t6 − 11t5 + 27t4 − 37t3 + 27t2 − 11t + 2

= 1

Therefore, by using only Alexander polynomial, we cannot

prove existence of an epimorphism from G(11a38) onto

G(11a8) or not.

6 Wada Invariant• p : a prime number

• Fp = Z/pZ : the finite prime field of characteristic p

• SL(2; Fp) : the 2-dimensional special linear group over Fp

SL(2; Fp) 3

a b

c d

!

; a, b, c, d ∈ Fp, ad − bc ≡ 1 mod p

6 Wada Invariant• p : a prime number

• Fp = Z/pZ : the finite prime field of characteristic p

• SL(2; Fp) : the 2-dimensional special linear group over Fp

SL(2; Fp) 3

a b

c d

!

; a, b, c, d ∈ Fp, ad − bc ≡ 1 mod p

The Wada invariant of K for a representation

ρ : G(K) → SL(2; Fp) is defined as a rational expression

∆K,ρ(t) =∆N

K,ρ(t)

∆DK,ρ(t)

.

Here ∆NK,ρ(t) and ∆D

K,ρ(t) are Laurent polynomials of one

variable t over Fp.

Theorem[Kitano-Suzuki-Wada]

If there exists a representation ρ2 : G(K2) → SL(2; Fp)

such that for any ρ1 : G(K1) → SL(2; Fp)

·∆NK1,ρ1(t) is not divisible

by ∆NK2,ρ2(t)

or

·∆DK2,ρ2(t) 6= ∆D

K1,ρ1(t),

G(K1) 99K G(K2)

ρ1 ↘ ↙ ρ2

SL(2; Fp)

⇒ there exist no epimorphism.

Theorem[Kitano-Suzuki-Wada]

If there exists a representation ρ2 : G(K2) → SL(2; Fp)

such that for any ρ1 : G(K1) → SL(2; Fp)

·∆NK1,ρ1(t) is not divisible

by ∆NK2,ρ2(t)

or

·∆DK2,ρ2(t) 6= ∆D

K1,ρ1(t),

G(K1) 99K G(K2)

ρ1 ↘ ↙ ρ2

SL(2; Fp)

⇒ there exist no epimorphism.

We can check nonexistence of an epimorphism by using

Wada invariant.

We can reduce the number of cases

3, 072 → 88.

Example 6.1 11a8 and 11a38

For a certain representation ρ : G(11a38) → SL(2; F2)

∆N11a38, ρ(t) = t12 + t10 + t2 + 1, ∆D

11a38, ρ(t) = t2 + 1

All twisted Alexander polynomials of 11a8 are as follows:

∆N11a8,ρi

∆D11a8,ρi

ρ1 t8 + t6 + t4 + t2 + 1 t2 + 1

ρ2 t8 + t7 + t5 + t4 + t3 + t + 1 t2 + t + 1

ρ3 t8 + t6 + t4 + t2 + t + 1 t2 + 1

ρ4 t12 + 1 t2 + 1

Therefore ,11a8 ˜ 11a38.

7 Searching Epimorphisms

Alexander polynomial 579,048→3,072

twisted Alexander polynomial 3,072→88

We want to find an epimorphism for all pairs of knots which

belong to these 88 cases.

It is done by using Mathematica.

Example 7.1 11a245 and 31

11a245 31

G(11a245) =

*

x1, x2, x3, x9x2x−19 x−1

1 , x8x3x−18 x−1

2 , x7x4x−17 x−1

3 ,

x4, x5, x6, x6x5x−16 x−1

4 , x11x6x−111 x−1

5 , x3x7x−13 x−1

6 ,

x7, x8, x9, x4x8x−14 x−1

7 , x1x9x−11 x−1

8 , x2x10x−12 x−1

9 ,

x10, x11 x5x11x−15 x−1

10

+

G(31) = 〈y1, y2, y3 | y3y1y−13 y

−12 , y1y2y

−11 y

−13 〉.

If generators are mapped to the following words :

x1 7→ y1, x2 7→ y2, x3 7→ y2, x4 7→ y1, x5 7→ y1, x6 7→ y1,

x7 7→ y3, x8 7→ y2, x9 7→ y−11 y2y1, x10 7→ y1, x11 7→ y1.

The image of any relator in G(11a245) becomes trivial in

G(31).

x5x11x−15 x−1

10 7→ y1y1y−11 y−1

1 = 1

x8x3x−18 x−1

2 7→ y2y2y−12 y−1

2 = 1. . .

Then there exists an epimorphism from G(11a245) onto

G(31). Therefore we can prove

11a245 ≥ 31

8 ResultsAll of knots satisfying the partial order relation are as follows:

Knots with up to 10 crossings

85, 810, 815, 818, 819, 820, 821, 91, 96, 916,923, 924, 928, 940, 105, 109, 1032, 1040, 1061,1062, 1063, 1064, 1065, 1066, 1076, 1077,1078, 1082, 1084, 1085, 1087, 1098, 1099,10103, 10106, 10112, 10114, 10139, 10140,10141, 10142, 10143, 10144, 10159, 10164

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≥ 31

818, 937, 940, 1058, 1059, 1060,10122, 10136, 10137, 10138

ff

≥ 41

1074, 10120, 10122 ≥ 52

Theorem 8.1

11a43, 11a44, 11a46, 11a47, 11a57, 11a58,11a71, 11a72, 11a73, 11a100, 11a106, 11a107,11a108, 11a109, 11a117, 11a134, 11a139,11a157, 11a165, 11a171, 11a175, 11a176,11a194, 11a196, 11a203, 11a212, 11a216,11a223, 11a231, 11a232, 11a236, 11a244,11a245, 11a261, 11a263, 11a264, 11a286,11a305, 11a306, 11a318, 11a332, 11a338,11a340, 11a351, 11a352, 11a355, 11n71,11n72, 11n73, 11n74, 11n75, 11n76, 11n77,11n78, 11n81, 11n85, 11n86, 11n87, 11n94,11n104, 11n105, 11n106, 11n107, 11n136,11n164, 11n183, 11n184, 11n185,

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≥ 31

11a5, 11a6, 11a51, 11a132, 11a239, 11a297, 11a348,11a349, 11n100, 11n148, 11n157, 11n165

ff

≥ 41

11n78, 11n148 ≥ 51

11n71, 11n185 ≥ 52

11a352 ≥ 61

11a351 ≥ 62

11a47, 11a239 ≥ 63

9 Observation and RemarksObservation(Kitano-Suzuki)

• For any knots K1, K2, there exists a knot K such that

K ≥ K1 and K ≥ K2.

• 31, 41, 51, 52, 61, 62 are local minimum.

Remark

There doesn’t exist an epimorphism from one of

31, 41, 51, 52, 61, 62 to one with higher crossing number.

Problem

If K1 ≥ K2, then the crossing number of K1 is greater than

that of K2?

Simon’s Conjecture

For any knot K, G(K) surjects onto only finitely many knot

groups.

Theorem(Boileau-Boyer-Reid-Wang)

Simon’s conjecture holds for all two bridge knots.

As one result of this theory,

Proposition 9.1 (BBRW)

Any epimorphism between 2-bridge hyperbolic knots is always

induced from a non zero degree map.

On the other hand, there are some interesting example.

Example 9.2

1059, 10137 are 3-bridge hyperbolic knots.

• 1059, 10137 ≥ 41.

• any epimorshism between them is induced from a degree zero

map.

• 1059, 10137 are Montesinos knots.

Thank you!

a partial order on the set of prime knots with up to 11

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