8
16
(K8a
15
)
A knot diagram
1
Linearized knot diagam
4 5 1 7 8 3 2 6
Solving Sequence
4,7 2,5
8 1 3 6
c
4
c
7
c
1
c
3
c
6
c
2
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
4
+ b + 2u − 2, −5u
4
− 2u
3
− u
2
+ a + 10u − 11, u
5
− 2u
2
+ 3u − 1i
I
u
2
= h816u
11
+ 1706u
10
+ ··· + 605b − 492, −596u
11
− 1838u
10
+ ··· + 121a − 1235,
u
12
+ 3u
11
+ 6u
10
+ 9u
9
+ 20u
8
+ 31u
7
+ 41u
6
+ 39u
5
+ 34u
4
+ 22u
3
+ 12u
2
+ 4u + 1i
* 2 irreducible components of dim
C
= 0, with total 17 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
4
+ b + 2u − 2, −5u
4
− 2u
3
− u
2
+ a + 10u − 11, u
5
− 2u
2
+ 3u − 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
2
=
5u
4
+ 2u
3
+ u
2
− 10u + 11
u
4
− 2u + 2
a
5
=
1
u
2
a
8
=
17u
4
+ 8u
3
+ 4u
2
− 33u + 36
3u
4
+ u
3
+ u
2
− 5u + 6
a
1
=
4u
4
+ 2u
3
+ u
2
− 8u + 9
u
4
− 2u + 2
a
3
=
5u
4
+ 2u
3
+ u
2
− 9u + 11
u
4
+ u
3
− 2u + 2
a
6
=
19u
4
+ 9u
3
+ 4u
2
− 35u + 40
3u
4
+ 2u
3
+ u
2
− 6u + 7
(ii) Obstruction class = −1
(iii) Cusp Shapes = −12u
4
− 4u
3
+ 20u − 26
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
u
5
+ 2u
4
− 2u
2
− u − 1
c
2
, c
4
u
5
+ 2u
2
+ 3u + 1
c
6
u
5
+ 7u
4
+ 19u
3
+ 30u
2
+ 24u + 8
c
7
u
5
+ 7u
4
+ 18u
3
+ 23u
2
+ 14u + 4
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
y
5
− 4y
4
+ 6y
3
− 3y −1
c
2
, c
4
y
5
+ 6y
3
− 4y
2
+ 5y −1
c
6
y
5
− 11y
4
− 11y
3
− 100y
2
+ 96y −64
c
7
y
5
− 13y
4
+ 30y
3
− 81y
2
+ 12y −16
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.761218 + 0.545187I
a = 0.148341 − 0.707998I
b = −0.131705 − 0.621876I
−1.32133 − 1.30034I −2.51370 + 2.13902I
u = 0.761218 − 0.545187I
a = 0.148341 + 0.707998I
b = −0.131705 + 0.621876I
−1.32133 + 1.30034I −2.51370 − 2.13902I
u = 0.476529
a = 6.93603
b = 1.09851
3.68417 −17.5210
u = −0.99948 + 1.18099I
a = −0.116359 − 1.043350I
b = −1.41755 − 0.49337I
6.88145 + 10.57900I 6.27422 − 6.37200I
u = −0.99948 − 1.18099I
a = −0.116359 + 1.043350I
b = −1.41755 + 0.49337I
6.88145 − 10.57900I 6.27422 + 6.37200I
5
II. I
u
2
= h816u
11
+ 1706u
10
+ · · · + 605b − 492, −596u
11
− 1838u
10
+ · · · +
121a − 1235, u
12
+ 3u
11
+ · · · + 4u + 1i
(i) Arc colorings
a
4
=
1
0
a
7
=
0
u
a
2
=
4.92562u
11
+ 15.1901u
10
+ ··· + 37.2893u + 10.2066
−1.34876u
11
− 2.81983u
10
+ ··· − 3.02149u + 0.813223
a
5
=
1
u
2
a
8
=
−7.14050u
11
− 21.7521u
10
+ ··· − 47.2314u − 12.1653
1.29256u
11
+ 2.11901u
10
+ ··· − 3.27107u − 3.27934
a
1
=
6.27438u
11
+ 18.0099u
10
+ ··· + 40.3107u + 9.39339
−1.34876u
11
− 2.81983u
10
+ ··· − 3.02149u + 0.813223
a
3
=
4.42314u
11
+ 13.6298u
10
+ ··· + 33.7322u + 8.98017
−1.14380u
11
− 2.49917u
10
+ ··· − 2.30744u + 0.866116
a
6
=
−3.42314u
11
− 10.6298u
10
+ ··· − 21.7322u − 4.98017
−0.0826446u
11
− 1.67769u
10
+ ··· − 6.90083u − 3.21488
(ii) Obstruction class = −1
(iii) Cusp Shapes =
128
605
u
11
−
112
605
u
10
−
8
605
u
9
−
632
605
u
8
−
436
605
u
7
−
744
121
u
6
−
1512
605
u
5
−
7904
605
u
4
−
156
11
u
3
−
544
55
u
2
−
1896
605
u +
2414
605
6
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
u
12
− u
11
+ ··· − 4u + 1
c
2
, c
4
u
12
− 3u
11
+ ··· − 4u + 1
c
6
(u
2
− u + 1)
6
c
7
(u
3
− u
2
+ 1)
4
7
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
y
12
− 9y
11
+ ··· + 80y
2
+ 1
c
2
, c
4
y
12
+ 3y
11
+ ··· + 8y + 1
c
6
(y
2
+ y + 1)
6
c
7
(y
3
− y
2
+ 2y −1)
4
8
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.654045 + 0.759899I
a = 0.007824 + 1.147940I
b = 0.167732 + 1.153850I
1.91067 + 4.85801I 4.49024 − 6.44355I
u = −0.654045 − 0.759899I
a = 0.007824 − 1.147940I
b = 0.167732 − 1.153850I
1.91067 − 4.85801I 4.49024 + 6.44355I
u = 0.204191 + 0.813066I
a = 0.219331 − 0.873352I
b = 1.52069 − 0.58643I
6.04826 − 2.02988I 11.01951 + 3.46410I
u = 0.204191 − 0.813066I
a = 0.219331 + 0.873352I
b = 1.52069 + 0.58643I
6.04826 + 2.02988I 11.01951 − 3.46410I
u = −0.438452 + 0.525580I
a = 1.65687 + 0.28727I
b = 0.210547 − 0.250904I
1.91067 − 0.79824I 4.49024 − 0.48465I
u = −0.438452 − 0.525580I
a = 1.65687 − 0.28727I
b = 0.210547 + 0.250904I
1.91067 + 0.79824I 4.49024 + 0.48465I
u = −0.217317 + 0.536846I
a = −0.62366 + 1.88689I
b = 1.029010 + 0.216402I
1.91067 + 0.79824I 4.49024 + 0.48465I
u = −0.217317 − 0.536846I
a = −0.62366 − 1.88689I
b = 1.029010 − 0.216402I
1.91067 − 0.79824I 4.49024 − 0.48465I
u = 0.97217 + 1.33344I
a = 0.051487 + 0.695562I
b = −1.192210 + 0.314018I
1.91067 − 4.85801I 4.49024 + 6.44355I
u = 0.97217 − 1.33344I
a = 0.051487 − 0.695562I
b = −1.192210 − 0.314018I
1.91067 + 4.85801I 4.49024 − 6.44355I
9
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.36655 + 1.20020I
a = −0.311849 − 0.273888I
b = −1.235770 + 0.092938I
6.04826 − 2.02988I 11.01951 + 3.46410I
u = −1.36655 − 1.20020I
a = −0.311849 + 0.273888I
b = −1.235770 − 0.092938I
6.04826 + 2.02988I 11.01951 − 3.46410I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
c
8
(u
5
+ 2u
4
− 2u
2
− u − 1)(u
12
− u
11
+ ··· − 4u + 1)
c
2
, c
4
(u
5
+ 2u
2
+ 3u + 1)(u
12
− 3u
11
+ ··· − 4u + 1)
c
6
(u
2
− u + 1)
6
(u
5
+ 7u
4
+ 19u
3
+ 30u
2
+ 24u + 8)
c
7
(u
3
− u
2
+ 1)
4
(u
5
+ 7u
4
+ 18u
3
+ 23u
2
+ 14u + 4)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
5
c
8
(y
5
− 4y
4
+ 6y
3
− 3y −1)(y
12
− 9y
11
+ ··· + 80y
2
+ 1)
c
2
, c
4
(y
5
+ 6y
3
− 4y
2
+ 5y −1)(y
12
+ 3y
11
+ ··· + 8y + 1)
c
6
(y
2
+ y + 1)
6
(y
5
− 11y
4
− 11y
3
− 100y
2
+ 96y −64)
c
7
(y
3
− y
2
+ 2y −1)
4
(y
5
− 13y
4
+ 30y
3
− 81y
2
+ 12y −16)
12
