11a
58
(K11a
58
)
A knot diagram
1
Linearized knot diagam
4 1 8 2 11 10 9 3 7 5 6
Solving Sequence
5,10
11 6 7
1,2
3 4 9 8
c
10
c
5
c
6
c
11
c
2
c
4
c
9
c
8
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
25
− 9u
23
+ ··· + b + u, −u
28
− u
27
+ ··· + a − 3, u
29
+ 2u
28
+ ··· + 3u + 1i
I
u
2
= hu
2
+ b, a − 1, u
12
− 4u
10
+ u
9
+ 6u
8
− 3u
7
− u
6
+ 3u
5
− 5u
4
+ u
3
+ 3u
2
− 2u + 1i
I
u
3
= hb + 1, a − 1, u − 1i
* 3 irreducible components of dim
C
= 0, with total 42 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
25
−9u
23
+· · ·+b+u, −u
28
−u
27
+· · ·+a−3, u
29
+2u
28
+· · ·+3u+1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
u
−u
3
+ u
a
7
=
−u
3
+ 2u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
u
28
+ u
27
+ ··· − 5u + 3
−u
25
+ 9u
23
+ ··· + 4u
2
− u
a
3
=
−u
28
+ 11u
26
+ ··· − 8u + 1
u
28
+ u
27
+ ··· + u + 1
a
4
=
−u
27
+ 11u
25
+ ··· + 6u − 2
−u
28
− u
27
+ ··· − u − 1
a
9
=
u
6
− 3u
4
+ 2u
2
+ 1
u
6
− 2u
4
+ u
2
a
8
=
−u
9
+ 4u
7
− 5u
5
+ 3u
−u
9
+ 3u
7
− 3u
5
+ u
a
8
=
−u
9
+ 4u
7
− 5u
5
+ 3u
−u
9
+ 3u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −4u
28
− 6u
27
+ 40u
26
+ 58u
25
− 176u
24
− 236u
23
+ 428u
22
+ 482u
21
− 568u
20
−
370u
19
+ 236u
18
− 414u
17
+ 460u
16
+ 1092u
15
− 788u
14
− 532u
13
+ 356u
12
− 584u
11
+
236u
10
+ 608u
9
− 308u
8
+ 84u
7
+ 40u
6
− 178u
5
+ 60u
4
− 26u
3
+ 8u
2
+ 6u − 8
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
29
− 2u
28
+ ··· − u + 1
c
2
u
29
+ 16u
28
+ ··· + 7u + 1
c
3
, c
8
u
29
+ 2u
28
+ ··· + 2u + 2
c
5
, c
10
, c
11
u
29
+ 2u
28
+ ··· + 3u + 1
c
6
, c
7
, c
9
u
29
− 6u
28
+ ··· + 8u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
29
− 16y
28
+ ··· + 7y −1
c
2
y
29
− 4y
28
+ ··· − 17y −1
c
3
, c
8
y
29
+ 6y
28
+ ··· + 8y −4
c
5
, c
10
, c
11
y
29
− 24y
28
+ ··· + 23y −1
c
6
, c
7
, c
9
y
29
+ 30y
28
+ ··· + 504y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.050913 + 0.910185I
a = −0.45193 − 1.39576I
b = 0.00992 − 2.35228I
−10.46170 + 8.03356I −4.76249 − 5.59744I
u = 0.050913 − 0.910185I
a = −0.45193 + 1.39576I
b = 0.00992 + 2.35228I
−10.46170 − 8.03356I −4.76249 + 5.59744I
u = −0.008721 + 0.887960I
a = −0.49123 + 1.42056I
b = 0.14380 + 2.36020I
−10.71030 − 1.52343I −5.35413 + 0.68771I
u = −0.008721 − 0.887960I
a = −0.49123 − 1.42056I
b = 0.14380 − 2.36020I
−10.71030 + 1.52343I −5.35413 − 0.68771I
u = 1.189730 + 0.056062I
a = −0.370086 − 0.255115I
b = −1.23899 − 0.69502I
2.39907 + 0.12369I 3.50407 + 1.07759I
u = 1.189730 − 0.056062I
a = −0.370086 + 0.255115I
b = −1.23899 + 0.69502I
2.39907 − 0.12369I 3.50407 − 1.07759I
u = 1.242320 + 0.189774I
a = −0.715591 + 0.438399I
b = 0.20240 + 2.88734I
0.91595 + 3.56420I 0.67873 − 4.99863I
u = 1.242320 − 0.189774I
a = −0.715591 − 0.438399I
b = 0.20240 − 2.88734I
0.91595 − 3.56420I 0.67873 + 4.99863I
u = 0.230236 + 0.672244I
a = −0.11327 − 1.56979I
b = −0.07670 − 1.55700I
−1.85869 + 5.19499I −2.04173 − 8.30480I
u = 0.230236 − 0.672244I
a = −0.11327 + 1.56979I
b = −0.07670 + 1.55700I
−1.85869 − 5.19499I −2.04173 + 8.30480I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.249690 + 0.417811I
a = −0.215564 − 0.608911I
b = −0.927812 − 0.165212I
−3.02235 + 1.47420I 1.47993 − 0.60903I
u = 1.249690 − 0.417811I
a = −0.215564 + 0.608911I
b = −0.927812 + 0.165212I
−3.02235 − 1.47420I 1.47993 + 0.60903I
u = −1.311940 + 0.179476I
a = −0.339083 + 0.475947I
b = −0.644137 + 0.471467I
4.98921 − 3.78682I 7.27007 + 4.16727I
u = −1.311940 − 0.179476I
a = −0.339083 − 0.475947I
b = −0.644137 − 0.471467I
4.98921 + 3.78682I 7.27007 − 4.16727I
u = −1.342150 + 0.040293I
a = −0.537408 − 0.444706I
b = −0.23164 − 1.41782I
6.61715 − 2.27209I 8.89752 + 3.80982I
u = −1.342150 − 0.040293I
a = −0.537408 + 0.444706I
b = −0.23164 + 1.41782I
6.61715 + 2.27209I 8.89752 − 3.80982I
u = 1.286000 + 0.418935I
a = −0.762208 + 0.613998I
b = 1.69138 + 2.74611I
−6.68464 + 6.20004I −1.73580 − 3.81481I
u = 1.286000 − 0.418935I
a = −0.762208 − 0.613998I
b = 1.69138 − 2.74611I
−6.68464 − 6.20004I −1.73580 + 3.81481I
u = −1.333980 + 0.244603I
a = −0.683762 − 0.524847I
b = 0.83387 − 2.33932I
3.04589 − 8.42692I 3.52830 + 8.66921I
u = −1.333980 − 0.244603I
a = −0.683762 + 0.524847I
b = 0.83387 + 2.33932I
3.04589 + 8.42692I 3.52830 − 8.66921I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.301320 + 0.407588I
a = −0.248387 + 0.609615I
b = −0.872384 + 0.107277I
−2.63518 − 7.77071I 2.10858 + 5.30383I
u = −1.301320 − 0.407588I
a = −0.248387 − 0.609615I
b = −0.872384 − 0.107277I
−2.63518 + 7.77071I 2.10858 − 5.30383I
u = 0.529946 + 0.329108I
a = 0.667313 − 0.986592I
b = −0.347726 − 0.588274I
0.93542 + 1.41053I 5.39446 − 5.74020I
u = 0.529946 − 0.329108I
a = 0.667313 + 0.986592I
b = −0.347726 + 0.588274I
0.93542 − 1.41053I 5.39446 + 5.74020I
u = −1.320520 + 0.424615I
a = −0.743481 − 0.622059I
b = 1.72687 − 2.60904I
−6.1783 − 12.8069I −0.92308 + 8.12569I
u = −1.320520 − 0.424615I
a = −0.743481 + 0.622059I
b = 1.72687 + 2.60904I
−6.1783 + 12.8069I −0.92308 − 8.12569I
u = −0.063245 + 0.516212I
a = −0.28684 + 2.09363I
b = 0.49644 + 1.38676I
−3.02142 − 1.01433I −6.77496 + 0.83339I
u = −0.063245 − 0.516212I
a = −0.28684 − 2.09363I
b = 0.49644 − 1.38676I
−3.02142 + 1.01433I −6.77496 − 0.83339I
u = −0.193938
a = 4.58305
b = 0.469396
−1.29813 −8.53890
7
II. I
u
2
= hu
2
+ b, a − 1, u
12
− 4u
10
+ · · · − 2u + 1i
(i) Arc colorings
a
5
=
0
u
a
10
=
1
0
a
11
=
1
−u
2
a
6
=
u
−u
3
+ u
a
7
=
−u
3
+ 2u
−u
3
+ u
a
1
=
−u
2
+ 1
u
4
− 2u
2
a
2
=
1
−u
2
a
3
=
u
4
− u
2
+ 1
−u
6
+ 2u
4
− u
2
a
4
=
u
−u
3
+ u
a
9
=
u
6
− 3u
4
+ 2u
2
+ 1
u
6
− 2u
4
+ u
2
a
8
=
−u
9
+ 4u
7
− 5u
5
+ 3u
−u
9
+ 3u
7
− 3u
5
+ u
a
8
=
−u
9
+ 4u
7
− 5u
5
+ 3u
−u
9
+ 3u
7
− 3u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
9
− 12u
7
+ 4u
6
+ 12u
5
− 8u
4
+ 8u
3
+ 4u
2
− 12u + 6
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
5
c
10
, c
11
u
12
− 4u
10
+ u
9
+ 6u
8
− 3u
7
− u
6
+ 3u
5
− 5u
4
+ u
3
+ 3u
2
− 2u + 1
c
2
u
12
+ 8u
11
+ ··· − 2u + 1
c
3
, c
8
(u
4
− u
3
+ u
2
+ 1)
3
c
6
, c
7
, c
9
(u
4
− u
3
+ 3u
2
− 2u + 1)
3
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
5
c
10
, c
11
y
12
− 8y
11
+ ··· + 2y + 1
c
2
y
12
− 8y
11
+ ··· + 2y + 1
c
3
, c
8
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
3
c
6
, c
7
, c
9
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
3
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.944825 + 0.321917I
a = 1.00000
b = −0.789064 − 0.608311I
0.21101 − 1.41510I 1.82674 + 4.90874I
u = 0.944825 − 0.321917I
a = 1.00000
b = −0.789064 + 0.608311I
0.21101 + 1.41510I 1.82674 − 4.90874I
u = 0.031664 + 0.878090I
a = 1.00000
b = 0.770039 − 0.055609I
−6.79074 + 3.16396I −1.82674 − 2.56480I
u = 0.031664 − 0.878090I
a = 1.00000
b = 0.770039 + 0.055609I
−6.79074 − 3.16396I −1.82674 + 2.56480I
u = −1.186690 + 0.158407I
a = 1.00000
b = −1.38315 + 0.37596I
0.21101 − 1.41510I 1.82674 + 4.90874I
u = −1.186690 − 0.158407I
a = 1.00000
b = −1.38315 − 0.37596I
0.21101 + 1.41510I 1.82674 − 4.90874I
u = 1.240280 + 0.455646I
a = 1.00000
b = −1.33067 − 1.13025I
−6.79074 − 3.16396I −1.82674 + 2.56480I
u = 1.240280 − 0.455646I
a = 1.00000
b = −1.33067 + 1.13025I
−6.79074 + 3.16396I −1.82674 − 2.56480I
u = −1.271940 + 0.422443I
a = 1.00000
b = −1.43937 + 1.07464I
−6.79074 − 3.16396I −1.82674 + 2.56480I
u = −1.271940 − 0.422443I
a = 1.00000
b = −1.43937 − 1.07464I
−6.79074 + 3.16396I −1.82674 − 2.56480I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.241868 + 0.480324I
a = 1.00000
b = 0.172212 − 0.232350I
0.21101 + 1.41510I 1.82674 − 4.90874I
u = 0.241868 − 0.480324I
a = 1.00000
b = 0.172212 + 0.232350I
0.21101 − 1.41510I 1.82674 + 4.90874I
12
III. I
u
3
= hb + 1, a − 1, u − 1i
(i) Arc colorings
a
5
=
0
1
a
10
=
1
0
a
11
=
1
−1
a
6
=
1
0
a
7
=
1
0
a
1
=
0
−1
a
2
=
1
−1
a
3
=
1
0
a
4
=
1
0
a
9
=
1
0
a
8
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
10
, c
11
u − 1
c
2
, c
4
, c
5
u + 1
c
3
, c
6
, c
7
c
8
, c
9
u
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
5
, c
10
, c
11
y −1
c
3
, c
6
, c
7
c
8
, c
9
y
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 1.00000
b = −1.00000
0 0
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u − 1)(u
12
− 4u
10
+ ··· − 2u + 1)
· (u
29
− 2u
28
+ ··· − u + 1)
c
2
(u + 1)(u
12
+ 8u
11
+ ··· − 2u + 1)(u
29
+ 16u
28
+ ··· + 7u + 1)
c
3
, c
8
u(u
4
− u
3
+ u
2
+ 1)
3
(u
29
+ 2u
28
+ ··· + 2u + 2)
c
4
(u + 1)(u
12
− 4u
10
+ ··· − 2u + 1)
· (u
29
− 2u
28
+ ··· − u + 1)
c
5
(u + 1)(u
12
− 4u
10
+ ··· − 2u + 1)
· (u
29
+ 2u
28
+ ··· + 3u + 1)
c
6
, c
7
, c
9
u(u
4
− u
3
+ 3u
2
− 2u + 1)
3
(u
29
− 6u
28
+ ··· + 8u + 4)
c
10
, c
11
(u − 1)(u
12
− 4u
10
+ ··· − 2u + 1)
· (u
29
+ 2u
28
+ ··· + 3u + 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y −1)(y
12
− 8y
11
+ ··· + 2y + 1)(y
29
− 16y
28
+ ··· + 7y −1)
c
2
(y −1)(y
12
− 8y
11
+ ··· + 2y + 1)(y
29
− 4y
28
+ ··· − 17y −1)
c
3
, c
8
y(y
4
+ y
3
+ 3y
2
+ 2y + 1)
3
(y
29
+ 6y
28
+ ··· + 8y −4)
c
5
, c
10
, c
11
(y −1)(y
12
− 8y
11
+ ··· + 2y + 1)(y
29
− 24y
28
+ ··· + 23y −1)
c
6
, c
7
, c
9
y(y
4
+ 5y
3
+ ··· + 2y + 1)
3
(y
29
+ 30y
28
+ ··· + 504y −16)
18
