12n
0684
(K12n
0684
)
A knot diagram
1
Linearized knot diagam
4 5 6 8 10 12 1 2 5 6 3 7
Solving Sequence
6,12
7 1
4,8
3 11 10 5 2 9
c
6
c
12
c
7
c
3
c
11
c
10
c
5
c
2
c
8
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h2.79905 × 10
36
u
48
− 3.47735 × 10
36
u
47
+ ··· + 2.46397 × 10
35
b − 8.54140 × 10
36
,
− 1.52263 × 10
37
u
48
+ 2.01595 × 10
37
u
47
+ ··· + 2.46397 × 10
35
a + 4.41343 × 10
37
, u
49
− u
48
+ ··· − 11u − 1i
I
u
2
= hu
8
− 6u
6
− u
5
+ 11u
4
+ 4u
3
− 6u
2
+ b − 3u,
− u
9
− u
8
+ 7u
7
+ 7u
6
− 16u
5
− 16u
4
+ 13u
3
+ 13u
2
+ a − 3u − 2,
u
11
− 8u
9
− u
8
+ 23u
7
+ 6u
6
− 28u
5
− 11u
4
+ 12u
3
+ 6u
2
− 1i
* 2 irreducible components of dim
C
= 0, with total 60 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
= h2.80×10
36
u
48
−3.48×10
36
u
47
+· · ·+2.46×10
35
b−8.54×10
36
, −1.52×
10
37
u
48
+2.02×10
37
u
47
+· · ·+2.46×10
35
a+4.41×10
37
, u
49
−u
48
+· · ·−11u−1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
7
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
61.7957u
48
− 81.8168u
47
+ ··· − 1478.08u − 179.118
−11.3599u
48
+ 14.1128u
47
+ ··· + 260.816u + 34.6651
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
50.4358u
48
− 67.7041u
47
+ ··· − 1217.27u − 144.453
−11.3599u
48
+ 14.1128u
47
+ ··· + 260.816u + 34.6651
a
11
=
0.0440751u
48
+ 0.409727u
47
+ ··· + 27.7427u + 11.4620
−14.3871u
48
+ 19.4508u
47
+ ··· + 353.858u + 45.6249
a
10
=
14.4312u
48
− 19.0411u
47
+ ··· − 326.115u − 34.1629
−14.3871u
48
+ 19.4508u
47
+ ··· + 353.858u + 45.6249
a
5
=
64.0525u
48
− 85.7056u
47
+ ··· − 1545.71u − 185.562
−12.0629u
48
+ 15.1894u
47
+ ··· + 279.471u + 36.8940
a
2
=
49.9570u
48
− 67.3760u
47
+ ··· − 1201.28u − 146.399
−23.5956u
48
+ 30.8548u
47
+ ··· + 561.583u + 71.2213
a
9
=
4.08898u
48
− 6.07759u
47
+ ··· − 126.997u − 13.5152
20.0723u
48
− 26.3130u
47
+ ··· − 472.737u − 58.6870
(ii) Obstruction class = −1
(iii) Cusp Shapes = 16.0931u
48
− 20.2673u
47
+ ··· − 400.029u − 58.9748
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
49
+ u
48
+ ··· + 25u − 7
c
2
u
49
+ 2u
48
+ ··· − 33u + 1
c
3
u
49
− 3u
48
+ ··· + 210u − 19
c
4
u
49
− 9u
47
+ ··· − 36u + 8
c
5
, c
9
, c
10
u
49
+ u
48
+ ··· − 13u + 1
c
6
, c
7
, c
12
u
49
− u
48
+ ··· − 11u − 1
c
8
u
49
− 20u
47
+ ··· + 8u − 1
c
11
u
49
− 3u
48
+ ··· + 288u − 32
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
49
− 5y
48
+ ··· + 4153y −49
c
2
y
49
+ 48y
48
+ ··· + 37y −1
c
3
y
49
− 35y
48
+ ··· + 18032y −361
c
4
y
49
− 18y
48
+ ··· + 1232y −64
c
5
, c
9
, c
10
y
49
− 3y
48
+ ··· + 43y −1
c
6
, c
7
, c
12
y
49
− 55y
48
+ ··· + 81y −1
c
8
y
49
− 40y
48
+ ··· + 180y −1
c
11
y
49
+ 35y
48
+ ··· + 56832y −1024
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.641764 + 0.767757I
a = 0.212708 − 1.025840I
b = 1.42317 + 0.51659I
−5.75634 − 10.26200I 6.00000 + 7.47867I
u = −0.641764 − 0.767757I
a = 0.212708 + 1.025840I
b = 1.42317 − 0.51659I
−5.75634 + 10.26200I 6.00000 − 7.47867I
u = 0.651655 + 0.692689I
a = 0.330766 − 1.178850I
b = −1.300300 − 0.081727I
−5.91198 + 1.73898I 4.34337 − 2.71429I
u = 0.651655 − 0.692689I
a = 0.330766 + 1.178850I
b = −1.300300 + 0.081727I
−5.91198 − 1.73898I 4.34337 + 2.71429I
u = −0.418476 + 0.844249I
a = 0.040141 − 0.607878I
b = 1.308240 − 0.286159I
−6.42599 + 4.95557I 3.91217 − 3.09562I
u = −0.418476 − 0.844249I
a = 0.040141 + 0.607878I
b = 1.308240 + 0.286159I
−6.42599 − 4.95557I 3.91217 + 3.09562I
u = 0.916750 + 0.118980I
a = 0.496846 − 0.222435I
b = 0.612158 − 0.106927I
0.735978 + 0.014240I 7.49141 − 0.27992I
u = 0.916750 − 0.118980I
a = 0.496846 + 0.222435I
b = 0.612158 + 0.106927I
0.735978 − 0.014240I 7.49141 + 0.27992I
u = −0.885292
a = −1.45711
b = 0.545766
5.54572 19.1080
u = 0.394498 + 0.733725I
a = −0.353920 − 0.514157I
b = −1.47752 + 0.36490I
−6.67269 + 3.03560I 3.18463 − 3.23924I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.394498 − 0.733725I
a = −0.353920 + 0.514157I
b = −1.47752 − 0.36490I
−6.67269 − 3.03560I 3.18463 + 3.23924I
u = 0.403337 + 0.674822I
a = 0.837586 + 0.999517I
b = 0.918674 − 0.482145I
0.07476 + 3.86279I 10.31758 − 8.58603I
u = 0.403337 − 0.674822I
a = 0.837586 − 0.999517I
b = 0.918674 + 0.482145I
0.07476 − 3.86279I 10.31758 + 8.58603I
u = −0.493733 + 0.506424I
a = −0.022452 + 1.205660I
b = −1.239100 − 0.295194I
−2.18381 − 1.77583I 1.28363 + 3.40944I
u = −0.493733 − 0.506424I
a = −0.022452 − 1.205660I
b = −1.239100 + 0.295194I
−2.18381 + 1.77583I 1.28363 − 3.40944I
u = 0.596049 + 0.332001I
a = 0.0930989 + 0.0322612I
b = 0.645047 + 0.351272I
0.939381 − 0.004377I 11.68329 − 2.14560I
u = 0.596049 − 0.332001I
a = 0.0930989 − 0.0322612I
b = 0.645047 − 0.351272I
0.939381 + 0.004377I 11.68329 + 2.14560I
u = 0.652271
a = 0.372389
b = 0.495923
0.846303 11.2940
u = −1.401180 + 0.071282I
a = −0.228982 − 1.296890I
b = −0.744826 + 0.130454I
2.86436 − 3.85776I 0
u = −1.401180 − 0.071282I
a = −0.228982 + 1.296890I
b = −0.744826 − 0.130454I
2.86436 + 3.85776I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.488996 + 0.327400I
a = 0.094924 + 1.056710I
b = −0.305008 − 1.271100I
−0.54405 − 4.24517I 9.7728 + 10.6066I
u = −0.488996 − 0.327400I
a = 0.094924 − 1.056710I
b = −0.305008 + 1.271100I
−0.54405 + 4.24517I 9.7728 − 10.6066I
u = 1.44515
a = −0.954855
b = 1.92486
8.23306 0
u = −1.45917 + 0.05319I
a = −0.69275 + 1.30705I
b = 0.771612 − 1.000240I
7.20281 − 1.16880I 0
u = −1.45917 − 0.05319I
a = −0.69275 − 1.30705I
b = 0.771612 + 1.000240I
7.20281 + 1.16880I 0
u = 1.41436 + 0.36426I
a = −0.617433 + 0.658074I
b = 1.074250 + 0.051023I
−0.604182 − 0.592066I 0
u = 1.41436 − 0.36426I
a = −0.617433 − 0.658074I
b = 1.074250 − 0.051023I
−0.604182 + 0.592066I 0
u = −1.44925 + 0.23283I
a = 0.84275 + 1.38899I
b = −1.63496 − 0.71058I
−0.76793 − 6.48546I 0
u = −1.44925 − 0.23283I
a = 0.84275 − 1.38899I
b = −1.63496 + 0.71058I
−0.76793 + 6.48546I 0
u = 1.46849 + 0.04497I
a = 0.55860 − 1.84941I
b = −1.027200 + 0.837713I
4.92852 + 3.18189I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.46849 − 0.04497I
a = 0.55860 + 1.84941I
b = −1.027200 − 0.837713I
4.92852 − 3.18189I 0
u = −1.49018 + 0.23223I
a = −0.13850 − 1.62478I
b = 0.993261 + 0.685172I
6.26979 − 7.15802I 0
u = −1.49018 − 0.23223I
a = −0.13850 + 1.62478I
b = 0.993261 − 0.685172I
6.26979 + 7.15802I 0
u = 1.51747 + 0.10193I
a = −0.02195 − 2.18037I
b = −0.11588 + 1.86186I
6.15386 + 5.81406I 0
u = 1.51747 − 0.10193I
a = −0.02195 + 2.18037I
b = −0.11588 − 1.86186I
6.15386 − 5.81406I 0
u = 0.070104 + 0.447962I
a = 0.86248 + 2.37421I
b = −0.112925 + 0.222459I
−1.85819 + 2.41374I 0.89088 − 1.57246I
u = 0.070104 − 0.447962I
a = 0.86248 − 2.37421I
b = −0.112925 − 0.222459I
−1.85819 − 2.41374I 0.89088 + 1.57246I
u = 1.54796 + 0.13800I
a = 0.89692 − 1.56299I
b = −1.24570 + 0.84885I
4.67206 + 4.01107I 0
u = 1.54796 − 0.13800I
a = 0.89692 + 1.56299I
b = −1.24570 − 0.84885I
4.67206 − 4.01107I 0
u = −1.57059
a = −0.818373
b = 1.53662
8.63484 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.58313 + 0.26404I
a = −0.66377 + 1.57029I
b = 1.47297 − 0.74698I
1.5668 + 14.1154I 0
u = 1.58313 − 0.26404I
a = −0.66377 − 1.57029I
b = 1.47297 + 0.74698I
1.5668 − 14.1154I 0
u = −1.59972 + 0.24025I
a = 0.852446 + 1.040690I
b = −1.079970 − 0.166315I
1.59039 − 5.25681I 0
u = −1.59972 − 0.24025I
a = 0.852446 − 1.040690I
b = −1.079970 + 0.166315I
1.59039 + 5.25681I 0
u = −0.330191 + 0.002846I
a = 0.86697 + 4.04000I
b = −0.815216 − 0.660545I
−1.14597 − 2.72643I 9.81282 + 0.77699I
u = −0.330191 − 0.002846I
a = 0.86697 − 4.04000I
b = −0.815216 + 0.660545I
−1.14597 + 2.72643I 9.81282 − 0.77699I
u = 1.69294
a = −0.633603
b = 0.160994
14.6945 0
u = −1.78378
a = −0.529461
b = 0.690019
11.4992 0
u = −0.132976
a = 5.52802
b = 1.40427
2.79880 −5.65840
9
II. I
u
2
= hu
8
− 6u
6
− u
5
+ 11u
4
+ 4u
3
− 6u
2
+ b − 3u, −u
9
− u
8
+ · · · + a −
2, u
11
− 8u
9
+ · · · + 6u
2
− 1i
(i) Arc colorings
a
6
=
1
0
a
12
=
0
u
a
7
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
u
9
+ u
8
− 7u
7
− 7u
6
+ 16u
5
+ 16u
4
− 13u
3
− 13u
2
+ 3u + 2
−u
8
+ 6u
6
+ u
5
− 11u
4
− 4u
3
+ 6u
2
+ 3u
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
u
9
− 7u
7
− u
6
+ 17u
5
+ 5u
4
− 17u
3
− 7u
2
+ 6u + 2
−u
8
+ 6u
6
+ u
5
− 11u
4
− 4u
3
+ 6u
2
+ 3u
a
11
=
−u
4
+ 4u
2
− 3
u
7
− 5u
5
− u
4
+ 7u
3
+ 3u
2
− 2u − 1
a
10
=
−u
7
+ 5u
5
− 7u
3
+ u
2
+ 2u − 2
u
7
− 5u
5
− u
4
+ 7u
3
+ 3u
2
− 2u − 1
a
5
=
u
9
− 7u
7
− 2u
6
+ 17u
5
+ 9u
4
− 16u
3
− 11u
2
+ 4u + 3
u
6
− 4u
4
− u
3
+ 4u
2
+ 2u
a
2
=
2u
9
+ u
8
− 13u
7
− 7u
6
+ 28u
5
+ 15u
4
− 23u
3
− 10u
2
+ 7u
−u
9
− u
8
+ 6u
7
+ 7u
6
− 11u
5
− 15u
4
+ 5u
3
+ 10u
2
+ u − 1
a
9
=
−u
8
− u
7
+ 6u
6
+ 7u
5
− 12u
4
− 14u
3
+ 9u
2
+ 8u − 2
u
10
− 7u
8
− u
7
+ 17u
6
+ 4u
5
− 16u
4
− 4u
3
+ 4u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes
= 2u
10
+ 5u
9
− 13u
8
− 35u
7
+ 22u
6
+ 88u
5
+ 5u
4
− 94u
3
− 29u
2
+ 33u + 21
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
11
− 8u
10
+ ··· + 6u − 1
c
2
u
11
+ 3u
10
+ ··· − 4u + 1
c
3
u
11
− 4u
9
+ 6u
8
+ 4u
7
− 15u
6
+ 8u
5
+ 9u
4
− 11u
3
+ u
2
+ 3u − 1
c
4
u
11
− u
10
− 5u
9
+ 5u
8
+ 8u
7
− 9u
6
− 6u
5
+ 8u
4
+ 4u
3
− 4u
2
− u + 1
c
5
u
11
− 4u
9
+ u
8
+ u
7
+ 3u
5
+ u
4
− u
2
− 1
c
6
, c
7
u
11
− 8u
9
− u
8
+ 23u
7
+ 6u
6
− 28u
5
− 11u
4
+ 12u
3
+ 6u
2
− 1
c
8
u
11
+ u
10
− 4u
9
− 4u
8
+ 8u
7
+ 6u
6
− 9u
5
− 8u
4
+ 5u
3
+ 5u
2
− u − 1
c
9
, c
10
u
11
− 4u
9
− u
8
+ u
7
+ 3u
5
− u
4
+ u
2
+ 1
c
11
u
11
+ u
9
− u
7
− 3u
6
− u
4
− u
3
+ 4u
2
− 1
c
12
u
11
− 8u
9
+ u
8
+ 23u
7
− 6u
6
− 28u
5
+ 11u
4
+ 12u
3
− 6u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
11
− 2y
10
+ ··· + 40y −1
c
2
y
11
+ 3y
10
+ ··· + 44y −1
c
3
y
11
− 8y
10
+ ··· + 11y −1
c
4
y
11
− 11y
10
+ ··· + 9y −1
c
5
, c
9
, c
10
y
11
− 8y
10
+ 18y
9
− 3y
8
− 23y
7
+ 4y
6
+ 11y
5
+ y
4
+ 2y
3
+ y
2
− 2y −1
c
6
, c
7
, c
12
y
11
− 16y
10
+ ··· + 12y −1
c
8
y
11
− 9y
10
+ ··· + 11y −1
c
11
y
11
+ 2y
10
− y
9
− 2y
8
− y
7
− 11y
6
− 4y
5
+ 23y
4
+ 3y
3
− 18y
2
+ 8y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.878786
a = −1.68666
b = 0.926856
4.86824 6.36840
u = −1.076610 + 0.315115I
a = −0.290331 + 0.209919I
b = −0.724726 + 0.256674I
0.954503 + 0.928333I 11.11734 − 6.67941I
u = −1.076610 − 0.315115I
a = −0.290331 − 0.209919I
b = −0.724726 − 0.256674I
0.954503 − 0.928333I 11.11734 + 6.67941I
u = −0.334220 + 0.350205I
a = −0.55984 + 2.81156I
b = −0.785078 − 0.651739I
−1.45690 − 3.34942I 3.58906 + 9.96048I
u = −0.334220 − 0.350205I
a = −0.55984 − 2.81156I
b = −0.785078 + 0.651739I
−1.45690 + 3.34942I 3.58906 − 9.96048I
u = −1.52390
a = −1.26943
b = 2.03629
9.60351 18.6690
u = 1.52509 + 0.12133I
a = 0.66157 − 1.92570I
b = −0.785157 + 1.100430I
4.99526 + 5.07300I 8.69107 − 6.17699I
u = 1.52509 − 0.12133I
a = 0.66157 + 1.92570I
b = −0.785157 − 1.100430I
4.99526 − 5.07300I 8.69107 + 6.17699I
u = 0.357380
a = 1.15323
b = 1.49451
3.06451 25.4100
u = −1.71050
a = −0.920514
b = 0.631482
14.2218 4.29470
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.76972
a = 0.100596
b = −0.499214
11.8941 20.4630
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
11
− 8u
10
+ ··· + 6u − 1)(u
49
+ u
48
+ ··· + 25u − 7)
c
2
(u
11
+ 3u
10
+ ··· − 4u + 1)(u
49
+ 2u
48
+ ··· − 33u + 1)
c
3
(u
11
− 4u
9
+ 6u
8
+ 4u
7
− 15u
6
+ 8u
5
+ 9u
4
− 11u
3
+ u
2
+ 3u − 1)
· (u
49
− 3u
48
+ ··· + 210u − 19)
c
4
(u
11
− u
10
− 5u
9
+ 5u
8
+ 8u
7
− 9u
6
− 6u
5
+ 8u
4
+ 4u
3
− 4u
2
− u + 1)
· (u
49
− 9u
47
+ ··· − 36u + 8)
c
5
(u
11
− 4u
9
+ ··· − u
2
− 1)(u
49
+ u
48
+ ··· − 13u + 1)
c
6
, c
7
(u
11
− 8u
9
− u
8
+ 23u
7
+ 6u
6
− 28u
5
− 11u
4
+ 12u
3
+ 6u
2
− 1)
· (u
49
− u
48
+ ··· − 11u − 1)
c
8
(u
11
+ u
10
− 4u
9
− 4u
8
+ 8u
7
+ 6u
6
− 9u
5
− 8u
4
+ 5u
3
+ 5u
2
− u − 1)
· (u
49
− 20u
47
+ ··· + 8u − 1)
c
9
, c
10
(u
11
− 4u
9
+ ··· + u
2
+ 1)(u
49
+ u
48
+ ··· − 13u + 1)
c
11
(u
11
+ u
9
+ ··· + 4u
2
− 1)(u
49
− 3u
48
+ ··· + 288u − 32)
c
12
(u
11
− 8u
9
+ u
8
+ 23u
7
− 6u
6
− 28u
5
+ 11u
4
+ 12u
3
− 6u
2
+ 1)
· (u
49
− u
48
+ ··· − 11u − 1)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
11
− 2y
10
+ ··· + 40y −1)(y
49
− 5y
48
+ ··· + 4153y −49)
c
2
(y
11
+ 3y
10
+ ··· + 44y −1)(y
49
+ 48y
48
+ ··· + 37y −1)
c
3
(y
11
− 8y
10
+ ··· + 11y −1)(y
49
− 35y
48
+ ··· + 18032y −361)
c
4
(y
11
− 11y
10
+ ··· + 9y −1)(y
49
− 18y
48
+ ··· + 1232y −64)
c
5
, c
9
, c
10
(y
11
− 8y
10
+ 18y
9
− 3y
8
− 23y
7
+ 4y
6
+ 11y
5
+ y
4
+ 2y
3
+ y
2
− 2y −1)
· (y
49
− 3y
48
+ ··· + 43y −1)
c
6
, c
7
, c
12
(y
11
− 16y
10
+ ··· + 12y −1)(y
49
− 55y
48
+ ··· + 81y −1)
c
8
(y
11
− 9y
10
+ ··· + 11y −1)(y
49
− 40y
48
+ ··· + 180y −1)
c
11
(y
11
+ 2y
10
− y
9
− 2y
8
− y
7
− 11y
6
− 4y
5
+ 23y
4
+ 3y
3
− 18y
2
+ 8y −1)
· (y
49
+ 35y
48
+ ··· + 56832y −1024)
16
