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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)
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
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
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
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
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
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.
• Alexander polynomial
• Twisted Alexander polynomial [=Wada invariant]
3 Approach
(i) To check nonexistence of epimorphism.
• Alexander polynomial
• Twisted Alexander polynomial [=Wada invariant]
(ii) We search epimorphism for the rest of cases.
3 Approach
(i) To check nonexistence of epimorphism.
• Alexander polynomial
• 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.
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?
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?
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
(ii) Modify the regular diagram so that any two points of
crossings, local maximum points or local minimum points
don’t have same height.
(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).
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).
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
9
>
>
>
>
>
>
=
>
>
>
>
>
>
;
≥ 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,
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
≥ 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.