11a
186
(K11a
186
)
A knot diagram
1
Linearized knot diagam
6 1 8 10 7 2 3 4 11 5 9
Solving Sequence
4,10
5 11 9 1 8 3 2 7 6
c
4
c
10
c
9
c
11
c
8
c
3
c
2
c
7
c
6
c
1
, c
5
Ideals for irreducible components
2
of X
par
I
u
1
= hu
11
− 2u
9
+ 4u
7
− u
6
− 4u
5
+ u
4
+ 3u
3
− 2u
2
− 2u + 1i
I
u
2
= hu
36
+ u
35
+ ··· + u
3
+ 1i
* 2 irreducible components of dim
C
= 0, with total 47 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
11
− 2u
9
+ 4u
7
− u
6
− 4u
5
+ u
4
+ 3u
3
− 2u
2
− 2u + 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
11
=
−u
−u
3
+ u
a
9
=
u
3
u
5
− u
3
+ u
a
1
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
8
=
u
5
+ u
u
5
− u
3
+ u
a
3
=
−u
10
+ u
8
− 2u
6
+ u
4
− u
2
+ 1
−u
10
+ 2u
8
− 3u
6
+ 2u
4
− u
2
a
2
=
−u
10
+ u
8
+ u
7
− 2u
6
+ u
4
+ u
3
− u
2
+ 1
−u
10
+ u
9
+ 2u
8
− u
7
− 3u
6
+ 2u
5
+ 2u
4
− u
3
− u
2
a
7
=
−u
10
+ u
8
− u
7
− 2u
6
+ 2u
5
+ u
4
− 2u
3
+ 2u
−u
10
− u
9
+ 2u
8
+ u
7
− 3u
6
− u
5
+ 3u
4
− u
2
+ u
a
6
=
−u
10
− u
9
+ 2u
8
+ u
7
− 3u
6
− u
5
+ 3u
4
− 2u
2
+ u + 1
−u
9
+ u
8
+ 2u
7
− 2u
6
− 3u
5
+ 3u
4
+ 2u
3
− 3u
2
− u + 1
a
6
=
−u
10
− u
9
+ 2u
8
+ u
7
− 3u
6
− u
5
+ 3u
4
− 2u
2
+ u + 1
−u
9
+ u
8
+ 2u
7
− 2u
6
− 3u
5
+ 3u
4
+ 2u
3
− 3u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
9
− 4u
8
− 8u
7
+ 4u
6
+ 12u
5
− 12u
4
− 8u
3
+ 8u
2
+ 4u − 18
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
10
u
11
− 2u
9
+ 4u
7
+ u
6
− 4u
5
− u
4
+ 3u
3
+ 2u
2
− 2u − 1
c
2
, c
5
, c
9
c
11
u
11
+ 4u
10
+ ··· + 8u + 1
c
3
, c
7
, c
8
u
11
+ 5u
10
+ 8u
9
+ 5u
8
+ 9u
7
+ 19u
6
+ 8u
5
− 2u
4
+ 9u
3
+ u
2
− 12u − 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
y
11
− 4y
10
+ ··· + 8y −1
c
2
, c
5
, c
9
c
11
y
11
+ 8y
10
+ ··· + 28y −1
c
3
, c
7
, c
8
y
11
− 9y
10
+ ··· + 152y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.574057 + 0.778762I
−0.32700 + 2.62828I −9.00950 − 0.39606I
u = 0.574057 − 0.778762I
−0.32700 − 2.62828I −9.00950 + 0.39606I
u = −0.786275 + 0.725485I
5.13423 + 2.26440I −5.35075 − 2.78673I
u = −0.786275 − 0.725485I
5.13423 − 2.26440I −5.35075 + 2.78673I
u = −0.903688
−4.12325 −21.6840
u = 1.13447
−11.8669 −21.5190
u = 0.937682 + 0.702007I
4.20048 − 8.65870I −8.03545 + 9.01618I
u = 0.937682 − 0.702007I
4.20048 + 8.65870I −8.03545 − 9.01618I
u = −1.053250 + 0.672906I
−3.16344 + 13.64350I −13.1560 − 9.4873I
u = −1.053250 − 0.672906I
−3.16344 − 13.64350I −13.1560 + 9.4873I
u = 0.424792
−0.633212 −15.6940
5
II. I
u
2
= hu
36
+ u
35
+ · · · + u
3
+ 1i
(i) Arc colorings
a
4
=
1
0
a
10
=
0
u
a
5
=
1
u
2
a
11
=
−u
−u
3
+ u
a
9
=
u
3
u
5
− u
3
+ u
a
1
=
−u
5
− u
−u
7
+ u
5
− 2u
3
+ u
a
8
=
u
5
+ u
u
5
− u
3
+ u
a
3
=
−u
10
+ u
8
− 2u
6
+ u
4
− u
2
+ 1
−u
10
+ 2u
8
− 3u
6
+ 2u
4
− u
2
a
2
=
u
22
− 3u
20
+ ··· − 3u
4
+ 1
u
24
− 4u
22
+ ··· + 8u
4
− 2u
2
a
7
=
−u
15
+ 2u
13
− 4u
11
+ 4u
9
− 4u
7
+ 4u
5
− 2u
3
+ 2u
−u
15
+ 3u
13
− 6u
11
+ 7u
9
− 6u
7
+ 4u
5
− 2u
3
+ u
a
6
=
−u
32
+ 5u
30
+ ··· + 2u
2
+ 1
−u
32
+ 6u
30
+ ··· − 6u
4
+ 2u
2
a
6
=
−u
32
+ 5u
30
+ ··· + 2u
2
+ 1
−u
32
+ 6u
30
+ ··· − 6u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
32
− 24u
30
+ 88u
28
− 224u
26
+ 440u
24
− 700u
22
+ 928u
20
−
1060u
18
+ 4u
17
+ 1048u
16
−16u
15
−912u
14
+ 36u
13
+ 692u
12
−52u
11
−452u
10
+ 52u
9
+
256u
8
− 44u
7
− 116u
6
+ 32u
5
+ 44u
4
− 20u
3
− 8u
2
+ 8u − 14
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
10
u
36
− u
35
+ ··· − u
3
+ 1
c
2
, c
5
, c
9
c
11
u
36
+ 13u
35
+ ··· − 10u
2
+ 1
c
3
, c
7
, c
8
(u
18
− 2u
17
+ ··· + 2u + 1)
2
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
y
36
− 13y
35
+ ··· − 10y
2
+ 1
c
2
, c
5
, c
9
c
11
y
36
+ 19y
35
+ ··· − 20y + 1
c
3
, c
7
, c
8
(y
18
− 18y
17
+ ··· − 10y + 1)
2
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.568398 + 0.797612I
−1.72161 − 8.10595I −11.08535 + 5.00657I
u = −0.568398 − 0.797612I
−1.72161 + 8.10595I −11.08535 − 5.00657I
u = 0.759891 + 0.733182I
4.73704 + 3.18642I −6.45994 − 3.31717I
u = 0.759891 − 0.733182I
4.73704 − 3.18642I −6.45994 + 3.31717I
u = −0.527375 + 0.775874I
−6.14948 − 1.48503I −15.5689 + 0.3788I
u = −0.527375 − 0.775874I
−6.14948 + 1.48503I −15.5689 − 0.3788I
u = −0.853258 + 0.641261I
1.83259 + 2.50180I −6.41929 − 3.81694I
u = −0.853258 − 0.641261I
1.83259 − 2.50180I −6.41929 + 3.81694I
u = −0.898798 + 0.229050I
−0.88834 + 4.72205I −15.5195 − 7.2621I
u = −0.898798 − 0.229050I
−0.88834 − 4.72205I −15.5195 + 7.2621I
u = 0.720307 + 0.524101I
−0.218096 + 0.036628I −13.43748 − 0.95651I
u = 0.720307 − 0.524101I
−0.218096 − 0.036628I −13.43748 + 0.95651I
u = −1.115130 + 0.024468I
−6.14948 + 1.48503I −15.5689 − 0.3788I
u = −1.115130 − 0.024468I
−6.14948 − 1.48503I −15.5689 + 0.3788I
u = 0.936753 + 0.611605I
−0.88834 − 4.72205I −15.5195 + 7.2621I
u = 0.936753 − 0.611605I
−0.88834 + 4.72205I −15.5195 − 7.2621I
u = −0.475172 + 0.740129I
−2.30993 + 5.17624I −11.82231 − 5.02355I
u = −0.475172 − 0.740129I
−2.30993 − 5.17624I −11.82231 + 5.02355I
u = 0.510565 + 0.712216I
−0.822851 −9.60076 + 0.I
u = 0.510565 − 0.712216I
−0.822851 −9.60076 + 0.I
u = 1.129810 + 0.032613I
−7.69896 − 6.87816I −17.6593 + 5.1131I
u = 1.129810 − 0.032613I
−7.69896 + 6.87816I −17.6593 − 5.1131I
u = −0.917289 + 0.702643I
4.73704 + 3.18642I −6.45994 − 3.31717I
u = −0.917289 − 0.702643I
4.73704 − 3.18642I −6.45994 + 3.31717I
u = 0.772239 + 0.333861I
−0.218096 − 0.036628I −13.43748 + 0.95651I
u = 0.772239 − 0.333861I
−0.218096 + 0.036628I −13.43748 − 0.95651I
u = 1.038670 + 0.636561I
−2.30993 − 5.17624I −11.82231 + 5.02355I
u = 1.038670 − 0.636561I
−2.30993 + 5.17624I −11.82231 − 5.02355I
u = −1.051520 + 0.626704I
−3.95239 −14.4550 + 0.I
u = −1.051520 − 0.626704I
−3.95239 −14.4550 + 0.I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.045370 + 0.669009I
−1.72161 − 8.10595I −11.08535 + 5.00657I
u = 1.045370 − 0.669009I
−1.72161 + 8.10595I −11.08535 − 5.00657I
u = −1.056180 + 0.652350I
−7.69896 + 6.87816I −17.6593 − 5.1131I
u = −1.056180 − 0.652350I
−7.69896 − 6.87816I −17.6593 + 5.1131I
u = 0.049508 + 0.478803I
1.83259 − 2.50180I −6.41929 + 3.81694I
u = 0.049508 − 0.478803I
1.83259 + 2.50180I −6.41929 − 3.81694I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
10
(u
11
− 2u
9
+ 4u
7
+ u
6
− 4u
5
− u
4
+ 3u
3
+ 2u
2
− 2u − 1)
· (u
36
− u
35
+ ··· − u
3
+ 1)
c
2
, c
5
, c
9
c
11
(u
11
+ 4u
10
+ ··· + 8u + 1)(u
36
+ 13u
35
+ ··· − 10u
2
+ 1)
c
3
, c
7
, c
8
(u
11
+ 5u
10
+ 8u
9
+ 5u
8
+ 9u
7
+ 19u
6
+ 8u
5
− 2u
4
+ 9u
3
+ u
2
− 12u − 4)
· (u
18
− 2u
17
+ ··· + 2u + 1)
2
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
10
(y
11
− 4y
10
+ ··· + 8y −1)(y
36
− 13y
35
+ ··· − 10y
2
+ 1)
c
2
, c
5
, c
9
c
11
(y
11
+ 8y
10
+ ··· + 28y −1)(y
36
+ 19y
35
+ ··· − 20y + 1)
c
3
, c
7
, c
8
(y
11
− 9y
10
+ ··· + 152y −16)(y
18
− 18y
17
+ ··· − 10y + 1)
2
12
