11n
105
(K11n
105
)
A knot diagram
1
Linearized knot diagam
5 1 11 10 2 11 1 3 4 9 7
Solving Sequence
1,7 3,8
9 2 11 4 6 5 10
c
7
c
8
c
2
c
11
c
3
c
6
c
5
c
10
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
17
+ u
16
+ ··· + 8b − 1, a + u, u
19
− u
18
+ ··· + 2u − 1i
I
u
2
= h6724436u
21
− 6900373u
20
+ ··· + 11494529b − 7797736,
3098806u
21
+ 3180019u
20
+ ··· + 57472645a + 15394503, u
22
− u
21
+ ··· − 2u + 5i
I
u
3
= hb
4
− 4b
3
+ 8b
2
− 8b + 5, a − 1, u + 1i
I
u
4
= hb
3
+ 3b
2
+ 3b + 1, a + 1, u − 1i
* 4 irreducible components of dim
C
= 0, with total 48 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−u
17
+ u
16
+ · · · + 8b − 1, a + u, u
19
− u
18
+ · · · + 2u − 1i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
3
=
−u
1
8
u
17
−
1
8
u
16
+ ··· +
3
4
u +
1
8
a
8
=
1
u
2
a
9
=
1
8
u
18
−
1
8
u
17
+ ··· +
1
8
u + 1
−
3
4
u
18
+
7
8
u
17
+ ··· −
5
4
u +
1
8
a
2
=
−u
1
8
u
17
−
1
8
u
16
+ ··· +
3
4
u +
1
8
a
11
=
u
u
a
4
=
1
8
u
17
−
1
8
u
16
+ ··· −
5
4
u +
1
8
1
4
u
17
−
1
4
u
16
+ ··· +
1
2
u +
1
4
a
6
=
−u
2
+ 1
−u
2
a
5
=
1
−
1
8
u
18
+
1
8
u
17
+ ··· −
7
4
u
2
−
1
8
u
a
10
=
−
1
2
u
18
+
9
8
u
17
+ ··· −
5
2
u +
15
8
−
7
8
u
18
+
17
8
u
17
+ ··· −
9
8
u + 1
a
10
=
−
1
2
u
18
+
9
8
u
17
+ ··· −
5
2
u +
15
8
−
7
8
u
18
+
17
8
u
17
+ ··· −
9
8
u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
5
2
u
18
+
11
4
u
17
+
15
4
u
16
−
25
4
u
15
−
37
2
u
14
+
93
4
u
13
+
27
2
u
12
−
123
4
u
11
−
139
4
u
10
+
191
4
u
9
+
21
4
u
8
−
135
4
u
7
−
41
4
u
6
+
55
4
u
5
−
21
4
u
4
−
17
4
u
3
+ 8u
2
+
7
2
u −
53
4
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
u
19
+ u
18
+ ··· + 2u + 1
c
2
u
19
+ 5u
18
+ ··· + 10u + 1
c
3
u
19
+ 9u
18
+ ··· + 106u + 14
c
4
, c
9
u
19
+ 3u
18
+ ··· + 6u + 2
c
8
u
19
− 3u
18
+ ··· − 70u + 26
c
10
u
19
+ 9u
18
+ ··· + 4u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
y
19
− 5y
18
+ ··· + 10y −1
c
2
y
19
+ 27y
18
+ ··· + 38y −1
c
3
y
19
+ 3y
18
+ ··· + 1044y −196
c
4
, c
9
y
19
− 9y
18
+ ··· + 4y −4
c
8
y
19
− 9y
18
+ ··· − 14444y −676
c
10
y
19
+ 3y
18
+ ··· + 144y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.777977 + 0.409956I
a = 0.777977 − 0.409956I
b = 0.369181 + 0.923791I
−4.54946 + 5.47873I −11.07465 − 8.55667I
u = −0.777977 − 0.409956I
a = 0.777977 + 0.409956I
b = 0.369181 − 0.923791I
−4.54946 − 5.47873I −11.07465 + 8.55667I
u = 0.133918 + 0.761043I
a = −0.133918 − 0.761043I
b = 0.139898 + 0.619393I
1.53985 − 2.22177I −2.53368 + 4.26379I
u = 0.133918 − 0.761043I
a = −0.133918 + 0.761043I
b = 0.139898 − 0.619393I
1.53985 + 2.22177I −2.53368 − 4.26379I
u = 0.647319 + 0.397441I
a = −0.647319 − 0.397441I
b = 0.063657 + 0.711783I
−1.23847 − 1.46671I −6.68531 + 4.74531I
u = 0.647319 − 0.397441I
a = −0.647319 + 0.397441I
b = 0.063657 − 0.711783I
−1.23847 + 1.46671I −6.68531 − 4.74531I
u = −0.697477 + 0.278835I
a = 0.697477 − 0.278835I
b = −0.339503 + 1.077420I
−4.36768 − 2.62850I −9.97134 − 1.16882I
u = −0.697477 − 0.278835I
a = 0.697477 + 0.278835I
b = −0.339503 − 1.077420I
−4.36768 + 2.62850I −9.97134 + 1.16882I
u = 1.033800 + 0.730109I
a = −1.033800 − 0.730109I
b = −0.98428 − 1.15874I
−0.86154 − 5.74817I −11.63360 + 4.50327I
u = 1.033800 − 0.730109I
a = −1.033800 + 0.730109I
b = −0.98428 + 1.15874I
−0.86154 + 5.74817I −11.63360 − 4.50327I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.887480 + 0.926590I
a = −0.887480 − 0.926590I
b = 0.401639 − 1.047330I
4.22972 + 0.35072I −6.13837 − 0.22490I
u = 0.887480 − 0.926590I
a = −0.887480 + 0.926590I
b = 0.401639 + 1.047330I
4.22972 − 0.35072I −6.13837 + 0.22490I
u = −0.968832 + 0.893275I
a = 0.968832 − 0.893275I
b = −0.058802 − 1.386650I
5.51436 + 5.01792I −4.62542 − 4.88249I
u = −0.968832 − 0.893275I
a = 0.968832 + 0.893275I
b = −0.058802 + 1.386650I
5.51436 − 5.01792I −4.62542 + 4.88249I
u = −1.107490 + 0.812678I
a = 1.107490 − 0.812678I
b = 0.84460 − 1.89768I
4.47255 + 8.26447I −5.93446 − 4.83162I
u = −1.107490 − 0.812678I
a = 1.107490 + 0.812678I
b = 0.84460 + 1.89768I
4.47255 − 8.26447I −5.93446 + 4.83162I
u = 1.151650 + 0.788448I
a = −1.151650 − 0.788448I
b = −1.17629 − 2.06224I
2.31925 − 13.64210I −8.92588 + 8.81475I
u = 1.151650 − 0.788448I
a = −1.151650 + 0.788448I
b = −1.17629 + 2.06224I
2.31925 + 13.64210I −8.92588 − 8.81475I
u = 0.395222
a = −0.395222
b = 0.479812
−0.957690 −10.9550
6
II. I
u
2
= h6.72 × 10
6
u
21
− 6.90 × 10
6
u
20
+ · · · + 1.15 × 10
7
b − 7.80 × 10
6
, 3.10 ×
10
6
u
21
+ 3.18 × 10
6
u
20
+ · · · + 5.75 × 10
7
a + 1.54 × 10
7
, u
22
− u
21
+ · · · − 2u + 5i
(i) Arc colorings
a
1
=
0
u
a
7
=
1
0
a
3
=
−0.0539179u
21
− 0.0553310u
20
+ ··· − 5.96818u − 0.267858
−0.585012u
21
+ 0.600318u
20
+ ··· − 1.51708u + 0.678387
a
8
=
1
u
2
a
9
=
0.0828205u
21
− 0.00564449u
20
+ ··· + 0.238599u − 2.78491
−0.124555u
21
+ 0.416476u
20
+ ··· + 0.115943u − 2.65547
a
2
=
−0.0539179u
21
− 0.0553310u
20
+ ··· − 5.96818u − 0.267858
−0.253918u
21
+ 0.144669u
20
+ ··· − 1.56818u + 0.132142
a
11
=
u
u
a
4
=
−0.345839u
21
+ 0.500433u
20
+ ··· − 3.06360u − 0.890633
−0.876933u
21
+ 1.15608u
20
+ ··· + 1.38750u + 0.0556113
a
6
=
−u
2
+ 1
−u
2
a
5
=
−0.0264284u
21
− 0.227490u
20
+ ··· − 0.137095u − 2.51532
−1
a
10
=
−0.628809u
21
+ 0.408950u
20
+ ··· − 0.851245u − 1.74742
−1.23704u
21
+ 1.45113u
20
+ ··· + 3.35522u + 0.270710
a
10
=
−0.628809u
21
+ 0.408950u
20
+ ··· − 0.851245u − 1.74742
−1.23704u
21
+ 1.45113u
20
+ ··· + 3.35522u + 0.270710
(ii) Obstruction class = −1
(iii) Cusp Shapes =
492576
11494529
u
21
+
21062536
11494529
u
20
+ ··· −
92150964
11494529
u −
289179942
11494529
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
u
22
+ u
21
+ ··· + 2u + 5
c
2
u
22
+ 9u
21
+ ··· + 224u + 25
c
3
(u
11
− 3u
10
+ 4u
9
− u
8
+ 2u
7
− 8u
6
+ 8u
5
+ 5u
4
− 3u
3
− u
2
+ 4u − 1)
2
c
4
, c
9
(u
11
− u
10
− 2u
9
+ 3u
8
+ 2u
7
− 4u
6
+ 3u
4
− u
3
− u
2
+ 1)
2
c
8
(u
11
+ u
10
− 6u
9
− 5u
8
+ 12u
7
+ 6u
6
− 10u
5
+ u
4
+ 5u
3
− u
2
+ 1)
2
c
10
(u
11
+ 5u
10
+ ··· + 2u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
y
22
− 9y
21
+ ··· − 224y + 25
c
2
y
22
+ 7y
21
+ ··· + 5624y + 625
c
3
(y
11
− y
10
+ ··· + 14y −1)
2
c
4
, c
9
(y
11
− 5y
10
+ ··· + 2y −1)
2
c
8
(y
11
− 13y
10
+ ··· + 2y −1)
2
c
10
(y
11
+ 3y
10
+ ··· − 10y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.968725 + 0.342171I
a = −1.14483 + 0.84471I
b = −1.72734 + 0.45538I
−4.92613 + 1.27541I −13.47945 − 0.80097I
u = 0.968725 − 0.342171I
a = −1.14483 − 0.84471I
b = −1.72734 − 0.45538I
−4.92613 − 1.27541I −13.47945 + 0.80097I
u = 0.729583 + 0.772577I
a = 0.813042 + 0.684201I
b = 0.635345 + 0.872369I
0.0927065 −9.81428 + 0.I
u = 0.729583 − 0.772577I
a = 0.813042 − 0.684201I
b = 0.635345 − 0.872369I
0.0927065 −9.81428 + 0.I
u = 1.182920 + 0.018546I
a = −0.183030 − 0.210319I
b = 0.415301 − 0.525828I
−1.99990 − 0.45477I −4.80492 + 1.36957I
u = 1.182920 − 0.018546I
a = −0.183030 + 0.210319I
b = 0.415301 + 0.525828I
−1.99990 + 0.45477I −4.80492 − 1.36957I
u = 0.624756 + 1.026890I
a = 1.16034 + 0.84838I
b = −0.266486 + 1.063010I
3.97498 + 7.02220I −6.49946 − 4.88619I
u = 0.624756 − 1.026890I
a = 1.16034 − 0.84838I
b = −0.266486 − 1.063010I
3.97498 − 7.02220I −6.49946 + 4.88619I
u = −1.195770 + 0.178364I
a = 0.591030 + 0.642561I
b = 0.87334 + 1.16676I
−4.92613 + 1.27541I −13.47945 − 0.80097I
u = −1.195770 − 0.178364I
a = 0.591030 − 0.642561I
b = 0.87334 − 1.16676I
−4.92613 − 1.27541I −13.47945 + 0.80097I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.620308 + 0.489049I
a = −1.65639 + 1.26230I
b = −1.66929 − 0.34266I
−3.66655 − 4.75030I −9.35891 + 6.77690I
u = 0.620308 − 0.489049I
a = −1.65639 − 1.26230I
b = −1.66929 + 0.34266I
−3.66655 + 4.75030I −9.35891 − 6.77690I
u = −0.703026 + 0.993334I
a = −1.120470 + 0.746267I
b = −0.001089 + 1.272950I
5.74879 − 1.64593I −3.95012 + 0.24481I
u = −0.703026 − 0.993334I
a = −1.120470 − 0.746267I
b = −0.001089 − 1.272950I
5.74879 + 1.64593I −3.95012 − 0.24481I
u = −0.892154 + 0.917804I
a = −1.050620 + 0.484687I
b = −0.75266 + 1.62251I
5.74879 + 1.64593I −3.95012 − 0.24481I
u = −0.892154 − 0.917804I
a = −1.050620 − 0.484687I
b = −0.75266 − 1.62251I
5.74879 − 1.64593I −3.95012 + 0.24481I
u = −1.282540 + 0.010543I
a = 0.117415 − 0.472097I
b = −0.481877 − 1.258190I
−3.66655 + 4.75030I −9.35891 − 6.77690I
u = −1.282540 − 0.010543I
a = 0.117415 + 0.472097I
b = −0.481877 + 1.258190I
−3.66655 − 4.75030I −9.35891 + 6.77690I
u = 0.967997 + 0.889244I
a = 1.032500 + 0.377033I
b = 1.09468 + 1.69502I
3.97498 − 7.02220I −6.49946 + 4.88619I
u = 0.967997 − 0.889244I
a = 1.032500 − 0.377033I
b = 1.09468 − 1.69502I
3.97498 + 7.02220I −6.49946 − 4.88619I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.520797 + 0.242115I
a = 2.24102 + 0.95759I
b = 1.380080 − 0.137991I
−1.99990 + 0.45477I −4.80492 − 1.36957I
u = −0.520797 − 0.242115I
a = 2.24102 − 0.95759I
b = 1.380080 + 0.137991I
−1.99990 − 0.45477I −4.80492 + 1.36957I
12
III. I
u
3
= hb
4
− 4b
3
+ 8b
2
− 8b + 5, a − 1, u + 1i
(i) Arc colorings
a
1
=
0
−1
a
7
=
1
0
a
3
=
1
b
a
8
=
1
1
a
9
=
−b + 2
−b
2
+ b + 1
a
2
=
1
b − 1
a
11
=
−1
−1
a
4
=
b
2b − 1
a
6
=
0
−1
a
5
=
−1
−b
a
10
=
b
3
− 4b
2
+ 5b − 3
b
3
− 6b
2
+ 9b − 7
a
10
=
b
3
− 4b
2
+ 5b − 3
b
3
− 6b
2
+ 9b − 7
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4b
2
+ 8b − 24
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
(u + 1)
4
c
3
, c
8
u
4
+ 2u
2
+ 2
c
4
, c
9
u
4
− 2u
2
+ 2
c
5
, c
11
(u − 1)
4
c
10
(u
2
+ 2u + 2)
2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
11
(y −1)
4
c
3
, c
8
(y
2
+ 2y + 2)
2
c
4
, c
9
(y
2
− 2y + 2)
2
c
10
(y
2
+ 4)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −1.00000
a = 1.00000
b = 0.544910 + 1.098680I
−5.75727 − 3.66386I −16.0000 + 4.0000I
u = −1.00000
a = 1.00000
b = 0.544910 − 1.098680I
−5.75727 + 3.66386I −16.0000 − 4.0000I
u = −1.00000
a = 1.00000
b = 1.45509 + 1.09868I
−5.75727 + 3.66386I −16.0000 − 4.0000I
u = −1.00000
a = 1.00000
b = 1.45509 − 1.09868I
−5.75727 − 3.66386I −16.0000 + 4.0000I
16
IV. I
u
4
= hb
3
+ 3b
2
+ 3b + 1, a + 1, u − 1i
(i) Arc colorings
a
1
=
0
1
a
7
=
1
0
a
3
=
−1
b
a
8
=
1
1
a
9
=
b + 2
−b
2
− b + 1
a
2
=
−1
b + 1
a
11
=
1
1
a
4
=
b
2b + 1
a
6
=
0
−1
a
5
=
−1
b
a
10
=
b
2
+ 2b + 2
b
2
+ 2b + 2
a
10
=
b
2
+ 2b + 2
b
2
+ 2b + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4b
2
+ 8b − 8
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
(u − 1)
3
c
2
, c
5
, c
11
(u + 1)
3
c
3
, c
4
, c
8
c
9
, c
10
u
3
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
, c
7
, c
11
(y −1)
3
c
3
, c
4
, c
8
c
9
, c
10
y
3
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
u = 1.00000
a = −1.00000
b = −1.00000
−3.28987 −12.0000
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
, c
7
((u − 1)
3
)(u + 1)
4
(u
19
+ u
18
+ ··· + 2u + 1)(u
22
+ u
21
+ ··· + 2u + 5)
c
2
((u + 1)
7
)(u
19
+ 5u
18
+ ··· + 10u + 1)(u
22
+ 9u
21
+ ··· + 224u + 25)
c
3
u
3
(u
4
+ 2u
2
+ 2)
· (u
11
− 3u
10
+ 4u
9
− u
8
+ 2u
7
− 8u
6
+ 8u
5
+ 5u
4
− 3u
3
− u
2
+ 4u − 1)
2
· (u
19
+ 9u
18
+ ··· + 106u + 14)
c
4
, c
9
u
3
(u
4
− 2u
2
+ 2)(u
11
− u
10
+ ··· − u
2
+ 1)
2
· (u
19
+ 3u
18
+ ··· + 6u + 2)
c
5
, c
11
((u − 1)
4
)(u + 1)
3
(u
19
+ u
18
+ ··· + 2u + 1)(u
22
+ u
21
+ ··· + 2u + 5)
c
8
u
3
(u
4
+ 2u
2
+ 2)
· (u
11
+ u
10
− 6u
9
− 5u
8
+ 12u
7
+ 6u
6
− 10u
5
+ u
4
+ 5u
3
− u
2
+ 1)
2
· (u
19
− 3u
18
+ ··· − 70u + 26)
c
10
u
3
(u
2
+ 2u + 2)
2
(u
11
+ 5u
10
+ ··· + 2u + 1)
2
· (u
19
+ 9u
18
+ ··· + 4u + 4)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
, c
6
c
7
, c
11
((y −1)
7
)(y
19
− 5y
18
+ ··· + 10y −1)(y
22
− 9y
21
+ ··· − 224y + 25)
c
2
((y −1)
7
)(y
19
+ 27y
18
+ ··· + 38y −1)(y
22
+ 7y
21
+ ··· + 5624y + 625)
c
3
y
3
(y
2
+ 2y + 2)
2
(y
11
− y
10
+ ··· + 14y −1)
2
· (y
19
+ 3y
18
+ ··· + 1044y −196)
c
4
, c
9
y
3
(y
2
− 2y + 2)
2
(y
11
− 5y
10
+ ··· + 2y −1)
2
· (y
19
− 9y
18
+ ··· + 4y −4)
c
8
y
3
(y
2
+ 2y + 2)
2
(y
11
− 13y
10
+ ··· + 2y −1)
2
· (y
19
− 9y
18
+ ··· − 14444y −676)
c
10
y
3
(y
2
+ 4)
2
(y
11
+ 3y
10
+ ··· − 10y −1)
2
· (y
19
+ 3y
18
+ ··· + 144y −16)
22
