10
57
(K10a
6
)
A knot diagram
1
Linearized knot diagam
3 9 1 7 4 10 5 2 6 8
Solving Sequence
3,9
2 1
4,5
6 8 7 10
c
2
c
1
c
3
c
5
c
8
c
7
c
10
c
4
, c
6
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
40
+ u
39
+ ··· + 2u
2
+ b, u
25
− 4u
23
+ ··· + a − 3u, u
42
− 2u
41
+ ··· + 2u − 1i
I
u
2
= hb − 1, a − u, u
3
+ u
2
− 1i
* 2 irreducible components of dim
C
= 0, with total 45 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
40
+u
39
+· · ·+2u
2
+b, u
25
−4u
23
+· · ·+a−3u, u
42
−2u
41
+· · ·+2u−1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
u
2
a
1
=
−u
2
+ 1
u
2
a
4
=
u
4
− u
2
+ 1
−u
4
a
5
=
−u
25
+ 4u
23
+ ··· − 4u
2
+ 3u
u
40
− u
39
+ ··· + 5u
3
− 2u
2
a
6
=
−u
41
+ u
40
+ ··· + 4u − 1
u
41
+ u
40
+ ··· − u
2
− u
a
8
=
−u
−u
3
+ u
a
7
=
−u
41
+ u
40
+ ··· + 2u
2
− 2u
u
41
− u
40
+ ··· − 6u
3
+ 3u
2
a
10
=
−u
6
+ u
4
− 2u
2
+ 1
−u
8
+ 2u
6
− 2u
4
+ 2u
2
(ii) Obstruction class = −1
(iii) Cusp Shapes = 9u
41
− 10u
40
+ ··· − 3u + 11
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
42
− 14u
41
+ ··· + 2u + 1
c
2
, c
8
u
42
− 2u
41
+ ··· + 2u − 1
c
4
, c
7
u
42
− 4u
41
+ ··· + 7u − 1
c
5
u
42
+ 20u
41
+ ··· + 39u + 1
c
6
, c
9
u
42
− u
41
+ ··· − 28u + 8
c
10
u
42
+ 2u
41
+ ··· − 168u − 49
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
y
42
+ 30y
41
+ ··· + 2y + 1
c
2
, c
8
y
42
− 14y
41
+ ··· + 2y + 1
c
4
, c
7
y
42
− 20y
41
+ ··· − 39y + 1
c
5
y
42
+ 8y
41
+ ··· − 999y + 1
c
6
, c
9
y
42
− 21y
41
+ ··· − 784y + 64
c
10
y
42
− 6y
41
+ ··· − 7154y + 2401
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.991138 + 0.067760I
a = 0.72613 − 1.72423I
b = −0.599813 + 0.692072I
1.67988 + 2.03798I 8.18964 − 3.67578I
u = 0.991138 − 0.067760I
a = 0.72613 + 1.72423I
b = −0.599813 − 0.692072I
1.67988 − 2.03798I 8.18964 + 3.67578I
u = 0.645452 + 0.781684I
a = −0.611186 − 0.493033I
b = 0.814133 − 0.823314I
0.56632 − 2.39851I 5.00404 + 0.87866I
u = 0.645452 − 0.781684I
a = −0.611186 + 0.493033I
b = 0.814133 + 0.823314I
0.56632 + 2.39851I 5.00404 − 0.87866I
u = −0.703889 + 0.756112I
a = 2.19949 + 0.78549I
b = −0.50504 − 2.77745I
−3.91253 + 1.78828I 0.036224 − 1.373729I
u = −0.703889 − 0.756112I
a = 2.19949 − 0.78549I
b = −0.50504 + 2.77745I
−3.91253 − 1.78828I 0.036224 + 1.373729I
u = −0.794934 + 0.673703I
a = −0.870200 + 0.235772I
b = 0.92529 + 1.29854I
−2.06220 − 2.20756I 3.08817 + 4.39193I
u = −0.794934 − 0.673703I
a = −0.870200 − 0.235772I
b = 0.92529 − 1.29854I
−2.06220 + 2.20756I 3.08817 − 4.39193I
u = 0.745202 + 0.733734I
a = −1.136060 − 0.593599I
b = 1.064410 + 0.315955I
−4.59267 + 0.70618I 0.622977 + 0.556758I
u = 0.745202 − 0.733734I
a = −1.136060 + 0.593599I
b = 1.064410 − 0.315955I
−4.59267 − 0.70618I 0.622977 − 0.556758I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.938084
a = 0.506699
b = 1.24884
0.325164 11.1790
u = 0.670918 + 0.832205I
a = 2.15907 − 0.24239I
b = −1.29724 + 2.23565I
−1.77790 − 7.76497I 1.88925 + 4.74518I
u = 0.670918 − 0.832205I
a = 2.15907 + 0.24239I
b = −1.29724 − 2.23565I
−1.77790 + 7.76497I 1.88925 − 4.74518I
u = −1.074440 + 0.080759I
a = 0.469289 − 1.085500I
b = −0.649806 + 0.505264I
6.58974 − 1.93798I 11.95326 + 1.38361I
u = −1.074440 − 0.080759I
a = 0.469289 + 1.085500I
b = −0.649806 − 0.505264I
6.58974 + 1.93798I 11.95326 − 1.38361I
u = −1.083350 + 0.141922I
a = 0.18294 + 1.60896I
b = −0.381965 − 0.537269I
4.86295 − 7.53350I 9.04295 + 6.51119I
u = −1.083350 − 0.141922I
a = 0.18294 − 1.60896I
b = −0.381965 + 0.537269I
4.86295 + 7.53350I 9.04295 − 6.51119I
u = 0.988336 + 0.481239I
a = 0.224561 − 0.665612I
b = 1.61306 + 0.54768I
2.85726 − 1.06689I 7.69538 + 0.36183I
u = 0.988336 − 0.481239I
a = 0.224561 + 0.665612I
b = 1.61306 − 0.54768I
2.85726 + 1.06689I 7.69538 − 0.36183I
u = −0.932953 + 0.658227I
a = −0.274599 + 1.130300I
b = 2.18013 − 0.41245I
−1.62453 − 2.94974I 4.00088 + 1.92478I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.932953 − 0.658227I
a = −0.274599 − 1.130300I
b = 2.18013 + 0.41245I
−1.62453 + 2.94974I 4.00088 − 1.92478I
u = 0.999660 + 0.570752I
a = −0.434336 + 1.089620I
b = −0.92287 − 1.72945I
3.66366 + 4.35155I 8.59858 − 5.33139I
u = 0.999660 − 0.570752I
a = −0.434336 − 1.089620I
b = −0.92287 + 1.72945I
3.66366 − 4.35155I 8.59858 + 5.33139I
u = −0.836375 + 0.809644I
a = −0.881965 + 0.568772I
b = 1.240720 − 0.165182I
−4.73966 − 4.32552I 1.66531 + 7.57694I
u = −0.836375 − 0.809644I
a = −0.881965 − 0.568772I
b = 1.240720 + 0.165182I
−4.73966 + 4.32552I 1.66531 − 7.57694I
u = 0.962070 + 0.695356I
a = −0.789231 − 0.908899I
b = 0.697224 − 0.036762I
−3.92956 + 4.75718I 2.72048 − 5.86296I
u = 0.962070 − 0.695356I
a = −0.789231 + 0.908899I
b = 0.697224 + 0.036762I
−3.92956 − 4.75718I 2.72048 + 5.86296I
u = −0.923145 + 0.781924I
a = −0.761880 + 0.755330I
b = 0.897855 + 0.246991I
−4.47229 − 1.63203I 2.91298 − 2.62995I
u = −0.923145 − 0.781924I
a = −0.761880 − 0.755330I
b = 0.897855 − 0.246991I
−4.47229 + 1.63203I 2.91298 + 2.62995I
u = −0.988556 + 0.699620I
a = −0.71509 − 2.16040I
b = −2.19567 + 2.94125I
−3.05223 − 7.32917I 2.09146 + 6.67478I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.988556 − 0.699620I
a = −0.71509 + 2.16040I
b = −2.19567 − 2.94125I
−3.05223 + 7.32917I 2.09146 − 6.67478I
u = 1.020290 + 0.695366I
a = −0.318656 − 0.733367I
b = 1.83800 + 0.30183I
1.68665 + 7.98804I 6.75545 − 5.63639I
u = 1.020290 − 0.695366I
a = −0.318656 + 0.733367I
b = 1.83800 − 0.30183I
1.68665 − 7.98804I 6.75545 + 5.63639I
u = 1.028180 + 0.723271I
a = −0.13761 + 2.12451I
b = −2.62947 − 2.13483I
−0.68940 + 13.58860I 3.64913 − 9.29837I
u = 1.028180 − 0.723271I
a = −0.13761 − 2.12451I
b = −2.62947 + 2.13483I
−0.68940 − 13.58860I 3.64913 + 9.29837I
u = 0.368496 + 0.622797I
a = 1.324790 − 0.246617I
b = −0.344960 + 0.719696I
2.03700 + 0.16365I 5.74023 − 0.29295I
u = 0.368496 − 0.622797I
a = 1.324790 + 0.246617I
b = −0.344960 − 0.719696I
2.03700 − 0.16365I 5.74023 + 0.29295I
u = 0.209332 + 0.676070I
a = −0.682383 − 0.891170I
b = 0.848519 − 0.570944I
0.63189 + 5.08816I 2.51962 − 5.57765I
u = 0.209332 − 0.676070I
a = −0.682383 + 0.891170I
b = 0.848519 + 0.570944I
0.63189 − 5.08816I 2.51962 + 5.57765I
u = 0.647067
a = 0.662985
b = −0.112358
0.883120 11.7260
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.145920 + 0.358325I
a = −0.25789 + 1.99901I
b = 0.839252 + 0.324615I
−1.72875 − 0.76607I −3.12845 + 1.30178I
u = −0.145920 − 0.358325I
a = −0.25789 − 1.99901I
b = 0.839252 − 0.324615I
−1.72875 + 0.76607I −3.12845 − 1.30178I
9
II. I
u
2
= hb − 1, a − u, u
3
+ u
2
− 1i
(i) Arc colorings
a
3
=
1
0
a
9
=
0
u
a
2
=
1
u
2
a
1
=
−u
2
+ 1
u
2
a
4
=
u
−u
2
− u + 1
a
5
=
u
1
a
6
=
0
u
2
+ u
a
8
=
−u
u
2
+ u − 1
a
7
=
0
u
2
+ u
a
10
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −2u
2
+ u + 2
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
3
+ u
2
+ 2u + 1
c
2
u
3
+ u
2
− 1
c
3
, c
10
u
3
− u
2
+ 2u − 1
c
4
(u − 1)
3
c
5
, c
7
(u + 1)
3
c
6
, c
9
u
3
c
8
u
3
− u
2
+ 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
10
y
3
+ 3y
2
+ 2y −1
c
2
, c
8
y
3
− y
2
+ 2y −1
c
4
, c
5
, c
7
(y −1)
3
c
6
, c
9
y
3
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.877439 + 0.744862I
b = 1.00000
−4.66906 − 2.82812I 0.69240 + 3.35914I
u = −0.877439 − 0.744862I
a = −0.877439 − 0.744862I
b = 1.00000
−4.66906 + 2.82812I 0.69240 − 3.35914I
u = 0.754878
a = 0.754878
b = 1.00000
−0.531480 1.61520
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
3
+ u
2
+ 2u + 1)(u
42
− 14u
41
+ ··· + 2u + 1)
c
2
(u
3
+ u
2
− 1)(u
42
− 2u
41
+ ··· + 2u − 1)
c
3
(u
3
− u
2
+ 2u − 1)(u
42
− 14u
41
+ ··· + 2u + 1)
c
4
((u − 1)
3
)(u
42
− 4u
41
+ ··· + 7u − 1)
c
5
((u + 1)
3
)(u
42
+ 20u
41
+ ··· + 39u + 1)
c
6
, c
9
u
3
(u
42
− u
41
+ ··· − 28u + 8)
c
7
((u + 1)
3
)(u
42
− 4u
41
+ ··· + 7u − 1)
c
8
(u
3
− u
2
+ 1)(u
42
− 2u
41
+ ··· + 2u − 1)
c
10
(u
3
− u
2
+ 2u − 1)(u
42
+ 2u
41
+ ··· − 168u − 49)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
(y
3
+ 3y
2
+ 2y −1)(y
42
+ 30y
41
+ ··· + 2y + 1)
c
2
, c
8
(y
3
− y
2
+ 2y −1)(y
42
− 14y
41
+ ··· + 2y + 1)
c
4
, c
7
((y −1)
3
)(y
42
− 20y
41
+ ··· − 39y + 1)
c
5
((y −1)
3
)(y
42
+ 8y
41
+ ··· − 999y + 1)
c
6
, c
9
y
3
(y
42
− 21y
41
+ ··· − 784y + 64)
c
10
(y
3
+ 3y
2
+ 2y −1)(y
42
− 6y
41
+ ··· − 7154y + 2401)
15
