12n
0078
(K12n
0078
)
A knot diagram
1
Linearized knot diagam
3 5 8 2 8 11 3 1 12 6 10 9
Solving Sequence
1,8 4,9
3 2 5 7 12 10 11 6
c
8
c
3
c
1
c
4
c
7
c
12
c
9
c
11
c
6
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h37387481608u
36
− 262071379227u
35
+ ··· + 62312469363b + 37745018210,
39879980401u
36
− 353616847554u
35
+ ··· + 62312469363a + 919059301277,
u
37
− 8u
36
+ ··· + 19u − 1i
I
u
2
= hb, −u
4
− u
3
− 4u
2
+ a − 3u − 3, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
* 2 irreducible components of dim
C
= 0, with total 42 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
=
h3.74×10
10
u
36
−2.62×10
11
u
35
+· · ·+6.23×10
10
b+3.77×10
10
, 3.99×10
10
u
36
−
3.54 × 10
11
u
35
+ · · · + 6.23 × 10
10
a + 9.19 × 10
11
, u
37
− 8u
36
+ · · · + 19u − 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
−0.640000u
36
+ 5.67490u
35
+ ··· + 118.964u − 14.7492
−0.600000u
36
+ 4.20576u
35
+ ··· + 2.58902u − 0.605738
a
9
=
1
−u
2
a
3
=
−1.24000u
36
+ 9.88066u
35
+ ··· + 121.553u − 15.3549
−0.600000u
36
+ 4.20576u
35
+ ··· + 2.58902u − 0.605738
a
2
=
−0.440000u
36
+ 4.28395u
35
+ ··· + 104.318u − 12.1296
−0.400000u
36
+ 2.79424u
35
+ ··· + 2.41098u − 0.394262
a
5
=
−0.600000u
36
+ 4.20165u
35
+ ··· + 21.8826u − 3.88735
−0.400000u
36
+ 2.79424u
35
+ ··· + 2.41098u − 0.394262
a
7
=
−0.394262u
36
+ 2.75410u
35
+ ··· + 35.8645u − 5.08000
−0.598354u
36
+ 4.78683u
35
+ ··· + 7.51265u − 0.600000
a
12
=
−u
u
3
+ u
a
10
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
−u
3
− 2u
u
5
+ 3u
3
+ u
a
6
=
−1.00000u
36
+ 6.99588u
35
+ ··· + 24.2936u − 4.28162
−0.400000u
36
+ 2.79424u
35
+ ··· + 2.41098u − 0.394262
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
126456874109
20770823121
u
36
+
325863614327
6923607707
u
35
+ ··· +
5835323526623
20770823121
u −
551008395754
20770823121
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
37
+ 12u
36
+ ··· + 5u + 1
c
2
, c
4
u
37
− 6u
36
+ ··· − 3u + 1
c
3
, c
7
u
37
− u
36
+ ··· + 120u
2
+ 32
c
5
u
37
+ 2u
36
+ ··· + 3u + 1
c
6
, c
10
u
37
+ 2u
36
+ ··· + 3u + 1
c
8
, c
9
, c
11
c
12
u
37
− 8u
36
+ ··· + 19u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
37
+ 32y
36
+ ··· + 5y −1
c
2
, c
4
y
37
− 12y
36
+ ··· + 5y −1
c
3
, c
7
y
37
+ 33y
36
+ ··· − 7680y −1024
c
5
y
37
− 40y
36
+ ··· + 19y −1
c
6
, c
10
y
37
− 8y
36
+ ··· + 19y −1
c
8
, c
9
, c
11
c
12
y
37
+ 44y
36
+ ··· + 99y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.192278 + 0.981446I
a = 0.020576 − 0.394070I
b = 0.028151 + 0.615473I
−2.37817 + 2.37893I 2.00000 − 4.24354I
u = 0.192278 − 0.981446I
a = 0.020576 + 0.394070I
b = 0.028151 − 0.615473I
−2.37817 − 2.37893I 2.00000 + 4.24354I
u = 0.954498 + 0.071334I
a = −0.13625 + 3.34385I
b = 0.22449 − 1.57573I
7.67323 + 3.34146I 7.54708 − 2.94673I
u = 0.954498 − 0.071334I
a = −0.13625 − 3.34385I
b = 0.22449 + 1.57573I
7.67323 − 3.34146I 7.54708 + 2.94673I
u = 0.739019 + 0.820681I
a = −0.80030 − 2.64862I
b = −0.02142 + 1.55446I
5.46691 + 2.17037I 0
u = 0.739019 − 0.820681I
a = −0.80030 + 2.64862I
b = −0.02142 − 1.55446I
5.46691 − 2.17037I 0
u = 0.488581 + 0.716463I
a = −0.400880 + 0.540394I
b = 0.929621 − 0.055392I
−0.60086 + 3.42978I 1.46957 − 8.06302I
u = 0.488581 − 0.716463I
a = −0.400880 − 0.540394I
b = 0.929621 + 0.055392I
−0.60086 − 3.42978I 1.46957 + 8.06302I
u = 0.694171 + 0.952511I
a = 0.57988 + 2.67460I
b = 0.43869 − 1.52488I
4.63566 + 8.76465I 0
u = 0.694171 − 0.952511I
a = 0.57988 − 2.67460I
b = 0.43869 + 1.52488I
4.63566 − 8.76465I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.301349 + 0.644967I
a = −0.44045 + 2.61190I
b = −0.171200 − 0.710425I
−2.48892 + 1.77260I 1.52089 − 2.95783I
u = 0.301349 − 0.644967I
a = −0.44045 − 2.61190I
b = −0.171200 + 0.710425I
−2.48892 − 1.77260I 1.52089 + 2.95783I
u = −0.337610 + 0.584660I
a = −1.40228 + 3.03617I
b = −0.36360 − 1.42665I
2.83785 − 4.34213I −1.01036 + 2.76263I
u = −0.337610 − 0.584660I
a = −1.40228 − 3.03617I
b = −0.36360 + 1.42665I
2.83785 + 4.34213I −1.01036 − 2.76263I
u = 0.530968 + 0.127903I
a = −0.409091 − 0.456941I
b = 0.495562 + 0.299085I
1.138660 + 0.126784I 8.90691 − 0.21757I
u = 0.530968 − 0.127903I
a = −0.409091 + 0.456941I
b = 0.495562 − 0.299085I
1.138660 − 0.126784I 8.90691 + 0.21757I
u = −0.361376 + 0.397065I
a = 1.91674 − 3.19790I
b = −0.08394 + 1.41819I
3.38035 + 1.79092I −0.30116 − 2.50097I
u = −0.361376 − 0.397065I
a = 1.91674 + 3.19790I
b = −0.08394 − 1.41819I
3.38035 − 1.79092I −0.30116 + 2.50097I
u = −0.05375 + 1.50789I
a = 0.41135 − 1.87901I
b = 0.31409 + 1.45132I
−2.96990 + 0.48951I 0
u = −0.05375 − 1.50789I
a = 0.41135 + 1.87901I
b = 0.31409 − 1.45132I
−2.96990 − 0.48951I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.121575 + 0.445561I
a = −0.633975 + 0.683934I
b = −0.744443 + 0.181579I
−1.49379 + 0.01486I −3.88474 − 1.23232I
u = 0.121575 − 0.445561I
a = −0.633975 − 0.683934I
b = −0.744443 − 0.181579I
−1.49379 − 0.01486I −3.88474 + 1.23232I
u = 0.04039 + 1.56760I
a = 0.458186 − 0.041159I
b = −1.179650 + 0.205890I
−8.59444 + 0.63473I 0
u = 0.04039 − 1.56760I
a = 0.458186 + 0.041159I
b = −1.179650 − 0.205890I
−8.59444 − 0.63473I 0
u = −0.08929 + 1.58859I
a = −0.45250 + 1.92403I
b = −0.60720 − 1.42349I
−4.65231 − 5.85535I 0
u = −0.08929 − 1.58859I
a = −0.45250 − 1.92403I
b = −0.60720 + 1.42349I
−4.65231 + 5.85535I 0
u = 0.08760 + 1.59752I
a = −0.01097 + 2.27117I
b = −0.029458 − 1.097490I
−10.20710 + 3.22026I 0
u = 0.08760 − 1.59752I
a = −0.01097 − 2.27117I
b = −0.029458 + 1.097490I
−10.20710 − 3.22026I 0
u = 0.14093 + 1.60627I
a = −0.520990 + 0.031249I
b = 1.201680 + 0.152602I
−8.46942 + 5.78175I 0
u = 0.14093 − 1.60627I
a = −0.520990 − 0.031249I
b = 1.201680 − 0.152602I
−8.46942 − 5.78175I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.23304 + 1.62264I
a = −0.36642 − 2.02217I
b = −0.30906 + 1.52120I
−2.61045 + 5.89112I 0
u = 0.23304 − 1.62264I
a = −0.36642 + 2.02217I
b = −0.30906 − 1.52120I
−2.61045 − 5.89112I 0
u = 0.21905 + 1.68997I
a = 0.36567 + 1.99824I
b = 0.62038 − 1.45655I
−4.28516 + 12.41430I 0
u = 0.21905 − 1.68997I
a = 0.36567 − 1.99824I
b = 0.62038 + 1.45655I
−4.28516 − 12.41430I 0
u = 0.05172 + 1.71102I
a = 0.004613 − 0.307882I
b = −0.004397 + 0.596568I
−11.96180 + 3.35478I 0
u = 0.05172 − 1.71102I
a = 0.004613 + 0.307882I
b = −0.004397 − 0.596568I
−11.96180 − 3.35478I 0
u = 0.0937286
a = −7.36583
b = −0.476600
−1.21783 −10.0460
8
II. I
u
2
= hb, −u
4
− u
3
− 4u
2
+ a − 3u − 3, u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1i
(i) Arc colorings
a
1
=
0
u
a
8
=
1
0
a
4
=
u
4
+ u
3
+ 4u
2
+ 3u + 3
0
a
9
=
1
−u
2
a
3
=
u
4
+ u
3
+ 4u
2
+ 3u + 3
0
a
2
=
u
4
+ u
3
+ 4u
2
+ 3u + 3
u
a
5
=
0
−u
a
7
=
1
0
a
12
=
−u
u
3
+ u
a
10
=
u
2
+ 1
−u
4
− 2u
2
a
11
=
−u
3
− 2u
−u
4
− u
3
− 3u
2
− 2u − 1
a
6
=
−u
−u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
4
+ 6u
3
+ 28u
2
+ 17u + 12
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
5
c
3
, c
7
u
5
c
4
(u + 1)
5
c
5
, c
8
, c
9
u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1
c
6
u
5
+ u
4
− u
2
+ u + 1
c
10
u
5
− u
4
+ u
2
+ u − 1
c
11
, c
12
u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
5
c
3
, c
7
y
5
c
5
, c
8
, c
9
c
11
, c
12
y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1
c
6
, c
10
y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.233677 + 0.885557I
a = 0.278580 + 1.055720I
b = 0
−3.46474 − 2.21397I −6.65223 + 4.39723I
u = −0.233677 − 0.885557I
a = 0.278580 − 1.055720I
b = 0
−3.46474 + 2.21397I −6.65223 − 4.39723I
u = −0.416284
a = 2.40221
b = 0
−0.762751 9.55270
u = −0.05818 + 1.69128I
a = 0.020316 + 0.590570I
b = 0
−12.60320 − 3.33174I −9.12414 + 2.18947I
u = −0.05818 − 1.69128I
a = 0.020316 − 0.590570I
b = 0
−12.60320 + 3.33174I −9.12414 − 2.18947I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
37
+ 12u
36
+ ··· + 5u + 1)
c
2
((u − 1)
5
)(u
37
− 6u
36
+ ··· − 3u + 1)
c
3
, c
7
u
5
(u
37
− u
36
+ ··· + 120u
2
+ 32)
c
4
((u + 1)
5
)(u
37
− 6u
36
+ ··· − 3u + 1)
c
5
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
37
+ 2u
36
+ ··· + 3u + 1)
c
6
(u
5
+ u
4
− u
2
+ u + 1)(u
37
+ 2u
36
+ ··· + 3u + 1)
c
8
, c
9
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)(u
37
− 8u
36
+ ··· + 19u − 1)
c
10
(u
5
− u
4
+ u
2
+ u − 1)(u
37
+ 2u
36
+ ··· + 3u + 1)
c
11
, c
12
(u
5
− u
4
+ 4u
3
− 3u
2
+ 3u − 1)(u
37
− 8u
36
+ ··· + 19u − 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
5
)(y
37
+ 32y
36
+ ··· + 5y −1)
c
2
, c
4
((y −1)
5
)(y
37
− 12y
36
+ ··· + 5y −1)
c
3
, c
7
y
5
(y
37
+ 33y
36
+ ··· − 7680y −1024)
c
5
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
37
− 40y
36
+ ··· + 19y −1)
c
6
, c
10
(y
5
− y
4
+ 4y
3
− 3y
2
+ 3y −1)(y
37
− 8y
36
+ ··· + 19y −1)
c
8
, c
9
, c
11
c
12
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y −1)(y
37
+ 44y
36
+ ··· + 99y −1)
14
