12a
1161
(K12a
1161
)
A knot diagram
1
Linearized knot diagam
4 9 8 10 12 11 1 3 2 7 6 5
Solving Sequence
2,9
3 10 8 4 5 1 7 11 6 12
c
2
c
9
c
8
c
3
c
4
c
1
c
7
c
10
c
6
c
12
c
5
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
32
− u
31
+ ··· + 3u
2
+ 1i
I
u
2
= hu
5
+ 3u
3
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 37 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
32
− u
31
+ · · · + 3u
2
+ 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
10
=
u
u
a
8
=
−u
u
3
+ u
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
5
=
u
6
+ 3u
4
+ 2u
2
+ 1
u
6
+ 2u
4
− u
2
a
1
=
u
6
+ 3u
4
+ 2u
2
+ 1
−u
8
− 4u
6
− 4u
4
a
7
=
u
11
+ 6u
9
+ 12u
7
+ 10u
5
+ 5u
3
−u
13
− 7u
11
− 17u
9
− 16u
7
− 4u
5
+ u
3
+ u
a
11
=
−u
25
− 14u
23
+ ··· + 5u
5
+ u
u
27
+ 15u
25
+ ··· + u
3
+ u
a
6
=
u
24
+ 13u
22
+ ··· − u
2
− u
−u
31
− 18u
29
+ ··· − 2u
2
− u
a
12
=
−u
20
− 11u
18
+ ··· + 3u
2
+ 1
−u
20
− 10u
18
− 38u
16
− 66u
14
− 47u
12
− 4u
10
+ 6u
8
+ 2u
6
− 5u
4
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
30
+4u
29
−76u
28
+68u
27
−632u
26
+500u
25
−3012u
24
+2072u
23
−9052u
22
+5284u
21
−
17812u
20
+ 8508u
19
−23164u
18
+ 8580u
17
−19788u
16
+ 5292u
15
−10868u
14
+ 1960u
13
−
3384u
12
+344u
11
−36u
10
−80u
9
+376u
8
−44u
7
+156u
6
−96u
5
+36u
4
−12u
3
−28u
2
+4u−10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
32
− 9u
31
+ ··· − 110u + 33
c
2
, c
3
, c
8
c
9
u
32
+ u
31
+ ··· + 3u
2
+ 1
c
4
, c
7
u
32
− 4u
31
+ ··· − 108u + 36
c
5
, c
6
, c
10
c
11
, c
12
u
32
+ u
31
+ ··· + 2u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
32
− 7y
31
+ ··· − 3982y + 1089
c
2
, c
3
, c
8
c
9
y
32
+ 37y
31
+ ··· + 6y + 1
c
4
, c
7
y
32
− 20y
31
+ ··· + 19080y + 1296
c
5
, c
6
, c
10
c
11
, c
12
y
32
+ 41y
31
+ ··· + 6y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.300810 + 0.836112I
6.82102 − 1.42764I −4.51155 − 0.98064I
u = 0.300810 − 0.836112I
6.82102 + 1.42764I −4.51155 + 0.98064I
u = 0.489031 + 0.734239I
8.08609 + 8.05482I −2.31310 − 6.73676I
u = 0.489031 − 0.734239I
8.08609 − 8.05482I −2.31310 + 6.73676I
u = −0.454853 + 0.736272I
−0.91691 − 6.31171I −4.12363 + 8.39972I
u = −0.454853 − 0.736272I
−0.91691 + 6.31171I −4.12363 − 8.39972I
u = 0.408575 + 0.750241I
−3.78730 + 3.13913I −10.07435 − 5.21729I
u = 0.408575 − 0.750241I
−3.78730 − 3.13913I −10.07435 + 5.21729I
u = −0.508887 + 0.453816I
12.96460 − 1.76928I 2.67390 + 3.90594I
u = −0.508887 − 0.453816I
12.96460 + 1.76928I 2.67390 − 3.90594I
u = 0.431538 + 0.445802I
3.53431 + 1.54706I 2.87219 − 5.01991I
u = 0.431538 − 0.445802I
3.53431 − 1.54706I 2.87219 + 5.01991I
u = 0.598654 + 0.128367I
9.86359 − 4.36101I 1.36134 + 2.03096I
u = 0.598654 − 0.128367I
9.86359 + 4.36101I 1.36134 − 2.03096I
u = −0.554088 + 0.092280I
0.95148 + 2.86543I 0.05561 − 3.87784I
u = −0.554088 − 0.092280I
0.95148 − 2.86543I 0.05561 + 3.87784I
u = −0.09682 + 1.49663I
6.60174 − 3.81122I 0. + 2.89590I
u = −0.09682 − 1.49663I
6.60174 + 3.81122I 0. − 2.89590I
u = −0.188661 + 0.441864I
−0.184695 − 0.792115I −5.07140 + 8.68136I
u = −0.188661 − 0.441864I
−0.184695 + 0.792115I −5.07140 − 8.68136I
u = 0.06894 + 1.52078I
−2.97865 + 3.12510I −4.00000 − 3.93405I
u = 0.06894 − 1.52078I
−2.97865 − 3.12510I −4.00000 + 3.93405I
u = −0.02317 + 1.55273I
−7.06597 − 1.36583I 0. + 4.57803I
u = −0.02317 − 1.55273I
−7.06597 + 1.36583I 0. − 4.57803I
u = −0.13063 + 1.61792I
−8.94709 − 8.50994I 0
u = −0.13063 − 1.61792I
−8.94709 + 8.50994I 0
u = 0.14214 + 1.61709I
0.08629 + 10.42610I 0
u = 0.14214 − 1.61709I
0.08629 − 10.42610I 0
u = 0.11670 + 1.62115I
−11.89960 + 5.11913I 0
u = 0.11670 − 1.62115I
−11.89960 − 5.11913I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.09928 + 1.62252I
−9.83905 − 1.68285I 0
u = −0.09928 − 1.62252I
−9.83905 + 1.68285I 0
6
II. I
u
2
= hu
5
+ 3u
3
+ u − 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
0
u
a
3
=
1
−u
2
a
10
=
u
u
a
8
=
−u
u
3
+ u
a
4
=
u
2
+ 1
−u
4
− 2u
2
a
5
=
u
2
+ u + 1
−u
4
− 2u
2
+ u
a
1
=
u
2
+ u + 1
−u
3
+ u
2
− u
a
7
=
u
2
+ 1
u
2
a
11
=
−u
3
−u
3
+ u
a
6
=
u
4
+ u
2
+ 1
u
4
a
12
=
−u
3
+ 1
−2u
3
− 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
5
− 2u
4
− u
3
+ 4u
2
+ 3u − 3
c
2
, c
3
, c
5
c
6
, c
8
, c
9
c
10
, c
11
, c
12
u
5
+ 3u
3
+ u + 1
c
4
, c
7
(u + 1)
5
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
5
− 6y
4
+ 23y
3
− 34y
2
+ 33y −9
c
2
, c
3
, c
5
c
6
, c
8
, c
9
c
10
, c
11
, c
12
y
5
+ 6y
4
+ 11y
3
+ 6y
2
+ y −1
c
4
, c
7
(y −1)
5
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.343105 + 0.770791I
−1.64493 −6.00000
u = −0.343105 − 0.770791I
−1.64493 −6.00000
u = 0.525261
−1.64493 −6.00000
u = 0.08047 + 1.63341I
−1.64493 −6.00000
u = 0.08047 − 1.63341I
−1.64493 −6.00000
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
5
− 2u
4
− u
3
+ 4u
2
+ 3u − 3)(u
32
− 9u
31
+ ··· − 110u + 33)
c
2
, c
3
, c
8
c
9
(u
5
+ 3u
3
+ u + 1)(u
32
+ u
31
+ ··· + 3u
2
+ 1)
c
4
, c
7
((u + 1)
5
)(u
32
− 4u
31
+ ··· − 108u + 36)
c
5
, c
6
, c
10
c
11
, c
12
(u
5
+ 3u
3
+ u + 1)(u
32
+ u
31
+ ··· + 2u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
5
− 6y
4
+ 23y
3
− 34y
2
+ 33y −9)(y
32
− 7y
31
+ ··· − 3982y + 1089)
c
2
, c
3
, c
8
c
9
(y
5
+ 6y
4
+ 11y
3
+ 6y
2
+ y −1)(y
32
+ 37y
31
+ ··· + 6y + 1)
c
4
, c
7
((y −1)
5
)(y
32
− 20y
31
+ ··· + 19080y + 1296)
c
5
, c
6
, c
10
c
11
, c
12
(y
5
+ 6y
4
+ 11y
3
+ 6y
2
+ y −1)(y
32
+ 41y
31
+ ··· + 6y + 1)
12
