12n
0138
(K12n
0138
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 12 4 10 12 7 8 6 9
Solving Sequence
7,10
8
4,11
3 6 12 5 2 9 1
c
7
c
10
c
3
c
6
c
11
c
5
c
2
c
9
c
12
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h1.30613 × 10
24
u
22
+ 1.93530 × 10
25
u
21
+ ··· + 3.16864 × 10
26
b − 1.61340 × 10
26
,
1.48301 × 10
26
u
22
+ 2.07464 × 10
27
u
21
+ ··· + 3.16864 × 10
26
a + 3.75365 × 10
28
,
u
23
+ 14u
22
+ ··· + 247u − 1i
I
u
2
= h−676a
8
− 5525a
7
+ 10837a
6
− 7123a
5
− 92a
4
+ 4655a
3
− 6197a
2
+ 717b − 295a + 1497,
a
9
+ 7a
8
− 25a
7
+ 34a
6
− 25a
5
+ 9a
4
+ 5a
3
− 6a
2
+ 1, u − 1i
I
u
3
= h5a
2
u − 3a
2
+ 12au + b − 7a + 3u − 1, a
3
− a
2
u + a
2
+ 3au + 6a + 3u + 5, u
2
+ u − 1i
I
u
4
= hb, −3u
2
+ a − 5u − 4, u
3
+ u
2
− 1i
* 4 irreducible components of dim
C
= 0, with total 41 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
=
h1.31×10
24
u
22
+1.94×10
25
u
21
+· · ·+3.17×10
26
b−1.61×10
26
, 1.48×10
26
u
22
+
2.07 × 10
27
u
21
+ · · · + 3.17 × 10
26
a + 3.75 × 10
28
, u
23
+ 14u
22
+ · · · + 247u − 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
8
=
1
u
2
a
4
=
−0.468025u
22
− 6.54739u
21
+ ··· + 247.949u − 118.462
−0.00412206u
22
− 0.0610765u
21
+ ··· + 2.33980u + 0.509176
a
11
=
−u
−u
3
+ u
a
3
=
−0.472147u
22
− 6.60847u
21
+ ··· + 250.289u − 117.953
−0.00412206u
22
− 0.0610765u
21
+ ··· + 2.33980u + 0.509176
a
6
=
0.253629u
22
+ 3.54938u
21
+ ··· − 137.567u + 62.4798
8.27189 × 10
−6
u
22
+ 0.00149756u
21
+ ··· + 1.46879u − 0.275118
a
12
=
0.0618338u
22
+ 0.867080u
21
+ ··· − 30.7615u + 15.7102
−0.00140785u
22
− 0.0182940u
21
+ ··· − 0.437254u − 0.0618338
a
5
=
0.273382u
22
+ 3.82624u
21
+ ··· − 147.561u + 66.4442
−0.000313866u
22
− 0.00313831u
21
+ ··· + 2.38335u − 0.294871
a
2
=
−0.264797u
22
− 3.70794u
21
+ ··· + 143.659u − 63.5528
0.000313866u
22
+ 0.00313831u
21
+ ··· − 2.38335u + 0.294871
a
9
=
u
u
a
1
=
0.0632652u
22
+ 0.887387u
21
+ ··· − 30.6962u + 15.7102
0.0000236385u
22
+ 0.00201246u
21
+ ··· − 0.372026u − 0.0618418
(ii) Obstruction class = −1
(iii) Cusp Shapes =
12439096516079350509692
1237751551862707466683597
u
22
+
24959142431741094135677015
158432198638426555735500416
u
21
+
··· +
1416393727615768311794426071
158432198638426555735500416
u −
1337939601650771137272682937
158432198638426555735500416
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
23
+ 23u
22
+ ··· + 12783u + 1
c
2
, c
4
u
23
− 7u
22
+ ··· − 113u − 1
c
3
, c
6
u
23
− 4u
22
+ ··· − 36u + 8
c
5
, c
11
u
23
+ 3u
22
+ ··· − 32u − 64
c
7
, c
9
, c
10
u
23
− 14u
22
+ ··· + 247u + 1
c
8
, c
12
u
23
+ 5u
22
+ ··· + 4608u − 512
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
23
− 39y
22
+ ··· + 163240279y −1
c
2
, c
4
y
23
− 23y
22
+ ··· + 12783y −1
c
3
, c
6
y
23
− 12y
22
+ ··· + 7568y −64
c
5
, c
11
y
23
+ 37y
22
+ ··· + 234496y −4096
c
7
, c
9
, c
10
y
23
− 48y
22
+ ··· + 59963y −1
c
8
, c
12
y
23
− 111y
22
+ ··· + 71041024y −262144
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.970382
a = 8.07238
b = 0.439625
−2.87501 −99.4720
u = −0.718687 + 0.638413I
a = 0.403992 − 0.145437I
b = 0.282905 − 0.561433I
1.45854 + 3.25209I −3.51442 − 11.82565I
u = −0.718687 − 0.638413I
a = 0.403992 + 0.145437I
b = 0.282905 + 0.561433I
1.45854 − 3.25209I −3.51442 + 11.82565I
u = 0.820787 + 0.297606I
a = 3.29844 − 2.74934I
b = 0.271589 + 0.441556I
−2.85899 − 0.09109I −11.2448 + 8.7640I
u = 0.820787 − 0.297606I
a = 3.29844 + 2.74934I
b = 0.271589 − 0.441556I
−2.85899 + 0.09109I −11.2448 − 8.7640I
u = −0.989873 + 0.547667I
a = −0.262757 − 0.269042I
b = −0.904186 − 1.051940I
−5.12106 − 6.15902I −10.50715 + 1.63362I
u = −0.989873 − 0.547667I
a = −0.262757 + 0.269042I
b = −0.904186 + 1.051940I
−5.12106 + 6.15902I −10.50715 − 1.63362I
u = 0.736463
a = −0.794473
b = −0.0940545
−1.10354 −8.74790
u = −0.077756 + 0.538901I
a = −0.928505 + 0.171334I
b = 0.810706 + 0.505931I
−0.87687 − 1.52898I −6.60742 + 3.54271I
u = −0.077756 − 0.538901I
a = −0.928505 − 0.171334I
b = 0.810706 − 0.505931I
−0.87687 + 1.52898I −6.60742 − 3.54271I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.404196 + 0.182896I
a = 0.32709 + 2.82080I
b = 0.273102 − 1.253150I
2.20419 + 2.68521I 2.70136 + 6.44368I
u = 0.404196 − 0.182896I
a = 0.32709 − 2.82080I
b = 0.273102 + 1.253150I
2.20419 − 2.68521I 2.70136 − 6.44368I
u = −1.63114
a = −1.85070
b = −0.603575
−9.92701 35.8110
u = 0.00408400
a = −117.446
b = 0.518673
−1.19404 −8.40790
u = −1.89245 + 0.70982I
a = −1.043110 − 0.396575I
b = −1.16222 + 1.51464I
15.7088 + 13.9110I −11.35191 − 5.40734I
u = −1.89245 − 0.70982I
a = −1.043110 + 0.396575I
b = −1.16222 − 1.51464I
15.7088 − 13.9110I −11.35191 + 5.40734I
u = −2.26833 + 0.53777I
a = 0.840120 + 0.152223I
b = 1.41200 − 1.76863I
19.7178 + 6.1351I −10.22986 − 1.96379I
u = −2.26833 − 0.53777I
a = 0.840120 − 0.152223I
b = 1.41200 + 1.76863I
19.7178 − 6.1351I −10.22986 + 1.96379I
u = 1.99410 + 1.87801I
a = −0.410446 + 0.386878I
b = −2.39957 − 0.70874I
−14.2988 − 3.5584I 0
u = 1.99410 − 1.87801I
a = −0.410446 − 0.386878I
b = −2.39957 + 0.70874I
−14.2988 + 3.5584I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.78866 + 0.30349I
a = −0.537585 − 0.114355I
b = −1.97498 − 1.71262I
14.2364 + 2.5672I 0
u = −2.78866 − 0.30349I
a = −0.537585 + 0.114355I
b = −1.97498 + 1.71262I
14.2364 − 2.5672I 0
u = −3.04646
a = 0.644751
b = 2.52063
18.9120 0
7
II. I
u
2
= h−676a
8
+ 717b + · · · − 295a + 1497, a
9
+ 7a
8
+ · · · − 6a
2
+ 1, u − 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
1
a
8
=
1
1
a
4
=
a
0.942817a
8
+ 7.70572a
7
+ ··· + 0.411437a − 2.08787
a
11
=
−1
0
a
3
=
0.942817a
8
+ 7.70572a
7
+ ··· + 1.41144a − 2.08787
0.942817a
8
+ 7.70572a
7
+ ··· + 0.411437a − 2.08787
a
6
=
−1.10600a
8
− 8.45607a
7
+ ··· + 2.08787a + 1.94282
−2.28870a
8
− 17.4045a
7
+ ··· + 4.19107a + 2.41004
a
12
=
−1.53556a
8
− 11.8131a
7
+ ··· + 2.37378a + 0.0794979
−1.53556a
8
− 11.8131a
7
+ ··· + 2.37378a + 0.0794979
a
5
=
2.51743a
8
+ 18.9149a
7
+ ··· − 5.17015a − 3.72524
1.33473a
8
+ 9.96653a
7
+ ··· − 3.06695a − 3.25802
a
2
=
2.34589a
8
+ 17.6987a
7
+ ··· − 4.60251a − 4.32218
1.33473a
8
+ 9.96653a
7
+ ··· − 3.06695a − 3.25802
a
9
=
1
1
a
1
=
−1.53556a
8
− 11.8131a
7
+ ··· + 2.37378a + 0.0794979
−1.53556a
8
− 11.8131a
7
+ ··· + 2.37378a + 0.0794979
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
10493
717
a
8
+
26713
239
a
7
−
210605
717
a
6
+
75659
239
a
5
−
133631
717
a
4
+
11474
239
a
3
+
50845
717
a
2
−
6563
239
a −
6150
239
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1
c
2
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
3
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
4
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
5
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
c
6
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
7
(u − 1)
9
c
8
, c
12
u
9
c
9
, c
10
(u + 1)
9
c
11
u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
c
2
, c
4
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
3
, c
6
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
5
, c
11
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
c
7
, c
9
, c
10
(y −1)
9
c
8
, c
12
y
9
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.162031 + 0.927542I
b = −0.140343 + 0.966856I
0.13850 + 2.09337I −6.65973 − 4.50528I
u = 1.00000
a = 0.162031 − 0.927542I
b = −0.140343 − 0.966856I
0.13850 − 2.09337I −6.65973 + 4.50528I
u = 1.00000
a = 0.990590 + 0.515152I
b = 0.796005 − 0.733148I
−6.01628 − 1.33617I −13.00050 + 1.13735I
u = 1.00000
a = 0.990590 − 0.515152I
b = 0.796005 + 0.733148I
−6.01628 + 1.33617I −13.00050 − 1.13735I
u = 1.00000
a = 0.702315 + 0.150499I
b = 0.728966 + 0.986295I
−5.24306 − 7.08493I −11.6081 + 10.4867I
u = 1.00000
a = 0.702315 − 0.150499I
b = 0.728966 − 0.986295I
−5.24306 + 7.08493I −11.6081 − 10.4867I
u = 1.00000
a = −0.405386 + 0.113252I
b = −0.628449 − 0.875112I
−2.26187 − 2.45442I −9.69685 + 4.13179I
u = 1.00000
a = −0.405386 − 0.113252I
b = −0.628449 + 0.875112I
−2.26187 + 2.45442I −9.69685 − 4.13179I
u = 1.00000
a = −9.89910
b = −0.512358
−2.84338 193.930
11
III. I
u
3
=
h5a
2
u−3a
2
+12au+b−7a+3u−1, a
3
−a
2
u+a
2
+3au+6a+3u+5, u
2
+u−1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
8
=
1
−u + 1
a
4
=
a
−5a
2
u + 3a
2
− 12au + 7a − 3u + 1
a
11
=
−u
−u + 1
a
3
=
−5a
2
u + 3a
2
− 12au + 8a − 3u + 1
−5a
2
u + 3a
2
− 12au + 7a − 3u + 1
a
6
=
−a
2
u + a
2
− 3au + 2a − u + 1
−2a
2
u + a
2
− 5au + 3a − 2u + 1
a
12
=
−u
−u + 1
a
5
=
−a
2
u + a
2
− 3au + 2a − u + 1
−2a
2
u + a
2
− 5au + 3a − 2u + 1
a
2
=
−3a
2
u + 2a
2
− 8au + 5a − 3u
−2a
2
u + a
2
− 5au + 3a − 2u + 1
a
9
=
u
u
a
1
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 17a
2
u − 9a
2
+ 24au − 10a + 3u − 18
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
(u
3
− u
2
+ 2u − 1)
2
c
2
(u
3
+ u
2
− 1)
2
c
4
(u
3
− u
2
+ 1)
2
c
5
, c
11
u
6
c
6
(u
3
+ u
2
+ 2u + 1)
2
c
7
, c
8
(u
2
+ u − 1)
3
c
9
, c
10
, c
12
(u
2
− u − 1)
3
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
6
(y
3
+ 3y
2
+ 2y −1)
2
c
2
, c
4
(y
3
− y
2
+ 2y −1)
2
c
5
, c
11
y
6
c
7
, c
8
, c
9
c
10
, c
12
(y
2
− 3y + 1)
3
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −0.832857
b = −0.569840
−2.10041 −19.1260
u = 0.618034
a = 0.22545 + 2.85986I
b = −0.215080 − 1.307140I
2.03717 − 2.82812I −27.3018 + 15.7639I
u = 0.618034
a = 0.22545 − 2.85986I
b = −0.215080 + 1.307140I
2.03717 + 2.82812I −27.3018 − 15.7639I
u = −1.61803
a = −0.255488 + 0.062996I
b = −0.215080 + 1.307140I
−5.85852 + 2.82812I −12.61597 − 1.90115I
u = −1.61803
a = −0.255488 − 0.062996I
b = −0.215080 − 1.307140I
−5.85852 − 2.82812I −12.61597 + 1.90115I
u = −1.61803
a = −2.10706
b = −0.569840
−9.99610 −82.0390
15
IV. I
u
4
= hb, −3u
2
+ a − 5u − 4, u
3
+ u
2
− 1i
(i) Arc colorings
a
7
=
1
0
a
10
=
0
u
a
8
=
1
u
2
a
4
=
3u
2
+ 5u + 4
0
a
11
=
−u
u
2
+ u − 1
a
3
=
3u
2
+ 5u + 4
0
a
6
=
1
0
a
12
=
u
2
− 1
u
2
+ u − 1
a
5
=
−2u
2
+ 2
−2u
2
− u + 2
a
2
=
5u
2
+ 5u + 2
2u
2
+ u − 2
a
9
=
u
u
a
1
=
2u
2
− 2
2u
2
+ u − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21u
2
+ 45u + 27
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
6
u
3
c
4
(u + 1)
3
c
5
u
3
+ 3u
2
+ 2u − 1
c
7
u
3
+ u
2
− 1
c
8
u
3
− u
2
+ 2u − 1
c
9
, c
10
u
3
− u
2
+ 1
c
11
u
3
− 3u
2
+ 2u + 1
c
12
u
3
+ u
2
+ 2u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
6
y
3
c
5
, c
11
y
3
− 5y
2
+ 10y −1
c
7
, c
9
, c
10
y
3
− y
2
+ 2y −1
c
8
, c
12
y
3
+ 3y
2
+ 2y −1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.258045 − 0.197115I
b = 0
1.37919 + 2.82812I −7.96807 + 6.06881I
u = −0.877439 − 0.744862I
a = 0.258045 + 0.197115I
b = 0
1.37919 − 2.82812I −7.96807 − 6.06881I
u = 0.754878
a = 9.48391
b = 0
−2.75839 72.9360
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)
3
(u
3
− u
2
+ 2u − 1)
2
· (u
9
− 5u
8
+ 12u
7
− 15u
6
+ 9u
5
+ u
4
− 4u
3
+ 2u
2
+ u − 1)
· (u
23
+ 23u
22
+ ··· + 12783u + 1)
c
2
(u − 1)
3
(u
3
+ u
2
− 1)
2
(u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1)
· (u
23
− 7u
22
+ ··· − 113u − 1)
c
3
u
3
(u
3
− u
2
+ 2u − 1)
2
(u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1)
· (u
23
− 4u
22
+ ··· − 36u + 8)
c
4
(u + 1)
3
(u
3
− u
2
+ 1)
2
(u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1)
· (u
23
− 7u
22
+ ··· − 113u − 1)
c
5
u
6
(u
3
+ 3u
2
+ 2u − 1)
· (u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
23
+ 3u
22
+ ··· − 32u − 64)
c
6
u
3
(u
3
+ u
2
+ 2u + 1)
2
(u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1)
· (u
23
− 4u
22
+ ··· − 36u + 8)
c
7
((u − 1)
9
)(u
2
+ u − 1)
3
(u
3
+ u
2
− 1)(u
23
− 14u
22
+ ··· + 247u + 1)
c
8
u
9
(u
2
+ u − 1)
3
(u
3
− u
2
+ 2u − 1)(u
23
+ 5u
22
+ ··· + 4608u − 512)
c
9
, c
10
((u + 1)
9
)(u
2
− u − 1)
3
(u
3
− u
2
+ 1)(u
23
− 14u
22
+ ··· + 247u + 1)
c
11
u
6
(u
3
− 3u
2
+ 2u + 1)
· (u
9
+ 3u
8
+ 8u
7
+ 13u
6
+ 17u
5
+ 17u
4
+ 12u
3
+ 6u
2
+ u − 1)
· (u
23
+ 3u
22
+ ··· − 32u − 64)
c
12
u
9
(u
2
− u − 1)
3
(u
3
+ u
2
+ 2u + 1)(u
23
+ 5u
22
+ ··· + 4608u − 512)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
3
(y
3
+ 3y
2
+ 2y −1)
2
· (y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
23
− 39y
22
+ ··· + 163240279y −1)
c
2
, c
4
(y −1)
3
(y
3
− y
2
+ 2y −1)
2
· (y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1)
· (y
23
− 23y
22
+ ··· + 12783y −1)
c
3
, c
6
y
3
(y
3
+ 3y
2
+ 2y −1)
2
· (y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1)
· (y
23
− 12y
22
+ ··· + 7568y −64)
c
5
, c
11
y
6
(y
3
− 5y
2
+ 10y −1)
· (y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
23
+ 37y
22
+ ··· + 234496y −4096)
c
7
, c
9
, c
10
(y −1)
9
(y
2
− 3y + 1)
3
(y
3
− y
2
+ 2y −1)
· (y
23
− 48y
22
+ ··· + 59963y −1)
c
8
, c
12
y
9
(y
2
− 3y + 1)
3
(y
3
+ 3y
2
+ 2y −1)
· (y
23
− 111y
22
+ ··· + 71041024y −262144)
21
