11n
85
(K11n
85
)
A knot diagram
1
Linearized knot diagam
6 1 8 11 2 3 9 5 3 8 9
Solving Sequence
2,5
6 1
3,9
10 8 7 11 4
c
5
c
1
c
2
c
9
c
8
c
7
c
11
c
4
c
3
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−2u
22
− 2u
21
+ ··· + 4b − 2, 2u
22
+ u
21
+ ··· + 4a − 2, u
23
+ 2u
22
+ ··· + 4u + 2i
I
u
2
= hb + 1, u
3
+ 2u
2
+ 2a + 4, u
4
+ 2u
2
+ 2i
I
u
3
= h−a
2
u + 2b + a − 2, a
3
+ 2a
2
u − au + 2a + 2, u
2
− u + 1i
I
v
1
= ha, b − 1, v −1i
* 4 irreducible components of dim
C
= 0, with total 34 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
= h−2u
22
−2u
21
+· · ·+4b−2, 2u
22
+u
21
+· · ·+4a−2, u
23
+2u
22
+· · ·+4u+2i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
1
=
−u
u
3
+ u
a
3
=
−u
3
u
5
+ u
3
+ u
a
9
=
−
1
2
u
22
−
1
4
u
21
+ ··· −
1
2
u +
1
2
1
2
u
22
+
1
2
u
21
+ ··· +
1
2
u +
1
2
a
10
=
−
1
4
u
18
− u
16
+ ··· +
1
2
u +
1
2
−
1
2
u
13
−
3
2
u
11
+ ··· − u
3
− u
a
8
=
1
4
u
21
+ u
19
+ ··· −
1
2
u
3
+ 1
1
2
u
22
+
1
2
u
21
+ ··· +
1
2
u +
1
2
a
7
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
11
=
1
2
u
22
+
3
4
u
21
+ ··· −
1
2
u −
1
2
1
2
u
22
+
1
2
u
21
+ ··· +
3
2
u +
1
2
a
4
=
−
1
4
u
21
− u
19
+ ··· −
3
2
u
3
− u
−
1
4
u
21
−
5
4
u
19
+ ··· +
1
2
u
2
+
1
2
u
a
4
=
−
1
4
u
21
− u
19
+ ··· −
3
2
u
3
− u
−
1
4
u
21
−
5
4
u
19
+ ··· +
1
2
u
2
+
1
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
22
+ 4u
21
+ 10u
20
+ 16u
19
+ 24u
18
+ 34u
17
+ 30u
16
+ 40u
15
+
16u
14
+ 30u
13
+ 32u
11
+ 8u
10
+ 46u
9
+ 14u
8
+ 42u
7
−8u
6
+ 2u
5
−24u
4
+ 2u
3
+ 4u
2
+ 10u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
23
+ 2u
22
+ ··· + 4u + 2
c
2
u
23
+ 10u
22
+ ··· + 8u − 4
c
3
u
23
+ 27u
21
+ ··· − 7u + 1
c
4
, c
9
u
23
− 2u
22
+ ··· + 9u + 1
c
6
u
23
− 2u
22
+ ··· − 88u + 16
c
7
u
23
+ 2u
22
+ ··· − 11u + 1
c
8
u
23
+ 2u
22
+ ··· − 3u + 1
c
10
u
23
− 5u
22
+ ··· + 128u + 1706
c
11
u
23
+ 8u
22
+ ··· + 1035u + 297
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
23
+ 10y
22
+ ··· + 8y −4
c
2
y
23
+ 6y
22
+ ··· + 160y −16
c
3
y
23
+ 54y
22
+ ··· + 25y −1
c
4
, c
9
y
23
− 34y
22
+ ··· − 27y −1
c
6
y
23
+ 2y
22
+ ··· + 960y −256
c
7
y
23
+ 46y
22
+ ··· − 135y −1
c
8
y
23
− 2y
22
+ ··· − 11y −1
c
10
y
23
+ 41y
22
+ ··· − 12287288y −2910436
c
11
y
23
− 26y
22
+ ··· + 935793y −88209
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.887093 + 0.448454I
a = 0.492271 + 1.285580I
b = −1.14839 − 1.01565I
11.55830 − 6.04378I 2.90457 + 2.40956I
u = 0.887093 − 0.448454I
a = 0.492271 − 1.285580I
b = −1.14839 + 1.01565I
11.55830 + 6.04378I 2.90457 − 2.40956I
u = −0.865908 + 0.562605I
a = 0.29789 + 1.55245I
b = −0.94225 − 1.16788I
12.26940 − 1.86843I 3.51927 + 2.09858I
u = −0.865908 − 0.562605I
a = 0.29789 − 1.55245I
b = −0.94225 + 1.16788I
12.26940 + 1.86843I 3.51927 − 2.09858I
u = −0.126252 + 0.927958I
a = 0.472434 + 0.373694I
b = 0.816023 − 0.401741I
−1.83455 + 1.28121I −6.39377 − 3.70883I
u = −0.126252 − 0.927958I
a = 0.472434 − 0.373694I
b = 0.816023 + 0.401741I
−1.83455 − 1.28121I −6.39377 + 3.70883I
u = −0.687410 + 0.551797I
a = 0.93433 − 1.33726I
b = −0.566101 + 0.784858I
2.63493 + 2.12803I 3.22069 − 2.55962I
u = −0.687410 − 0.551797I
a = 0.93433 + 1.33726I
b = −0.566101 − 0.784858I
2.63493 − 2.12803I 3.22069 + 2.55962I
u = −0.439313 + 1.087580I
a = −1.12704 + 1.03997I
b = −1.028740 − 0.075248I
−4.16811 − 3.61856I −9.97032 + 4.29272I
u = −0.439313 − 1.087580I
a = −1.12704 − 1.03997I
b = −1.028740 + 0.075248I
−4.16811 + 3.61856I −9.97032 − 4.29272I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.611535 + 1.029680I
a = 0.57884 − 1.78933I
b = 0.744103 + 0.840632I
1.22653 − 7.16348I −0.15345 + 7.54828I
u = −0.611535 − 1.029680I
a = 0.57884 + 1.78933I
b = 0.744103 − 0.840632I
1.22653 + 7.16348I −0.15345 − 7.54828I
u = 0.470162 + 1.125140I
a = −1.139790 − 0.221923I
b = −0.174623 + 0.267387I
−0.75408 + 3.78076I 1.64329 − 3.83078I
u = 0.470162 − 1.125140I
a = −1.139790 + 0.221923I
b = −0.174623 − 0.267387I
−0.75408 − 3.78076I 1.64329 + 3.83078I
u = 0.066964 + 1.228960I
a = 0.900031 + 0.128011I
b = 0.988654 + 0.944921I
5.54974 − 3.50228I −1.93120 + 2.15966I
u = 0.066964 − 1.228960I
a = 0.900031 − 0.128011I
b = 0.988654 − 0.944921I
5.54974 + 3.50228I −1.93120 − 2.15966I
u = −0.694097 + 1.072020I
a = −1.125380 + 0.108515I
b = 0.83601 − 1.20931I
10.72870 − 3.91001I 1.79235 + 2.50229I
u = −0.694097 − 1.072020I
a = −1.125380 − 0.108515I
b = 0.83601 + 1.20931I
10.72870 + 3.91001I 1.79235 − 2.50229I
u = 0.652491 + 1.132530I
a = 1.13140 + 1.85064I
b = 1.20493 − 0.95597I
9.4833 + 11.7267I 0.34491 − 6.55767I
u = 0.652491 − 1.132530I
a = 1.13140 − 1.85064I
b = 1.20493 + 0.95597I
9.4833 − 11.7267I 0.34491 + 6.55767I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.601530 + 0.285314I
a = 1.041030 − 0.706848I
b = −0.179556 + 0.418301I
1.74084 + 0.44680I 5.28361 − 1.38333I
u = 0.601530 − 0.285314I
a = 1.041030 + 0.706848I
b = −0.179556 − 0.418301I
1.74084 − 0.44680I 5.28361 + 1.38333I
u = −0.507450
a = 0.0879637
b = 0.899884
−1.46388 −6.51990
7
II. I
u
2
= hb + 1, u
3
+ 2u
2
+ 2a + 4, u
4
+ 2u
2
+ 2i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u
2
a
1
=
−u
u
3
+ u
a
3
=
−u
3
−u
3
− u
a
9
=
−
1
2
u
3
− u
2
− 2
−1
a
10
=
1
2
u
3
− u
2
− 2
u
3
+ u − 1
a
8
=
−
1
2
u
3
− u
2
− 3
−1
a
7
=
−1
0
a
11
=
−
3
2
u
3
− u
2
− u − 3
−1
a
4
=
3
2
u
3
+ u
2
+ u + 4
1
a
4
=
3
2
u
3
+ u
2
+ u + 4
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
− 8
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
4
+ 2u
2
+ 2
c
2
(u
2
+ 2u + 2)
2
c
3
u
4
+ 4u
3
+ 4u
2
+ 1
c
4
(u + 1)
4
c
6
, c
10
u
4
− 2u
2
+ 2
c
7
, c
8
, c
9
(u − 1)
4
c
11
u
4
− 4u
3
+ 4u
2
+ 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
2
+ 2y + 2)
2
c
2
(y
2
+ 4)
2
c
3
, c
11
y
4
− 8y
3
+ 18y
2
+ 8y + 1
c
4
, c
7
, c
8
c
9
(y −1)
4
c
6
, c
10
(y
2
− 2y + 2)
2
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.455090 + 1.098680I
a = −0.223113 − 0.678203I
b = −1.00000
−2.46740 + 3.66386I −4.00000 − 4.00000I
u = 0.455090 − 1.098680I
a = −0.223113 + 0.678203I
b = −1.00000
−2.46740 − 3.66386I −4.00000 + 4.00000I
u = −0.455090 + 1.098680I
a = −1.77689 + 1.32180I
b = −1.00000
−2.46740 − 3.66386I −4.00000 + 4.00000I
u = −0.455090 − 1.098680I
a = −1.77689 − 1.32180I
b = −1.00000
−2.46740 + 3.66386I −4.00000 − 4.00000I
11
III. I
u
3
= h−a
2
u + 2b + a − 2, a
3
+ 2a
2
u − au + 2a + 2, u
2
− u + 1i
(i) Arc colorings
a
2
=
0
u
a
5
=
1
0
a
6
=
1
−u + 1
a
1
=
−u
u − 1
a
3
=
1
0
a
9
=
a
1
2
a
2
u −
1
2
a + 1
a
10
=
1
2
a
2
u +
1
2
a + 1
1
2
a
2
u −
1
2
a + 1
a
8
=
1
2
a
2
u +
1
2
a + 1
1
2
a
2
u −
1
2
a + 1
a
7
=
u
−u + 1
a
11
=
−
1
2
a
2
u −
1
2
a − 1
−
1
2
a
2
u +
1
2
a − 1
a
4
=
−
1
2
a
2
u −
1
2
a + 1
1
2
a
2
u −
1
2
a
2
+
1
2
au − a + u
a
4
=
−
1
2
a
2
u −
1
2
a + 1
1
2
a
2
u −
1
2
a
2
+
1
2
au − a + u
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u + 2
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
2
− u + 1)
3
c
2
, c
6
(u
2
+ u + 1)
3
c
3
, c
4
, c
8
c
9
u
6
− 2u
4
− u
3
+ u
2
+ u + 1
c
7
u
6
+ 4u
5
+ 6u
4
+ 3u
3
− u
2
− u + 1
c
10
u
6
c
11
u
6
− 4u
5
+ 6u
4
− 3u
3
− u
2
+ u + 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
(y
2
+ y + 1)
3
c
3
, c
4
, c
8
c
9
y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1
c
7
, c
11
y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1
c
10
y
6
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.412728 + 1.011420I
b = 0.218964 − 0.666188I
2.02988I 0. − 3.46410I
u = 0.500000 + 0.866025I
a = −0.562490 − 0.528127I
b = 1.033350 + 0.428825I
2.02988I 0. − 3.46410I
u = 0.500000 + 0.866025I
a = −0.85024 − 2.21534I
b = −1.252310 + 0.237364I
2.02988I 0. − 3.46410I
u = 0.500000 − 0.866025I
a = 0.412728 − 1.011420I
b = 0.218964 + 0.666188I
− 2.02988I 0. + 3.46410I
u = 0.500000 − 0.866025I
a = −0.562490 + 0.528127I
b = 1.033350 − 0.428825I
− 2.02988I 0. + 3.46410I
u = 0.500000 − 0.866025I
a = −0.85024 + 2.21534I
b = −1.252310 − 0.237364I
− 2.02988I 0. + 3.46410I
15
IV. I
v
1
= ha, b − 1, v − 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
1
0
a
6
=
1
0
a
1
=
1
0
a
3
=
1
0
a
9
=
0
1
a
10
=
1
1
a
8
=
1
1
a
7
=
1
0
a
11
=
1
1
a
4
=
2
1
a
4
=
2
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
10
u
c
3
, c
4
, c
7
c
11
u − 1
c
8
, c
9
u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
10
y
c
3
, c
4
, c
7
c
8
, c
9
, c
11
y −1
18
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
0 0
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u(u
2
− u + 1)
3
(u
4
+ 2u
2
+ 2)(u
23
+ 2u
22
+ ··· + 4u + 2)
c
2
u(u
2
+ u + 1)
3
(u
2
+ 2u + 2)
2
(u
23
+ 10u
22
+ ··· + 8u − 4)
c
3
(u − 1)(u
4
+ 4u
3
+ 4u
2
+ 1)(u
6
− 2u
4
− u
3
+ u
2
+ u + 1)
· (u
23
+ 27u
21
+ ··· − 7u + 1)
c
4
(u − 1)(u + 1)
4
(u
6
− 2u
4
+ ··· + u + 1)(u
23
− 2u
22
+ ··· + 9u + 1)
c
6
u(u
2
+ u + 1)
3
(u
4
− 2u
2
+ 2)(u
23
− 2u
22
+ ··· − 88u + 16)
c
7
((u − 1)
5
)(u
6
+ 4u
5
+ ··· − u + 1)(u
23
+ 2u
22
+ ··· − 11u + 1)
c
8
((u − 1)
4
)(u + 1)(u
6
− 2u
4
+ ··· + u + 1)(u
23
+ 2u
22
+ ··· − 3u + 1)
c
9
((u − 1)
4
)(u + 1)(u
6
− 2u
4
+ ··· + u + 1)(u
23
− 2u
22
+ ··· + 9u + 1)
c
10
u
7
(u
4
− 2u
2
+ 2)(u
23
− 5u
22
+ ··· + 128u + 1706)
c
11
(u − 1)(u
4
− 4u
3
+ 4u
2
+ 1)(u
6
− 4u
5
+ 6u
4
− 3u
3
− u
2
+ u + 1)
· (u
23
+ 8u
22
+ ··· + 1035u + 297)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y(y
2
+ y + 1)
3
(y
2
+ 2y + 2)
2
(y
23
+ 10y
22
+ ··· + 8y −4)
c
2
y(y
2
+ 4)
2
(y
2
+ y + 1)
3
(y
23
+ 6y
22
+ ··· + 160y −16)
c
3
(y −1)(y
4
− 8y
3
+ ··· + 8y + 1)(y
6
− 4y
5
+ ··· + y + 1)
· (y
23
+ 54y
22
+ ··· + 25y −1)
c
4
, c
9
(y −1)
5
(y
6
− 4y
5
+ 6y
4
− 3y
3
− y
2
+ y + 1)
· (y
23
− 34y
22
+ ··· − 27y −1)
c
6
y(y
2
− 2y + 2)
2
(y
2
+ y + 1)
3
(y
23
+ 2y
22
+ ··· + 960y −256)
c
7
(y −1)
5
(y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1)
· (y
23
+ 46y
22
+ ··· − 135y −1)
c
8
((y −1)
5
)(y
6
− 4y
5
+ ··· + y + 1)(y
23
− 2y
22
+ ··· − 11y −1)
c
10
y
7
(y
2
− 2y + 2)
2
(y
23
+ 41y
22
+ ··· − 1.22873 × 10
7
y −2910436)
c
11
(y −1)(y
4
− 8y
3
+ 18y
2
+ 8y + 1)
· (y
6
− 4y
5
+ 10y
4
− 11y
3
+ 19y
2
− 3y + 1)
· (y
23
− 26y
22
+ ··· + 935793y −88209)
21
