12n
0673
(K12n
0673
)
A knot diagram
1
Linearized knot diagam
4 5 7 2 9 3 11 12 5 7 8 10
Solving Sequence
3,5
2 4
1,10
9 6 7 11 12 8
c
2
c
4
c
1
c
9
c
5
c
6
c
10
c
12
c
8
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h7073u
11
+ 66159u
10
+ ··· + 129872b + 81956,
− 114251u
11
− 1031968u
10
+ ··· + 259744a − 4523833,
u
12
+ 9u
11
+ 25u
10
− 103u
8
− 97u
7
+ 152u
6
+ 251u
5
+ 27u
4
− 144u
3
− 101u
2
+ 45u − 1i
I
u
2
= ha
5
+ a
4
+ 3a
3
+ 2a
2
+ b + 3a + 1, a
6
+ a
5
+ 3a
4
+ 2a
3
+ 2a
2
+ a − 1, u − 1i
I
u
3
= hb + u + 2, a, u
2
+ u − 1i
I
u
4
= hb − 1, a, u
2
+ u − 1i
* 4 irreducible components of dim
C
= 0, with total 22 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
= h7073u
11
+ 66159u
10
+ · · · + 129872b + 81956, −1.14 × 10
5
u
11
−
1.03 × 10
6
u
10
+ · · · + 2.60 × 10
5
a − 4.52 × 10
6
, u
12
+ 9u
11
+ · · · + 45u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
−u
2
a
4
=
u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
10
=
0.439860u
11
+ 3.97302u
10
+ ··· − 47.5490u + 17.4165
−0.0544613u
11
− 0.509417u
10
+ ··· + 6.62604u − 0.631052
a
9
=
0.439860u
11
+ 3.97302u
10
+ ··· − 47.5490u + 17.4165
−0.0450405u
11
− 0.419729u
10
+ ··· + 6.42332u − 0.616773
a
6
=
0.133728u
11
+ 1.20365u
10
+ ··· − 13.6953u + 6.55144
0.0197733u
11
+ 0.165055u
10
+ ··· − 1.07934u − 0.126956
a
7
=
0.113955u
11
+ 1.03859u
10
+ ··· − 12.6159u + 6.67839
0.0197733u
11
+ 0.165055u
10
+ ··· − 1.07934u − 0.126956
a
11
=
0.315345u
11
+ 2.83075u
10
+ ··· − 32.5667u + 13.8030
−0.0286436u
11
− 0.268888u
10
+ ··· + 3.78724u − 0.484901
a
12
=
−0.113955u
11
− 1.03859u
10
+ ··· + 12.6159u − 6.67839
−0.0649601u
11
− 0.595448u
10
+ ··· + 7.01396u + 0.0444168
a
8
=
0.315345u
11
+ 2.83075u
10
+ ··· − 32.5667u + 13.8030
0.0488288u
11
+ 0.442590u
10
+ ··· − 6.15990u − 0.207200
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
1683
8117
u
11
−
255349
129872
u
10
+ ··· +
3115965
129872
u +
1650111
129872
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
12
− 9u
11
+ ··· − 45u − 1
c
3
, c
6
u
12
+ 15u
11
+ ··· + 320u − 64
c
5
, c
9
u
12
+ 3u
11
+ ··· + 32u + 16
c
7
, c
8
, c
10
c
11
u
12
− 4u
11
+ ··· − 10u + 1
c
12
u
12
− 4u
11
+ ··· + 4188u − 167
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
12
− 31y
11
+ ··· − 1823y + 1
c
3
, c
6
y
12
− 45y
11
+ ··· − 200704y + 4096
c
5
, c
9
y
12
+ 25y
11
+ ··· − 5248y + 256
c
7
, c
8
, c
10
c
11
y
12
− 12y
11
+ ··· − 50y + 1
c
12
y
12
+ 124y
11
+ ··· − 13537022y + 27889
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.568188 + 0.801699I
a = 1.49494 − 0.68223I
b = 1.272770 + 0.614644I
2.73506 − 3.39089I 4.46250 + 1.24643I
u = −0.568188 − 0.801699I
a = 1.49494 + 0.68223I
b = 1.272770 − 0.614644I
2.73506 + 3.39089I 4.46250 − 1.24643I
u = 0.823127
a = −0.456967
b = 1.12827
−1.14502 −10.9850
u = 1.41132 + 0.46960I
a = −0.770194 + 0.438317I
b = 2.04382 + 1.53387I
−1.311700 + 0.306316I 0.43565 − 2.29946I
u = 1.41132 − 0.46960I
a = −0.770194 − 0.438317I
b = 2.04382 − 1.53387I
−1.311700 − 0.306316I 0.43565 + 2.29946I
u = 0.310507
a = −1.57680
b = 1.86809
8.15010 18.2770
u = −1.69127
a = −0.389523
b = 0.988886
−7.58478 10.3110
u = 0.0235034
a = 16.2652
b = −0.471383
0.765123 13.2690
u = −2.10231 + 0.69985I
a = −0.295636 − 1.021660I
b = 4.49200 + 0.95627I
−16.1694 + 9.7565I 2.18491 − 3.24065I
u = −2.10231 − 0.69985I
a = −0.295636 + 1.021660I
b = 4.49200 − 0.95627I
−16.1694 − 9.7565I 2.18491 + 3.24065I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −2.97376 + 0.73623I
a = 0.149929 + 1.199420I
b = −6.56552 − 7.10055I
14.6533 + 4.1219I 0.48044 − 1.82182I
u = −2.97376 − 0.73623I
a = 0.149929 − 1.199420I
b = −6.56552 + 7.10055I
14.6533 − 4.1219I 0.48044 + 1.82182I
6
II.
I
u
2
= ha
5
+ a
4
+ 3a
3
+ 2a
2
+ b + 3a + 1, a
6
+ a
5
+ 3a
4
+ 2a
3
+ 2a
2
+ a − 1, u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
1
a
2
=
1
−1
a
4
=
1
0
a
1
=
0
−1
a
10
=
a
−a
5
− a
4
− 3a
3
− 2a
2
− 3a − 1
a
9
=
a
−a
5
− a
4
− 3a
3
− 2a
2
− 2a − 1
a
6
=
a
2
0
a
7
=
a
2
0
a
11
=
−a
5
− a
3
+ a
−a
5
− a
4
− 3a
3
− 2a
2
− 3a − 1
a
12
=
a
2
−a
2
− 2
a
8
=
−a
5
− a
3
+ a
−a
4
− 2a
2
− 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7a
5
+ 15a
4
+ 29a
3
+ 33a
2
+ 28a + 20
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
6
u
6
c
4
(u + 1)
6
c
5
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
c
7
, c
8
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
9
, c
12
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
10
, c
11
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
6
y
6
c
5
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
c
7
, c
8
, c
10
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −0.873214
b = 2.01841
6.01515 6.57090
u = 1.00000
a = 0.138835 + 1.234450I
b = −0.228804 − 0.434483I
−4.60518 + 1.97241I −0.89950 − 4.53432I
u = 1.00000
a = 0.138835 − 1.234450I
b = −0.228804 + 0.434483I
−4.60518 − 1.97241I −0.89950 + 4.53432I
u = 1.00000
a = −0.408802 + 1.276380I
b = 0.636388 − 0.565801I
2.05064 − 4.59213I 1.73030 + 5.96315I
u = 1.00000
a = −0.408802 − 1.276380I
b = 0.636388 + 0.565801I
2.05064 + 4.59213I 1.73030 − 5.96315I
u = 1.00000
a = 0.413150
b = −2.83358
−0.906083 39.7680
10
III. I
u
3
= hb + u + 2, a, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u − 1
a
4
=
u
−u + 1
a
1
=
u
−u
a
10
=
0
−u − 2
a
9
=
0
−u − 2
a
6
=
0
u
a
7
=
−u
u
a
11
=
−1
−u − 1
a
12
=
u
u + 3
a
8
=
1
2u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −16
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
u
2
− u − 1
c
5
, c
9
u
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0
b = −2.61803
7.89568 −16.0000
u = −1.61803
a = 0
b = −0.381966
−7.89568 −16.0000
14
IV. I
u
4
= hb − 1, a, u
2
+ u − 1i
(i) Arc colorings
a
3
=
1
0
a
5
=
0
u
a
2
=
1
u − 1
a
4
=
u
−u + 1
a
1
=
u
−u
a
10
=
0
1
a
9
=
0
1
a
6
=
0
u
a
7
=
−u
u
a
11
=
−u + 1
u
a
12
=
u
0
a
8
=
u − 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −1
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
3
c
10
, c
11
, c
12
u
2
+ u − 1
c
4
, c
6
, c
7
c
8
u
2
− u − 1
c
5
, c
9
u
2
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
6
, c
7
c
8
, c
10
, c
11
c
12
y
2
− 3y + 1
c
5
, c
9
y
2
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 0
b = 1.00000
0 −1.00000
u = −1.61803
a = 0
b = 1.00000
0 −1.00000
18
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
6
)(u
2
+ u − 1)
2
(u
12
− 9u
11
+ ··· − 45u − 1)
c
3
u
6
(u
2
+ u − 1)
2
(u
12
+ 15u
11
+ ··· + 320u − 64)
c
4
((u + 1)
6
)(u
2
− u − 1)
2
(u
12
− 9u
11
+ ··· − 45u − 1)
c
5
u
4
(u
6
− u
5
+ ··· − u − 1)(u
12
+ 3u
11
+ ··· + 32u + 16)
c
6
u
6
(u
2
− u − 1)
2
(u
12
+ 15u
11
+ ··· + 320u − 64)
c
7
, c
8
(u
2
− u − 1)
2
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)
· (u
12
− 4u
11
+ ··· − 10u + 1)
c
9
u
4
(u
6
+ u
5
+ ··· + u − 1)(u
12
+ 3u
11
+ ··· + 32u + 16)
c
10
, c
11
(u
2
+ u − 1)
2
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
12
− 4u
11
+ ··· − 10u + 1)
c
12
(u
2
+ u − 1)
2
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)
· (u
12
− 4u
11
+ ··· + 4188u − 167)
19
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
6
)(y
2
− 3y + 1)
2
(y
12
− 31y
11
+ ··· − 1823y + 1)
c
3
, c
6
y
6
(y
2
− 3y + 1)
2
(y
12
− 45y
11
+ ··· − 200704y + 4096)
c
5
, c
9
y
4
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
12
+ 25y
11
+ ··· − 5248y + 256)
c
7
, c
8
, c
10
c
11
(y
2
− 3y + 1)
2
(y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1)
· (y
12
− 12y
11
+ ··· − 50y + 1)
c
12
(y
2
− 3y + 1)
2
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
12
+ 124y
11
+ ··· − 13537022y + 27889)
20
