10
73
(K10a
3
)
A knot diagram
1
Linearized knot diagam
8 4 7 6 9 3 10 1 5 2
Solving Sequence
3,6
7 4
5,10
8 2 1 9
c
6
c
3
c
4
c
7
c
2
c
1
c
9
c
5
, c
8
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
42
− u
41
+ ··· + u
2
+ b, −11u
42
− 26u
41
+ ··· + 2a + 15, u
43
+ 3u
42
+ ··· − 3u − 1i
I
u
2
= hb, a
2
− a + 1, u − 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
42
−u
41
+· · ·+u
2
+b, −11u
42
−26u
41
+· · ·+2a+15, u
43
+3u
42
+· · ·−3u−1i
(i) Arc colorings
a
3
=
0
u
a
6
=
1
0
a
7
=
1
−u
2
a
4
=
u
−u
3
+ u
a
5
=
u
3
−u
3
+ u
a
10
=
11
2
u
42
+ 13u
41
+ ··· − 13u −
15
2
u
42
+ u
41
+ ··· − u
3
− u
2
a
8
=
−
1
2
u
42
− u
41
+ ··· + u +
1
2
u
11
− 3u
9
− 2u
8
+ 4u
7
+ 4u
6
− u
5
− 4u
4
− u
3
+ u
a
2
=
−u
3
u
5
− u
3
+ u
a
1
=
5u
42
+ 11u
41
+ ··· − 11u − 6
3
2
u
42
+ 3u
41
+ ··· − 2u −
3
2
a
9
=
11
2
u
42
+ 11u
41
+ ··· − 11u −
13
2
u
41
+ 2u
40
+ ··· − u − 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = −6u
42
− 15u
41
+ ··· + 19u + 13
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
8
u
43
+ 2u
42
+ ··· + 4u
2
− 1
c
2
u
43
− 23u
42
+ ··· + 3u − 1
c
3
, c
6
u
43
+ 3u
42
+ ··· − 3u − 1
c
4
u
43
+ 15u
42
+ ··· − 136u − 16
c
5
, c
9
u
43
− u
42
+ ··· + 8u − 4
c
7
u
43
− 2u
42
+ ··· + 54u − 9
c
10
u
43
+ 20u
42
+ ··· + 8u − 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
8
y
43
+ 20y
42
+ ··· + 8y −1
c
2
y
43
− 3y
42
+ ··· + 23y −1
c
3
, c
6
y
43
− 23y
42
+ ··· + 3y −1
c
4
y
43
+ 23y
42
+ ··· + 4128y −256
c
5
, c
9
y
43
+ 15y
42
+ ··· − 136y −16
c
7
y
43
− 4y
42
+ ··· + 2520y −81
c
10
y
43
+ 8y
42
+ ··· + 140y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.702205 + 0.692426I
a = −0.216763 + 1.063530I
b = −0.000164 − 0.427737I
−5.79250 + 1.33127I −3.13829 − 0.68119I
u = −0.702205 − 0.692426I
a = −0.216763 − 1.063530I
b = −0.000164 + 0.427737I
−5.79250 − 1.33127I −3.13829 + 0.68119I
u = −0.781262 + 0.586254I
a = 0.083513 − 0.751531I
b = −0.318284 + 0.078334I
−2.37041 − 2.31340I 1.61332 + 3.65794I
u = −0.781262 − 0.586254I
a = 0.083513 + 0.751531I
b = −0.318284 − 0.078334I
−2.37041 + 2.31340I 1.61332 − 3.65794I
u = 0.983429 + 0.401988I
a = −0.18299 + 1.51178I
b = 0.95580 − 1.21635I
0.117899 + 0.694763I 2.77512 − 0.93635I
u = 0.983429 − 0.401988I
a = −0.18299 − 1.51178I
b = 0.95580 + 1.21635I
0.117899 − 0.694763I 2.77512 + 0.93635I
u = −0.856054 + 0.662832I
a = −0.459912 + 0.582589I
b = 0.671689 − 0.315858I
−5.34910 − 6.48185I −1.72488 + 7.04551I
u = −0.856054 − 0.662832I
a = −0.459912 − 0.582589I
b = 0.671689 + 0.315858I
−5.34910 + 6.48185I −1.72488 − 7.04551I
u = −0.225042 + 0.862192I
a = −0.295884 − 0.330542I
b = 1.27608 − 1.18759I
−1.69061 + 8.49752I 0.86281 − 6.51033I
u = −0.225042 − 0.862192I
a = −0.295884 + 0.330542I
b = 1.27608 + 1.18759I
−1.69061 − 8.49752I 0.86281 + 6.51033I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.344780 + 0.758252I
a = −0.130381 − 0.811333I
b = 0.921218 − 0.514197I
−4.05606 + 1.05891I −2.88403 − 0.52575I
u = −0.344780 − 0.758252I
a = −0.130381 + 0.811333I
b = 0.921218 + 0.514197I
−4.05606 − 1.05891I −2.88403 + 0.52575I
u = −0.199953 + 0.800457I
a = 0.051467 + 0.372242I
b = −0.93072 + 1.17107I
0.47003 + 3.49797I 3.95877 − 2.64358I
u = −0.199953 − 0.800457I
a = 0.051467 − 0.372242I
b = −0.93072 − 1.17107I
0.47003 − 3.49797I 3.95877 + 2.64358I
u = 1.178400 + 0.107020I
a = 0.396216 − 0.024715I
b = −0.090847 − 0.484893I
0.86535 + 1.34877I 0.663414 + 0.523198I
u = 1.178400 − 0.107020I
a = 0.396216 + 0.024715I
b = −0.090847 + 0.484893I
0.86535 − 1.34877I 0.663414 − 0.523198I
u = −1.113040 + 0.411275I
a = −1.45881 + 1.05016I
b = −0.432177 − 1.242820I
3.14686 − 0.23394I 6.25545 + 1.76917I
u = −1.113040 − 0.411275I
a = −1.45881 − 1.05016I
b = −0.432177 + 1.242820I
3.14686 + 0.23394I 6.25545 − 1.76917I
u = 1.140070 + 0.437496I
a = −0.80318 − 1.92969I
b = −0.35123 + 1.92571I
4.51030 + 2.45703I 9.07720 − 1.39524I
u = 1.140070 − 0.437496I
a = −0.80318 + 1.92969I
b = −0.35123 − 1.92571I
4.51030 − 2.45703I 9.07720 + 1.39524I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.126240 + 0.485857I
a = 0.62977 + 2.31079I
b = 0.62085 − 2.17017I
2.59778 + 7.42216I 5.67783 − 6.16302I
u = 1.126240 − 0.485857I
a = 0.62977 − 2.31079I
b = 0.62085 + 2.17017I
2.59778 − 7.42216I 5.67783 + 6.16302I
u = −1.139980 + 0.455119I
a = 1.12604 − 1.46501I
b = 0.68367 + 1.40137I
4.38701 − 5.48645I 8.10083 + 6.46210I
u = −1.139980 − 0.455119I
a = 1.12604 + 1.46501I
b = 0.68367 − 1.40137I
4.38701 + 5.48645I 8.10083 − 6.46210I
u = 0.769344
a = −0.716816
b = 0.736269
1.12210 9.24310
u = −1.116400 + 0.556388I
a = −0.04192 + 1.41433I
b = −1.30408 − 1.10564I
−1.77735 − 6.01104I 0. + 4.92263I
u = −1.116400 − 0.556388I
a = −0.04192 − 1.41433I
b = −1.30408 + 1.10564I
−1.77735 + 6.01104I 0. − 4.92263I
u = 1.207770 + 0.337329I
a = −1.27016 − 1.08338I
b = 0.27337 + 1.44443I
4.74199 + 0.21154I 9.28481 + 0.I
u = 1.207770 − 0.337329I
a = −1.27016 + 1.08338I
b = 0.27337 − 1.44443I
4.74199 − 0.21154I 9.28481 + 0.I
u = 0.596984 + 0.406248I
a = 1.37593 − 0.78349I
b = −1.384640 − 0.071781I
−1.01538 + 2.84865I 1.39768 − 5.43636I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.596984 − 0.406248I
a = 1.37593 + 0.78349I
b = −1.384640 + 0.071781I
−1.01538 − 2.84865I 1.39768 + 5.43636I
u = 1.253070 + 0.301863I
a = 1.51757 + 0.68122I
b = −0.561583 − 1.231910I
3.01994 − 4.67918I 0. + 5.37573I
u = 1.253070 − 0.301863I
a = 1.51757 − 0.68122I
b = −0.561583 + 1.231910I
3.01994 + 4.67918I 0. − 5.37573I
u = −1.177230 + 0.535254I
a = 0.33951 − 2.06230I
b = 1.28426 + 1.59104I
3.34693 − 8.44363I 0
u = −1.177230 − 0.535254I
a = 0.33951 + 2.06230I
b = 1.28426 − 1.59104I
3.34693 + 8.44363I 0
u = −0.674002 + 0.118500I
a = −0.26643 − 1.82830I
b = −0.006255 − 0.381927I
0.77235 − 2.35753I −0.15632 + 5.03988I
u = −0.674002 − 0.118500I
a = −0.26643 + 1.82830I
b = −0.006255 + 0.381927I
0.77235 + 2.35753I −0.15632 − 5.03988I
u = −1.191540 + 0.559537I
a = −0.06666 + 2.26638I
b = −1.51037 − 1.66131I
1.20075 − 13.70690I 0
u = −1.191540 − 0.559537I
a = −0.06666 − 2.26638I
b = −1.51037 + 1.66131I
1.20075 + 13.70690I 0
u = −0.025551 + 0.621606I
a = −0.721932 + 0.454368I
b = 0.017381 + 1.186670I
1.39154 + 1.44262I 5.16219 − 3.23191I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.025551 − 0.621606I
a = −0.721932 − 0.454368I
b = 0.017381 − 1.186670I
1.39154 − 1.44262I 5.16219 + 3.23191I
u = 0.176403 + 0.591173I
a = 1.253380 − 0.348319I
b = −0.682100 − 1.225820I
−0.03125 − 3.16118I 2.63487 + 2.30647I
u = 0.176403 − 0.591173I
a = 1.253380 + 0.348319I
b = −0.682100 + 1.225820I
−0.03125 + 3.16118I 2.63487 − 2.30647I
9
II. I
u
2
= hb, a
2
− a + 1, u − 1i
(i) Arc colorings
a
3
=
0
1
a
6
=
1
0
a
7
=
1
−1
a
4
=
1
0
a
5
=
1
0
a
10
=
a
0
a
8
=
a
−1
a
2
=
−1
1
a
1
=
0
a
a
9
=
a
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4a + 7
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
, c
10
u
2
− u + 1
c
2
, c
3
(u + 1)
2
c
4
, c
5
, c
9
u
2
c
6
(u − 1)
2
c
8
u
2
+ u + 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
, c
8
c
10
y
2
+ y + 1
c
2
, c
3
, c
6
(y − 1)
2
c
4
, c
5
, c
9
y
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.00000
a = 0.500000 + 0.866025I
b = 0
1.64493 − 2.02988I 9.00000 + 3.46410I
u = 1.00000
a = 0.500000 − 0.866025I
b = 0
1.64493 + 2.02988I 9.00000 − 3.46410I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
2
− u + 1)(u
43
+ 2u
42
+ ··· + 4u
2
− 1)
c
2
((u + 1)
2
)(u
43
− 23u
42
+ ··· + 3u − 1)
c
3
((u + 1)
2
)(u
43
+ 3u
42
+ ··· − 3u − 1)
c
4
u
2
(u
43
+ 15u
42
+ ··· − 136u − 16)
c
5
, c
9
u
2
(u
43
− u
42
+ ··· + 8u − 4)
c
6
((u − 1)
2
)(u
43
+ 3u
42
+ ··· − 3u − 1)
c
7
(u
2
− u + 1)(u
43
− 2u
42
+ ··· + 54u − 9)
c
8
(u
2
+ u + 1)(u
43
+ 2u
42
+ ··· + 4u
2
− 1)
c
10
(u
2
− u + 1)(u
43
+ 20u
42
+ ··· + 8u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
8
(y
2
+ y + 1)(y
43
+ 20y
42
+ ··· + 8y − 1)
c
2
((y − 1)
2
)(y
43
− 3y
42
+ ··· + 23y − 1)
c
3
, c
6
((y − 1)
2
)(y
43
− 23y
42
+ ··· + 3y − 1)
c
4
y
2
(y
43
+ 23y
42
+ ··· + 4128y − 256)
c
5
, c
9
y
2
(y
43
+ 15y
42
+ ··· − 136y − 16)
c
7
(y
2
+ y + 1)(y
43
− 4y
42
+ ··· + 2520y − 81)
c
10
(y
2
+ y + 1)(y
43
+ 8y
42
+ ··· + 140y − 1)
15
