11n
87
(K11n
87
)
A knot diagram
1
Linearized knot diagam
6 1 10 8 2 3 4 1 5 7 8
Solving Sequence
2,5
6 1
3,8
9 10 4 7 11
c
5
c
1
c
2
c
8
c
9
c
4
c
7
c
11
c
3
, c
6
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
13
− u
12
+ 4u
11
− 3u
10
+ 7u
9
− 6u
8
+ 7u
7
− 7u
6
+ 4u
5
− 5u
4
+ 2u
3
− 2u
2
+ b + 2u − 1,
− u
15
+ 3u
14
+ ··· + 2a − 4u, u
16
− 3u
15
+ ··· − 6u + 2i
I
u
2
= h−u
6
a − u
5
a − 3u
4
a + u
5
− 2u
3
a + u
4
− 3u
2
a + 3u
3
− 2au + 2u
2
+ b − a + 3u + 2,
− 2u
7
a + 2u
7
− 4u
5
a + 2u
6
+ 5u
5
− 3u
3
a + 3u
4
− 2u
2
a + 4u
3
+ a
2
+ 2au + 3u
2
− 2a − 2u,
u
8
+ u
7
+ 3u
6
+ 2u
5
+ 3u
4
+ 2u
3
− 1i
I
u
3
= hb − 1, u
3
− 2u
2
+ 2a − 2, u
4
+ 2u
2
+ 2i
I
v
1
= ha, b + 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 37 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
13
−u
12
+· · ·+b−1, −u
15
+3u
14
+· · ·+2a−4u, u
16
−3u
15
+· · ·−6u+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
8
=
1
2
u
15
−
3
2
u
14
+ ··· − 3u
2
+ 2u
−u
13
+ u
12
+ ··· − 2u + 1
a
9
=
−
1
2
u
15
+
3
2
u
14
+ ··· − 3u + 2
u
13
− u
12
+ ··· + u − 1
a
10
=
−
1
2
u
15
+
3
2
u
14
+ ··· − 2u + 1
u
13
− u
12
+ ··· + u − 1
a
4
=
−
1
2
u
15
+
3
2
u
14
+ ··· − 2u + 1
−u
15
+ 2u
14
+ ··· − 2u + 1
a
7
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
11
=
−
1
2
u
15
+
3
2
u
14
+ ··· − 5u + 2
u
13
− u
12
+ ··· + 3u − 1
a
11
=
−
1
2
u
15
+
3
2
u
14
+ ··· − 5u + 2
u
13
− u
12
+ ··· + 3u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
14
− 6u
13
+ 16u
12
− 24u
11
+ 38u
10
− 40u
9
+ 50u
8
− 42u
7
+
40u
6
− 30u
5
+ 20u
4
− 18u
3
+ 12u
2
− 14u + 12
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
u
16
− 3u
15
+ ··· − 6u + 2
c
2
u
16
+ 9u
15
+ ··· + 4u + 4
c
3
, c
4
, c
7
u
16
− u
15
+ ··· − 4u
2
+ 1
c
6
u
16
+ 3u
15
+ ··· − 22u + 10
c
8
, c
11
u
16
− 3u
15
+ ··· − 8u + 1
c
9
u
16
+ u
15
+ ··· + 2u
2
+ 1
c
10
u
16
+ 14u
15
+ ··· + 1024u + 256
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
y
16
+ 9y
15
+ ··· + 4y + 4
c
2
y
16
− 3y
15
+ ··· + 144y + 16
c
3
, c
4
, c
7
y
16
− 3y
15
+ ··· − 8y + 1
c
6
y
16
− 15y
15
+ ··· − 364y + 100
c
8
, c
11
y
16
+ 29y
15
+ ··· + 4y + 1
c
9
y
16
+ 33y
15
+ ··· + 4y + 1
c
10
y
16
− 12y
15
+ ··· + 65536y + 65536
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.908785 + 0.099623I
a = −0.572641 − 0.673580I
b = 1.10609 − 0.91078I
−6.01654 − 7.18776I 5.49115 + 4.28840I
u = 0.908785 − 0.099623I
a = −0.572641 + 0.673580I
b = 1.10609 + 0.91078I
−6.01654 + 7.18776I 5.49115 − 4.28840I
u = −0.142689 + 1.132380I
a = 0.430492 + 1.197390I
b = 0.424706 + 0.862829I
−3.75188 + 0.61754I −1.35608 − 1.57553I
u = −0.142689 − 1.132380I
a = 0.430492 − 1.197390I
b = 0.424706 − 0.862829I
−3.75188 − 0.61754I −1.35608 + 1.57553I
u = −0.569839 + 0.991415I
a = −1.15783 − 1.20126I
b = 0.827978 − 0.641852I
−0.48607 − 7.00413I 4.93065 + 8.89860I
u = −0.569839 − 0.991415I
a = −1.15783 + 1.20126I
b = 0.827978 + 0.641852I
−0.48607 + 7.00413I 4.93065 − 8.89860I
u = 0.482015 + 1.060220I
a = −0.245187 + 0.549239I
b = 0.592760 − 0.123653I
−0.74617 + 3.29967I 2.58175 − 1.95258I
u = 0.482015 − 1.060220I
a = −0.245187 − 0.549239I
b = 0.592760 + 0.123653I
−0.74617 − 3.29967I 2.58175 + 1.95258I
u = −0.641580 + 0.478671I
a = 0.852394 + 0.173024I
b = −0.683716 − 0.565826I
0.96609 + 2.28706I 6.88422 − 4.18311I
u = −0.641580 − 0.478671I
a = 0.852394 − 0.173024I
b = −0.683716 + 0.565826I
0.96609 − 2.28706I 6.88422 + 4.18311I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.531551 + 0.451405I
a = 0.924124 + 0.014074I
b = −0.514524 − 0.293804I
1.066480 + 0.823485I 7.17691 − 4.58909I
u = 0.531551 − 0.451405I
a = 0.924124 − 0.014074I
b = −0.514524 + 0.293804I
1.066480 − 0.823485I 7.17691 + 4.58909I
u = 0.409686 + 1.284700I
a = −1.11380 + 0.89168I
b = −1.07939 + 0.99616I
−10.32590 − 2.59855I 1.50083 + 1.34763I
u = 0.409686 − 1.284700I
a = −1.11380 − 0.89168I
b = −1.07939 − 0.99616I
−10.32590 + 2.59855I 1.50083 − 1.34763I
u = 0.522071 + 1.247140I
a = 0.38245 − 2.21900I
b = −1.17390 − 0.90333I
−9.4923 + 12.3434I 2.79056 − 7.18778I
u = 0.522071 − 1.247140I
a = 0.38245 + 2.21900I
b = −1.17390 + 0.90333I
−9.4923 − 12.3434I 2.79056 + 7.18778I
6
II. I
u
2
= h−u
6
a − u
5
a + · · · − a + 2, −2u
7
a + 2u
7
+ · · · + a
2
− 2a, u
8
+ u
7
+
3u
6
+ 2u
5
+ 3u
4
+ 2u
3
− 1i
(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
8
=
a
u
6
a + u
5
a + ··· + a − 2
a
9
=
−u
7
− u
6
+ u
4
a − 3u
5
− 2u
4
+ 2u
2
a − 3u
3
− 2u
2
+ 2a
u
7
+ u
5
a + u
6
+ 2u
5
+ 2u
3
a + u
4
+ 2au − 2u − 2
a
10
=
u
5
a + u
4
a − u
5
+ 2u
3
a − u
4
+ 2u
2
a − 3u
3
+ 2au − 2u
2
+ 2a − 2u − 2
u
7
+ u
5
a + u
6
+ 2u
5
+ 2u
3
a + u
4
+ 2au − 2u − 2
a
4
=
u
5
a + u
4
a − u
5
+ 2u
3
a − u
4
+ 2u
2
a − 3u
3
+ 2au − 2u
2
+ 2a − 2u − 2
−u
7
a − u
6
a + ··· + 2u + 2
a
7
=
−u
6
− u
4
+ 1
−u
7
− u
6
− 2u
5
− u
4
− 2u
3
+ 1
a
11
=
−u
5
a + u
6
+ ··· − 2a + 1
−u
5
a − 2u
3
a + 2u
3
− 2au + 2u + 1
a
11
=
−u
5
a + u
6
+ ··· − 2a + 1
−u
5
a − 2u
3
a + 2u
3
− 2au + 2u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −4u
7
− 4u
6
− 8u
5
− 4u
4
− 4u
3
− 4u
2
+ 4u + 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u
8
+ u
7
+ 3u
6
+ 2u
5
+ 3u
4
+ 2u
3
− 1)
2
c
2
(u
8
+ 5u
7
+ 11u
6
+ 10u
5
− u
4
− 10u
3
− 6u
2
+ 1)
2
c
3
, c
4
, c
7
u
16
− u
15
+ ··· − 2u − 1
c
6
(u
8
− u
7
− 5u
6
+ 4u
5
+ 7u
4
− 4u
3
− 2u
2
+ 2u − 1)
2
c
8
, c
11
u
16
− 5u
15
+ ··· − 4u + 1
c
9
u
16
+ u
15
+ ··· + 376u + 419
c
10
(u − 1)
16
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
5
(y
8
+ 5y
7
+ 11y
6
+ 10y
5
− y
4
− 10y
3
− 6y
2
+ 1)
2
c
2
(y
8
− 3y
7
+ 19y
6
− 34y
5
+ 71y
4
− 66y
3
+ 34y
2
− 12y + 1)
2
c
3
, c
4
, c
7
y
16
− 5y
15
+ ··· − 4y + 1
c
6
(y
8
− 11y
7
+ 47y
6
− 98y
5
+ 103y
4
− 50y
3
+ 6y
2
+ 1)
2
c
8
, c
11
y
16
+ 11y
15
+ ··· − 88y + 1
c
9
y
16
+ 15y
15
+ ··· − 846972y + 175561
c
10
(y −1)
16
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.914675
a = −0.212645 + 0.481348I
b = 0.839959 + 1.056690I
−6.88602 4.17790
u = −0.914675
a = −0.212645 − 0.481348I
b = 0.839959 − 1.056690I
−6.88602 4.17790
u = −0.252896 + 0.819281I
a = 1.063650 − 0.266196I
b = −1.168220 − 0.139969I
2.79859 − 1.27532I 6.81947 + 5.08518I
u = −0.252896 + 0.819281I
a = 0.44780 − 2.90112I
b = 0.928039 − 0.286587I
2.79859 − 1.27532I 6.81947 + 5.08518I
u = −0.252896 − 0.819281I
a = 1.063650 + 0.266196I
b = −1.168220 + 0.139969I
2.79859 + 1.27532I 6.81947 − 5.08518I
u = −0.252896 − 0.819281I
a = 0.44780 + 2.90112I
b = 0.928039 + 0.286587I
2.79859 + 1.27532I 6.81947 − 5.08518I
u = 0.394459 + 1.112500I
a = −0.562009 − 0.850115I
b = −0.114249 − 0.439221I
−1.05533 + 3.63283I 1.57760 − 4.51802I
u = 0.394459 + 1.112500I
a = 0.101648 + 1.236760I
b = 1.123030 + 0.184302I
−1.05533 + 3.63283I 1.57760 − 4.51802I
u = 0.394459 − 1.112500I
a = −0.562009 + 0.850115I
b = −0.114249 + 0.439221I
−1.05533 − 3.63283I 1.57760 + 4.51802I
u = 0.394459 − 1.112500I
a = 0.101648 − 1.236760I
b = 1.123030 − 0.184302I
−1.05533 − 3.63283I 1.57760 + 4.51802I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.473514 + 1.273020I
a = −1.25140 − 0.85751I
b = −0.783945 − 1.141180I
−10.78260 − 4.93524I 1.01557 + 2.99422I
u = −0.473514 + 1.273020I
a = 0.21489 + 2.01964I
b = −0.93848 + 1.07490I
−10.78260 − 4.93524I 1.01557 + 2.99422I
u = −0.473514 − 1.273020I
a = −1.25140 + 0.85751I
b = −0.783945 + 1.141180I
−10.78260 + 4.93524I 1.01557 − 2.99422I
u = −0.473514 − 1.273020I
a = 0.21489 − 2.01964I
b = −0.93848 − 1.07490I
−10.78260 + 4.93524I 1.01557 − 2.99422I
u = 0.578577
a = 1.01161
b = −1.12958
1.93558 4.99680
u = 0.578577
a = 1.38452
b = 0.357319
1.93558 4.99680
11
III. I
u
3
= hb − 1, u
3
− 2u
2
+ 2a − 2, 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
8
=
−
1
2
u
3
+ u
2
+ 1
1
a
9
=
−
1
2
u
3
+ u
2
− u + 1
u
3
+ u + 1
a
10
=
1
2
u
3
+ u
2
+ 2
u
3
+ u + 1
a
4
=
−
1
2
u
3
+ u
2
+ 2
1
a
7
=
−1
0
a
11
=
−
1
2
u
3
+ u
2
− u + 1
u
3
+ u + 1
a
11
=
−
1
2
u
3
+ u
2
− u + 1
u
3
+ u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u
2
+ 4
12
(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
, c
7
, c
8
(u + 1)
4
c
4
, c
11
(u − 1)
4
c
6
u
4
− 2u
2
+ 2
c
9
u
4
+ 4u
3
+ 4u
2
+ 1
c
10
u
4
− 4u
3
+ 4u
2
+ 1
13
(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
4
, c
7
c
8
, c
11
(y −1)
4
c
6
(y
2
− 2y + 2)
2
c
9
, c
10
y
4
− 8y
3
+ 18y
2
+ 8y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 0.455090 + 1.098680I
a = 0.77689 + 1.32180I
b = 1.00000
0.82247 + 3.66386I 8.00000 − 4.00000I
u = 0.455090 − 1.098680I
a = 0.77689 − 1.32180I
b = 1.00000
0.82247 − 3.66386I 8.00000 + 4.00000I
u = −0.455090 + 1.098680I
a = −0.776887 − 0.678203I
b = 1.00000
0.82247 − 3.66386I 8.00000 + 4.00000I
u = −0.455090 − 1.098680I
a = −0.776887 + 0.678203I
b = 1.00000
0.82247 + 3.66386I 8.00000 − 4.00000I
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
8
=
0
−1
a
9
=
−1
−1
a
10
=
−2
−1
a
4
=
1
1
a
7
=
1
0
a
11
=
−1
−1
a
11
=
−1
−1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 12
16
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
5
c
6
u
c
3
, c
7
, c
9
c
10
, c
11
u − 1
c
4
, c
8
u + 1
17
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
c
6
y
c
3
, c
4
, c
7
c
8
, c
9
, c
10
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
3.28987 12.0000
19
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
5
u(u
4
+ 2u
2
+ 2)(u
8
+ u
7
+ 3u
6
+ 2u
5
+ 3u
4
+ 2u
3
− 1)
2
· (u
16
− 3u
15
+ ··· − 6u + 2)
c
2
u(u
2
+ 2u + 2)
2
(u
8
+ 5u
7
+ 11u
6
+ 10u
5
− u
4
− 10u
3
− 6u
2
+ 1)
2
· (u
16
+ 9u
15
+ ··· + 4u + 4)
c
3
, c
7
(u − 1)(u + 1)
4
(u
16
− u
15
+ ··· − 2u − 1)(u
16
− u
15
+ ··· − 4u
2
+ 1)
c
4
((u − 1)
4
)(u + 1)(u
16
− u
15
+ ··· − 2u − 1)(u
16
− u
15
+ ··· − 4u
2
+ 1)
c
6
u(u
4
− 2u
2
+ 2)(u
8
− u
7
− 5u
6
+ 4u
5
+ 7u
4
− 4u
3
− 2u
2
+ 2u − 1)
2
· (u
16
+ 3u
15
+ ··· − 22u + 10)
c
8
((u + 1)
5
)(u
16
− 5u
15
+ ··· − 4u + 1)(u
16
− 3u
15
+ ··· − 8u + 1)
c
9
(u − 1)(u
4
+ 4u
3
+ 4u
2
+ 1)(u
16
+ u
15
+ ··· + 376u + 419)
· (u
16
+ u
15
+ ··· + 2u
2
+ 1)
c
10
((u − 1)
17
)(u
4
− 4u
3
+ 4u
2
+ 1)(u
16
+ 14u
15
+ ··· + 1024u + 256)
c
11
((u − 1)
5
)(u
16
− 5u
15
+ ··· − 4u + 1)(u
16
− 3u
15
+ ··· − 8u + 1)
20
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
5
y(y
2
+ 2y + 2)
2
(y
8
+ 5y
7
+ 11y
6
+ 10y
5
− y
4
− 10y
3
− 6y
2
+ 1)
2
· (y
16
+ 9y
15
+ ··· + 4y + 4)
c
2
y(y
2
+ 4)
2
· (y
8
− 3y
7
+ 19y
6
− 34y
5
+ 71y
4
− 66y
3
+ 34y
2
− 12y + 1)
2
· (y
16
− 3y
15
+ ··· + 144y + 16)
c
3
, c
4
, c
7
((y −1)
5
)(y
16
− 5y
15
+ ··· − 4y + 1)(y
16
− 3y
15
+ ··· − 8y + 1)
c
6
y(y
2
− 2y + 2)
2
· (y
8
− 11y
7
+ 47y
6
− 98y
5
+ 103y
4
− 50y
3
+ 6y
2
+ 1)
2
· (y
16
− 15y
15
+ ··· − 364y + 100)
c
8
, c
11
((y −1)
5
)(y
16
+ 11y
15
+ ··· − 88y + 1)(y
16
+ 29y
15
+ ··· + 4y + 1)
c
9
(y −1)(y
4
− 8y
3
+ 18y
2
+ 8y + 1)
· (y
16
+ 15y
15
+ ··· − 846972y + 175561)(y
16
+ 33y
15
+ ··· + 4y + 1)
c
10
(y −1)
17
(y
4
− 8y
3
+ 18y
2
+ 8y + 1)
· (y
16
− 12y
15
+ ··· + 65536y + 65536)
21
