12n
0336
(K12n
0336
)
A knot diagram
1
Linearized knot diagam
3 6 8 7 11 2 9 4 12 6 9 10
Solving Sequence
9,11
12 10
1,4
8 3 7 5 6 2
c
11
c
9
c
12
c
8
c
3
c
7
c
4
c
5
c
2
c
1
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−1706550177091u
18
− 23512487300441u
17
+ ··· + 39336273597696b − 69157159108721,
− 32608816754461u
18
− 456051534293894u
17
+ ··· + 19668136798848a − 1093408920959144,
u
19
+ 14u
18
+ ··· + 52u + 1i
I
u
2
= h−a
8
+ a
7
+ 3a
6
− 2a
5
− 3a
4
+ 2a
3
+ b + a + 2, a
9
− a
8
− 2a
7
+ 3a
6
+ a
5
− 3a
4
+ 2a
3
− a + 1, u − 1i
I
u
3
= h−3a
3
u + 2a
3
− a
2
u + b + a, a
4
− a
2
u − 2a
2
+ 3u + 5, u
2
+ u − 1i
* 3 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−1.71 × 10
12
u
18
− 2.35 × 10
13
u
17
+ · · · + 3.93 × 10
13
b − 6.92 ×
10
13
, −3.26 × 10
13
u
18
− 4.56 × 10
14
u
17
+ · · · + 1.97 × 10
13
a − 1.09 ×
10
15
, u
19
+ 14u
18
+ · · · + 52u + 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
10
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
4
=
1.65795u
18
+ 23.1873u
17
+ ··· + 1589.07u + 55.5929
0.0433836u
18
+ 0.597730u
17
+ ··· + 39.8060u + 1.75810
a
8
=
1.84251u
18
+ 25.7334u
17
+ ··· + 1669.72u + 43.2886
0.0542335u
18
+ 0.767279u
17
+ ··· + 55.0156u + 1.70577
a
3
=
0.243745u
18
+ 3.43967u
17
+ ··· + 332.641u + 35.7067
−0.0193717u
18
− 0.267431u
17
+ ··· − 10.7266u + 0.631068
a
7
=
1.84251u
18
+ 25.7334u
17
+ ··· + 1669.72u + 43.2886
0.0598519u
18
+ 0.839590u
17
+ ··· + 56.3822u + 1.76749
a
5
=
0.0513825u
18
+ 0.731152u
17
+ ··· + 83.7307u + 7.87385
−0.00184480u
18
− 0.0331746u
17
+ ··· − 3.98487u + 0.0723727
a
6
=
0.0532273u
18
+ 0.764327u
17
+ ··· + 87.7156u + 7.80147
−0.00184480u
18
− 0.0331746u
17
+ ··· − 3.98487u + 0.0723727
a
2
=
−0.136733u
18
− 1.87566u
17
+ ··· − 19.9103u + 26.8869
−0.0281326u
18
− 0.392506u
17
+ ··· − 21.4809u + 0.261960
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
390713901803
1639011399904
u
18
−
21917006139285
6556045599616
u
17
+ ··· −
430703611764529
1639011399904
u −
74684759366323
6556045599616
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
19
− 9u
18
+ ··· + 9432u − 1296
c
2
, c
6
u
19
− 7u
18
+ ··· − 36u + 36
c
3
, c
8
u
19
− 2u
18
+ ··· − 12u + 9
c
4
u
19
− 6u
18
+ ··· + 19710u − 2349
c
5
, c
10
u
19
+ u
18
+ ··· + 1536u + 512
c
7
u
19
− 16u
18
+ ··· + 540u − 81
c
9
, c
11
, c
12
u
19
+ 14u
18
+ ··· + 52u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
19
− y
18
+ ··· + 159068448y −1679616
c
2
, c
6
y
19
− 9y
18
+ ··· + 9432y −1296
c
3
, c
8
y
19
− 16y
18
+ ··· + 540y −81
c
4
y
19
+ 56y
18
+ ··· + 419265396y −5517801
c
5
, c
10
y
19
+ 69y
18
+ ··· + 15204352y −262144
c
7
y
19
− 20y
18
+ ··· + 220968y −6561
c
9
, c
11
, c
12
y
19
− 72y
18
+ ··· + 696y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.090810 + 0.087492I
a = 0.864947 + 0.453435I
b = 1.25966 − 4.27188I
0.12746 + 2.07953I 0.79710 + 6.78619I
u = 1.090810 − 0.087492I
a = 0.864947 − 0.453435I
b = 1.25966 + 4.27188I
0.12746 − 2.07953I 0.79710 − 6.78619I
u = 1.12421
a = −0.369206
b = −0.262337
2.16627 2.38650
u = −0.866404 + 0.903603I
a = 0.403452 + 1.041090I
b = −1.260210 − 0.033364I
4.84224 + 5.75329I 6.23581 − 2.60069I
u = −0.866404 − 0.903603I
a = 0.403452 − 1.041090I
b = −1.260210 + 0.033364I
4.84224 − 5.75329I 6.23581 + 2.60069I
u = 0.264112 + 0.511238I
a = 0.998671 + 0.095546I
b = 0.655201 − 0.830329I
0.335625 + 1.223470I 4.13138 − 4.81848I
u = 0.264112 − 0.511238I
a = 0.998671 − 0.095546I
b = 0.655201 + 0.830329I
0.335625 − 1.223470I 4.13138 + 4.81848I
u = −1.54015 + 0.03306I
a = −0.616311 + 0.224492I
b = 0.39526 − 1.71073I
8.24788 − 1.49342I 14.3201 − 0.4966I
u = −1.54015 − 0.03306I
a = −0.616311 − 0.224492I
b = 0.39526 + 1.71073I
8.24788 + 1.49342I 14.3201 + 0.4966I
u = −0.272498
a = −3.42293
b = −0.561462
1.52167 5.90680
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.0292040 + 0.0188465I
a = 11.6667 + 24.4338I
b = 0.653657 + 0.621796I
−1.74489 + 2.05954I −4.07768 − 4.15264I
u = −0.0292040 − 0.0188465I
a = 11.6667 − 24.4338I
b = 0.653657 − 0.621796I
−1.74489 − 2.05954I −4.07768 + 4.15264I
u = −2.00138 + 0.76457I
a = −0.689796 + 0.112485I
b = −0.56693 − 3.57760I
−15.5608 − 12.7812I 6.72063 + 5.11794I
u = −2.00138 − 0.76457I
a = −0.689796 − 0.112485I
b = −0.56693 + 3.57760I
−15.5608 + 12.7812I 6.72063 − 5.11794I
u = −2.42283 + 0.72006I
a = 0.017324 − 0.538247I
b = 2.30004 + 3.34538I
−18.5440 − 5.6494I 4.00000 + 0.I
u = −2.42283 − 0.72006I
a = 0.017324 + 0.538247I
b = 2.30004 − 3.34538I
−18.5440 + 5.6494I 4.00000 + 0.I
u = −3.92517 + 0.42402I
a = 0.495462 − 0.122568I
b = −4.38040 + 4.23334I
−11.60890 − 1.83503I 0
u = −3.92517 − 0.42402I
a = 0.495462 + 0.122568I
b = −4.38040 − 4.23334I
−11.60890 + 1.83503I 0
u = 4.00873
a = 0.511273
b = −5.28877
11.4850 0
6
II. I
u
2
= h−a
8
+ a
7
+ 3a
6
− 2a
5
− 3a
4
+ 2a
3
+ b + a + 2, a
9
− a
8
− 2a
7
+
3a
6
+ a
5
− 3a
4
+ 2a
3
− a + 1, u − 1i
(i) Arc colorings
a
9
=
0
1
a
11
=
1
0
a
12
=
1
−1
a
10
=
1
0
a
1
=
0
−1
a
4
=
a
a
8
− a
7
− 3a
6
+ 2a
5
+ 3a
4
− 2a
3
− a − 2
a
8
=
−a
2
a
7
+ a
6
− 2a
5
− a
4
+ 2a
3
+ a
2
+ a + 2
a
3
=
−a
3
+ a
2a
8
− 5a
6
+ a
5
+ 5a
4
− a
3
+ a
2
+ a − 2
a
7
=
−a
2
a
7
+ a
6
− 2a
5
− a
4
+ 2a
3
+ a + 2
a
5
=
a
5
− a
3
+ a
0
a
6
=
a
5
− a
3
+ a
0
a
2
=
a
6
− 2a
4
+ a
2
2a
8
− 5a
6
+ a
5
+ 5a
4
− a
3
+ a
2
+ a − 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6a
8
− 3a
7
− 10a
6
+ 8a
5
+ 2a
4
− 8a
3
+ 12a
2
+ 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1
c
2
u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1
c
3
u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1
c
5
, c
10
u
9
c
6
u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1
c
7
u
9
+ 5u
8
+ 12u
7
+ 15u
6
+ 9u
5
− u
4
− 4u
3
− 2u
2
+ u + 1
c
8
u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1
c
9
(u + 1)
9
c
11
, c
12
(u − 1)
9
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1
c
2
, c
6
y
9
+ 3y
8
+ 8y
7
+ 13y
6
+ 17y
5
+ 17y
4
+ 12y
3
+ 6y
2
+ y −1
c
3
, c
8
y
9
− 5y
8
+ 12y
7
− 15y
6
+ 9y
5
+ y
4
− 4y
3
+ 2y
2
+ y −1
c
5
, c
10
y
9
c
7
y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1
c
9
, c
11
, c
12
(y −1)
9
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.772920 + 0.510351I
b = −3.25219 + 0.42284I
−0.13850 + 2.09337I 11.64247 + 5.88316I
u = 1.00000
a = 0.772920 − 0.510351I
b = −3.25219 − 0.42284I
−0.13850 − 2.09337I 11.64247 − 5.88316I
u = 1.00000
a = −0.825933
b = 0.106533
2.84338 15.4610
u = 1.00000
a = −1.173910 + 0.391555I
b = −0.121911 − 0.782086I
6.01628 − 1.33617I 7.94914 + 0.75351I
u = 1.00000
a = −1.173910 − 0.391555I
b = −0.121911 + 0.782086I
6.01628 + 1.33617I 7.94914 − 0.75351I
u = 1.00000
a = 0.141484 + 0.739668I
b = −0.217279 − 0.962736I
2.26187 − 2.45442I 4.75622 + 3.91612I
u = 1.00000
a = 0.141484 − 0.739668I
b = −0.217279 + 0.962736I
2.26187 + 2.45442I 4.75622 − 3.91612I
u = 1.00000
a = 1.172470 + 0.500383I
b = 0.038112 − 1.195250I
5.24306 + 7.08493I 7.92182 − 8.89461I
u = 1.00000
a = 1.172470 − 0.500383I
b = 0.038112 + 1.195250I
5.24306 − 7.08493I 7.92182 + 8.89461I
10
III. I
u
3
= h−3a
3
u + 2a
3
− a
2
u + b + a, a
4
− a
2
u − 2a
2
+ 3u + 5, u
2
+ u − 1i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
0
a
12
=
1
u − 1
a
10
=
u
−u + 1
a
1
=
u
−u
a
4
=
a
3a
3
u − 2a
3
+ a
2
u − a
a
8
=
−a
2
u
a
3
u − a
3
+ 3a
2
u − a
2
a
3
=
a
3
u − a
3
+ a
−a
3
u + a
3
− a + u + 1
a
7
=
−a
2
u
a
3
u − a
3
+ a
2
u
a
5
=
0
3a
3
u − 2a
3
a
6
=
−3a
3
u + 2a
3
3a
3
u − 2a
3
a
2
=
a
3
u − a
3
+ a + u
−a
3
u + a
3
− a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a
2
u − 4a
2
+ 8
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
8
c
2
, c
6
(u
2
+ 1)
4
c
3
, c
4
, c
8
(u
4
− u
2
+ 1)
2
c
5
, c
10
(u
4
+ 3u
2
+ 1)
2
c
7
(u
2
+ u + 1)
4
c
9
(u
2
− u − 1)
4
c
11
, c
12
(u
2
+ u − 1)
4
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
(y −1)
8
c
2
, c
6
(y + 1)
8
c
3
, c
4
, c
8
(y
2
− y + 1)
4
c
5
, c
10
(y
2
+ 3y + 1)
4
c
7
(y
2
+ y + 1)
4
c
9
, c
11
, c
12
(y
2
− 3y + 1)
4
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.40126 + 0.80902I
b = −0.592242 − 0.025792I
−0.65797 + 2.02988I 6.00000 − 3.46410I
u = 0.618034
a = 1.40126 − 0.80902I
b = −0.592242 + 0.025792I
−0.65797 − 2.02988I 6.00000 + 3.46410I
u = 0.618034
a = −1.40126 + 0.80902I
b = 2.21028 − 2.82831I
−0.65797 − 2.02988I 6.00000 + 3.46410I
u = 0.618034
a = −1.40126 − 0.80902I
b = 2.21028 + 2.82831I
−0.65797 + 2.02988I 6.00000 − 3.46410I
u = −1.61803
a = −0.535233 + 0.309017I
b = 0.226216 − 1.391820I
7.23771 + 2.02988I 6.00000 − 3.46410I
u = −1.61803
a = −0.535233 − 0.309017I
b = 0.226216 + 1.391820I
7.23771 − 2.02988I 6.00000 + 3.46410I
u = −1.61803
a = 0.535233 + 0.309017I
b = −0.84425 − 2.46228I
7.23771 − 2.02988I 6.00000 + 3.46410I
u = −1.61803
a = 0.535233 − 0.309017I
b = −0.84425 + 2.46228I
7.23771 + 2.02988I 6.00000 − 3.46410I
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
8
)(u
9
− 3u
8
+ ··· + u + 1)
· (u
19
− 9u
18
+ ··· + 9432u − 1296)
c
2
(u
2
+ 1)
4
(u
9
− u
8
+ 2u
7
− u
6
+ 3u
5
− u
4
+ 2u
3
+ u + 1)
· (u
19
− 7u
18
+ ··· − 36u + 36)
c
3
(u
4
− u
2
+ 1)
2
(u
9
− u
8
− 2u
7
+ 3u
6
+ u
5
− 3u
4
+ 2u
3
− u + 1)
· (u
19
− 2u
18
+ ··· − 12u + 9)
c
4
(u
4
− u
2
+ 1)
2
· (u
9
− 3u
8
+ 8u
7
− 13u
6
+ 17u
5
− 17u
4
+ 12u
3
− 6u
2
+ u + 1)
· (u
19
− 6u
18
+ ··· + 19710u − 2349)
c
5
, c
10
u
9
(u
4
+ 3u
2
+ 1)
2
(u
19
+ u
18
+ ··· + 1536u + 512)
c
6
(u
2
+ 1)
4
(u
9
+ u
8
+ 2u
7
+ u
6
+ 3u
5
+ u
4
+ 2u
3
+ u − 1)
· (u
19
− 7u
18
+ ··· − 36u + 36)
c
7
((u
2
+ u + 1)
4
)(u
9
+ 5u
8
+ ··· + u + 1)
· (u
19
− 16u
18
+ ··· + 540u − 81)
c
8
(u
4
− u
2
+ 1)
2
(u
9
+ u
8
− 2u
7
− 3u
6
+ u
5
+ 3u
4
+ 2u
3
− u − 1)
· (u
19
− 2u
18
+ ··· − 12u + 9)
c
9
((u + 1)
9
)(u
2
− u − 1)
4
(u
19
+ 14u
18
+ ··· + 52u + 1)
c
11
, c
12
((u − 1)
9
)(u
2
+ u − 1)
4
(u
19
+ 14u
18
+ ··· + 52u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)
8
(y
9
+ 7y
8
+ 20y
7
+ 25y
6
+ 5y
5
− 15y
4
+ 22y
2
+ 13y −1)
· (y
19
− y
18
+ ··· + 159068448y −1679616)
c
2
, c
6
((y + 1)
8
)(y
9
+ 3y
8
+ ··· + y −1)
· (y
19
− 9y
18
+ ··· + 9432y −1296)
c
3
, c
8
((y
2
− y + 1)
4
)(y
9
− 5y
8
+ ··· + y −1)
· (y
19
− 16y
18
+ ··· + 540y −81)
c
4
((y
2
− y + 1)
4
)(y
9
+ 7y
8
+ ··· + 13y −1)
· (y
19
+ 56y
18
+ ··· + 419265396y −5517801)
c
5
, c
10
y
9
(y
2
+ 3y + 1)
4
(y
19
+ 69y
18
+ ··· + 1.52044 × 10
7
y −262144)
c
7
(y
2
+ y + 1)
4
(y
9
− y
8
+ 12y
7
− 7y
6
+ 37y
5
+ y
4
− 10y
2
+ 5y −1)
· (y
19
− 20y
18
+ ··· + 220968y −6561)
c
9
, c
11
, c
12
((y −1)
9
)(y
2
− 3y + 1)
4
(y
19
− 72y
18
+ ··· + 696y −1)
16
