12n
0075
(K12n
0075
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 8 11 3 12 1 6 10 9
Solving Sequence
8,12
9 1
4,10
3 2 7 11 6 5
c
8
c
12
c
9
c
3
c
1
c
7
c
11
c
6
c
5
c
2
, c
4
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−4.17318 × 10
15
u
51
+ 1.53597 × 10
16
u
50
+ ··· + 1.39326 × 10
15
b + 3.61251 × 10
15
,
3.45613 × 10
16
u
51
− 1.28036 × 10
17
u
50
+ ··· + 2.78652 × 10
15
a + 6.93027 × 10
15
, u
52
− 5u
51
+ ··· + 14u + 1i
I
u
2
= hb, u
7
− 2u
6
− 2u
5
+ 4u
4
+ 2u
3
− u
2
+ a − u − 3, u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1i
I
u
3
= ha
2
+ 5b + 3a + 5, a
3
+ a
2
+ 4a + 5, u + 1i
* 3 irreducible components of dim
C
= 0, with total 63 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
= h−4.17 × 10
15
u
51
+ 1.54 × 10
16
u
50
+ · · · + 1.39 × 10
15
b + 3.61 ×
10
15
, 3.46 × 10
16
u
51
− 1.28 × 10
17
u
50
+ · · · + 2.79 × 10
15
a + 6.93 ×
10
15
, u
52
− 5u
51
+ · · · + 14u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−12.4030u
51
+ 45.9484u
50
+ ··· + 81.3098u − 2.48707
2.99526u
51
− 11.0243u
50
+ ··· − 30.7361u − 2.59285
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−9.40778u
51
+ 34.9242u
50
+ ··· + 50.5737u − 5.07992
2.99526u
51
− 11.0243u
50
+ ··· − 30.7361u − 2.59285
a
2
=
17.4460u
51
− 64.6564u
50
+ ··· − 149.987u − 4.05564
−3.75474u
51
+ 13.9757u
50
+ ··· + 42.5139u + 3.15715
a
7
=
−9.55271u
51
+ 35.8772u
50
+ ··· + 88.4408u + 2.91964
8.60172u
51
− 33.2652u
50
+ ··· − 104.422u − 7.28044
a
11
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
6
=
8.36062u
51
− 30.6096u
50
+ ··· − 105.182u − 10.2144
3.75474u
51
− 13.9757u
50
+ ··· − 42.5139u − 3.15715
a
5
=
4.60588u
51
− 16.6339u
50
+ ··· − 62.6681u − 7.05722
3.75474u
51
− 13.9757u
50
+ ··· − 42.5139u − 3.15715
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
11065552867124641
1393259322402082
u
51
+
42789532669637567
1393259322402082
u
50
+ ···+
1199032770084567
1393259322402082
u −
7200606156302692
696629661201041
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
52
+ 14u
51
+ ··· + 43u + 1
c
2
, c
4
u
52
− 10u
51
+ ··· − u + 1
c
3
, c
7
u
52
− 2u
51
+ ··· + 384u − 256
c
5
u
52
+ 3u
51
+ ··· − u − 1
c
6
, c
10
u
52
+ 2u
51
+ ··· − 28u − 8
c
8
, c
9
, c
12
u
52
+ 5u
51
+ ··· − 14u + 1
c
11
u
52
− 24u
51
+ ··· − 1488u + 64
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
52
+ 58y
51
+ ··· − 447y + 1
c
2
, c
4
y
52
− 14y
51
+ ··· − 43y + 1
c
3
, c
7
y
52
+ 54y
51
+ ··· + 606208y + 65536
c
5
y
52
− 61y
51
+ ··· − 19y + 1
c
6
, c
10
y
52
− 24y
51
+ ··· − 1488y + 64
c
8
, c
9
, c
12
y
52
− 47y
51
+ ··· − 104y + 1
c
11
y
52
+ 4y
51
+ ··· − 498944y + 4096
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.864111 + 0.613274I
a = 0.72971 − 2.28554I
b = 0.08277 + 1.65392I
7.67792 − 2.14785I 7.33727 + 2.34757I
u = −0.864111 − 0.613274I
a = 0.72971 + 2.28554I
b = 0.08277 − 1.65392I
7.67792 + 2.14785I 7.33727 − 2.34757I
u = −0.278369 + 0.886469I
a = 0.81489 − 1.68991I
b = 0.52357 + 1.58617I
5.03572 − 9.87508I 3.10372 + 7.08066I
u = −0.278369 − 0.886469I
a = 0.81489 + 1.68991I
b = 0.52357 − 1.58617I
5.03572 + 9.87508I 3.10372 − 7.08066I
u = −0.344404 + 0.851796I
a = −0.85095 + 1.66493I
b = −0.08288 − 1.66614I
6.09934 − 2.94801I 4.84338 + 2.76292I
u = −0.344404 − 0.851796I
a = −0.85095 − 1.66493I
b = −0.08288 + 1.66614I
6.09934 + 2.94801I 4.84338 − 2.76292I
u = −0.959966 + 0.581836I
a = −0.84889 + 2.03520I
b = 0.40253 − 1.60177I
7.11104 + 4.74276I 0
u = −0.959966 − 0.581836I
a = −0.84889 − 2.03520I
b = 0.40253 + 1.60177I
7.11104 − 4.74276I 0
u = −1.149080 + 0.112159I
a = −0.11168 + 3.35174I
b = 0.257735 − 0.531872I
0.064366 − 0.675431I 0
u = −1.149080 − 0.112159I
a = −0.11168 − 3.35174I
b = 0.257735 + 0.531872I
0.064366 + 0.675431I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.078334 + 0.779627I
a = −0.0018202 − 0.0329755I
b = −0.000332 − 0.629614I
−2.90851 − 2.74298I 4.20673 + 3.96739I
u = −0.078334 − 0.779627I
a = −0.0018202 + 0.0329755I
b = −0.000332 + 0.629614I
−2.90851 + 2.74298I 4.20673 − 3.96739I
u = 1.219950 + 0.081401I
a = 0.28979 + 1.63338I
b = −0.347717 − 1.160600I
5.43256 − 2.37277I 0
u = 1.219950 − 0.081401I
a = 0.28979 − 1.63338I
b = −0.347717 + 1.160600I
5.43256 + 2.37277I 0
u = −0.282889 + 0.711805I
a = 0.178611 − 0.588706I
b = 1.029030 + 0.101314I
−0.42821 − 3.83727I 2.28173 + 6.95386I
u = −0.282889 − 0.711805I
a = 0.178611 + 0.588706I
b = 1.029030 − 0.101314I
−0.42821 + 3.83727I 2.28173 − 6.95386I
u = −1.193430 + 0.324240I
a = 0.669986 − 0.648263I
b = −0.015350 + 0.613459I
0.485168 − 1.259080I 0
u = −1.193430 − 0.324240I
a = 0.669986 + 0.648263I
b = −0.015350 − 0.613459I
0.485168 + 1.259080I 0
u = −0.680369 + 0.289746I
a = 0.012168 − 0.147432I
b = 0.638639 − 0.199115I
1.074090 + 0.016258I 8.77032 − 1.10969I
u = −0.680369 − 0.289746I
a = 0.012168 + 0.147432I
b = 0.638639 + 0.199115I
1.074090 − 0.016258I 8.77032 + 1.10969I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.280180 + 0.138912I
a = 1.48485 + 1.45215I
b = −0.906569 − 0.186892I
2.21484 − 2.00952I 0
u = −1.280180 − 0.138912I
a = 1.48485 − 1.45215I
b = −0.906569 + 0.186892I
2.21484 + 2.00952I 0
u = −0.710230
a = 0.329379
b = 0.453636
1.01816 10.6560
u = −0.202574 + 0.614660I
a = −0.41637 − 1.80557I
b = −0.212659 + 0.790964I
−2.49821 − 1.98647I 1.56783 + 2.89756I
u = −0.202574 − 0.614660I
a = −0.41637 + 1.80557I
b = −0.212659 − 0.790964I
−2.49821 + 1.98647I 1.56783 − 2.89756I
u = 1.316330 + 0.333742I
a = −0.408950 − 0.703118I
b = −0.036069 + 0.646778I
1.46111 + 6.76040I 0
u = 1.316330 − 0.333742I
a = −0.408950 + 0.703118I
b = −0.036069 − 0.646778I
1.46111 − 6.76040I 0
u = 0.240670 + 0.591040I
a = −1.50879 − 1.89532I
b = −0.38554 + 1.46577I
2.88733 + 4.52304I −0.03636 − 3.32255I
u = 0.240670 − 0.591040I
a = −1.50879 + 1.89532I
b = −0.38554 − 1.46577I
2.88733 − 4.52304I −0.03636 + 3.32255I
u = 1.371880 + 0.190150I
a = 0.51449 − 1.70845I
b = −0.965667 + 0.552199I
3.35341 + 2.44910I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.371880 − 0.190150I
a = 0.51449 + 1.70845I
b = −0.965667 − 0.552199I
3.35341 − 2.44910I 0
u = 1.381790 + 0.242632I
a = 0.70682 + 2.59552I
b = −0.377861 − 1.015230I
2.55895 + 5.12601I 0
u = 1.381790 − 0.242632I
a = 0.70682 − 2.59552I
b = −0.377861 + 1.015230I
2.55895 − 5.12601I 0
u = −1.39431 + 0.23988I
a = −0.79430 + 4.03936I
b = −0.47035 − 1.61370I
8.11020 − 7.60143I 0
u = −1.39431 − 0.23988I
a = −0.79430 − 4.03936I
b = −0.47035 + 1.61370I
8.11020 + 7.60143I 0
u = −1.40452 + 0.18151I
a = 1.20569 − 4.10928I
b = 0.00116 + 1.68427I
8.93491 − 0.61807I 0
u = −1.40452 − 0.18151I
a = 1.20569 + 4.10928I
b = 0.00116 − 1.68427I
8.93491 + 0.61807I 0
u = 1.42142 + 0.11111I
a = −1.39366 − 1.43200I
b = 0.813091 + 0.784592I
7.33747 + 1.34403I 0
u = 1.42142 − 0.11111I
a = −1.39366 + 1.43200I
b = 0.813091 − 0.784592I
7.33747 − 1.34403I 0
u = 1.41493 + 0.27859I
a = −1.41802 + 1.14138I
b = 1.235590 − 0.150942I
4.99311 + 7.43624I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41493 − 0.27859I
a = −1.41802 − 1.14138I
b = 1.235590 + 0.150942I
4.99311 − 7.43624I 0
u = 0.314249 + 0.443599I
a = 1.74200 + 2.01299I
b = −0.08624 − 1.45158I
3.45819 − 1.75596I 0.22172 + 1.56110I
u = 0.314249 − 0.443599I
a = 1.74200 − 2.01299I
b = −0.08624 + 1.45158I
3.45819 + 1.75596I 0.22172 − 1.56110I
u = 1.43910 + 0.36041I
a = 0.67098 + 3.42215I
b = 0.61263 − 1.61586I
10.5096 + 14.3678I 0
u = 1.43910 − 0.36041I
a = 0.67098 − 3.42215I
b = 0.61263 + 1.61586I
10.5096 − 14.3678I 0
u = 1.46087 + 0.32708I
a = −1.00755 − 3.48031I
b = −0.19562 + 1.75720I
11.89100 + 7.20080I 0
u = 1.46087 − 0.32708I
a = −1.00755 + 3.48031I
b = −0.19562 − 1.75720I
11.89100 − 7.20080I 0
u = −0.111943 + 0.439985I
a = −1.241660 − 0.569711I
b = −0.772681 − 0.211532I
−1.51612 − 0.05304I −2.94270 + 1.44002I
u = −0.111943 − 0.439985I
a = −1.241660 + 0.569711I
b = −0.772681 + 0.211532I
−1.51612 + 0.05304I −2.94270 − 1.44002I
u = 1.54580 + 0.02807I
a = −0.26431 + 4.14163I
b = 0.27010 − 1.83700I
16.1487 + 3.7905I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.54580 − 0.02807I
a = −0.26431 − 4.14163I
b = 0.27010 + 1.83700I
16.1487 − 3.7905I 0
u = −0.0947755
a = −7.83548
b = −0.476249
−1.21791 −10.0970
10
II. I
u
2
=
hb, u
7
−2u
6
−2u
5
+4u
4
+2u
3
−u
2
+a−u−3, u
8
−u
7
−3u
6
+2u
5
+3u
4
−2u−1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
u
a
9
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
−u
7
+ 2u
6
+ 2u
5
− 4u
4
− 2u
3
+ u
2
+ u + 3
0
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
3
=
−u
7
+ 2u
6
+ 2u
5
− 4u
4
− 2u
3
+ u
2
+ u + 3
0
a
2
=
−u
7
+ 2u
6
+ 2u
5
− 4u
4
− 2u
3
+ u
2
+ 2u + 3
−u
3
+ u
a
7
=
1
0
a
11
=
−u
5
+ 2u
3
− u
u
7
− 3u
5
+ 2u
3
+ u
a
6
=
u
3
− 2u
u
3
− u
a
5
=
−u
u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
7
+ 10u
6
+ 7u
5
− 25u
4
− 9u
3
+ 12u
2
+ 8u + 13
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
8
c
3
, c
7
u
8
c
4
(u + 1)
8
c
5
u
8
+ 3u
7
+ 7u
6
+ 10u
5
+ 11u
4
+ 10u
3
+ 6u
2
+ 4u + 1
c
6
u
8
+ u
7
− u
6
− 2u
5
+ u
4
+ 2u
3
− 2u − 1
c
8
, c
9
u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1
c
10
u
8
− u
7
− u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u − 1
c
11
u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1
c
12
u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
8
c
3
, c
7
y
8
c
5
, c
11
y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1
c
6
, c
10
y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1
c
8
, c
9
, c
12
y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.180120 + 0.268597I
a = −0.281371 − 1.128550I
b = 0
−0.604279 − 1.131230I −2.43193 + 0.79885I
u = −1.180120 − 0.268597I
a = −0.281371 + 1.128550I
b = 0
−0.604279 + 1.131230I −2.43193 − 0.79885I
u = −0.108090 + 0.747508I
a = 0.208670 + 0.825203I
b = 0
−3.80435 − 2.57849I −5.57469 + 3.25625I
u = −0.108090 − 0.747508I
a = 0.208670 − 0.825203I
b = 0
−3.80435 + 2.57849I −5.57469 − 3.25625I
u = 1.37100
a = 0.829189
b = 0
4.85780 8.00600
u = 1.334530 + 0.318930I
a = 0.284386 − 0.605794I
b = 0
0.73474 + 6.44354I −0.28408 − 3.92092I
u = 1.334530 − 0.318930I
a = 0.284386 + 0.605794I
b = 0
0.73474 − 6.44354I −0.28408 + 3.92092I
u = −0.463640
a = 2.74744
b = 0
−0.799899 11.5750
14
III. I
u
3
= ha
2
+ 5b + 3a + 5, a
3
+ a
2
+ 4a + 5, u + 1i
(i) Arc colorings
a
8
=
1
0
a
12
=
0
−1
a
9
=
1
−1
a
1
=
−1
0
a
4
=
a
−
1
5
a
2
−
3
5
a − 1
a
10
=
0
−1
a
3
=
−
1
5
a
2
+
2
5
a − 1
−
1
5
a
2
−
3
5
a − 1
a
2
=
−2
−
2
5
a
2
−
1
5
a
a
7
=
0
−
2
5
a
2
−
1
5
a
a
11
=
0
−1
a
6
=
0
−
2
5
a
2
−
1
5
a
a
5
=
2
5
a
2
+
1
5
a
−
2
5
a
2
−
1
5
a
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
9
5
a
2
+
13
5
a − 5
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
3
− u
2
+ 2u − 1
c
2
u
3
+ u
2
− 1
c
4
u
3
− u
2
+ 1
c
5
u
3
+ 3u
2
+ 2u − 1
c
6
, c
10
, c
11
u
3
c
7
u
3
+ u
2
+ 2u + 1
c
8
, c
9
(u + 1)
3
c
12
(u − 1)
3
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
y
3
+ 3y
2
+ 2y −1
c
2
, c
4
y
3
− y
2
+ 2y −1
c
5
y
3
− 5y
2
+ 10y −1
c
6
, c
10
, c
11
y
3
c
8
, c
9
, c
12
(y −1)
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = −1.18504
b = −0.569840
0.531480 −10.6090
u = −1.00000
a = 0.09252 + 2.05200I
b = −0.215080 − 1.307140I
4.66906 − 2.82812I 2.80443 + 4.65175I
u = −1.00000
a = 0.09252 − 2.05200I
b = −0.215080 + 1.307140I
4.66906 + 2.82812I 2.80443 − 4.65175I
18
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
3
− u
2
+ 2u − 1)(u
52
+ 14u
51
+ ··· + 43u + 1)
c
2
((u − 1)
8
)(u
3
+ u
2
− 1)(u
52
− 10u
51
+ ··· − u + 1)
c
3
u
8
(u
3
− u
2
+ 2u − 1)(u
52
− 2u
51
+ ··· + 384u − 256)
c
4
((u + 1)
8
)(u
3
− u
2
+ 1)(u
52
− 10u
51
+ ··· − u + 1)
c
5
(u
3
+ 3u
2
+ 2u − 1)(u
8
+ 3u
7
+ ··· + 4u + 1)
· (u
52
+ 3u
51
+ ··· − u − 1)
c
6
u
3
(u
8
+ u
7
+ ··· − 2u − 1)(u
52
+ 2u
51
+ ··· − 28u − 8)
c
7
u
8
(u
3
+ u
2
+ 2u + 1)(u
52
− 2u
51
+ ··· + 384u − 256)
c
8
, c
9
(u + 1)
3
(u
8
− u
7
− 3u
6
+ 2u
5
+ 3u
4
− 2u − 1)
· (u
52
+ 5u
51
+ ··· − 14u + 1)
c
10
u
3
(u
8
− u
7
+ ··· + 2u − 1)(u
52
+ 2u
51
+ ··· − 28u − 8)
c
11
u
3
(u
8
− 3u
7
+ 7u
6
− 10u
5
+ 11u
4
− 10u
3
+ 6u
2
− 4u + 1)
· (u
52
− 24u
51
+ ··· − 1488u + 64)
c
12
(u − 1)
3
(u
8
+ u
7
− 3u
6
− 2u
5
+ 3u
4
+ 2u − 1)
· (u
52
+ 5u
51
+ ··· − 14u + 1)
19
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
8
)(y
3
+ 3y
2
+ 2y −1)(y
52
+ 58y
51
+ ··· − 447y + 1)
c
2
, c
4
((y −1)
8
)(y
3
− y
2
+ 2y −1)(y
52
− 14y
51
+ ··· − 43y + 1)
c
3
, c
7
y
8
(y
3
+ 3y
2
+ 2y −1)(y
52
+ 54y
51
+ ··· + 606208y + 65536)
c
5
(y
3
− 5y
2
+ 10y −1)
· (y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
52
− 61y
51
+ ··· − 19y + 1)
c
6
, c
10
y
3
(y
8
− 3y
7
+ 7y
6
− 10y
5
+ 11y
4
− 10y
3
+ 6y
2
− 4y + 1)
· (y
52
− 24y
51
+ ··· − 1488y + 64)
c
8
, c
9
, c
12
(y −1)
3
(y
8
− 7y
7
+ 19y
6
− 22y
5
+ 3y
4
+ 14y
3
− 6y
2
− 4y + 1)
· (y
52
− 47y
51
+ ··· − 104y + 1)
c
11
y
3
(y
8
+ 5y
7
+ 11y
6
+ 6y
5
− 17y
4
− 34y
3
− 22y
2
− 4y + 1)
· (y
52
+ 4y
51
+ ··· − 498944y + 4096)
20
