12a
0835
(K12a
0835
)
A knot diagram
1
Linearized knot diagam
4 5 9 2 10 11 12 3 1 6 7 8
Solving Sequence
7,11
12
3,8
9 4 1 6 10 5 2
c
11
c
7
c
8
c
3
c
12
c
6
c
10
c
5
c
2
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−3u
37
+ 76u
35
+ ··· + b + 1, 2u
37
+ u
36
+ ··· + a − 3, u
38
+ 2u
37
+ ··· − 4u − 1i
I
u
2
= hu
2
+ b − 1, a − 1, u
3
− u
2
− 2u + 1i
* 2 irreducible components of dim
C
= 0, with total 41 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−3u
37
+76u
35
+· · ·+b+1, 2u
37
+u
36
+· · ·+a−3, u
38
+2u
37
+· · ·−4u−1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
3
=
−2u
37
− u
36
+ ··· − 15u
2
+ 3
3u
37
− 76u
35
+ ··· − 6u − 1
a
8
=
u
−u
3
+ u
a
9
=
u
8
− 5u
6
+ 7u
4
− 4u
2
+ 1
−u
10
+ 6u
8
− 11u
6
+ 6u
4
+ u
2
a
4
=
−u
36
+ 25u
34
+ ··· − u + 2
−u
37
+ 26u
35
+ ··· + u + 1
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
6
=
−u
u
a
10
=
−u
2
+ 1
u
2
a
5
=
u
3
− 2u
−u
3
+ u
a
2
=
−u
37
− u
36
+ ··· − u + 3
u
37
− 25u
35
+ ··· + u
2
− 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = u
37
+ 4u
36
+ ··· + 31u − 7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
38
− 4u
37
+ ··· − u − 1
c
3
, c
8
u
38
+ u
37
+ ··· + 12u + 8
c
5
, c
6
, c
7
c
10
, c
11
, c
12
u
38
+ 2u
37
+ ··· − 4u − 1
c
9
u
38
− 6u
37
+ ··· − 476u + 55
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
38
− 36y
37
+ ··· + 35y + 1
c
3
, c
8
y
38
− 21y
37
+ ··· − 848y + 64
c
5
, c
6
, c
7
c
10
, c
11
, c
12
y
38
− 54y
37
+ ··· − 28y + 1
c
9
y
38
+ 6y
37
+ ··· − 132856y + 3025
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.879915 + 0.280083I
a = −0.422412 + 0.001593I
b = 1.259570 + 0.041859I
−4.46594 − 0.62155I 1.23906 − 1.00554I
u = 0.879915 − 0.280083I
a = −0.422412 − 0.001593I
b = 1.259570 − 0.041859I
−4.46594 + 0.62155I 1.23906 + 1.00554I
u = 1.107830 + 0.111217I
a = 1.127560 + 0.838416I
b = −0.893007 − 0.188118I
2.67127 + 0.72425I 3.65418 − 0.86134I
u = 1.107830 − 0.111217I
a = 1.127560 − 0.838416I
b = −0.893007 + 0.188118I
2.67127 − 0.72425I 3.65418 + 0.86134I
u = −1.132560 + 0.171155I
a = 0.264697 − 1.114460I
b = 0.795265 − 0.506019I
1.45072 − 3.23668I 4.64812 + 3.17787I
u = −1.132560 − 0.171155I
a = 0.264697 + 1.114460I
b = 0.795265 + 0.506019I
1.45072 + 3.23668I 4.64812 − 3.17787I
u = 1.174750 + 0.191050I
a = −0.51302 − 1.48671I
b = 0.349369 + 0.196389I
4.18688 + 5.41975I 6.64086 − 6.14271I
u = 1.174750 − 0.191050I
a = −0.51302 + 1.48671I
b = 0.349369 − 0.196389I
4.18688 − 5.41975I 6.64086 + 6.14271I
u = −1.211660 + 0.074429I
a = −0.357902 + 0.705905I
b = −0.289689 + 0.133201I
6.39460 − 0.99370I 11.55440 + 0.I
u = −1.211660 − 0.074429I
a = −0.357902 − 0.705905I
b = −0.289689 − 0.133201I
6.39460 + 0.99370I 11.55440 + 0.I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.201030 + 0.253869I
a = −0.05721 + 1.56031I
b = 0.060414 + 0.164516I
−1.60053 + 9.43906I 3.22111 − 6.40713I
u = 1.201030 − 0.253869I
a = −0.05721 − 1.56031I
b = 0.060414 − 0.164516I
−1.60053 − 9.43906I 3.22111 + 6.40713I
u = −0.477067 + 0.497622I
a = 0.243665 + 0.668012I
b = −0.819079 − 1.087840I
−6.95978 − 6.84371I −0.65587 + 7.18457I
u = −0.477067 − 0.497622I
a = 0.243665 − 0.668012I
b = −0.819079 + 1.087840I
−6.95978 + 6.84371I −0.65587 − 7.18457I
u = −1.36368
a = 1.08623
b = −0.216111
3.27250 0
u = −0.423369 + 0.400375I
a = −0.726383 − 0.423149I
b = 0.291431 + 0.956910I
−0.93228 − 3.40297I 2.14199 + 8.41753I
u = −0.423369 − 0.400375I
a = −0.726383 + 0.423149I
b = 0.291431 − 0.956910I
−0.93228 + 3.40297I 2.14199 − 8.41753I
u = −0.195413 + 0.544437I
a = −1.31438 − 1.37135I
b = 0.069081 − 0.125786I
−7.79779 + 3.42232I −3.36452 − 0.77365I
u = −0.195413 − 0.544437I
a = −1.31438 + 1.37135I
b = 0.069081 + 0.125786I
−7.79779 − 3.42232I −3.36452 + 0.77365I
u = 0.341654 + 0.397148I
a = −1.48910 − 0.45750I
b = −0.079740 + 0.997605I
−3.23774 + 1.34870I −0.24766 − 4.74966I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.341654 − 0.397148I
a = −1.48910 + 0.45750I
b = −0.079740 − 0.997605I
−3.23774 − 1.34870I −0.24766 + 4.74966I
u = 0.477554 + 0.133920I
a = 0.643576 + 0.290655I
b = −0.342626 − 0.376975I
0.891274 + 0.223442I 10.83666 − 1.68176I
u = 0.477554 − 0.133920I
a = 0.643576 − 0.290655I
b = −0.342626 + 0.376975I
0.891274 − 0.223442I 10.83666 + 1.68176I
u = −0.229708 + 0.385709I
a = 1.41072 + 0.84547I
b = 0.063553 − 0.396538I
−1.49424 + 0.71629I −1.75673 + 0.24825I
u = −0.229708 − 0.385709I
a = 1.41072 − 0.84547I
b = 0.063553 + 0.396538I
−1.49424 − 0.71629I −1.75673 − 0.24825I
u = −1.70748
a = 0.768656
b = −0.625166
4.46722 0
u = −0.234823
a = 2.58144
b = 0.732163
−1.29292 −12.5790
u = −1.76514 + 0.02788I
a = 0.56054 − 1.49467I
b = −1.66540 + 3.03614I
13.15060 − 1.31489I 0
u = −1.76514 − 0.02788I
a = 0.56054 + 1.49467I
b = −1.66540 − 3.03614I
13.15060 + 1.31489I 0
u = 1.76796 + 0.04114I
a = 0.92517 + 2.50893I
b = −1.34181 − 4.80741I
12.00080 + 4.13178I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.76796 − 0.04114I
a = 0.92517 − 2.50893I
b = −1.34181 + 4.80741I
12.00080 − 4.13178I 0
u = −1.77750 + 0.04775I
a = −0.01149 + 2.45709I
b = 0.24659 − 4.97450I
14.9410 − 6.4586I 0
u = −1.77750 − 0.04775I
a = −0.01149 − 2.45709I
b = 0.24659 + 4.97450I
14.9410 + 6.4586I 0
u = −1.78301 + 0.06547I
a = −0.63670 − 2.77800I
b = 1.33294 + 5.47020I
9.2442 − 10.8528I 0
u = −1.78301 − 0.06547I
a = −0.63670 + 2.77800I
b = 1.33294 − 5.47020I
9.2442 + 10.8528I 0
u = 1.78759 + 0.01827I
a = −0.73970 − 1.68325I
b = 1.31660 + 3.30669I
17.4040 + 1.4058I 0
u = 1.78759 − 0.01827I
a = −0.73970 + 1.68325I
b = 1.31660 − 3.30669I
17.4040 − 1.4058I 0
u = 1.82025
a = 1.74840
b = −3.59779
15.0989 0
8
II. I
u
2
= hu
2
+ b − 1, a − 1, u
3
− u
2
− 2u + 1i
(i) Arc colorings
a
7
=
0
u
a
11
=
1
0
a
12
=
1
−u
2
a
3
=
1
−u
2
+ 1
a
8
=
u
−u
2
− u + 1
a
9
=
u
−u
2
− u + 1
a
4
=
1
−u
2
+ 1
a
1
=
−u
2
+ 1
u
2
+ u − 1
a
6
=
−u
u
a
10
=
−u
2
+ 1
u
2
a
5
=
u
2
− 1
−u
2
− u + 1
a
2
=
−u
2
+ 2
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
2
+ u + 11
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
8
u
3
c
4
(u + 1)
3
c
5
, c
6
, c
7
c
9
u
3
+ u
2
− 2u − 1
c
10
, c
11
, c
12
u
3
− u
2
− 2u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
3
c
3
, c
8
y
3
c
5
, c
6
, c
7
c
9
, c
10
, c
11
c
12
y
3
− 5y
2
+ 6y −1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.24698
a = 1.00000
b = −0.554958
4.69981 8.19810
u = 0.445042
a = 1.00000
b = 0.801938
−0.939962 11.2470
u = 1.80194
a = 1.00000
b = −2.24698
15.9794 9.55500
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
3
)(u
38
− 4u
37
+ ··· − u − 1)
c
3
, c
8
u
3
(u
38
+ u
37
+ ··· + 12u + 8)
c
4
((u + 1)
3
)(u
38
− 4u
37
+ ··· − u − 1)
c
5
, c
6
, c
7
(u
3
+ u
2
− 2u − 1)(u
38
+ 2u
37
+ ··· − 4u − 1)
c
9
(u
3
+ u
2
− 2u − 1)(u
38
− 6u
37
+ ··· − 476u + 55)
c
10
, c
11
, c
12
(u
3
− u
2
− 2u + 1)(u
38
+ 2u
37
+ ··· − 4u − 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
3
)(y
38
− 36y
37
+ ··· + 35y + 1)
c
3
, c
8
y
3
(y
38
− 21y
37
+ ··· − 848y + 64)
c
5
, c
6
, c
7
c
10
, c
11
, c
12
(y
3
− 5y
2
+ 6y −1)(y
38
− 54y
37
+ ··· − 28y + 1)
c
9
(y
3
− 5y
2
+ 6y −1)(y
38
+ 6y
37
+ ··· − 132856y + 3025)
14
