12n
0271
(K12n
0271
)
A knot diagram
1
Linearized knot diagam
3 4 10 2 9 4 12 3 6 7 8 11
Solving Sequence
3,10 4,6
7 11 2 1 9 5 8 12
c
3
c
6
c
10
c
2
c
1
c
9
c
5
c
8
c
12
c
4
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h2.10894 × 10
29
u
39
+ 1.76613 × 10
30
u
38
+ ··· + 3.90907 × 10
30
b + 1.29800 × 10
31
,
− 5.78629 × 10
30
u
39
− 1.35625 × 10
31
u
38
+ ··· + 1.95453 × 10
30
a − 1.20066 × 10
30
, u
40
+ 2u
39
+ ··· − 2u + 1i
I
u
2
= hb
4
− 8b
3
u + 4b
3
+ 2b
2
u − 18b
2
+ 20bu − 4b − 4u + 7, a + u − 1, u
2
− u + 1i
I
u
3
= hb
3
+ 6b
2
u + 3b
2
− 9b − 6u − 3, a − u − 1, u
2
+ u + 1i
* 3 irreducible components of dim
C
= 0, with total 54 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I.
I
u
1
= h2.11×10
29
u
39
+1.77×10
30
u
38
+· · ·+3.91×10
30
b+1.30×10
31
, −5.79×
10
30
u
39
−1.36×10
31
u
38
+· · ·+1.95×10
30
a−1.20×10
30
, u
40
+2u
39
+· · ·−2u+1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
u
2
a
6
=
2.96044u
39
+ 6.93899u
38
+ ··· + 26.8232u + 0.614297
−0.0539500u
39
− 0.451804u
38
+ ··· − 0.947110u − 3.32050
a
7
=
2.92788u
39
+ 6.66255u
38
+ ··· + 26.8003u − 1.68810
−0.0424279u
39
− 0.410687u
38
+ ··· − 1.33715u − 3.10919
a
11
=
3.86426u
39
+ 7.66563u
38
+ ··· + 30.4928u − 10.5738
0.320422u
39
+ 0.529059u
38
+ ··· − 0.339238u − 2.95540
a
2
=
u
2
+ 1
u
4
a
1
=
u
4
+ u
2
+ 1
u
4
a
9
=
1.97656u
39
+ 3.87161u
38
+ ··· + 21.4803u − 4.58523
0.944726u
39
+ 1.94465u
38
+ ··· + 5.22209u − 2.88878
a
5
=
u
4
+ u
2
+ 1
u
6
+ u
2
a
8
=
2.92129u
39
+ 5.81625u
38
+ ··· + 26.7024u − 7.47401
0.944726u
39
+ 1.94465u
38
+ ··· + 5.22209u − 2.88878
a
12
=
1.93457u
39
+ 2.86642u
38
+ ··· + 4.61941u − 15.8609
−0.942390u
39
− 2.36759u
38
+ ··· − 11.6667u − 2.35706
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3.91026u
39
+ 8.38086u
38
+ ··· + 29.6027u + 1.57860
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
40
+ 56u
39
+ ··· + 84u + 1
c
2
, c
4
u
40
− 8u
39
+ ··· − 28u + 1
c
3
u
40
+ 2u
39
+ ··· − 2u + 1
c
5
, c
9
u
40
− 3u
39
+ ··· + 43u + 13
c
6
u
40
+ 4u
39
+ ··· + 18344u + 4339
c
7
, c
11
u
40
− u
39
+ ··· − 12u + 4
c
8
u
40
− 44u
38
+ ··· − 2449090u + 232661
c
10
u
40
+ u
39
+ ··· − 36u + 4
c
12
u
40
+ 25u
39
+ ··· + 80u + 16
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
40
− 136y
39
+ ··· + 23220y + 1
c
2
, c
4
y
40
+ 56y
39
+ ··· + 84y + 1
c
3
y
40
+ 8y
39
+ ··· + 28y + 1
c
5
, c
9
y
40
− 5y
39
+ ··· + 2779y + 169
c
6
y
40
+ 40y
39
+ ··· − 166465604y + 18826921
c
7
, c
11
y
40
+ 25y
39
+ ··· + 80y + 16
c
8
y
40
− 88y
39
+ ··· − 1066872899128y + 54131140921
c
10
y
40
− 55y
39
+ ··· − 112y + 16
c
12
y
40
− 15y
39
+ ··· − 2816y + 256
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.764187 + 0.653787I
a = 0.315530 − 0.761484I
b = 0.186332 + 0.279615I
−1.47509 + 2.25056I 0.06495 − 2.95440I
u = −0.764187 − 0.653787I
a = 0.315530 + 0.761484I
b = 0.186332 − 0.279615I
−1.47509 − 2.25056I 0.06495 + 2.95440I
u = −0.213036 + 0.949581I
a = −0.140457 + 1.057230I
b = −0.04726 − 2.29506I
1.94553 + 4.51368I 6.38970 − 7.82355I
u = −0.213036 − 0.949581I
a = −0.140457 − 1.057230I
b = −0.04726 + 2.29506I
1.94553 − 4.51368I 6.38970 + 7.82355I
u = 0.955003 + 0.516879I
a = 0.487517 + 0.951136I
b = −0.146437 + 0.072845I
−6.22578 + 0.90518I −4.74646 − 0.36762I
u = 0.955003 − 0.516879I
a = 0.487517 − 0.951136I
b = −0.146437 − 0.072845I
−6.22578 − 0.90518I −4.74646 + 0.36762I
u = −0.444049 + 0.991733I
a = 0.526547 + 0.513973I
b = −0.30344 − 1.55709I
0.55266 + 1.44737I −0.667154 + 0.069332I
u = −0.444049 − 0.991733I
a = 0.526547 − 0.513973I
b = −0.30344 + 1.55709I
0.55266 − 1.44737I −0.667154 − 0.069332I
u = 0.670524 + 0.887860I
a = 0.748302 − 0.870483I
b = 0.20959 + 1.99021I
0.22448 − 2.62080I −1.03330 + 3.61519I
u = 0.670524 − 0.887860I
a = 0.748302 + 0.870483I
b = 0.20959 − 1.99021I
0.22448 + 2.62080I −1.03330 − 3.61519I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.808131 + 0.773706I
a = 0.177775 + 0.864533I
b = 0.140485 − 0.763458I
−3.58036 − 7.21978I −1.70984 + 7.47042I
u = 0.808131 − 0.773706I
a = 0.177775 − 0.864533I
b = 0.140485 + 0.763458I
−3.58036 + 7.21978I −1.70984 − 7.47042I
u = −0.500435 + 1.020290I
a = −0.288714 + 0.431513I
b = 0.79094 − 1.35172I
−0.11396 + 2.62702I 1.21166 − 3.80016I
u = −0.500435 − 1.020290I
a = −0.288714 − 0.431513I
b = 0.79094 + 1.35172I
−0.11396 − 2.62702I 1.21166 + 3.80016I
u = 0.601637 + 0.967840I
a = −0.452431 − 0.174923I
b = 0.779446 + 0.975335I
−2.80937 + 1.79823I −2.49074 − 1.24106I
u = 0.601637 − 0.967840I
a = −0.452431 + 0.174923I
b = 0.779446 − 0.975335I
−2.80937 − 1.79823I −2.49074 + 1.24106I
u = −0.737881 + 0.290228I
a = 0.95910 + 1.11896I
b = 0.508089 − 0.650289I
−1.82966 + 2.58669I −3.19003 − 3.49376I
u = −0.737881 − 0.290228I
a = 0.95910 − 1.11896I
b = 0.508089 + 0.650289I
−1.82966 − 2.58669I −3.19003 + 3.49376I
u = 0.178235 + 0.755403I
a = −0.004535 − 1.355120I
b = −0.53972 + 1.87193I
2.74915 − 0.87130I 9.38374 − 0.76428I
u = 0.178235 − 0.755403I
a = −0.004535 + 1.355120I
b = −0.53972 − 1.87193I
2.74915 + 0.87130I 9.38374 + 0.76428I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.526875 + 1.146360I
a = −0.507182 − 0.577873I
b = 1.35004 + 1.47812I
−3.91478 − 6.46553I −1.90714 + 6.14891I
u = 0.526875 − 1.146360I
a = −0.507182 + 0.577873I
b = 1.35004 − 1.47812I
−3.91478 + 6.46553I −1.90714 − 6.14891I
u = −0.332002 + 0.640935I
a = 0.522256 − 0.170528I
b = 0.045479 − 0.256841I
−0.02195 + 1.48740I 0.17371 − 4.94146I
u = −0.332002 − 0.640935I
a = 0.522256 + 0.170528I
b = 0.045479 + 0.256841I
−0.02195 − 1.48740I 0.17371 + 4.94146I
u = 1.030620 + 0.897013I
a = −1.173260 − 0.172864I
b = 0.179150 − 1.015860I
−11.47200 + 0.66922I 0
u = 1.030620 − 0.897013I
a = −1.173260 + 0.172864I
b = 0.179150 + 1.015860I
−11.47200 − 0.66922I 0
u = −1.073240 + 0.852439I
a = −1.229830 + 0.153303I
b = −0.092752 + 1.188350I
−15.3931 − 6.0460I 0
u = −1.073240 − 0.852439I
a = −1.229830 − 0.153303I
b = −0.092752 − 1.188350I
−15.3931 + 6.0460I 0
u = 0.926096 + 1.056710I
a = 0.023831 + 1.156260I
b = −1.21185 − 2.24051I
−10.92760 − 7.82692I 0
u = 0.926096 − 1.056710I
a = 0.023831 − 1.156260I
b = −1.21185 + 2.24051I
−10.92760 + 7.82692I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.04261 + 0.96199I
a = −1.160190 + 0.237458I
b = 0.515590 + 1.140420I
−15.7491 + 4.5181I 0
u = −1.04261 − 0.96199I
a = −1.160190 − 0.237458I
b = 0.515590 − 1.140420I
−15.7491 − 4.5181I 0
u = −0.90713 + 1.09807I
a = −0.010802 − 1.172610I
b = −1.27643 + 2.56164I
−14.5518 + 13.2615I 0
u = −0.90713 − 1.09807I
a = −0.010802 + 1.172610I
b = −1.27643 − 2.56164I
−14.5518 − 13.2615I 0
u = −0.98478 + 1.04683I
a = 0.065240 − 1.184100I
b = −1.50476 + 1.94665I
−15.4588 + 2.8896I 0
u = −0.98478 − 1.04683I
a = 0.065240 + 1.184100I
b = −1.50476 − 1.94665I
−15.4588 − 2.8896I 0
u = 0.162466 + 0.376483I
a = 0.55787 − 2.40002I
b = −0.720369 + 0.864698I
1.74484 − 0.37727I 6.34033 + 0.02713I
u = 0.162466 − 0.376483I
a = 0.55787 + 2.40002I
b = −0.720369 − 0.864698I
1.74484 + 0.37727I 6.34033 − 0.02713I
u = 0.139759 + 0.272112I
a = −1.91657 + 2.94354I
b = −1.36214 − 0.47472I
−0.74436 − 3.76425I 2.15171 + 3.16942I
u = 0.139759 − 0.272112I
a = −1.91657 − 2.94354I
b = −1.36214 + 0.47472I
−0.74436 + 3.76425I 2.15171 − 3.16942I
8
II. I
u
2
= h−8b
3
u + 2b
2
u + · · · − 4b + 7, a + u − 1, u
2
− u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
u − 1
a
6
=
−u + 1
b
a
7
=
b − 2u + 1
bu + 1
a
11
=
−b
2
u + 2bu − 4b + 3u
−b
2
u + b
2
− 2bu − b + 2u − 2
a
2
=
u
−u
a
1
=
0
−u
a
9
=
u − 1
−b + u
a
5
=
0
u
a
8
=
−b + 2u − 1
−b + u
a
12
=
−2b
2
u + 4bu − 8b + 4u + 2
b
3
u − b
3
+ b
2
u + 4b
2
− 9bu + 3b + 2u − 4
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b
2
u − 4b
2
+ 8bu + 8b − 8u + 8
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
(u
2
− u + 1)
4
c
2
(u
2
+ u + 1)
4
c
5
(u − 1)
8
c
6
u
8
− 4u
7
+ 12u
6
− 16u
5
+ 15u
4
+ 8u
3
− 4u
2
+ 1
c
7
, c
11
(u
4
+ 2u
2
+ 2)
2
c
8
u
8
+ 4u
7
+ 12u
6
+ 16u
5
+ 15u
4
− 8u
3
− 4u
2
+ 1
c
9
(u + 1)
8
c
10
(u
4
− 2u
2
+ 2)
2
c
12
(u
2
+ 2u + 2)
4
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
(y
2
+ y + 1)
4
c
5
, c
9
(y − 1)
8
c
6
, c
8
y
8
+ 8y
7
+ 46y
6
+ 160y
5
+ 387y
4
− 160y
3
+ 46y
2
− 8y + 1
c
7
, c
11
(y
2
+ 2y + 2)
4
c
10
(y
2
− 2y + 2)
4
c
12
(y
2
+ 4)
4
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.500000 − 0.866025I
b = 0.943461 + 1.008110I
−0.82247 − 5.69375I 2.00000 + 7.46410I
u = 0.500000 + 0.866025I
a = 0.500000 − 0.866025I
b = 0.155223 + 0.553018I
−0.82247 + 1.63398I 2.00000 − 0.53590I
u = 0.500000 + 0.866025I
a = 0.500000 − 0.866025I
b = −0.94346 + 2.45599I
−0.82247 − 5.69375I 2.00000 + 7.46410I
u = 0.500000 + 0.866025I
a = 0.500000 − 0.866025I
b = −0.15522 + 2.91108I
−0.82247 + 1.63398I 2.00000 − 0.53590I
u = 0.500000 − 0.866025I
a = 0.500000 + 0.866025I
b = 0.943461 − 1.008110I
−0.82247 + 5.69375I 2.00000 − 7.46410I
u = 0.500000 − 0.866025I
a = 0.500000 + 0.866025I
b = 0.155223 − 0.553018I
−0.82247 − 1.63398I 2.00000 + 0.53590I
u = 0.500000 − 0.866025I
a = 0.500000 + 0.866025I
b = −0.94346 − 2.45599I
−0.82247 + 5.69375I 2.00000 − 7.46410I
u = 0.500000 − 0.866025I
a = 0.500000 + 0.866025I
b = −0.15522 − 2.91108I
−0.82247 − 1.63398I 2.00000 + 0.53590I
12
III. I
u
3
= hb
3
+ 6b
2
u + 3b
2
− 9b − 6u − 3, a − u − 1, u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
10
=
0
u
a
4
=
1
−u − 1
a
6
=
u + 1
b
a
7
=
b + 2u + 1
−bu + 1
a
11
=
−b
2
u + 2bu + 4b + 3u
−b
2
u − b
2
− 2bu + b + 2u + 2
a
2
=
−u
u
a
1
=
0
u
a
9
=
u + 1
b + u
a
5
=
0
−u
a
8
=
b + 2u + 1
b + u
a
12
=
0
−b
2
− 4bu − 2b + u + 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2b
2
u + 2b
2
+ 4bu − 4b − 10u − 2
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
8
(u
2
− u + 1)
3
c
2
, c
3
(u
2
+ u + 1)
3
c
5
(u + 1)
6
c
7
, c
10
, c
11
c
12
u
6
c
9
(u − 1)
6
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
8
(y
2
+ y + 1)
3
c
5
, c
9
(y − 1)
6
c
7
, c
10
, c
11
c
12
y
6
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = − 1.73205I
1.64493 + 2.02988I 6.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = − 1.73205I
1.64493 + 2.02988I 6.00000 − 3.46410I
u = −0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = − 1.73205I
1.64493 + 2.02988I 6.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 1.73205I
1.64493 − 2.02988I 6.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 1.73205I
1.64493 − 2.02988I 6.00000 + 3.46410I
u = −0.500000 − 0.866025I
a = 0.500000 − 0.866025I
b = 1.73205I
1.64493 − 2.02988I 6.00000 + 3.46410I
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
7
)(u
40
+ 56u
39
+ ··· + 84u + 1)
c
2
((u
2
+ u + 1)
7
)(u
40
− 8u
39
+ ··· − 28u + 1)
c
3
((u
2
− u + 1)
4
)(u
2
+ u + 1)
3
(u
40
+ 2u
39
+ ··· − 2u + 1)
c
4
((u
2
− u + 1)
7
)(u
40
− 8u
39
+ ··· − 28u + 1)
c
5
((u − 1)
8
)(u + 1)
6
(u
40
− 3u
39
+ ··· + 43u + 13)
c
6
(u
2
− u + 1)
3
(u
8
− 4u
7
+ 12u
6
− 16u
5
+ 15u
4
+ 8u
3
− 4u
2
+ 1)
· (u
40
+ 4u
39
+ ··· + 18344u + 4339)
c
7
, c
11
u
6
(u
4
+ 2u
2
+ 2)
2
(u
40
− u
39
+ ··· − 12u + 4)
c
8
(u
2
− u + 1)
3
(u
8
+ 4u
7
+ 12u
6
+ 16u
5
+ 15u
4
− 8u
3
− 4u
2
+ 1)
· (u
40
− 44u
38
+ ··· − 2449090u + 232661)
c
9
((u − 1)
6
)(u + 1)
8
(u
40
− 3u
39
+ ··· + 43u + 13)
c
10
u
6
(u
4
− 2u
2
+ 2)
2
(u
40
+ u
39
+ ··· − 36u + 4)
c
12
u
6
(u
2
+ 2u + 2)
4
(u
40
+ 25u
39
+ ··· + 80u + 16)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
7
)(y
40
− 136y
39
+ ··· + 23220y + 1)
c
2
, c
4
((y
2
+ y + 1)
7
)(y
40
+ 56y
39
+ ··· + 84y + 1)
c
3
((y
2
+ y + 1)
7
)(y
40
+ 8y
39
+ ··· + 28y + 1)
c
5
, c
9
((y − 1)
14
)(y
40
− 5y
39
+ ··· + 2779y + 169)
c
6
(y
2
+ y + 1)
3
· (y
8
+ 8y
7
+ 46y
6
+ 160y
5
+ 387y
4
− 160y
3
+ 46y
2
− 8y + 1)
· (y
40
+ 40y
39
+ ··· − 166465604y + 18826921)
c
7
, c
11
y
6
(y
2
+ 2y + 2)
4
(y
40
+ 25y
39
+ ··· + 80y + 16)
c
8
(y
2
+ y + 1)
3
· (y
8
+ 8y
7
+ 46y
6
+ 160y
5
+ 387y
4
− 160y
3
+ 46y
2
− 8y + 1)
· (y
40
− 88y
39
+ ··· − 1066872899128y + 54131140921)
c
10
y
6
(y
2
− 2y + 2)
4
(y
40
− 55y
39
+ ··· − 112y + 16)
c
12
y
6
(y
2
+ 4)
4
(y
40
− 15y
39
+ ··· − 2816y + 256)
18
