12n
0835
(K12n
0835
)
A knot diagram
1
Linearized knot diagam
4 6 12 9 2 10 12 6 4 8 3 7
Solving Sequence
3,12 4,8
7 1 11 10 6 2 5 9
c
3
c
7
c
12
c
11
c
10
c
6
c
2
c
5
c
9
c
1
, c
4
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−11962u
12
− 32982u
11
+ ··· + 180027b − 81583,
− 51167u
12
− 51141u
11
+ ··· + 180027a + 431806,
u
13
+ 2u
12
+ u
11
− 12u
10
− 10u
9
+ 12u
8
+ 39u
7
+ u
6
− 48u
5
− 18u
4
+ 21u
3
+ 18u
2
+ 3u − 1i
I
u
2
= hb, a + u, u
3
− u
2
+ 1i
I
u
3
= h34532u
11
− 8910u
10
+ ··· + 191471b − 108223,
126317u
11
− 132222u
10
+ ··· + 191471a − 196501,
u
12
− 2u
11
+ u
10
− 16u
8
+ 14u
7
+ 29u
6
− 12u
5
+ 4u
4
+ 16u
3
− 5u
2
− 2u + 1i
I
u
4
= h−188u
7
− 516u
6
− 1817u
5
− 4442u
4
− 8033u
3
− 9745u
2
+ 4095b − 9869u − 4774,
209u
7
+ 138u
6
+ 2216u
5
+ 5156u
4
+ 6164u
3
+ 20875u
2
+ 28665a + 9512u + 2737,
u
8
+ 2u
7
+ 10u
6
+ 24u
5
+ 45u
4
+ 76u
3
+ 92u
2
+ 70u + 49i
* 4 irreducible components of dim
C
= 0, with total 36 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−1.20 × 10
4
u
12
− 3.30 × 10
4
u
11
+ · · · + 1.80 × 10
5
b − 8.16 × 10
4
, −5.12 ×
10
4
u
12
−5.11×10
4
u
11
+· · · +1.80×10
5
a+4.32×10
5
, u
13
+2u
12
+· · · +3u −1i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
0.284218u
12
+ 0.284074u
11
+ ··· − 3.97384u − 2.39856
0.0664456u
12
+ 0.183206u
11
+ ··· + 0.314458u + 0.453171
a
7
=
0.284218u
12
+ 0.284074u
11
+ ··· − 3.97384u − 2.39856
0.0888700u
12
+ 0.283657u
11
+ ··· + 1.45177u + 0.168808
a
1
=
0.0237131u
12
− 0.0981353u
11
+ ··· − 2.62369u + 1.88978
0.105179u
12
+ 0.356713u
11
+ ··· − 0.902920u + 0.234726
a
11
=
u
u
a
10
=
−0.222456u
12
− 0.0646236u
11
+ ··· + 1.26037u − 1.50950
0.0856205u
12
+ 0.383542u
11
+ ··· + 1.88790u + 0.0960689
a
6
=
−0.509496u
12
− 1.24145u
11
+ ··· − 8.54774u − 0.268115
u
a
2
=
0.222456u
12
+ 0.0646236u
11
+ ··· − 1.26037u + 1.50950
−u
2
a
5
=
−0.889783u
12
− 1.80328u
11
+ ··· − 7.70561u − 0.0456598
−u
3
+ u
a
9
=
−0.506818u
12
− 0.610925u
11
+ ··· − 1.99085u − 1.22528
0.141223u
12
+ 0.341066u
11
+ ··· + 1.53626u + 0.118493
(ii) Obstruction class = −1
(iii) Cusp Shapes =
133277
60009
u
12
+
97412
20003
u
11
+ ··· +
2500157
60009
u +
279872
60009
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
13
+ 2u
12
+ ··· + 73u + 5
c
2
, c
3
, c
5
c
11
u
13
+ 2u
12
+ ··· + 3u − 1
c
4
, c
7
, c
9
c
12
u
13
+ 10u
11
+ ··· − 12u − 4
c
6
u
13
− 3u
12
+ ··· + 112u − 19
c
8
, c
10
u
13
+ 2u
12
+ ··· + 5u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
13
− 38y
12
+ ··· + 7329y −25
c
2
, c
3
, c
5
c
11
y
13
− 2y
12
+ ··· + 45y −1
c
4
, c
7
, c
9
c
12
y
13
+ 20y
12
+ ··· + 208y −16
c
6
y
13
+ 7y
12
+ ··· + 2094y −361
c
8
, c
10
y
13
− 8y
12
+ ··· + 7y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.122400 + 0.448785I
a = 0.292793 + 0.874754I
b = −0.109259 + 0.767460I
−6.52437 − 4.35431I −2.69323 + 5.57961I
u = 1.122400 − 0.448785I
a = 0.292793 − 0.874754I
b = −0.109259 − 0.767460I
−6.52437 + 4.35431I −2.69323 − 5.57961I
u = −0.749285 + 0.043693I
a = −0.509626 + 0.065149I
b = −0.738849 + 0.256795I
−1.302700 + 0.032683I −7.31648 − 0.16173I
u = −0.749285 − 0.043693I
a = −0.509626 − 0.065149I
b = −0.738849 − 0.256795I
−1.302700 − 0.032683I −7.31648 + 0.16173I
u = 1.265880 + 0.291483I
a = 0.18470 − 1.44719I
b = −0.35048 − 2.57213I
−14.5051 − 3.1980I −6.84015 + 2.15870I
u = 1.265880 − 0.291483I
a = 0.18470 + 1.44719I
b = −0.35048 + 2.57213I
−14.5051 + 3.1980I −6.84015 − 2.15870I
u = −0.438416 + 0.418996I
a = 1.83391 − 1.67711I
b = 0.675501 + 0.073236I
−8.52319 + 2.72289I −7.74001 − 0.35750I
u = −0.438416 − 0.418996I
a = 1.83391 + 1.67711I
b = 0.675501 − 0.073236I
−8.52319 − 2.72289I −7.74001 + 0.35750I
u = −0.86722 + 1.24197I
a = 0.518889 − 0.563420I
b = 0.04501 − 1.85139I
4.79571 + 2.82647I −1.37286 − 2.37043I
u = −0.86722 − 1.24197I
a = 0.518889 + 0.563420I
b = 0.04501 + 1.85139I
4.79571 − 2.82647I −1.37286 + 2.37043I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.158979
a = −2.81443
b = 0.486240
0.852515 11.9090
u = −1.41284 + 1.83586I
a = −0.413448 + 0.805929I
b = −0.26505 + 2.03068I
0.95941 + 10.95210I −1.49177 − 4.40872I
u = −1.41284 − 1.83586I
a = −0.413448 − 0.805929I
b = −0.26505 − 2.03068I
0.95941 − 10.95210I −1.49177 + 4.40872I
6
II. I
u
2
= hb, a + u, u
3
− u
2
+ 1i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
−u
0
a
7
=
−u
−u
2
+ 1
a
1
=
u
2
− 1
−1
a
11
=
u
u
a
10
=
−u
2
+ u + 1
u
a
6
=
2u
2
− u − 1
−u
a
2
=
u
2
− u − 1
−u
2
a
5
=
2u
2
u
2
− u − 1
a
9
=
u + 1
u
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
2
− 4u − 3
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
11
u
3
+ u
2
− 1
c
3
, c
5
u
3
− u
2
+ 1
c
4
, c
7
u
3
+ u
2
+ 2u + 1
c
6
u
3
+ 2u
2
+ 3u + 1
c
8
, c
9
, c
10
c
12
u
3
− u
2
+ 2u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
, c
11
y
3
− y
2
+ 2y −1
c
4
, c
7
, c
8
c
9
, c
10
, c
12
y
3
+ 3y
2
+ 2y −1
c
6
y
3
+ 2y
2
+ 5y −1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.877439 + 0.744862I
a = −0.877439 − 0.744862I
b = 0
−8.03068 − 3.77083I −5.21928 + 4.86340I
u = 0.877439 − 0.744862I
a = −0.877439 + 0.744862I
b = 0
−8.03068 + 3.77083I −5.21928 − 4.86340I
u = −0.754878
a = 0.754878
b = 0
−0.387983 3.43860
10
III. I
u
3
= h34532u
11
− 8910u
10
+ · · · + 191471b − 108223, 1.26 × 10
5
u
11
−
1.32 × 10
5
u
10
+ · · · + 1.91 × 10
5
a − 1.97 × 10
5
, u
12
− 2u
11
+ · · · − 2u + 1i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
−0.659719u
11
+ 0.690559u
10
+ ··· − 6.35697u + 1.02627
−0.180351u
11
+ 0.0465345u
10
+ ··· − 1.55862u + 0.565219
a
7
=
−0.659719u
11
+ 0.690559u
10
+ ··· − 6.35697u + 1.02627
−0.460054u
11
+ 0.535402u
10
+ ··· − 2.15666u + 1.19410
a
1
=
1.65524u
11
− 2.78398u
10
+ ··· − 1.00772u − 6.09660
0.224859u
11
− 0.296390u
10
+ ··· − 0.503115u − 1.64700
a
11
=
u
u
a
10
=
−1.52166u
11
+ 2.61616u
10
+ ··· + 1.10685u + 4.97611
−0.129111u
11
+ 0.261251u
10
+ ··· + 1.26558u + 1.19082
a
6
=
−1.16060u
11
+ 2.08065u
10
+ ··· + 1.32344u + 4.67364
−0.250257u
11
+ 0.475545u
10
+ ··· − 0.181777u + 1.44895
a
2
=
1.52166u
11
− 2.61616u
10
+ ··· − 1.10685u − 4.97611
0.112215u
11
− 0.111991u
10
+ ··· − 0.568222u − 1.54765
a
5
=
0.943589u
11
− 1.35098u
10
+ ··· + 0.790851u − 1.13909
0.512187u
11
− 0.959487u
10
+ ··· + 0.496691u − 1.18240
a
9
=
−1.37117u
11
+ 2.29907u
10
+ ··· + 0.508615u + 4.21244
−0.170658u
11
+ 0.355380u
10
+ ··· + 1.08285u + 1.20695
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
8084
11263
u
11
+
12088
11263
u
10
−
3464
11263
u
9
+
5702
11263
u
8
+
118154
11263
u
7
−
36546
11263
u
6
−
248588
11263
u
5
−
96058
11263
u
4
−
30924
11263
u
3
−
100428
11263
u
2
−
7174
1609
u +
29716
11263
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
6
− 7u
5
+ 14u
4
+ u
3
− 28u
2
+ 31u − 11)
2
c
2
, c
11
u
12
+ 2u
11
+ ··· + 2u + 1
c
3
, c
5
u
12
− 2u
11
+ ··· − 2u + 1
c
4
, c
7
u
12
− 2u
11
+ ··· + 8u + 4
c
6
(u
3
− u
2
+ u + 1)
4
c
8
, c
10
u
12
− 2u
11
+ ··· − 40u + 29
c
9
, c
12
u
12
+ 2u
11
+ ··· − 8u + 4
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
6
− 21y
5
+ 154y
4
− 373y
3
+ 414y
2
− 345y + 121)
2
c
2
, c
3
, c
5
c
11
y
12
− 2y
11
+ ··· − 14y + 1
c
4
, c
7
, c
9
c
12
y
12
+ 12y
11
+ ··· − 256y + 16
c
6
(y
3
+ y
2
+ 3y −1)
4
c
8
, c
10
y
12
+ 4y
11
+ ··· − 5892y + 841
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.911570
a = 0.298859
b = −0.487547
−0.365976 3.08740
u = 0.347067 + 0.717063I
a = 1.20188 + 1.50579I
b = 0.17433 + 2.36142I
−13.14100 − 3.17729I 0.45631 + 2.23029I
u = 0.347067 − 0.717063I
a = 1.20188 − 1.50579I
b = 0.17433 − 2.36142I
−13.14100 + 3.17729I 0.45631 − 2.23029I
u = −1.218800 + 0.206735I
a = −0.505212 + 0.220973I
b = 0.190428 + 0.825741I
−5.24529 + 3.17729I 0.45631 − 2.23029I
u = −1.218800 − 0.206735I
a = −0.505212 − 0.220973I
b = 0.190428 − 0.825741I
−5.24529 − 3.17729I 0.45631 + 2.23029I
u = −0.446425
a = 1.52685
b = 0.487547
−0.365976 3.08740
u = 0.341414 + 0.167973I
a = −2.13354 − 3.14045I
b = −0.190428 − 0.825741I
−5.24529 + 3.17729I 0.45631 − 2.23029I
u = 0.341414 − 0.167973I
a = −2.13354 + 3.14045I
b = −0.190428 + 0.825741I
−5.24529 − 3.17729I 0.45631 + 2.23029I
u = 1.94996 + 0.26394I
a = 0.121333 − 1.079840I
b = −0.17433 − 2.36142I
−13.14100 − 3.17729I 0.45631 + 2.23029I
u = 1.94996 − 0.26394I
a = 0.121333 + 1.079840I
b = −0.17433 + 2.36142I
−13.14100 + 3.17729I 0.45631 − 2.23029I
14
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.25935 + 2.11008I
a = 0.402681 − 0.542364I
b = − 1.68959I
7.52971 3.08738 + 0.I
u = 0.25935 − 2.11008I
a = 0.402681 + 0.542364I
b = 1.68959I
7.52971 3.08738 + 0.I
15
IV. I
u
4
= h−188u
7
− 516u
6
+ · · · + 4095b − 4774, 209u
7
+ 138u
6
+ · · · +
28665a + 2737, u
8
+ 2u
7
+ · · · + 70u + 49i
(i) Arc colorings
a
3
=
1
0
a
12
=
0
u
a
4
=
1
u
2
a
8
=
−0.00729112u
7
− 0.00481423u
6
+ ··· − 0.331833u − 0.0954823
0.0459096u
7
+ 0.126007u
6
+ ··· + 2.41001u + 1.16581
a
7
=
−0.00729112u
7
− 0.00481423u
6
+ ··· − 0.331833u − 0.0954823
0.0219780u
7
+ 0.0234432u
6
+ ··· + 2.08352u + 0.687179
a
1
=
0.0163265u
7
− 0.00544218u
6
+ ··· + 0.673469u + 0.304762
0.100611u
7
+ 0.141392u
6
+ ··· + 2.81343u + 1.52650
a
11
=
u
u
a
10
=
0.0176173u
7
+ 0.0135008u
6
+ ··· + 0.273295u − 0.644933
0.00366300u
7
+ 0.115018u
6
+ ··· + 2.68059u + 1.22564
a
6
=
−0.0278039u
7
− 0.0736787u
6
+ ··· − 1.81169u − 1.14383
0.0798535u
7
+ 0.134066u
6
+ ··· + 2.35678u − 0.174359
a
2
=
−0.0176173u
7
− 0.0135008u
6
+ ··· − 0.273295u + 0.644933
0.104029u
7
+ 0.0131868u
6
+ ··· + 0.288645u − 1.40513
a
5
=
−0.0431188u
7
− 0.0986918u
6
+ ··· + 0.194244u − 0.0923077
0.196337u
7
+ 0.351648u
6
+ ··· − 1.28059u − 3.35897
a
9
=
−0.0561312u
7
− 0.106646u
6
+ ··· − 1.74917u − 0.805617
0.189988u
7
+ 0.361172u
6
+ ··· + 4.37973u − 0.114530
(ii) Obstruction class = −1
(iii) Cusp Shapes = −2
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
4
− 2u
3
+ u
2
+ 5)
2
c
2
, c
3
, c
5
c
11
u
8
+ 2u
7
+ 10u
6
+ 24u
5
+ 45u
4
+ 76u
3
+ 92u
2
+ 70u + 49
c
4
, c
7
, c
9
c
12
u
8
− 2u
7
− 4u
5
+ 10u
4
+ 4u
3
+ 12u
2
+ 4
c
6
(u
2
+ 1)
4
c
8
, c
10
u
8
+ 2u
7
− 4u
6
− 18u
5
+ 10u
4
+ 98u
3
+ 156u
2
+ 118u + 41
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y
4
− 2y
3
+ 11y
2
+ 10y + 25)
2
c
2
, c
3
, c
5
c
11
y
8
+ 16y
7
+ ··· + 4116y + 2401
c
4
, c
7
, c
9
c
12
y
8
− 4y
7
+ 4y
6
+ 24y
5
+ 140y
4
+ 224y
3
+ 224y
2
+ 96y + 16
c
6
(y + 1)
8
c
8
, c
10
y
8
− 12y
7
+ ··· − 1132y + 1681
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.250028 + 1.085420I
a = 0.548797 + 0.000925I
b = 0.618034I
−2.30291 −2.00000
u = −0.250028 − 1.085420I
a = 0.548797 − 0.000925I
b = − 0.618034I
−2.30291 −2.00000
u = −1.36801 + 0.53261I
a = −0.779945 + 0.583385I
b = 0.618034I
−2.30291 −2.00000
u = −1.36801 − 0.53261I
a = −0.779945 − 0.583385I
b = − 0.618034I
−2.30291 −2.00000
u = 0.11336 + 1.75843I
a = 0.543770 − 0.231510I
b = − 1.61803I
5.59278 −2.00000
u = 0.11336 − 1.75843I
a = 0.543770 + 0.231510I
b = 1.61803I
5.59278 −2.00000
u = 0.50467 + 2.37647I
a = −0.455479 + 0.781371I
b = 1.61803I
5.59278 −2.00000
u = 0.50467 − 2.37647I
a = −0.455479 − 0.781371I
b = − 1.61803I
5.59278 −2.00000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
− 1)(u
4
− 2u
3
+ u
2
+ 5)
2
· (u
6
− 7u
5
+ 14u
4
+ u
3
− 28u
2
+ 31u − 11)
2
· (u
13
+ 2u
12
+ ··· + 73u + 5)
c
2
, c
11
(u
3
+ u
2
− 1)(u
8
+ 2u
7
+ ··· + 70u + 49)
· (u
12
+ 2u
11
+ ··· + 2u + 1)(u
13
+ 2u
12
+ ··· + 3u − 1)
c
3
, c
5
(u
3
− u
2
+ 1)(u
8
+ 2u
7
+ ··· + 70u + 49)
· (u
12
− 2u
11
+ ··· − 2u + 1)(u
13
+ 2u
12
+ ··· + 3u − 1)
c
4
, c
7
(u
3
+ u
2
+ 2u + 1)(u
8
− 2u
7
− 4u
5
+ 10u
4
+ 4u
3
+ 12u
2
+ 4)
· (u
12
− 2u
11
+ ··· + 8u + 4)(u
13
+ 10u
11
+ ··· − 12u − 4)
c
6
(u
2
+ 1)
4
(u
3
− u
2
+ u + 1)
4
(u
3
+ 2u
2
+ 3u + 1)
· (u
13
− 3u
12
+ ··· + 112u − 19)
c
8
, c
10
(u
3
− u
2
+ 2u − 1)
· (u
8
+ 2u
7
− 4u
6
− 18u
5
+ 10u
4
+ 98u
3
+ 156u
2
+ 118u + 41)
· (u
12
− 2u
11
+ ··· − 40u + 29)(u
13
+ 2u
12
+ ··· + 5u − 1)
c
9
, c
12
(u
3
− u
2
+ 2u − 1)(u
8
− 2u
7
− 4u
5
+ 10u
4
+ 4u
3
+ 12u
2
+ 4)
· (u
12
+ 2u
11
+ ··· − 8u + 4)(u
13
+ 10u
11
+ ··· − 12u − 4)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
3
− y
2
+ 2y −1)(y
4
− 2y
3
+ 11y
2
+ 10y + 25)
2
· (y
6
− 21y
5
+ 154y
4
− 373y
3
+ 414y
2
− 345y + 121)
2
· (y
13
− 38y
12
+ ··· + 7329y −25)
c
2
, c
3
, c
5
c
11
(y
3
− y
2
+ 2y −1)(y
8
+ 16y
7
+ ··· + 4116y + 2401)
· (y
12
− 2y
11
+ ··· − 14y + 1)(y
13
− 2y
12
+ ··· + 45y −1)
c
4
, c
7
, c
9
c
12
(y
3
+ 3y
2
+ 2y −1)
· (y
8
− 4y
7
+ 4y
6
+ 24y
5
+ 140y
4
+ 224y
3
+ 224y
2
+ 96y + 16)
· (y
12
+ 12y
11
+ ··· − 256y + 16)(y
13
+ 20y
12
+ ··· + 208y −16)
c
6
(y + 1)
8
(y
3
+ y
2
+ 3y −1)
4
(y
3
+ 2y
2
+ 5y −1)
· (y
13
+ 7y
12
+ ··· + 2094y −361)
c
8
, c
10
(y
3
+ 3y
2
+ 2y −1)(y
8
− 12y
7
+ ··· − 1132y + 1681)
· (y
12
+ 4y
11
+ ··· − 5892y + 841)(y
13
− 8y
12
+ ··· + 7y −1)
21
