10
71
(K10a
10
)
A knot diagram
1
Linearized knot diagam
3 5 7 2 10 8 4 1 6 9
Solving Sequence
6,9
10 1
3,5
2 4 8 7
c
9
c
10
c
5
c
2
c
4
c
8
c
6
c
1
, c
3
, c
7
Ideals for irreducible components
2
of X
par
I
u
1
= hu
27
+ 5u
25
+ ··· + b + u, −u
39
− u
38
+ ··· + a − 4u, u
40
+ 2u
39
+ ··· + 4u
2
+ 1i
I
u
2
= hb + u + 1, a, u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 42 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
27
+5u
25
+· · ·+b+u, −u
39
−u
38
+· · ·+a−4u, u
40
+2u
39
+· · ·+4u
2
+1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
u
2
a
1
=
u
2
+ 1
u
2
a
3
=
u
39
+ u
38
+ ··· − 5u
2
+ 4u
−u
27
− 5u
25
+ ··· + 4u
2
− u
a
5
=
u
u
3
+ u
a
2
=
u
21
+ 4u
19
+ ··· − 4u
2
+ 3u
−u
39
− 2u
38
+ ··· − u − 1
a
4
=
−u
39
− u
38
+ ··· + 3u − 1
−2u
39
− 4u
38
+ ··· − 4u
2
− 2
a
8
=
u
4
+ u
2
+ 1
u
4
a
7
=
−u
9
− 2u
7
− 3u
5
− 2u
3
− u
−u
9
− u
7
− u
5
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes
= u
39
+ 9u
38
+ 17u
37
+ 66u
36
+ 94u
35
+ 285u
34
+ 329u
33
+ 852u
32
+ 826u
31
+ 1962u
30
+
1576u
29
+3630u
28
+2362u
27
+5577u
26
+2755u
25
+7286u
24
+2421u
23
+8227u
22
+1323u
21
+
8198u
20
−136u
19
+7289u
18
−1361u
17
+5878u
16
−1976u
15
+4322u
14
−1882u
13
+2864u
12
−
1394u
11
+ 1700u
10
−826u
9
+ 838u
8
−385u
7
+ 329u
6
−157u
5
+ 92u
4
−42u
3
+ 7u
2
−5u + 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
40
− 21u
39
+ ··· − 3u + 1
c
2
, c
4
u
40
+ 3u
39
+ ··· + 3u + 1
c
3
, c
7
u
40
+ u
39
+ ··· − 8u + 4
c
5
, c
9
u
40
− 2u
39
+ ··· + 4u
2
+ 1
c
6
u
40
+ 15u
39
+ ··· + 120u + 16
c
8
, c
10
u
40
− 14u
39
+ ··· − 8u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
40
− y
39
+ ··· + 17y + 1
c
2
, c
4
y
40
− 21y
39
+ ··· − 3y + 1
c
3
, c
7
y
40
− 15y
39
+ ··· − 120y + 16
c
5
, c
9
y
40
+ 14y
39
+ ··· + 8y + 1
c
6
y
40
+ 17y
39
+ ··· + 2016y + 256
c
8
, c
10
y
40
+ 26y
39
+ ··· + 44y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.725993 + 0.653238I
a = −1.77855 + 1.81598I
b = −2.56295 − 0.04821I
0.63968 + 1.74616I 0.044303 − 1.257582I
u = 0.725993 − 0.653238I
a = −1.77855 − 1.81598I
b = −2.56295 + 0.04821I
0.63968 − 1.74616I 0.044303 + 1.257582I
u = 0.657117 + 0.787048I
a = 1.66831 + 0.40061I
b = 0.89334 + 1.41707I
−1.07354 − 2.17702I −2.16670 + 4.43587I
u = 0.657117 − 0.787048I
a = 1.66831 − 0.40061I
b = 0.89334 − 1.41707I
−1.07354 + 2.17702I −2.16670 − 4.43587I
u = 0.096376 + 1.028080I
a = 0.442341 + 0.052565I
b = −0.722317 + 0.146557I
2.32493 − 2.41163I 2.33571 + 3.34704I
u = 0.096376 − 1.028080I
a = 0.442341 − 0.052565I
b = −0.722317 − 0.146557I
2.32493 + 2.41163I 2.33571 − 3.34704I
u = −0.824710 + 0.626683I
a = −1.67414 − 1.41541I
b = −2.39518 + 0.13829I
−1.20323 − 7.65538I −1.63964 + 4.86252I
u = −0.824710 − 0.626683I
a = −1.67414 + 1.41541I
b = −2.39518 − 0.13829I
−1.20323 + 7.65538I −1.63964 − 4.86252I
u = −0.789408 + 0.675423I
a = 1.235380 + 0.007261I
b = 1.156310 − 0.509552I
−3.72005 − 2.44717I −4.96365 + 1.04542I
u = −0.789408 − 0.675423I
a = 1.235380 − 0.007261I
b = 1.156310 + 0.509552I
−3.72005 + 2.44717I −4.96365 − 1.04542I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.386153 + 0.965172I
a = −0.238506 + 0.455641I
b = −0.397991 + 0.639039I
0.74845 − 2.81821I −1.95524 + 6.55211I
u = 0.386153 − 0.965172I
a = −0.238506 − 0.455641I
b = −0.397991 − 0.639039I
0.74845 + 2.81821I −1.95524 − 6.55211I
u = −0.023616 + 1.041760I
a = −1.087150 − 0.838239I
b = 0.393277 − 1.005930I
6.00686 + 1.32070I 7.28134 − 0.72610I
u = −0.023616 − 1.041760I
a = −1.087150 + 0.838239I
b = 0.393277 + 1.005930I
6.00686 − 1.32070I 7.28134 + 0.72610I
u = −0.650732 + 0.672523I
a = 0.172779 + 0.250083I
b = −0.359504 − 0.987978I
1.25887 + 0.68759I −0.543601 + 0.759704I
u = −0.650732 − 0.672523I
a = 0.172779 − 0.250083I
b = −0.359504 + 0.987978I
1.25887 − 0.68759I −0.543601 − 0.759704I
u = 0.095598 + 1.116440I
a = −0.834103 + 0.849030I
b = 0.370183 + 0.684126I
5.21580 − 6.90989I 5.24227 + 6.39245I
u = 0.095598 − 1.116440I
a = −0.834103 − 0.849030I
b = 0.370183 − 0.684126I
5.21580 + 6.90989I 5.24227 − 6.39245I
u = 0.639866 + 0.934630I
a = −0.19632 + 1.43499I
b = −1.28155 + 0.60102I
−0.60920 − 2.86826I −1.22261 + 1.95241I
u = 0.639866 − 0.934630I
a = −0.19632 − 1.43499I
b = −1.28155 − 0.60102I
−0.60920 + 2.86826I −1.22261 − 1.95241I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.777168 + 0.837928I
a = −0.91272 − 1.60741I
b = −2.00597 − 0.19942I
−6.21108 + 0.22925I −5.84725 + 0.24543I
u = −0.777168 − 0.837928I
a = −0.91272 + 1.60741I
b = −2.00597 + 0.19942I
−6.21108 − 0.22925I −5.84725 − 0.24543I
u = −0.762796 + 0.899428I
a = 1.74060 + 0.53261I
b = 2.07413 − 1.05941I
−6.02457 + 5.56367I −5.18066 − 6.01609I
u = −0.762796 − 0.899428I
a = 1.74060 − 0.53261I
b = 2.07413 + 1.05941I
−6.02457 − 5.56367I −5.18066 + 6.01609I
u = −0.651476 + 0.987984I
a = 0.197553 + 0.008658I
b = −0.673463 − 1.012420I
2.21178 + 4.43619I 1.72906 − 5.48285I
u = −0.651476 − 0.987984I
a = 0.197553 − 0.008658I
b = −0.673463 + 1.012420I
2.21178 − 4.43619I 1.72906 + 5.48285I
u = 0.559538 + 1.043730I
a = 0.204110 + 0.051194I
b = −0.743272 + 0.884629I
2.37466 + 0.03317I 2.30074 − 1.92960I
u = 0.559538 − 1.043730I
a = 0.204110 − 0.051194I
b = −0.743272 − 0.884629I
2.37466 − 0.03317I 2.30074 + 1.92960I
u = 0.674430 + 1.003370I
a = 1.99901 − 1.47152I
b = 3.05304 + 1.06309I
1.68055 − 7.12390I 1.84913 + 6.13601I
u = 0.674430 − 1.003370I
a = 1.99901 + 1.47152I
b = 3.05304 − 1.06309I
1.68055 + 7.12390I 1.84913 − 6.13601I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.710235 + 0.337827I
a = 0.496293 − 0.392962I
b = −0.069099 + 0.837495I
0.39800 − 4.72692I −1.63267 + 6.05913I
u = 0.710235 − 0.337827I
a = 0.496293 + 0.392962I
b = −0.069099 − 0.837495I
0.39800 + 4.72692I −1.63267 − 6.05913I
u = −0.705098 + 1.010600I
a = −0.462757 − 1.241560I
b = −1.290960 − 0.160212I
−2.70648 + 8.09252I −2.94350 − 6.08172I
u = −0.705098 − 1.010600I
a = −0.462757 + 1.241560I
b = −1.290960 + 0.160212I
−2.70648 − 8.09252I −2.94350 + 6.08172I
u = −0.703890 + 1.042830I
a = 1.61753 + 1.43395I
b = 2.91214 − 0.75501I
0.05370 + 13.38520I 0.42075 − 9.35928I
u = −0.703890 − 1.042830I
a = 1.61753 − 1.43395I
b = 2.91214 + 0.75501I
0.05370 − 13.38520I 0.42075 + 9.35928I
u = 0.566007 + 0.177460I
a = 0.944370 + 0.216688I
b = 0.239645 − 0.184623I
−1.47568 − 0.52119I −6.28438 + 0.91978I
u = 0.566007 − 0.177460I
a = 0.944370 − 0.216688I
b = 0.239645 + 0.184623I
−1.47568 + 0.52119I −6.28438 − 0.91978I
u = −0.222419 + 0.359701I
a = −0.03404 + 1.68269I
b = −0.589808 − 0.653481I
1.75548 + 0.68997I 4.17661 + 0.16492I
u = −0.222419 − 0.359701I
a = −0.03404 − 1.68269I
b = −0.589808 + 0.653481I
1.75548 − 0.68997I 4.17661 − 0.16492I
8
II. I
u
2
= hb + u + 1, a, u
2
+ u + 1i
(i) Arc colorings
a
6
=
0
u
a
9
=
1
0
a
10
=
1
−u − 1
a
1
=
−u
−u − 1
a
3
=
0
−u − 1
a
5
=
u
u + 1
a
2
=
−u
−2u − 2
a
4
=
0
−u − 1
a
8
=
0
u
a
7
=
0
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4u + 1
9
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u + 1)
2
c
3
, c
6
, c
7
u
2
c
4
(u − 1)
2
c
5
, c
10
u
2
− u + 1
c
8
, c
9
u
2
+ u + 1
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
2
c
3
, c
6
, c
7
y
2
c
5
, c
8
, c
9
c
10
y
2
+ y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.500000 + 0.866025I
a = 0
b = −0.500000 − 0.866025I
1.64493 + 2.02988I 3.00000 − 3.46410I
u = −0.500000 − 0.866025I
a = 0
b = −0.500000 + 0.866025I
1.64493 − 2.02988I 3.00000 + 3.46410I
12
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u + 1)
2
)(u
40
− 21u
39
+ ··· − 3u + 1)
c
2
((u + 1)
2
)(u
40
+ 3u
39
+ ··· + 3u + 1)
c
3
, c
7
u
2
(u
40
+ u
39
+ ··· − 8u + 4)
c
4
((u − 1)
2
)(u
40
+ 3u
39
+ ··· + 3u + 1)
c
5
(u
2
− u + 1)(u
40
− 2u
39
+ ··· + 4u
2
+ 1)
c
6
u
2
(u
40
+ 15u
39
+ ··· + 120u + 16)
c
8
(u
2
+ u + 1)(u
40
− 14u
39
+ ··· − 8u + 1)
c
9
(u
2
+ u + 1)(u
40
− 2u
39
+ ··· + 4u
2
+ 1)
c
10
(u
2
− u + 1)(u
40
− 14u
39
+ ··· − 8u + 1)
13
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y −1)
2
)(y
40
− y
39
+ ··· + 17y + 1)
c
2
, c
4
((y −1)
2
)(y
40
− 21y
39
+ ··· − 3y + 1)
c
3
, c
7
y
2
(y
40
− 15y
39
+ ··· − 120y + 16)
c
5
, c
9
(y
2
+ y + 1)(y
40
+ 14y
39
+ ··· + 8y + 1)
c
6
y
2
(y
40
+ 17y
39
+ ··· + 2016y + 256)
c
8
, c
10
(y
2
+ y + 1)(y
40
+ 26y
39
+ ··· + 44y + 1)
14
