11a
259
(K11a
259
)
A knot diagram
1
Linearized knot diagam
4 7 1 2 10 9 3 11 5 6 8
Solving Sequence
5,10
6
2,11
4 1 3 9 7 8
c
5
c
10
c
4
c
1
c
3
c
9
c
6
c
8
c
2
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= hu
43
+ u
42
+ ··· + b − u, u
43
+ u
42
+ ··· + a − 2, u
44
+ 2u
43
+ ··· − 3u − 1i
I
u
2
= hb + 1, u
3
+ a − 2u + 1, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 49 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
43
+ u
42
+ · · · + b − u, u
43
+ u
42
+ · · · + a − 2, u
44
+ 2u
43
+ · · · − 3u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
2
=
−u
43
− u
42
+ ··· + 6u + 2
−u
43
− u
42
+ ··· + 3u
2
+ u
a
11
=
−u
−u
3
+ u
a
4
=
−2u
43
− 2u
42
+ ··· + 7u + 3
−u
43
− u
42
+ ··· + 11u
3
+ 2u
2
a
1
=
−u
9
+ 4u
7
− 5u
5
+ 2u
3
− u
−u
11
+ 5u
9
− 8u
7
+ 3u
5
+ u
3
+ u
a
3
=
−3u
43
− 3u
42
+ ··· + 6u + 3
−u
43
− u
42
+ ··· − 17u
4
+ 6u
3
a
9
=
u
u
a
7
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
8
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
8
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
43
+ 4u
42
+ ··· + u − 10
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
u
44
− 6u
43
+ ··· + 3u − 1
c
2
, c
7
u
44
− u
43
+ ··· + 32u + 32
c
5
, c
9
, c
10
u
44
+ 2u
43
+ ··· − 3u − 1
c
6
u
44
− 6u
43
+ ··· + 175u + 53
c
8
, c
11
u
44
+ 6u
43
+ ··· − 57u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
y
44
− 46y
43
+ ··· − 17y + 1
c
2
, c
7
y
44
− 33y
43
+ ··· − 512y + 1024
c
5
, c
9
, c
10
y
44
− 42y
43
+ ··· + y + 1
c
6
y
44
− 18y
43
+ ··· − 41755y + 2809
c
8
, c
11
y
44
+ 42y
43
+ ··· − 3555y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.899130 + 0.176754I
a = 0.521391 + 0.292332I
b = 1.43714 + 0.02392I
−6.64112 − 0.04136I −14.06391 − 0.57797I
u = −0.899130 − 0.176754I
a = 0.521391 − 0.292332I
b = 1.43714 − 0.02392I
−6.64112 + 0.04136I −14.06391 + 0.57797I
u = 0.425086 + 0.710647I
a = 1.05109 − 1.91783I
b = 1.56049 + 0.28855I
−11.4356 − 8.9717I −12.8397 + 6.3159I
u = 0.425086 − 0.710647I
a = 1.05109 + 1.91783I
b = 1.56049 − 0.28855I
−11.4356 + 8.9717I −12.8397 − 6.3159I
u = 0.553555 + 0.613338I
a = 1.044040 − 0.369018I
b = 1.57148 − 0.25916I
−11.90460 + 4.53656I −13.93261 − 0.49755I
u = 0.553555 − 0.613338I
a = 1.044040 + 0.369018I
b = 1.57148 + 0.25916I
−11.90460 − 4.53656I −13.93261 + 0.49755I
u = −0.460055 + 0.640339I
a = −1.73542 − 1.22014I
b = −1.47584 + 0.02352I
−6.79823 + 2.11103I −12.35870 − 3.20933I
u = −0.460055 − 0.640339I
a = −1.73542 + 1.22014I
b = −1.47584 − 0.02352I
−6.79823 − 2.11103I −12.35870 + 3.20933I
u = 0.433137 + 0.657379I
a = −0.966620 + 0.943319I
b = −0.550396 − 0.837192I
−4.52260 − 4.83043I −11.16325 + 6.23604I
u = 0.433137 − 0.657379I
a = −0.966620 − 0.943319I
b = −0.550396 + 0.837192I
−4.52260 + 4.83043I −11.16325 − 6.23604I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.479430 + 0.610552I
a = 0.236070 + 0.053486I
b = −0.609155 + 0.800806I
−4.71947 + 0.64919I −11.99618 + 0.22434I
u = 0.479430 − 0.610552I
a = 0.236070 − 0.053486I
b = −0.609155 − 0.800806I
−4.71947 − 0.64919I −11.99618 − 0.22434I
u = −1.233270 + 0.075682I
a = 0.824540 + 0.353873I
b = 0.058722 − 0.367772I
−2.14292 + 0.55015I −5.79202 + 0.I
u = −1.233270 − 0.075682I
a = 0.824540 − 0.353873I
b = 0.058722 + 0.367772I
−2.14292 − 0.55015I −5.79202 + 0.I
u = −0.156977 + 0.697187I
a = −0.81527 + 1.27945I
b = 1.42769 − 0.10914I
−4.26770 + 3.54538I −9.88379 − 4.33460I
u = −0.156977 − 0.697187I
a = −0.81527 − 1.27945I
b = 1.42769 + 0.10914I
−4.26770 − 3.54538I −9.88379 + 4.33460I
u = 1.315480 + 0.178595I
a = −0.371113 + 0.634542I
b = −0.262321 − 0.630465I
−3.43627 − 4.29473I 0
u = 1.315480 − 0.178595I
a = −0.371113 − 0.634542I
b = −0.262321 + 0.630465I
−3.43627 + 4.29473I 0
u = −0.359435 + 0.567287I
a = 0.738087 + 0.401907I
b = 0.328694 − 0.089361I
−0.76310 + 1.71420I −4.36493 − 4.23791I
u = −0.359435 − 0.567287I
a = 0.738087 − 0.401907I
b = 0.328694 + 0.089361I
−0.76310 − 1.71420I −4.36493 + 4.23791I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.334190 + 0.073613I
a = −0.545890 − 0.301912I
b = −0.878617 + 0.388577I
−5.28913 − 0.51604I 0
u = 1.334190 − 0.073613I
a = −0.545890 + 0.301912I
b = −0.878617 − 0.388577I
−5.28913 + 0.51604I 0
u = 1.323300 + 0.265801I
a = 1.05667 − 1.60928I
b = 1.43422 + 0.16711I
−8.90196 − 7.03042I 0
u = 1.323300 − 0.265801I
a = 1.05667 + 1.60928I
b = 1.43422 − 0.16711I
−8.90196 + 7.03042I 0
u = −1.345120 + 0.133766I
a = −2.19486 − 1.55848I
b = −1.237690 + 0.186516I
−6.11126 + 2.71120I 0
u = −1.345120 − 0.133766I
a = −2.19486 + 1.55848I
b = −1.237690 − 0.186516I
−6.11126 − 2.71120I 0
u = −0.107997 + 0.549878I
a = 0.469723 − 1.325100I
b = −0.192197 + 0.469108I
0.99729 + 1.64755I −2.31020 − 6.18875I
u = −0.107997 − 0.549878I
a = 0.469723 + 1.325100I
b = −0.192197 − 0.469108I
0.99729 − 1.64755I −2.31020 + 6.18875I
u = 1.43477 + 0.21962I
a = 1.086810 − 0.327306I
b = 0.432601 + 0.117164I
−6.51912 − 4.63399I 0
u = 1.43477 − 0.21962I
a = 1.086810 + 0.327306I
b = 0.432601 − 0.117164I
−6.51912 + 4.63399I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.46361
a = 2.48778
b = 1.56570
−13.6319 0
u = −1.47141 + 0.23987I
a = −1.43868 − 0.10244I
b = −0.545081 + 0.886664I
−10.66990 + 8.11106I 0
u = −1.47141 − 0.23987I
a = −1.43868 + 0.10244I
b = −0.545081 − 0.886664I
−10.66990 − 8.11106I 0
u = −1.47725 + 0.21569I
a = −0.386925 − 0.779700I
b = −0.663066 − 0.832758I
−11.03430 + 2.36395I 0
u = −1.47725 − 0.21569I
a = −0.386925 + 0.779700I
b = −0.663066 + 0.832758I
−11.03430 − 2.36395I 0
u = 1.47753 + 0.22911I
a = −2.97141 + 1.15676I
b = −1.50758 − 0.04579I
−13.05820 − 5.28645I 0
u = 1.47753 − 0.22911I
a = −2.97141 − 1.15676I
b = −1.50758 + 0.04579I
−13.05820 + 5.28645I 0
u = −1.47671 + 0.26167I
a = 2.44784 + 1.62296I
b = 1.56869 − 0.31082I
−17.5761 + 12.5201I 0
u = −1.47671 − 0.26167I
a = 2.44784 − 1.62296I
b = 1.56869 + 0.31082I
−17.5761 − 12.5201I 0
u = −1.50093 + 0.19538I
a = 2.44900 + 0.53318I
b = 1.60536 + 0.25067I
−18.5896 − 1.6346I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.50093 − 0.19538I
a = 2.44900 − 0.53318I
b = 1.60536 − 0.25067I
−18.5896 + 1.6346I 0
u = 0.132479 + 0.398765I
a = −0.14251 + 2.47920I
b = −1.082710 − 0.136687I
−1.44721 − 0.71558I −5.57948 − 1.23300I
u = 0.132479 − 0.398765I
a = −0.14251 − 2.47920I
b = −1.082710 + 0.136687I
−1.44721 + 0.71558I −5.57948 + 1.23300I
u = −0.304939
a = 0.799083
b = −0.406553
−0.758185 −13.9250
9
II. I
u
2
= hb + 1, u
3
+ a − 2u + 1, u
5
+ u
4
− 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
5
=
1
0
a
10
=
0
u
a
6
=
1
u
2
a
2
=
−u
3
+ 2u − 1
−1
a
11
=
−u
−u
3
+ u
a
4
=
−u
3
+ 2u
−1
a
1
=
−1
0
a
3
=
−u
3
+ 2u − 1
−1
a
9
=
u
u
a
7
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
8
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
a
8
=
−u
4
+ u
2
+ 1
−u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
3
+ u
2
+ 8u − 15
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
5
c
2
, c
7
u
5
c
3
, c
4
(u + 1)
5
c
5
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
6
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
8
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
9
, c
10
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
11
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
5
c
2
, c
7
y
5
c
5
, c
9
, c
10
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
6
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
c
8
, c
11
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.21774
a = −0.370286
b = −1.00000
−4.04602 −9.19250
u = 0.309916 + 0.549911I
a = −0.128779 + 1.107660I
b = −1.00000
−1.97403 − 1.53058I −11.97286 + 4.76366I
u = 0.309916 − 0.549911I
a = −0.128779 − 1.107660I
b = −1.00000
−1.97403 + 1.53058I −11.97286 − 4.76366I
u = −1.41878 + 0.21917I
a = −1.18608 − 0.87465I
b = −1.00000
−7.51750 + 4.40083I −16.4309 − 2.8075I
u = −1.41878 − 0.21917I
a = −1.18608 + 0.87465I
b = −1.00000
−7.51750 − 4.40083I −16.4309 + 2.8075I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
5
)(u
44
− 6u
43
+ ··· + 3u − 1)
c
2
, c
7
u
5
(u
44
− u
43
+ ··· + 32u + 32)
c
3
, c
4
((u + 1)
5
)(u
44
− 6u
43
+ ··· + 3u − 1)
c
5
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
44
+ 2u
43
+ ··· − 3u − 1)
c
6
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
44
− 6u
43
+ ··· + 175u + 53)
c
8
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
44
+ 6u
43
+ ··· − 57u − 9)
c
9
, c
10
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
44
+ 2u
43
+ ··· − 3u − 1)
c
11
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
44
+ 6u
43
+ ··· − 57u − 9)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
((y −1)
5
)(y
44
− 46y
43
+ ··· − 17y + 1)
c
2
, c
7
y
5
(y
44
− 33y
43
+ ··· − 512y + 1024)
c
5
, c
9
, c
10
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
44
− 42y
43
+ ··· + y + 1)
c
6
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)(y
44
− 18y
43
+ ··· − 41755y + 2809)
c
8
, c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
44
+ 42y
43
+ ··· − 3555y + 81)
15
