12n
0428
(K12n
0428
)
A knot diagram
1
Linearized knot diagam
3 6 8 12 2 10 5 12 7 8 4 10
Solving Sequence
2,6 3,10
7 1 5 8 9 12 4 11
c
2
c
6
c
1
c
5
c
7
c
9
c
12
c
4
c
11
c
3
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h3309761832986841u
28
− 484169443496906u
27
+ ··· + 2728407209023967b + 10346261032522342,
6.12347 × 10
15
u
28
− 3.65447 × 10
15
u
27
+ ··· + 2.72841 × 10
15
a + 3.93244 × 10
16
, u
29
− u
28
+ ··· + 11u − 1i
I
u
2
= h13u
13
+ 7u
12
+ ··· + b − 29, 224u
13
+ 105u
12
+ ··· + a − 480,
u
14
− 5u
12
+ u
11
+ 11u
10
− 4u
9
− 13u
8
+ 7u
7
+ 8u
6
− 10u
5
− u
4
+ 10u
3
− 2u
2
− 3u + 1i
* 2 irreducible components of dim
C
= 0, with total 43 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
= h3.31 × 10
15
u
28
− 4.84 × 10
14
u
27
+ · · · + 2.73 × 10
15
b + 1.03 × 10
16
, 6.12 ×
10
15
u
28
−3.65×10
15
u
27
+· · ·+2.73×10
15
a+3.93×10
16
, u
29
−u
28
+· · ·+11u−1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
−2.24434u
28
+ 1.33942u
27
+ ··· + 45.0118u − 14.4130
−1.21307u
28
+ 0.177455u
27
+ ··· + 17.9088u − 3.79205
a
7
=
3.72446u
28
− 1.92791u
27
+ ··· − 66.8278u + 18.8763
1.69382u
28
− 0.176481u
27
+ ··· − 18.8053u + 4.46366
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
8
=
3.69627u
28
− 1.96435u
27
+ ··· − 67.8684u + 19.1555
1.66563u
28
− 0.212920u
27
+ ··· − 19.8460u + 4.74286
a
9
=
−8.80217u
28
+ 2.99230u
27
+ ··· + 150.634u − 40.2378
−5.79123u
28
− 0.238662u
27
+ ··· + 64.3246u − 11.6085
a
12
=
−5.43646u
28
+ 2.41208u
27
+ ··· + 107.030u − 28.7016
−2.54933u
28
− 0.164013u
27
+ ··· + 33.7253u − 7.14216
a
4
=
2.50984u
28
− 6.17071u
27
+ ··· − 123.703u + 44.9330
−2.45081u
28
− 2.07795u
27
+ ··· − 2.61459u + 7.33026
a
11
=
11.3793u
28
− 5.86912u
27
+ ··· − 218.129u + 62.4454
6.01027u
28
− 0.499882u
27
+ ··· − 76.2473u + 16.4335
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
29606274479465
94083007207723
u
28
−
112181565603524
94083007207723
u
27
+ ··· −
79763733552791
94083007207723
u −
508359165202078
94083007207723
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
29
+ 23u
28
+ ··· + 57u + 1
c
2
, c
5
u
29
+ u
28
+ ··· + 11u + 1
c
3
u
29
+ 2u
28
+ ··· − 512u − 181
c
4
, c
11
u
29
+ 3u
28
+ ··· + 52u + 17
c
6
, c
9
u
29
+ 6u
28
+ ··· − 3u − 1
c
7
u
29
− 3u
28
+ ··· − 343u − 73
c
8
u
29
+ 3u
28
+ ··· − 802u − 61
c
10
u
29
− 2u
28
+ ··· + 306u + 17
c
12
u
29
− 13u
28
+ ··· − 419u − 617
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
29
− 27y
28
+ ··· + 3781y −1
c
2
, c
5
y
29
− 23y
28
+ ··· + 57y −1
c
3
y
29
− 66y
28
+ ··· + 773288y −32761
c
4
, c
11
y
29
− 47y
28
+ ··· + 3350y −289
c
6
, c
9
y
29
+ 30y
28
+ ··· + 5y −1
c
7
y
29
− 9y
28
+ ··· + 108013y −5329
c
8
y
29
− 53y
28
+ ··· + 159352y −3721
c
10
y
29
− 74y
28
+ ··· + 76942y −289
c
12
y
29
− 79y
28
+ ··· − 73773123y −380689
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.847490 + 0.607947I
a = −0.422315 − 0.373511I
b = −0.453967 − 0.141290I
1.85694 + 2.41024I −0.38198 − 1.83000I
u = −0.847490 − 0.607947I
a = −0.422315 + 0.373511I
b = −0.453967 + 0.141290I
1.85694 − 2.41024I −0.38198 + 1.83000I
u = 0.071949 + 0.929488I
a = 1.55015 + 0.37840I
b = −0.033988 + 0.404715I
−5.90516 − 1.34940I −11.77988 + 1.16724I
u = 0.071949 − 0.929488I
a = 1.55015 − 0.37840I
b = −0.033988 − 0.404715I
−5.90516 + 1.34940I −11.77988 − 1.16724I
u = 1.099740 + 0.372987I
a = 0.429259 − 0.314350I
b = −0.496131 + 0.172360I
−2.13143 − 3.61670I −10.87499 + 4.53147I
u = 1.099740 − 0.372987I
a = 0.429259 + 0.314350I
b = −0.496131 − 0.172360I
−2.13143 + 3.61670I −10.87499 − 4.53147I
u = −0.071629 + 1.166630I
a = −1.42014 + 0.38258I
b = 0.0590394 − 0.0700190I
−17.8156 + 5.4788I −11.43005 − 2.27554I
u = −0.071629 − 1.166630I
a = −1.42014 − 0.38258I
b = 0.0590394 + 0.0700190I
−17.8156 − 5.4788I −11.43005 + 2.27554I
u = −1.20779
a = 1.06012
b = 0.608535
−5.45443 −17.0500
u = −1.241400 + 0.177721I
a = 0.09350 − 1.46108I
b = −0.13368 − 2.49732I
−4.23135 + 3.90349I −13.5060 − 7.1974I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.241400 − 0.177721I
a = 0.09350 + 1.46108I
b = −0.13368 + 2.49732I
−4.23135 − 3.90349I −13.5060 + 7.1974I
u = 0.607700 + 0.416192I
a = 0.030334 + 0.170685I
b = 0.768475 − 0.109674I
−0.645013 + 0.114015I −8.30235 − 0.44217I
u = 0.607700 − 0.416192I
a = 0.030334 − 0.170685I
b = 0.768475 + 0.109674I
−0.645013 − 0.114015I −8.30235 + 0.44217I
u = 1.264620 + 0.097173I
a = 0.126127 + 0.975567I
b = 0.45216 + 2.48013I
−4.42915 + 0.22712I −13.77800 + 0.54237I
u = 1.264620 − 0.097173I
a = 0.126127 − 0.975567I
b = 0.45216 − 2.48013I
−4.42915 − 0.22712I −13.77800 − 0.54237I
u = 1.27084
a = −2.54450
b = −2.41619
−14.6938 −21.3900
u = −1.35314
a = −0.122473
b = 1.92297
−15.8953 −17.3790
u = 1.288070 + 0.432233I
a = −0.52667 − 1.40404I
b = −0.52972 − 2.41614I
−9.72617 − 3.55628I −15.1648 + 2.5004I
u = 1.288070 − 0.432233I
a = −0.52667 + 1.40404I
b = −0.52972 + 2.41614I
−9.72617 + 3.55628I −15.1648 − 2.5004I
u = −1.36698 + 0.39083I
a = 0.056520 + 1.082730I
b = −0.25816 + 2.57538I
−10.50180 + 6.08146I −14.9405 − 3.9753I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.36698 − 0.39083I
a = 0.056520 − 1.082730I
b = −0.25816 − 2.57538I
−10.50180 − 6.08146I −14.9405 + 3.9753I
u = 0.529556
a = −0.279573
b = 0.469661
−0.808015 −11.9270
u = 1.43624 + 0.53316I
a = 0.045983 + 1.342010I
b = −0.02188 + 2.71357I
16.8944 − 11.5066I −13.6242 + 4.6544I
u = 1.43624 − 0.53316I
a = 0.045983 − 1.342010I
b = −0.02188 − 2.71357I
16.8944 + 11.5066I −13.6242 − 4.6544I
u = −1.40063 + 0.62143I
a = 0.391966 − 0.973746I
b = 0.84600 − 2.06054I
17.5554 + 0.8894I −13.74018 − 0.79352I
u = −1.40063 − 0.62143I
a = 0.391966 + 0.973746I
b = 0.84600 + 2.06054I
17.5554 − 0.8894I −13.74018 + 0.79352I
u = −0.024136 + 0.378698I
a = −2.54654 − 0.61175I
b = −0.009603 + 0.352025I
−0.56395 − 1.71430I −3.16058 + 4.19709I
u = −0.024136 − 0.378698I
a = −2.54654 + 0.61175I
b = −0.009603 − 0.352025I
−0.56395 + 1.71430I −3.16058 − 4.19709I
u = 0.128440
a = −9.72992
b = −1.96207
−11.0443 −5.88640
7
II. I
u
2
= h13u
13
+ 7u
12
+ · · · + b − 29, 224u
13
+ 105u
12
+ · · · + a − 480, u
14
−
5u
12
+ · · · − 3u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
10
=
−224u
13
− 105u
12
+ ··· − 415u + 480
−13u
13
− 7u
12
+ ··· − 29u + 29
a
7
=
−143u
13
− 67u
12
+ ··· − 263u + 304
67u
13
+ 32u
12
+ ··· + 129u − 145
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
8
=
−189u
13
− 89u
12
+ ··· − 350u + 403
21u
13
+ 10u
12
+ ··· + 42u − 46
a
9
=
−143u
13
− 67u
12
+ ··· − 268u + 306
−8u
13
− 5u
12
+ ··· − 25u + 21
a
12
=
−98u
13
− 46u
12
+ ··· − 178u + 210
−22u
13
− 10u
12
+ ··· − 35u + 44
a
4
=
−130u
13
− 63u
12
+ ··· − 249u + 284
−2u
13
− 4u
12
+ ··· − 20u + 12
a
11
=
62u
13
+ 30u
12
+ ··· + 114u − 133
−13u
13
− 7u
12
+ ··· − 35u + 31
(ii) Obstruction class = 1
(iii) Cusp Shapes = 254u
13
+ 120u
12
− 1214u
11
− 319u
10
+ 2647u
9
+ 232u
8
−
3201u
7
+ 272u
6
+ 2171u
5
− 1523u
4
− 980u
3
+ 2088u
2
+ 476u − 559
8
(iv) u-Polynomials at the component
9
Crossings u-Polynomials at each crossing
c
1
u
14
− 10u
13
+ ··· − 13u + 1
c
2
u
14
− 5u
12
+ ··· − 3u + 1
c
3
u
14
+ u
13
+ ··· + 50u − 25
c
4
u
14
− 2u
13
+ ··· − 2u − 1
c
5
u
14
− 5u
12
+ ··· + 3u + 1
c
6
u
14
− u
13
+ ··· + 7u + 1
c
7
u
14
+ 2u
13
+ ··· + u − 1
c
8
u
14
+ 4u
13
+ ··· + 2u + 1
c
9
u
14
+ u
13
+ ··· − 7u + 1
c
10
u
14
+ 15u
13
+ ··· + 1248u + 169
c
11
u
14
+ 2u
13
+ ··· + 2u − 1
c
12
u
14
+ 6u
13
+ ··· + 215u + 25
10
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
14
− 6y
13
+ ··· − 45y + 1
c
2
, c
5
y
14
− 10y
13
+ ··· − 13y + 1
c
3
y
14
− 13y
13
+ ··· + 4600y + 625
c
4
, c
11
y
14
− 18y
13
+ ··· + 2y + 1
c
6
, c
9
y
14
+ 7y
13
+ ··· − 17y + 1
c
7
y
14
+ 8y
13
+ ··· + 7y + 1
c
8
y
14
− 16y
13
+ ··· − 8y + 1
c
10
y
14
− 9y
13
+ ··· − 67938y + 28561
c
12
y
14
− 22y
13
+ ··· − 5325y + 625
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.835923 + 0.681048I
a = −0.125313 − 0.206996I
b = −0.446348 + 0.397563I
1.26701 + 2.62945I −13.2524 − 5.2108I
u = −0.835923 − 0.681048I
a = −0.125313 + 0.206996I
b = −0.446348 − 0.397563I
1.26701 − 2.62945I −13.2524 + 5.2108I
u = 0.345551 + 0.838709I
a = −1.020240 − 0.715351I
b = −0.243705 − 0.098495I
−2.18681 − 1.21939I −11.09554 + 1.93458I
u = 0.345551 − 0.838709I
a = −1.020240 + 0.715351I
b = −0.243705 + 0.098495I
−2.18681 + 1.21939I −11.09554 − 1.93458I
u = −1.22266
a = 1.67303
b = 0.728923
−13.8048 −11.3860
u = 1.244710 + 0.138183I
a = −0.311608 − 0.937054I
b = 0.34348 − 2.09289I
−4.37404 − 2.52484I −15.1413 + 2.0385I
u = 1.244710 − 0.138183I
a = −0.311608 + 0.937054I
b = 0.34348 + 2.09289I
−4.37404 + 2.52484I −15.1413 − 2.0385I
u = −0.727304
a = 1.73201
b = 2.70193
−11.7805 −17.4240
u = 1.124850 + 0.672854I
a = 0.314703 + 0.747799I
b = 0.580657 + 1.233750I
−4.34890 − 4.36927I −12.83000 + 4.03520I
u = 1.124850 − 0.672854I
a = 0.314703 − 0.747799I
b = 0.580657 − 1.233750I
−4.34890 + 4.36927I −12.83000 − 4.03520I
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.372300 + 0.328498I
a = −0.239034 − 1.275240I
b = −0.26688 − 2.25518I
−7.44862 + 5.20740I −13.30820 − 4.11576I
u = −1.372300 − 0.328498I
a = −0.239034 + 1.275240I
b = −0.26688 + 2.25518I
−7.44862 − 5.20740I −13.30820 + 4.11576I
u = 0.468087 + 0.001290I
a = 0.67897 − 1.74706I
b = −0.682627 − 0.104415I
−1.36973 − 1.50897I −13.46764 + 1.98399I
u = 0.468087 − 0.001290I
a = 0.67897 + 1.74706I
b = −0.682627 + 0.104415I
−1.36973 + 1.50897I −13.46764 − 1.98399I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
14
− 10u
13
+ ··· − 13u + 1)(u
29
+ 23u
28
+ ··· + 57u + 1)
c
2
(u
14
− 5u
12
+ ··· − 3u + 1)(u
29
+ u
28
+ ··· + 11u + 1)
c
3
(u
14
+ u
13
+ ··· + 50u − 25)(u
29
+ 2u
28
+ ··· − 512u − 181)
c
4
(u
14
− 2u
13
+ ··· − 2u − 1)(u
29
+ 3u
28
+ ··· + 52u + 17)
c
5
(u
14
− 5u
12
+ ··· + 3u + 1)(u
29
+ u
28
+ ··· + 11u + 1)
c
6
(u
14
− u
13
+ ··· + 7u + 1)(u
29
+ 6u
28
+ ··· − 3u − 1)
c
7
(u
14
+ 2u
13
+ ··· + u − 1)(u
29
− 3u
28
+ ··· − 343u − 73)
c
8
(u
14
+ 4u
13
+ ··· + 2u + 1)(u
29
+ 3u
28
+ ··· − 802u − 61)
c
9
(u
14
+ u
13
+ ··· − 7u + 1)(u
29
+ 6u
28
+ ··· − 3u − 1)
c
10
(u
14
+ 15u
13
+ ··· + 1248u + 169)(u
29
− 2u
28
+ ··· + 306u + 17)
c
11
(u
14
+ 2u
13
+ ··· + 2u − 1)(u
29
+ 3u
28
+ ··· + 52u + 17)
c
12
(u
14
+ 6u
13
+ ··· + 215u + 25)(u
29
− 13u
28
+ ··· − 419u − 617)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
14
− 6y
13
+ ··· − 45y + 1)(y
29
− 27y
28
+ ··· + 3781y −1)
c
2
, c
5
(y
14
− 10y
13
+ ··· − 13y + 1)(y
29
− 23y
28
+ ··· + 57y −1)
c
3
(y
14
− 13y
13
+ ··· + 4600y + 625)
· (y
29
− 66y
28
+ ··· + 773288y −32761)
c
4
, c
11
(y
14
− 18y
13
+ ··· + 2y + 1)(y
29
− 47y
28
+ ··· + 3350y −289)
c
6
, c
9
(y
14
+ 7y
13
+ ··· − 17y + 1)(y
29
+ 30y
28
+ ··· + 5y −1)
c
7
(y
14
+ 8y
13
+ ··· + 7y + 1)(y
29
− 9y
28
+ ··· + 108013y −5329)
c
8
(y
14
− 16y
13
+ ··· − 8y + 1)(y
29
− 53y
28
+ ··· + 159352y −3721)
c
10
(y
14
− 9y
13
+ ··· − 67938y + 28561)
· (y
29
− 74y
28
+ ··· + 76942y −289)
c
12
(y
14
− 22y
13
+ ··· − 5325y + 625)
· (y
29
− 79y
28
+ ··· − 73773123y −380689)
16
