11a
245
(K11a
245
)
A knot diagram
1
Linearized knot diagam
7 1 11 10 9 8 2 5 6 3 4
Solving Sequence
3,10
11 4 5 1
2,6
9 8 7
c
10
c
3
c
4
c
11
c
2
c
9
c
8
c
7
c
1
, c
5
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hb − u, −u
12
+ u
11
+ 5u
10
− 4u
9
− 9u
8
+ 4u
7
+ 4u
6
+ 4u
5
+ 6u
4
− 7u
3
− 5u
2
+ a − u − 1,
u
14
− u
13
− 6u
12
+ 5u
11
+ 14u
10
− 8u
9
− 13u
8
− 2u
6
+ 11u
5
+ 11u
4
− 5u
3
− 4u
2
− 4u − 1i
I
u
2
= h−u
23
+ 8u
21
+ ··· + b + 1, u
22
− 7u
20
+ ··· + a − 1, u
24
− u
23
+ ··· + 4u
2
+ 1i
I
u
3
= hb + 1, a, u − 1i
* 3 irreducible components of dim
C
= 0, with total 39 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
= hb − u, −u
12
+ u
11
+ · · · + a − 1, u
14
− u
13
+ · · · − 4u − 1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
4
=
−u
−u
3
+ u
a
5
=
u
3
− 2u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
6
=
u
12
− u
11
+ ··· + u + 1
u
a
9
=
−u
13
+ u
12
+ ··· − u + 1
−u
2
a
8
=
−u
13
+ u
12
+ ··· − u + 1
u
4
− 2u
2
a
7
=
u
12
− u
11
− 5u
10
+ 4u
9
+ 9u
8
− 5u
7
− 4u
6
− 6u
4
+ 3u
3
+ 5u
2
+ 1
u
7
− 3u
5
+ 2u
3
+ u
a
7
=
u
12
− u
11
− 5u
10
+ 4u
9
+ 9u
8
− 5u
7
− 4u
6
− 6u
4
+ 3u
3
+ 5u
2
+ 1
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −2u
12
−2u
11
+14u
10
+12u
9
−34u
8
−28u
7
+24u
6
+24u
5
+28u
4
+10u
3
−38u
2
−24u −20
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
14
+ 3u
13
+ ··· + 4u + 2
c
2
, c
4
, c
6
u
14
+ 3u
13
+ ··· + 20u + 4
c
3
, c
5
, c
8
c
9
, c
10
, c
11
u
14
− u
13
+ ··· − 4u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
14
− 3y
13
+ ··· − 20y + 4
c
2
, c
4
, c
6
y
14
+ 13y
13
+ ··· − 168y + 16
c
3
, c
5
, c
8
c
9
, c
10
, c
11
y
14
− 13y
13
+ ··· − 8y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.029285 + 0.881113I
a = 0.03342 − 1.87376I
b = −0.029285 + 0.881113I
8.40861 + 3.17852I −5.65702 − 2.68027I
u = −0.029285 − 0.881113I
a = 0.03342 + 1.87376I
b = −0.029285 − 0.881113I
8.40861 − 3.17852I −5.65702 + 2.68027I
u = −1.276220 + 0.129179I
a = −2.13229 − 1.54329I
b = −1.276220 + 0.129179I
−5.86531 + 2.46178I −16.4162 − 2.9434I
u = −1.276220 − 0.129179I
a = −2.13229 + 1.54329I
b = −1.276220 − 0.129179I
−5.86531 − 2.46178I −16.4162 + 2.9434I
u = −1.284590 + 0.394747I
a = −0.34217 − 2.13508I
b = −1.284590 + 0.394747I
0.58736 + 5.97274I −12.69846 − 3.76747I
u = −1.284590 − 0.394747I
a = −0.34217 + 2.13508I
b = −1.284590 − 0.394747I
0.58736 − 5.97274I −12.69846 + 3.76747I
u = 1.364060 + 0.212940I
a = 1.10215 − 1.44780I
b = 1.364060 + 0.212940I
−8.69313 − 7.21786I −18.5779 + 6.6599I
u = 1.364060 − 0.212940I
a = 1.10215 + 1.44780I
b = 1.364060 − 0.212940I
−8.69313 + 7.21786I −18.5779 − 6.6599I
u = 1.38564
a = 1.62659
b = 1.38564
−11.4128 −21.8330
u = 1.329060 + 0.410124I
a = 0.27755 − 1.96683I
b = 1.329060 + 0.410124I
−0.11168 − 12.47310I −13.5601 + 7.9056I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.329060 − 0.410124I
a = 0.27755 + 1.96683I
b = 1.329060 − 0.410124I
−0.11168 + 12.47310I −13.5601 − 7.9056I
u = −0.150725 + 0.518889I
a = 0.285959 − 1.368390I
b = −0.150725 + 0.518889I
1.00801 + 1.75508I −6.01712 − 6.20279I
u = −0.150725 − 0.518889I
a = 0.285959 + 1.368390I
b = −0.150725 − 0.518889I
1.00801 − 1.75508I −6.01712 + 6.20279I
u = −0.290248
a = 0.924145
b = −0.290248
−0.639037 −16.3130
6
II.
I
u
2
= h−u
23
+8u
21
+· · ·+b+1, u
22
−7u
20
+· · ·+a−1, u
24
−u
23
+· · ·+4u
2
+1i
(i) Arc colorings
a
3
=
0
u
a
10
=
1
0
a
11
=
1
u
2
a
4
=
−u
−u
3
+ u
a
5
=
u
3
− 2u
−u
3
+ u
a
1
=
−u
2
+ 1
−u
4
+ 2u
2
a
2
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
6
=
−u
22
+ 7u
20
+ ··· + 5u + 1
u
23
− 8u
21
+ ··· − u − 1
a
9
=
−u
21
+ 8u
19
+ ··· + 5u + 2
u
23
− 7u
21
+ ··· − u − 2
a
8
=
u
19
− 6u
17
+ ··· + 4u + 1
2u
23
− 15u
21
+ ··· − u − 2
a
7
=
−u
16
+ 5u
14
+ ··· + 4u + 1
2u
23
− 16u
21
+ ··· − u − 2
a
7
=
−u
16
+ 5u
14
+ ··· + 4u + 1
2u
23
− 16u
21
+ ··· − u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
20
+ 28u
18
+ 4u
17
−80u
16
−24u
15
+ 100u
14
+ 56u
13
−4u
12
−
48u
11
− 124u
10
− 24u
9
+ 92u
8
+ 64u
7
+ 36u
6
− 12u
5
− 44u
4
− 24u
3
− 8u
2
− 10
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
(u
12
− u
11
− u
10
+ 2u
9
+ 3u
8
− 4u
7
− 2u
6
+ 4u
5
+ 2u
4
− 3u
3
− u
2
+ 1)
2
c
2
, c
4
, c
6
(u
12
+ 3u
11
+ ··· + 2u + 1)
2
c
3
, c
5
, c
8
c
9
, c
10
, c
11
u
24
− u
23
+ ··· + 4u
2
+ 1
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
(y
12
− 3y
11
+ ··· − 2y + 1)
2
c
2
, c
4
, c
6
(y
12
+ 13y
11
+ ··· + 6y + 1)
2
c
3
, c
5
, c
8
c
9
, c
10
, c
11
y
24
− 17y
23
+ ··· + 8y + 1
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.070751 + 0.894321I
a = −1.21139 + 1.52083I
b = 1.299300 − 0.409615I
4.26829 + 7.80134I −9.63389 − 5.63981I
u = −0.070751 − 0.894321I
a = −1.21139 − 1.52083I
b = 1.299300 + 0.409615I
4.26829 − 7.80134I −9.63389 + 5.63981I
u = −1.110590 + 0.134720I
a = 0.738153 + 0.451331I
b = 0.149210 − 0.343690I
−1.55013 + 0.71593I −8.04353 − 0.64874I
u = −1.110590 − 0.134720I
a = 0.738153 − 0.451331I
b = 0.149210 + 0.343690I
−1.55013 − 0.71593I −8.04353 + 0.64874I
u = −0.778878 + 0.387180I
a = −0.213014 + 0.440226I
b = 1.242510 + 0.071539I
−4.72717 − 0.35310I −18.6669 + 0.6298I
u = −0.778878 − 0.387180I
a = −0.213014 − 0.440226I
b = 1.242510 − 0.071539I
−4.72717 + 0.35310I −18.6669 − 0.6298I
u = 0.013292 + 0.856991I
a = 1.23384 + 1.58823I
b = −1.251930 − 0.421635I
4.62532 − 1.48234I −8.84742 + 0.67542I
u = 0.013292 − 0.856991I
a = 1.23384 − 1.58823I
b = −1.251930 + 0.421635I
4.62532 + 1.48234I −8.84742 − 0.67542I
u = 1.242510 + 0.071539I
a = −0.023283 − 0.340995I
b = −0.778878 + 0.387180I
−4.72717 − 0.35310I −18.6669 + 0.6298I
u = 1.242510 − 0.071539I
a = −0.023283 + 0.340995I
b = −0.778878 − 0.387180I
−4.72717 + 0.35310I −18.6669 − 0.6298I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.321894 + 0.643464I
a = −0.65810 + 1.42592I
b = 1.279920 − 0.182904I
−3.36661 + 4.24921I −14.1765 − 6.9831I
u = −0.321894 − 0.643464I
a = −0.65810 − 1.42592I
b = 1.279920 + 0.182904I
−3.36661 − 4.24921I −14.1765 + 6.9831I
u = 1.279920 + 0.182904I
a = −0.443771 + 0.752880I
b = −0.321894 − 0.643464I
−3.36661 − 4.24921I −14.1765 + 6.9831I
u = 1.279920 − 0.182904I
a = −0.443771 − 0.752880I
b = −0.321894 + 0.643464I
−3.36661 + 4.24921I −14.1765 − 6.9831I
u = −1.213270 + 0.447486I
a = −0.537563 − 0.128960I
b = 1.263090 + 0.396551I
0.75031 − 3.01307I −12.63175 + 2.63251I
u = −1.213270 − 0.447486I
a = −0.537563 + 0.128960I
b = 1.263090 − 0.396551I
0.75031 + 3.01307I −12.63175 − 2.63251I
u = −1.251930 + 0.421635I
a = 0.704102 + 1.098600I
b = 0.013292 − 0.856991I
4.62532 + 1.48234I −8.84742 − 0.67542I
u = −1.251930 − 0.421635I
a = 0.704102 − 1.098600I
b = 0.013292 + 0.856991I
4.62532 − 1.48234I −8.84742 + 0.67542I
u = 1.263090 + 0.396551I
a = 0.492596 − 0.221226I
b = −1.213270 + 0.447486I
0.75031 − 3.01307I −12.63175 + 2.63251I
u = 1.263090 − 0.396551I
a = 0.492596 + 0.221226I
b = −1.213270 − 0.447486I
0.75031 + 3.01307I −12.63175 − 2.63251I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.299300 + 0.409615I
a = −0.629315 + 1.115020I
b = −0.070751 − 0.894321I
4.26829 − 7.80134I −9.63389 + 5.63981I
u = 1.299300 − 0.409615I
a = −0.629315 − 1.115020I
b = −0.070751 + 0.894321I
4.26829 + 7.80134I −9.63389 − 5.63981I
u = 0.149210 + 0.343690I
a = 0.04774 + 2.58289I
b = −1.110590 − 0.134720I
−1.55013 − 0.71593I −8.04353 + 0.64874I
u = 0.149210 − 0.343690I
a = 0.04774 − 2.58289I
b = −1.110590 + 0.134720I
−1.55013 + 0.71593I −8.04353 − 0.64874I
12
III. I
u
3
= hb + 1, a, u − 1i
(i) Arc colorings
a
3
=
0
1
a
10
=
1
0
a
11
=
1
1
a
4
=
−1
0
a
5
=
−1
0
a
1
=
0
1
a
2
=
0
1
a
6
=
0
−1
a
9
=
1
−1
a
8
=
0
−1
a
7
=
0
−1
a
7
=
0
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
u
c
3
, c
8
, c
9
u + 1
c
5
, c
10
, c
11
u − 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
c
6
, c
7
y
c
3
, c
5
, c
8
c
9
, c
10
, c
11
y −1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0
b = −1.00000
−3.28987 −12.0000
16
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
7
u(u
12
− u
11
− u
10
+ 2u
9
+ 3u
8
− 4u
7
− 2u
6
+ 4u
5
+ 2u
4
− 3u
3
− u
2
+ 1)
2
· (u
14
+ 3u
13
+ ··· + 4u + 2)
c
2
, c
4
, c
6
u(u
12
+ 3u
11
+ ··· + 2u + 1)
2
(u
14
+ 3u
13
+ ··· + 20u + 4)
c
3
, c
8
, c
9
(u + 1)(u
14
− u
13
+ ··· − 4u − 1)(u
24
− u
23
+ ··· + 4u
2
+ 1)
c
5
, c
10
, c
11
(u − 1)(u
14
− u
13
+ ··· − 4u − 1)(u
24
− u
23
+ ··· + 4u
2
+ 1)
17
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
y(y
12
− 3y
11
+ ··· − 2y + 1)
2
(y
14
− 3y
13
+ ··· − 20y + 4)
c
2
, c
4
, c
6
y(y
12
+ 13y
11
+ ··· + 6y + 1)
2
(y
14
+ 13y
13
+ ··· − 168y + 16)
c
3
, c
5
, c
8
c
9
, c
10
, c
11
(y −1)(y
14
− 13y
13
+ ··· − 8y + 1)(y
24
− 17y
23
+ ··· + 8y + 1)
18
