10
48
(K10a
79
)
A knot diagram
1
Linearized knot diagam
4 7 1 2 9 10 3 5 6 8
Solving Sequence
5,8
9 6 10
1,3
7 2 4
c
8
c
5
c
9
c
10
c
7
c
2
c
4
c
1
, c
3
, c
6
Ideals for irreducible components
2
of X
par
I
u
1
= hu
25
− 14u
23
+ ··· + b − 1, −u
24
+ u
23
+ ··· + a + 2, u
26
− 2u
25
+ ··· + 3u + 1i
I
u
2
= hb, a − u − 1, u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 28 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
25
−14u
23
+· · ·+b−1, −u
24
+u
23
+· · ·+a+2, u
26
−2u
25
+· · ·+3u+1i
(i) Arc colorings
a
5
=
0
u
a
8
=
1
0
a
9
=
1
−u
2
a
6
=
u
−u
3
+ u
a
10
=
−u
2
+ 1
u
4
− 2u
2
a
1
=
u
4
− 3u
2
+ 1
u
4
− 2u
2
a
3
=
u
24
− u
23
+ ··· + 4u − 2
−u
25
+ 14u
23
+ ··· + 5u + 1
a
7
=
−u
3
+ 2u
u
5
− 3u
3
+ u
a
2
=
u
24
− u
23
+ ··· + u − 3
u
25
− 14u
23
+ ··· − 2u − 1
a
4
=
u
24
− u
23
+ ··· + 2u − 2
u
14
− 8u
12
+ ··· − u
2
+ 2u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 3u
24
− u
23
− 41u
22
+ 8u
21
+ 237u
20
− 2u
19
− 754u
18
− 180u
17
+
1435u
16
+ 802u
15
− 1634u
14
− 1608u
13
+ 954u
12
+ 1718u
11
+ 24u
10
− 1034u
9
− 412u
8
+
379u
7
+ 265u
6
− 70u
5
− 147u
4
+ 10u
3
+ 32u
2
+ 18u − 7
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
4
u
26
− 3u
25
+ ··· − 9u
2
− 1
c
2
, c
7
u
26
− u
25
+ ··· + 17u
2
− 4
c
5
, c
6
, c
8
c
9
u
26
− 2u
25
+ ··· + 3u + 1
c
10
u
26
+ 6u
25
+ ··· + 57u − 9
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
y
26
− 25y
25
+ ··· + 18y + 1
c
2
, c
7
y
26
− 15y
25
+ ··· − 136y + 16
c
5
, c
6
, c
8
c
9
y
26
− 30y
25
+ ··· − 19y + 1
c
10
y
26
+ 6y
25
+ ··· − 3159y + 81
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.668864 + 0.598628I
a = −1.61808 + 1.14418I
b = −1.31336 − 0.55274I
−6.97144 − 7.45946I −3.34661 + 6.43325I
u = −0.668864 − 0.598628I
a = −1.61808 − 1.14418I
b = −1.31336 + 0.55274I
−6.97144 + 7.45946I −3.34661 − 6.43325I
u = 1.198900 + 0.140779I
a = −0.217750 − 0.235577I
b = −1.276500 − 0.100979I
−3.39138 − 0.04941I −2.26185 − 0.23755I
u = 1.198900 − 0.140779I
a = −0.217750 + 0.235577I
b = −1.276500 + 0.100979I
−3.39138 + 0.04941I −2.26185 + 0.23755I
u = −0.586941 + 0.484436I
a = 1.86292 − 1.18885I
b = 1.051360 + 0.358584I
−0.89453 − 3.84444I −0.49259 + 7.28090I
u = −0.586941 − 0.484436I
a = 1.86292 + 1.18885I
b = 1.051360 − 0.358584I
−0.89453 + 3.84444I −0.49259 − 7.28090I
u = −0.283620 + 0.683381I
a = 1.65331 − 0.64793I
b = 1.330650 − 0.398492I
−8.11148 + 3.15061I −5.88075 − 0.93673I
u = −0.283620 − 0.683381I
a = 1.65331 + 0.64793I
b = 1.330650 + 0.398492I
−8.11148 − 3.15061I −5.88075 + 0.93673I
u = 0.486887 + 0.485193I
a = 0.722742 − 0.089263I
b = −0.151101 − 1.041920I
−3.27468 + 1.70414I −2.66466 − 3.89699I
u = 0.486887 − 0.485193I
a = 0.722742 + 0.089263I
b = −0.151101 + 1.041920I
−3.27468 − 1.70414I −2.66466 + 3.89699I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.622264 + 0.175930I
a = −0.280102 − 0.073287I
b = 0.338732 + 0.457385I
1.166700 + 0.399409I 7.28789 − 1.42640I
u = 0.622264 − 0.175930I
a = −0.280102 + 0.073287I
b = 0.338732 − 0.457385I
1.166700 − 0.399409I 7.28789 + 1.42640I
u = −0.345528 + 0.459270I
a = −2.06279 + 0.67901I
b = −0.939749 + 0.103171I
−1.59551 + 0.48344I −4.02832 + 0.08458I
u = −0.345528 − 0.459270I
a = −2.06279 − 0.67901I
b = −0.939749 − 0.103171I
−1.59551 − 0.48344I −4.02832 − 0.08458I
u = 1.51750 + 0.08533I
a = 1.165270 + 0.679850I
b = 1.069430 − 0.265042I
4.66926 + 1.10360I 0.162083 − 0.384354I
u = 1.51750 − 0.08533I
a = 1.165270 − 0.679850I
b = 1.069430 + 0.265042I
4.66926 − 1.10360I 0.162083 + 0.384354I
u = −1.53355 + 0.12553I
a = −0.351346 + 0.586908I
b = 0.393229 − 1.143620I
3.48759 − 3.82064I 0.89607 + 2.40126I
u = −1.53355 − 0.12553I
a = −0.351346 − 0.586908I
b = 0.393229 + 1.143620I
3.48759 + 3.82064I 0.89607 − 2.40126I
u = 1.56506 + 0.13930I
a = −0.92214 − 1.07989I
b = −1.134790 + 0.523946I
6.35171 + 6.10679I 2.97747 − 5.14405I
u = 1.56506 − 0.13930I
a = −0.92214 + 1.07989I
b = −1.134790 − 0.523946I
6.35171 − 6.10679I 2.97747 + 5.14405I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.58442 + 0.05231I
a = 0.214222 − 0.374662I
b = −0.346659 + 0.743600I
8.75625 − 1.27302I 7.34031 + 0.88258I
u = −1.58442 − 0.05231I
a = 0.214222 + 0.374662I
b = −0.346659 − 0.743600I
8.75625 + 1.27302I 7.34031 − 0.88258I
u = 1.59031 + 0.18503I
a = 0.687931 + 1.166390I
b = 1.28311 − 0.68193I
0.59945 + 10.37890I −0.23306 − 5.66856I
u = 1.59031 − 0.18503I
a = 0.687931 − 1.166390I
b = 1.28311 + 0.68193I
0.59945 − 10.37890I −0.23306 + 5.66856I
u = −1.67318
a = −0.394205
b = 0.856853
6.22196 −3.01540
u = −0.282820
a = −3.31419
b = −0.465579
−1.22611 −10.4970
7
II. I
u
2
= hb, a − u − 1, u
2
+ u − 1i
(i) Arc colorings
a
5
=
0
u
a
8
=
1
0
a
9
=
1
u − 1
a
6
=
u
−u + 1
a
10
=
u
−u
a
1
=
0
−u
a
3
=
u + 1
0
a
7
=
1
0
a
2
=
u + 1
0
a
4
=
u + 1
u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5
8
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u − 1)
2
c
2
, c
7
u
2
c
3
, c
4
(u + 1)
2
c
5
, c
6
u
2
− u − 1
c
8
, c
9
, c
10
u
2
+ u − 1
9
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
(y −1)
2
c
2
, c
7
y
2
c
5
, c
6
, c
8
c
9
, c
10
y
2
− 3y + 1
10
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.618034
a = 1.61803
b = 0
−0.657974 5.00000
u = −1.61803
a = −0.618034
b = 0
7.23771 5.00000
11
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
2
)(u
26
− 3u
25
+ ··· − 9u
2
− 1)
c
2
, c
7
u
2
(u
26
− u
25
+ ··· + 17u
2
− 4)
c
3
, c
4
((u + 1)
2
)(u
26
− 3u
25
+ ··· − 9u
2
− 1)
c
5
, c
6
(u
2
− u − 1)(u
26
− 2u
25
+ ··· + 3u + 1)
c
8
, c
9
(u
2
+ u − 1)(u
26
− 2u
25
+ ··· + 3u + 1)
c
10
(u
2
+ u − 1)(u
26
+ 6u
25
+ ··· + 57u − 9)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
((y −1)
2
)(y
26
− 25y
25
+ ··· + 18y + 1)
c
2
, c
7
y
2
(y
26
− 15y
25
+ ··· − 136y + 16)
c
5
, c
6
, c
8
c
9
(y
2
− 3y + 1)(y
26
− 30y
25
+ ··· − 19y + 1)
c
10
(y
2
− 3y + 1)(y
26
+ 6y
25
+ ··· − 3159y + 81)
13
