12a
1136
(K12a
1136
)
A knot diagram
1
Linearized knot diagam
4 8 9 10 11 12 1 3 2 7 6 5
Solving Sequence
4,9
3 8 2 10 5 1 7 11 12 6
c
3
c
8
c
2
c
9
c
4
c
1
c
7
c
10
c
12
c
6
c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
66
+ u
65
+ ··· + 2u
2
+ 1i
I
u
2
= hu
6
+ u
5
− 2u
4
− 2u
3
+ 1i
I
u
3
= hu − 1i
* 3 irreducible components of dim
C
= 0, with total 73 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
= hu
66
+ u
65
+ · · · + 2u
2
+ 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
8
=
−u
−u
3
+ u
a
2
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
5
=
u
12
− 5u
10
+ 9u
8
− 6u
6
+ u
2
+ 1
u
14
− 6u
12
+ 13u
10
− 10u
8
− 2u
6
+ 4u
4
+ u
2
a
1
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
7
=
−u
11
+ 4u
9
− 4u
7
− 2u
5
+ 3u
3
−u
11
+ 5u
9
− 8u
7
+ 3u
5
+ u
3
+ u
a
11
=
−u
29
+ 12u
27
+ ··· − 2u
3
+ u
−u
29
+ 13u
27
+ ··· + 3u
3
+ u
a
12
=
u
30
− 13u
28
+ ··· + 2u
2
+ 1
u
32
− 14u
30
+ ··· − 20u
8
+ 2u
2
a
6
=
u
65
− 28u
63
+ ··· + u + 2
u
65
− 29u
63
+ ··· + 3u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
63
+ 116u
61
+ ··· + 8u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
66
− 15u
65
+ ··· − 1396u + 113
c
2
, c
3
, c
8
u
66
− u
65
+ ··· + 2u
2
+ 1
c
4
, c
7
u
66
− 6u
65
+ ··· + 84u + 8
c
5
, c
6
, c
11
u
66
− u
65
+ ··· + 2u
2
+ 1
c
9
u
66
+ 3u
65
+ ··· − 6u − 1
c
10
, c
12
u
66
+ 3u
65
+ ··· − 2u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
66
+ 13y
65
+ ··· + 264176y + 12769
c
2
, c
3
, c
8
y
66
− 59y
65
+ ··· + 4y + 1
c
4
, c
7
y
66
− 42y
65
+ ··· − 8112y + 64
c
5
, c
6
, c
11
y
66
− 55y
65
+ ··· + 4y + 1
c
9
y
66
+ y
65
+ ··· − 52y + 1
c
10
, c
12
y
66
+ 37y
65
+ ··· − 4y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.990309 + 0.197371I
0.77149 − 3.51641I −4.00000 + 4.68932I
u = −0.990309 − 0.197371I
0.77149 + 3.51641I −4.00000 − 4.68932I
u = 0.983886 + 0.233964I
−4.05748 + 7.29975I 0. − 6.02392I
u = 0.983886 − 0.233964I
−4.05748 − 7.29975I 0. + 6.02392I
u = 0.917926
−1.63623 −6.25380
u = −0.855417 + 0.248728I
−8.04962 + 0.09735I −10.00686 + 0.77062I
u = −0.855417 − 0.248728I
−8.04962 − 0.09735I −10.00686 − 0.77062I
u = 0.738976 + 0.315928I
−3.71204 − 7.38704I −6.03668 + 3.97983I
u = 0.738976 − 0.315928I
−3.71204 + 7.38704I −6.03668 − 3.97983I
u = 1.20059
−1.54443 0
u = 0.262184 + 0.729860I
−5.40381 + 11.28260I −8.66884 − 8.74113I
u = 0.262184 − 0.729860I
−5.40381 − 11.28260I −8.66884 + 8.74113I
u = −0.710255 + 0.292534I
1.06112 + 3.49327I −1.18332 − 2.67462I
u = −0.710255 − 0.292534I
1.06112 − 3.49327I −1.18332 + 2.67462I
u = −0.262033 + 0.720142I
−0.59286 − 7.29825I −4.05645 + 7.44817I
u = −0.262033 − 0.720142I
−0.59286 + 7.29825I −4.05645 − 7.44817I
u = −0.224274 + 0.729292I
−10.09160 − 3.88890I −12.88295 + 4.18648I
u = −0.224274 − 0.729292I
−10.09160 + 3.88890I −12.88295 − 4.18648I
u = 0.256468 + 0.700542I
−3.25124 + 3.34292I −7.24196 − 3.75335I
u = 0.256468 − 0.700542I
−3.25124 − 3.34292I −7.24196 + 3.75335I
u = 0.176621 + 0.719075I
−6.50527 − 3.62888I −10.79295 + 1.37881I
u = 0.176621 − 0.719075I
−6.50527 + 3.62888I −10.79295 − 1.37881I
u = 0.224321 + 0.695502I
−3.61835 + 3.40694I −9.07650 − 5.66068I
u = 0.224321 − 0.695502I
−3.61835 − 3.40694I −9.07650 + 5.66068I
u = −0.354761 + 0.564052I
0.22062 − 5.60280I −3.25617 + 7.64930I
u = −0.354761 − 0.564052I
0.22062 + 5.60280I −3.25617 − 7.64930I
u = −1.317080 + 0.220326I
−0.01372 − 5.66365I 0
u = −1.317080 − 0.220326I
−0.01372 + 5.66365I 0
u = 0.610474 + 0.243113I
−1.66667 + 0.22451I −3.98223 − 1.07282I
u = 0.610474 − 0.243113I
−1.66667 − 0.22451I −3.98223 + 1.07282I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.368164 + 0.533247I
4.23434 + 1.69125I 1.84273 − 4.33152I
u = 0.368164 − 0.533247I
4.23434 − 1.69125I 1.84273 + 4.33152I
u = −1.354690 + 0.123144I
3.75022 − 0.64597I 0
u = −1.354690 − 0.123144I
3.75022 + 0.64597I 0
u = −0.392625 + 0.501531I
0.44010 + 2.19041I −2.25026 + 0.38117I
u = −0.392625 − 0.501531I
0.44010 − 2.19041I −2.25026 − 0.38117I
u = 1.355680 + 0.183514I
4.57952 + 3.25192I 0
u = 1.355680 − 0.183514I
4.57952 − 3.25192I 0
u = 0.102907 + 0.618996I
−4.43547 + 2.62457I −11.91060 − 4.28004I
u = 0.102907 − 0.618996I
−4.43547 − 2.62457I −11.91060 + 4.28004I
u = 1.367250 + 0.272605I
3.25888 + 3.51289I 0
u = 1.367250 − 0.272605I
3.25888 − 3.51289I 0
u = −1.408740 + 0.110369I
4.37917 − 1.51275I 0
u = −1.408740 − 0.110369I
4.37917 + 1.51275I 0
u = −1.38787 + 0.27581I
1.50869 − 6.93616I 0
u = −1.38787 − 0.27581I
1.50869 + 6.93616I 0
u = 1.41479 + 0.09089I
7.41801 − 2.39956I 0
u = 1.41479 − 0.09089I
7.41801 + 2.39956I 0
u = 1.38826 + 0.29191I
−4.97005 + 7.59047I 0
u = 1.38826 − 0.29191I
−4.97005 − 7.59047I 0
u = −1.42024 + 0.07916I
2.81245 + 6.36681I 0
u = −1.42024 − 0.07916I
2.81245 − 6.36681I 0
u = −1.40307 + 0.27814I
2.03846 − 6.90566I 0
u = −1.40307 − 0.27814I
2.03846 + 6.90566I 0
u = 1.42052 + 0.19384I
6.18270 + 0.37995I 0
u = 1.42052 − 0.19384I
6.18270 − 0.37995I 0
u = −1.42012 + 0.20531I
9.92026 − 4.41693I 0
u = −1.42012 − 0.20531I
9.92026 + 4.41693I 0
u = 1.40654 + 0.28588I
4.72530 + 10.95400I 0
u = 1.40654 − 0.28588I
4.72530 − 10.95400I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.42058 + 0.21544I
5.87848 + 8.46734I 0
u = 1.42058 − 0.21544I
5.87848 − 8.46734I 0
u = −1.40733 + 0.29033I
−0.0836 − 14.9873I 0
u = −1.40733 − 0.29033I
−0.0836 + 14.9873I 0
u = −0.148067 + 0.449066I
−0.202946 − 0.853104I −4.89270 + 7.93904I
u = −0.148067 − 0.449066I
−0.202946 + 0.853104I −4.89270 − 7.93904I
7
II. I
u
2
= hu
6
+ u
5
− 2u
4
− 2u
3
+ 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
u
a
3
=
1
u
2
a
8
=
−u
−u
3
+ u
a
2
=
−u
2
+ 1
−u
4
+ 2u
2
a
10
=
u
5
− 2u
3
+ u
1
a
5
=
u
5
− 2u
3
+ u + 1
1
a
1
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
7
=
−u
4
+ u
2
− u + 1
−u
4
− u
3
+ 2u
2
+ u
a
11
=
−u
4
+ u
2
+ u − 1
u
5
− 2u
3
− u
2
+ u + 1
a
12
=
u
3
− u
2
+ 1
−u
4
+ u
3
+ 2u
2
− u
a
6
=
−u
3
+ u
2
+ 1
−u
5
+ u
3
+ u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
6
− u
5
+ 2u
3
+ 2u
2
− 4u + 1
c
2
, c
3
, c
5
c
6
, c
8
, c
11
u
6
− u
5
− 2u
4
+ 2u
3
+ 1
c
4
, c
7
(u + 1)
6
c
9
, c
10
, c
12
u
6
+ u
4
− 2u
3
+ 2u
2
− 2u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
6
− y
5
+ 8y
4
− 10y
3
+ 20y
2
− 12y + 1
c
2
, c
3
, c
5
c
6
, c
8
, c
11
y
6
− 5y
5
+ 8y
4
− 2y
3
− 4y
2
+ 1
c
4
, c
7
(y −1)
6
c
9
, c
10
, c
12
y
6
+ 2y
5
+ 5y
4
− 2y
3
− 6y
2
− 8y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.733459
−1.64493 −6.00000
u = −0.181278 + 0.698849I
−1.64493 −6.00000
u = −0.181278 − 0.698849I
−1.64493 −6.00000
u = 1.35202
−1.64493 −6.00000
u = −1.361460 + 0.284643I
−1.64493 −6.00000
u = −1.361460 − 0.284643I
−1.64493 −6.00000
11
III. I
u
3
= hu − 1i
(i) Arc colorings
a
4
=
1
0
a
9
=
0
1
a
3
=
1
1
a
8
=
−1
0
a
2
=
0
1
a
10
=
0
1
a
5
=
1
1
a
1
=
1
1
a
7
=
0
1
a
11
=
0
1
a
12
=
1
1
a
6
=
1
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u − 1
c
2
, c
3
, c
4
c
5
, c
6
, c
7
c
8
, c
11
u + 1
c
9
, c
10
, c
12
u
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
4
, c
5
, c
6
c
7
, c
8
, c
11
y −1
c
9
, c
10
, c
12
y
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
−1.64493 −6.00000
15
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
6
− u
5
+ ··· − 4u + 1)(u
66
− 15u
65
+ ··· − 1396u + 113)
c
2
, c
3
, c
8
(u + 1)(u
6
− u
5
− 2u
4
+ 2u
3
+ 1)(u
66
− u
65
+ ··· + 2u
2
+ 1)
c
4
, c
7
((u + 1)
7
)(u
66
− 6u
65
+ ··· + 84u + 8)
c
5
, c
6
, c
11
(u + 1)(u
6
− u
5
− 2u
4
+ 2u
3
+ 1)(u
66
− u
65
+ ··· + 2u
2
+ 1)
c
9
u(u
6
+ u
4
+ ··· − 2u − 1)(u
66
+ 3u
65
+ ··· − 6u − 1)
c
10
, c
12
u(u
6
+ u
4
+ ··· − 2u − 1)(u
66
+ 3u
65
+ ··· − 2u − 1)
16
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y −1)(y
6
− y
5
+ 8y
4
− 10y
3
+ 20y
2
− 12y + 1)
· (y
66
+ 13y
65
+ ··· + 264176y + 12769)
c
2
, c
3
, c
8
(y −1)(y
6
− 5y
5
+ ··· − 4y
2
+ 1)(y
66
− 59y
65
+ ··· + 4y + 1)
c
4
, c
7
((y −1)
7
)(y
66
− 42y
65
+ ··· − 8112y + 64)
c
5
, c
6
, c
11
(y −1)(y
6
− 5y
5
+ ··· − 4y
2
+ 1)(y
66
− 55y
65
+ ··· + 4y + 1)
c
9
y(y
6
+ 2y
5
+ ··· − 8y + 1)(y
66
+ y
65
+ ··· − 52y + 1)
c
10
, c
12
y(y
6
+ 2y
5
+ ··· − 8y + 1)(y
66
+ 37y
65
+ ··· − 4y + 1)
17
