10
51
(K10a
16
)
A knot diagram
1
Linearized knot diagam
9 6 7 8 10 4 3 1 5 2
Solving Sequence
4,6
7 3 8
2,10
5 9 1
c
6
c
3
c
7
c
2
c
5
c
9
c
1
c
4
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= hu
35
+ 2u
34
+ ··· + b − 1, −2u
35
− 2u
34
+ ··· + a + 1, u
36
+ 2u
35
+ ··· − u − 1i
I
u
2
= hb, −u
2
+ a − 1, u
3
− u
2
+ 2u − 1i
* 2 irreducible components of dim
C
= 0, with total 39 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
35
+2u
34
+· · ·+b−1, −2u
35
−2u
34
+· · ·+a+1, u
36
+2u
35
+· · ·−u−1i
(i) Arc colorings
a
4
=
0
u
a
6
=
1
0
a
7
=
1
−u
2
a
3
=
−u
u
3
+ u
a
8
=
u
2
+ 1
−u
4
− 2u
2
a
2
=
−u
3
− 2u
u
3
+ u
a
10
=
2u
35
+ 2u
34
+ ··· − 4u − 1
−u
35
− 2u
34
+ ··· + 7u
2
+ 1
a
5
=
u
5
+ 2u
3
+ u
−u
7
− 3u
5
− 2u
3
+ u
a
9
=
u
23
+ 10u
21
+ ··· + 6u
2
− 4u
u
35
+ 2u
34
+ ··· − u − 1
a
1
=
u
35
+ u
34
+ ··· + 5u
2
− 5u
−u
24
− 10u
22
+ ··· − 6u
3
+ 4u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
35
−2u
34
−12u
33
−22u
32
−54u
31
−93u
30
−72u
29
−137u
28
+
303u
27
+ 296u
26
+ 1632u
25
+ 1737u
24
+ 3476u
23
+ 3488u
22
+ 3688u
21
+ 3414u
20
+
1095u
19
+ 880u
18
− 1598u
17
− 1475u
16
− 1414u
15
− 1484u
14
+ 24u
13
− 396u
12
+
150u
11
+ 124u
10
− 166u
9
+ 148u
8
+ 30u
7
+ 48u
6
+ 66u
5
− 17u
3
+ 24u
2
+ 14u + 1
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
36
− 4u
35
+ ··· + 8u − 1
c
2
, c
4
u
36
− 2u
35
+ ··· + 19u − 17
c
3
, c
6
, c
7
u
36
+ 2u
35
+ ··· − u − 1
c
5
, c
9
u
36
+ u
35
+ ··· + 12u + 8
c
10
u
36
+ 16u
35
+ ··· + 24u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
36
− 16y
35
+ ··· − 24y + 1
c
2
, c
4
y
36
− 26y
35
+ ··· + 2461y + 289
c
3
, c
6
, c
7
y
36
+ 30y
35
+ ··· + 5y + 1
c
5
, c
9
y
36
− 21y
35
+ ··· − 784y + 64
c
10
y
36
+ 12y
35
+ ··· − 516y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.836039 + 0.127083I
a = −2.32698 − 0.33462I
b = 1.30605 − 0.59694I
5.69474 − 8.30646I 7.90156 + 6.05994I
u = −0.836039 − 0.127083I
a = −2.32698 + 0.33462I
b = 1.30605 + 0.59694I
5.69474 + 8.30646I 7.90156 − 6.05994I
u = −0.837370 + 0.074490I
a = 2.44021 + 0.21899I
b = −1.346470 + 0.353306I
7.51295 − 2.38075I 10.48437 + 1.26314I
u = −0.837370 − 0.074490I
a = 2.44021 − 0.21899I
b = −1.346470 − 0.353306I
7.51295 + 2.38075I 10.48437 − 1.26314I
u = −0.393001 + 1.122730I
a = 0.839814 + 0.387760I
b = −1.315580 − 0.506223I
2.65006 + 3.86936I 5.24553 − 2.32285I
u = −0.393001 − 1.122730I
a = 0.839814 − 0.387760I
b = −1.315580 + 0.506223I
2.65006 − 3.86936I 5.24553 + 2.32285I
u = 0.773363 + 0.051034I
a = −0.111470 + 0.916399I
b = 0.224431 − 1.065040I
2.25781 + 2.29689I 7.21657 − 3.23152I
u = 0.773363 − 0.051034I
a = −0.111470 − 0.916399I
b = 0.224431 + 1.065040I
2.25781 − 2.29689I 7.21657 + 3.23152I
u = −0.388829 + 1.191850I
a = −1.062240 − 0.642876I
b = 1.360160 + 0.242055I
4.08196 − 2.02960I 7.16240 + 2.61607I
u = −0.388829 − 1.191850I
a = −1.062240 + 0.642876I
b = 1.360160 − 0.242055I
4.08196 + 2.02960I 7.16240 − 2.61607I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.741018
a = −2.98588
b = 0.930463
0.763718 8.86550
u = 0.316713 + 1.230230I
a = 0.731009 − 0.280668I
b = −0.073467 − 1.041860I
−1.35734 + 1.63914I 3.47794 − 0.38359I
u = 0.316713 − 1.230230I
a = 0.731009 + 0.280668I
b = −0.073467 + 1.041860I
−1.35734 − 1.63914I 3.47794 + 0.38359I
u = 0.110839 + 1.278840I
a = 0.199304 − 0.779639I
b = −0.585175 − 0.509756I
−3.23258 + 1.97104I 3.37344 − 3.58123I
u = 0.110839 − 1.278840I
a = 0.199304 + 0.779639I
b = −0.585175 + 0.509756I
−3.23258 − 1.97104I 3.37344 + 3.58123I
u = 0.444529 + 0.543366I
a = 0.840105 − 0.882584I
b = −1.105770 − 0.324662I
0.90728 + 4.09703I 5.30644 − 6.77310I
u = 0.444529 − 0.543366I
a = 0.840105 + 0.882584I
b = −1.105770 + 0.324662I
0.90728 − 4.09703I 5.30644 + 6.77310I
u = −0.027017 + 1.315680I
a = −0.29010 + 1.42344I
b = 0.625122 + 0.681126I
−6.47860 − 1.16610I −2.74685 + 0.24767I
u = −0.027017 − 1.315680I
a = −0.29010 − 1.42344I
b = 0.625122 − 0.681126I
−6.47860 + 1.16610I −2.74685 − 0.24767I
u = −0.311343 + 1.279420I
a = 1.87121 + 1.06972I
b = −0.965876 + 0.174407I
−3.22138 − 3.79621I 3.52420 + 4.06401I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.311343 − 1.279420I
a = 1.87121 − 1.06972I
b = −0.965876 − 0.174407I
−3.22138 + 3.79621I 3.52420 − 4.06401I
u = 0.335799 + 1.303370I
a = −0.797336 + 0.066999I
b = −0.336766 + 1.094920I
−1.97731 + 6.30262I 2.30057 − 5.66674I
u = 0.335799 − 1.303370I
a = −0.797336 − 0.066999I
b = −0.336766 − 1.094920I
−1.97731 − 6.30262I 2.30057 + 5.66674I
u = 0.543094 + 0.361071I
a = −0.752914 + 0.836491I
b = 1.016680 − 0.106012I
1.46636 − 0.53351I 7.64819 − 0.27613I
u = 0.543094 − 0.361071I
a = −0.752914 − 0.836491I
b = 1.016680 + 0.106012I
1.46636 + 0.53351I 7.64819 + 0.27613I
u = −0.372314 + 1.319560I
a = −1.30924 − 1.37083I
b = 1.323430 − 0.441863I
3.14977 − 6.72875I 6.21840 + 3.94329I
u = −0.372314 − 1.319560I
a = −1.30924 + 1.37083I
b = 1.323430 + 0.441863I
3.14977 + 6.72875I 6.21840 − 3.94329I
u = 0.210596 + 1.368850I
a = −0.424656 − 0.451211I
b = −0.759926 + 0.135831I
−3.87079 + 2.11524I 4.29140 + 1.12167I
u = 0.210596 − 1.368850I
a = −0.424656 + 0.451211I
b = −0.759926 − 0.135831I
−3.87079 − 2.11524I 4.29140 − 1.12167I
u = −0.365320 + 1.351690I
a = 1.28028 + 1.56464I
b = −1.28411 + 0.65656I
1.04241 − 12.63140I 3.42125 + 8.03158I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.365320 − 1.351690I
a = 1.28028 − 1.56464I
b = −1.28411 − 0.65656I
1.04241 + 12.63140I 3.42125 − 8.03158I
u = 0.096201 + 1.407940I
a = 0.333081 + 1.018580I
b = 0.995297 + 0.496043I
−5.27687 + 5.74916I 0. − 6.40491I
u = 0.096201 − 1.407940I
a = 0.333081 − 1.018580I
b = 0.995297 − 0.496043I
−5.27687 − 5.74916I 0. + 6.40491I
u = 0.456356
a = −0.741212
b = 0.450302
0.789103 12.7730
u = −0.157570 + 0.278904I
a = 0.40346 − 1.83069I
b = −0.268417 − 0.538256I
−1.65748 − 0.63628I −3.12504 + 1.61784I
u = −0.157570 − 0.278904I
a = 0.40346 + 1.83069I
b = −0.268417 + 0.538256I
−1.65748 + 0.63628I −3.12504 − 1.61784I
8
II. I
u
2
= hb, −u
2
+ a − 1, u
3
− u
2
+ 2u − 1i
(i) Arc colorings
a
4
=
0
u
a
6
=
1
0
a
7
=
1
−u
2
a
3
=
−u
u
2
− u + 1
a
8
=
u
2
+ 1
−u
2
+ u − 1
a
2
=
−u
2
− 1
u
2
− u + 1
a
10
=
u
2
+ 1
0
a
5
=
1
0
a
9
=
u
2
+ 1
0
a
1
=
0
u
2
− u + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
2
− 4u + 4
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u + 1)
3
c
2
, c
4
u
3
− u
2
+ 1
c
3
u
3
+ u
2
+ 2u + 1
c
5
, c
9
u
3
c
6
, c
7
u
3
− u
2
+ 2u − 1
c
8
, c
10
(u − 1)
3
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
, c
10
(y −1)
3
c
2
, c
4
y
3
− y
2
+ 2y −1
c
3
, c
6
, c
7
y
3
+ 3y
2
+ 2y −1
c
5
, c
9
y
3
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.215080 + 1.307140I
a = −0.662359 + 0.562280I
b = 0
−4.66906 + 2.82812I −1.84740 − 3.54173I
u = 0.215080 − 1.307140I
a = −0.662359 − 0.562280I
b = 0
−4.66906 − 2.82812I −1.84740 + 3.54173I
u = 0.569840
a = 1.32472
b = 0
−0.531480 2.69480
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u + 1)
3
)(u
36
− 4u
35
+ ··· + 8u − 1)
c
2
, c
4
(u
3
− u
2
+ 1)(u
36
− 2u
35
+ ··· + 19u − 17)
c
3
(u
3
+ u
2
+ 2u + 1)(u
36
+ 2u
35
+ ··· − u − 1)
c
5
, c
9
u
3
(u
36
+ u
35
+ ··· + 12u + 8)
c
6
, c
7
(u
3
− u
2
+ 2u − 1)(u
36
+ 2u
35
+ ··· − u − 1)
c
8
((u − 1)
3
)(u
36
− 4u
35
+ ··· + 8u − 1)
c
10
((u − 1)
3
)(u
36
+ 16u
35
+ ··· + 24u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
((y −1)
3
)(y
36
− 16y
35
+ ··· − 24y + 1)
c
2
, c
4
(y
3
− y
2
+ 2y −1)(y
36
− 26y
35
+ ··· + 2461y + 289)
c
3
, c
6
, c
7
(y
3
+ 3y
2
+ 2y −1)(y
36
+ 30y
35
+ ··· + 5y + 1)
c
5
, c
9
y
3
(y
36
− 21y
35
+ ··· − 784y + 64)
c
10
((y −1)
3
)(y
36
+ 12y
35
+ ··· − 516y + 1)
14
