12n
0522
(K12n
0522
)
A knot diagram
1
Linearized knot diagam
3 6 9 8 2 11 12 4 3 6 7 10
Solving Sequence
6,11
7 12
3,8
2 1 5 4 10 9
c
6
c
11
c
7
c
2
c
1
c
5
c
4
c
10
c
9
c
3
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−5889u
16
− 7060u
15
+ ··· + 12589b + 19445, 32189u
16
+ 36655u
15
+ ··· + 75534a − 90802,
u
17
+ 2u
16
+ ··· − 5u − 3i
I
u
2
= hb + 1, a − 1, u
2
− u − 1i
I
u
3
= hb − 1, a
2
+ 2a + 2u + 5, u
2
+ u − 1i
* 3 irreducible components of dim
C
= 0, with total 23 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−5889u
16
− 7060u
15
+ · · · + 12589b + 19445, 32189u
16
+ 36655u
15
+
· · · + 75534a − 90802, u
17
+ 2u
16
+ · · · − 5u − 3i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u
2
a
12
=
u
−u
3
+ u
a
3
=
−0.426152u
16
− 0.485278u
15
+ ··· + 2.12912u + 1.20213
0.467789u
16
+ 0.560807u
15
+ ··· − 0.689253u − 1.54460
a
8
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
0.0416369u
16
+ 0.0755289u
15
+ ··· + 1.43987u − 0.342468
0.467789u
16
+ 0.560807u
15
+ ··· − 0.689253u − 1.54460
a
1
=
−u
5
+ 2u
3
+ u
u
5
− 3u
3
+ u
a
5
=
−0.103278u
16
− 0.225501u
15
+ ··· + 1.46060u + 0.772526
0.345063u
16
+ 0.406545u
15
+ ··· − 0.432520u − 1.18063
a
4
=
−0.147841u
16
− 0.400932u
15
+ ··· + 1.50334u + 0.606627
0.232187u
16
+ 0.332949u
15
+ ··· + 0.0411470u − 0.962706
a
10
=
−u
u
a
9
=
−0.258864u
16
− 0.155586u
15
+ ··· − 0.170347u + 0.00406439
−0.0189451u
16
+ 0.0363810u
15
+ ··· + 0.256136u − 0.309834
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
11735
12589
u
16
−
15443
12589
u
15
+ ··· +
246832
12589
u +
106629
12589
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
+ 27u
16
+ ··· + 4064u + 121
c
2
, c
5
u
17
+ 3u
16
+ ··· − 40u + 11
c
3
, c
4
, c
8
c
9
u
17
− u
16
+ ··· − 8u + 4
c
6
, c
7
, c
10
c
11
u
17
+ 2u
16
+ ··· − 5u − 3
c
12
u
17
− 2u
16
+ ··· + 7u + 63
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
− 67y
16
+ ··· + 9958380y −14641
c
2
, c
5
y
17
− 27y
16
+ ··· + 4064y −121
c
3
, c
4
, c
8
c
9
y
17
+ 27y
16
+ ··· + 128y −16
c
6
, c
7
, c
10
c
11
y
17
− 18y
16
+ ··· + 43y −9
c
12
y
17
+ 54y
16
+ ··· + 28903y −3969
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.15416
a = 0.769400
b = −1.41340
0.566399 8.58110
u = 0.629765 + 0.993192I
a = 0.607738 − 0.970566I
b = −2.02831 + 0.15469I
18.5230 + 3.2436I 1.66321 − 2.07655I
u = 0.629765 − 0.993192I
a = 0.607738 + 0.970566I
b = −2.02831 − 0.15469I
18.5230 − 3.2436I 1.66321 + 2.07655I
u = −0.119587 + 0.703609I
a = 0.666875 + 0.456265I
b = 1.40522 − 0.37226I
−7.95189 − 0.77655I −0.292399 + 0.937296I
u = −0.119587 − 0.703609I
a = 0.666875 − 0.456265I
b = 1.40522 + 0.37226I
−7.95189 + 0.77655I −0.292399 − 0.937296I
u = −1.259020 + 0.292419I
a = −0.62819 − 2.15227I
b = 0.889145 + 0.880924I
−4.39263 − 2.75657I 5.16569 + 3.00882I
u = −1.259020 − 0.292419I
a = −0.62819 + 2.15227I
b = 0.889145 − 0.880924I
−4.39263 + 2.75657I 5.16569 − 3.00882I
u = 1.38400 + 0.32880I
a = −1.062080 + 0.704755I
b = 1.66229 − 0.01674I
−3.11311 + 4.57021I 4.59157 − 3.56675I
u = 1.38400 − 0.32880I
a = −1.062080 − 0.704755I
b = 1.66229 + 0.01674I
−3.11311 − 4.57021I 4.59157 + 3.56675I
u = 1.48295 + 0.02809I
a = 0.461098 − 1.264320I
b = −0.624675 + 0.638127I
4.78510 + 2.25631I 7.06459 − 4.01035I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.48295 − 0.02809I
a = 0.461098 + 1.264320I
b = −0.624675 − 0.638127I
4.78510 − 2.25631I 7.06459 + 4.01035I
u = −0.341751 + 0.353385I
a = −0.109209 + 1.262730I
b = −0.680413 − 0.329909I
−1.24584 − 1.09242I 0.17632 + 5.11244I
u = −0.341751 − 0.353385I
a = −0.109209 − 1.262730I
b = −0.680413 + 0.329909I
−1.24584 + 1.09242I 0.17632 − 5.11244I
u = 0.460304
a = 0.456311
b = 0.195899
0.651323 15.8300
u = −1.58318
a = −0.387801
b = 0.650314
7.81790 16.8850
u = −1.63784 + 0.36660I
a = 1.47814 + 1.42375I
b = −1.83966 − 0.39328I
−13.5898 − 8.3565I 3.98308 + 3.14605I
u = −1.63784 − 0.36660I
a = 1.47814 − 1.42375I
b = −1.83966 + 0.39328I
−13.5898 + 8.3565I 3.98308 − 3.14605I
6
II. I
u
2
= hb + 1, a − 1, u
2
− u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
−u − 1
a
12
=
u
−u − 1
a
3
=
1
−1
a
8
=
−u
u
a
2
=
0
−1
a
1
=
−1
0
a
5
=
1
−1
a
4
=
1
−1
a
10
=
−u
u
a
9
=
−u
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
2
c
3
, c
4
, c
8
c
9
u
2
c
5
(u + 1)
2
c
6
, c
7
u
2
− u − 1
c
10
, c
11
, c
12
u
2
+ u − 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
2
c
3
, c
4
, c
8
c
9
y
2
c
6
, c
7
, c
10
c
11
, c
12
y
2
− 3y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = 1.00000
b = −1.00000
−0.657974 2.00000
u = 1.61803
a = 1.00000
b = −1.00000
7.23771 2.00000
10
III. I
u
3
= hb − 1, a
2
+ 2a + 2u + 5, u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
7
=
1
u − 1
a
12
=
u
−u + 1
a
3
=
a
1
a
8
=
u
−u
a
2
=
a + 1
1
a
1
=
1
0
a
5
=
−a
−1
a
4
=
−au − u + 1
au − a + u − 2
a
10
=
−u
u
a
9
=
au + 2u + 2
−au
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
4
c
2
(u + 1)
4
c
3
, c
4
, c
8
c
9
(u
2
+ 2)
2
c
6
, c
7
, c
12
(u
2
+ u − 1)
2
c
10
, c
11
(u
2
− u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
4
c
3
, c
4
, c
8
c
9
(y + 2)
4
c
6
, c
7
, c
10
c
11
, c
12
(y
2
− 3y + 1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = −1.00000 + 2.28825I
b = 1.00000
−5.59278 4.00000
u = 0.618034
a = −1.00000 − 2.28825I
b = 1.00000
−5.59278 4.00000
u = −1.61803
a = −1.000000 + 0.874032I
b = 1.00000
2.30291 4.00000
u = −1.61803
a = −1.000000 − 0.874032I
b = 1.00000
2.30291 4.00000
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
6
)(u
17
+ 27u
16
+ ··· + 4064u + 121)
c
2
((u − 1)
2
)(u + 1)
4
(u
17
+ 3u
16
+ ··· − 40u + 11)
c
3
, c
4
, c
8
c
9
u
2
(u
2
+ 2)
2
(u
17
− u
16
+ ··· − 8u + 4)
c
5
((u − 1)
4
)(u + 1)
2
(u
17
+ 3u
16
+ ··· − 40u + 11)
c
6
, c
7
(u
2
− u − 1)(u
2
+ u − 1)
2
(u
17
+ 2u
16
+ ··· − 5u − 3)
c
10
, c
11
((u
2
− u − 1)
2
)(u
2
+ u − 1)(u
17
+ 2u
16
+ ··· − 5u − 3)
c
12
((u
2
+ u − 1)
3
)(u
17
− 2u
16
+ ··· + 7u + 63)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
6
)(y
17
− 67y
16
+ ··· + 9958380y −14641)
c
2
, c
5
((y −1)
6
)(y
17
− 27y
16
+ ··· + 4064y −121)
c
3
, c
4
, c
8
c
9
y
2
(y + 2)
4
(y
17
+ 27y
16
+ ··· + 128y −16)
c
6
, c
7
, c
10
c
11
((y
2
− 3y + 1)
3
)(y
17
− 18y
16
+ ··· + 43y −9)
c
12
((y
2
− 3y + 1)
3
)(y
17
+ 54y
16
+ ··· + 28903y −3969)
16
