10
81
(K10a
7
)
1
Arc Sequences
4 1 6 2 8 3 10 5 7 9
Solving Sequence
2,5
4 1
3,9
8 6 10 7
c
4
c
1
c
2
c
8
c
5
c
10
c
7
c
3
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
3
+ 2b
2
+ b + 1, u 1, b
2
2b + a 1i
I
u
2
= hu
3
+ u
2
1, a, u
2
+ b u 1i
I
u
3
= hu
48
+ 5u
47
+ ··· 10u + 1, 3.65379 × 10
15
u
47
+ 1.68839 × 10
16
u
46
+ ··· + 1.31625 × 10
15
a 2.67452 × 10
15
,
1.06095 × 10
16
u
47
+ 4.18753 × 10
16
u
46
+ ··· + 2.63249 × 10
15
b + 1.00335 × 10
16
i
There are 3 irreducible components with 54 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hb
3
+ 2b
2
+ b + 1, u 1, b
2
2b + a 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
1
a
4
=
1
1
a
1
=
0
1
a
3
=
1
1
a
9
=
b
2
+ 2b + 1
b
a
8
=
b
2
+ 2b + 1
b
2
+ 3b + 1
a
6
=
b
2
+ b 1
b
2
+ b 1
a
10
=
b
2
+ b 1
2
a
7
=
b
2
+ b 1
b
2
+ b 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = b
2
6b 3
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.569840
b = 1.75488
0.531480 10.6089
u = 1.00000
a = 0.215080 1.307141I
b = 0.122561 0.744862I
4.66906 2.82812I 2.80443 + 4.65175I
u = 1.00000
a = 0.215080 + 1.307141I
b = 0.122561 + 0.744862I
4.66906 + 2.82812I 2.80443 4.65175I
3
II. I
u
2
= hu
3
+ u
2
1, a, u
2
+ b u 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
u
2
+ u 1
a
9
=
0
u
2
+ u + 1
a
8
=
0
u
2
+ u + 1
a
6
=
0
u
a
10
=
u
2
+ 1
u + 1
a
7
=
u
2
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
2
8u 4
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.877439 0.744862I
a = 0
b = 0.337641 + 0.562280I
4.66906 2.82812I 2.80443 + 4.65175I
u = 0.877439 + 0.744862I
a = 0
b = 0.337641 0.562280I
4.66906 + 2.82812I 2.80443 4.65175I
u = 0.754878
a = 0
b = 2.32472
0.531480 10.6089
5
III. I
u
3
=
hu
48
+5u
47
+· · ·10u+1, 3.65×10
15
u
47
+1.69×10
16
u
46
+· · ·+1.32×10
15
a
2.67 × 10
15
, 1.06 × 10
16
u
47
+ 4.19 × 10
16
u
46
+ · · · + 2.63 × 10
15
b + 1.00 × 10
16
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
4
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
u
2
+ 1
u
4
a
9
=
2.77592u
47
12.8274u
46
+ ··· + 53.0196u + 2.03193
4.03021u
47
15.9071u
46
+ ··· + 31.5402u 3.81140
a
8
=
2.77592u
47
12.8274u
46
+ ··· + 53.0196u + 2.03193
3.93583u
47
16.7106u
46
+ ··· + 44.8386u 4.86364
a
6
=
0.200750u
47
1.29134u
46
+ ··· + 21 .5018u + 1.57008
4.13421u
47
+ 14.2589u
46
+ ··· 9.67206u + 0.828554
a
10
=
0.974080u
47
+ 5.17265u
46
+ ··· 28.2304u 1.21807
2.29529u
47
+ 10.2501u
46
+ ··· 26.5711u + 2.99681
a
7
=
0.974080u
47
5.17265u
46
+ ··· + 28.2304u + 1.21807
0.492456u
47
+ 2.32849u
46
+ ··· 3.35958u + 0.0322954
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
2572719667737379
1316245742897122
u
47
14240246736967463
1316245742897122
u
46
+ ···+
66418667522566225
1316245742897122
u +
3102450479488985
658122871448561
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.187100 0.637571I
a = 0.049780 1.058724I
b = 1.64839 1.75706I
14.4927I 9.26368I
u = 1.187100 + 0.637571I
a = 0.049780 + 1 .058724I
b = 1.64839 + 1.75706I
14.4927I 9.26368I
u = 1.158618 0.585509I
a = 0.042040 + 1 .067658I
b = 1.32130 + 1 .61006I
2.54767 8.71683I 2.88659 + 5.91299I
u = 1.158618 + 0.585509I
a = 0.042040 1.067658I
b = 1.32130 1.61006I
2.54767 + 8.71683I 2.88659 5.91299I
u = 1.098245 0.602404I
a = 0.967678 0.081308I
b = 0.147213 1.231224I
2.34804 7.85171I 1.49492 + 6.74189I
u = 1.098245 + 0.602404I
a = 0.967678 + 0 .081308I
b = 0.147213 + 1.231224I
2.34804 + 7.85171I 1.49492 6.74189I
u = 1.046731 0.598380I
a = 0.148835 0.892766I
b = 0.71948 2.04636I
3.29646 5.08791I 1.19601 + 5.66025I
u = 1.046731 + 0.598380I
a = 0.148835 + 0.892766I
b = 0.71948 + 2.04636I
3.29646 + 5.08791I 1.19601 5.66025I
u = 1.027891 0.366302I
a = 0.466177 + 1 .228826I
b = 0.323832 + 0 .905178I
5.20077 4.17900I 3.36906 + 7.53383I
u = 1.027891 + 0.366302I
a = 0.466177 1.228826I
b = 0.323832 0.905178I
5.20077 + 4.17900I 3.36906 7.53383I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.992673 0.539998I
a = 0.894990 + 0.318885I
b = 0.256174 + 0 .951147I
0.88639 2.97344I 0.29359 + 2.64448I
u = 0.992673 + 0.539998I
a = 0.894990 0.318885I
b = 0.256174 0.951147I
0.88639 + 2.97344I 0.29359 2.64448I
u = 0.964917 0.798665I
a = 0.538968 + 0 .243579I
b = 0.622086 0.661144I
4.55335 1.97419I 0.52111 3.87774I
u = 0.964917 + 0.798665I
a = 0.538968 0.243579I
b = 0.622086 + 0 .661144I
4.55335 + 1.97419I 0.52111 + 3.87774I
u = 0.935499 0.280058I
a = 0.73582 1.29839I
b = 0.048068 0.720474I
4.55335 + 1.97419I 0.52111 + 3.87774I
u = 0.935499 + 0.280058I
a = 0.73582 + 1.29839I
b = 0.048068 + 0.720474I
4.55335 1.97419I 0.52111 3.87774I
u = 0.747136 0.877281I
a = 0.234949 0.752941I
b = 1.115984 0.463169I
5.20077 4.17900I 3.36906 + 7.53383I
u = 0.747136 + 0.877281I
a = 0.234949 + 0.752941I
b = 1.115984 + 0.463169I
5.20077 + 4.17900I 3.36906 7.53383I
u = 0.628205 0.596659I
a = 0.354378 + 1.017384I
b = 0.431414 + 0 .884380I
2.00090 1.53468I 2.12390 + 3.90788I
u = 0.628205 + 0.596659I
a = 0.354378 1.017384I
b = 0.431414 0.884380I
2.00090 + 1.53468I 2.12390 3.90788I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.516978 0.722434I
a = 1.149628 + 0 .062167I
b = 0.612599 0.514059I
4.87326 + 0.03227I 4.84666 0.67896I
u = 0.516978 + 0.722434I
a = 1.149628 0.062167I
b = 0.612599 + 0.514059I
4.87326 0.03227I 4.84666 + 0.67896I
u = 0.425885 0.773654I
a = 0.051068 1.256804I
b = 0.62148 1.36994I
4.33954 + 2.65713I 5.08315 1.96927I
u = 0.425885 + 0.773654I
a = 0.051068 + 1.256804I
b = 0.62148 + 1.36994I
4.33954 2.65713I 5.08315 + 1.96927I
u = 0.351537 0.949778I
a = 1.285140 0.256220I
b = 0.362866 1.221306I
2.54767 + 8.71683I 2.88659 5.91299I
u = 0.351537 + 0.949778I
a = 1.285140 + 0 .256220I
b = 0.362866 + 1 .221306I
2.54767 8.71683I 2.88659 + 5.91299I
u = 0.295606 0.828875I
a = 1.368420 + 0.169038I
b = 0.299861 + 0.697053I
3.47198I 2.47118I
u = 0.295606 + 0.828875I
a = 1.368420 0.169038I
b = 0.299861 0.697053I
3.47198I 2.47118I
u = 0.0760954
a = 6.62320
b = 0.738020
1.20368 8.97039
u = 0.249189 0.602859I
a = 1.48355 + 0.53026I
b = 0.645003 0.646869I
2.00090 + 1.53468I 2.12390 3.90788I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.249189 + 0 .602859I
a = 1.48355 0.53026I
b = 0.645003 + 0.646869I
2.00090 1.53468I 2.12390 + 3.90788I
u = 0.547224 0.659382I
a = 1.253170 0.381509I
b = 0.61140 + 1 .30675I
0.88639 2.97344I 0.29359 + 2.64448I
u = 0.547224 + 0 .659382I
a = 1.253170 + 0 .381509I
b = 0.61140 1.30675I
0.88639 + 2.97344I 0.29359 2.64448I
u = 0.833904
a = 0.329839
b = 0.733994
1.20368 8.97039
u = 0.852801 0.288192I
a = 0.148093 + 0 .715516I
b = 1.72329 + 2.16023I
0.675636 + 0.515505I 2.57655 6.02720I
u = 0.852801 + 0 .288192I
a = 0.148093 0.715516I
b = 1.72329 2.16023I
0.675636 0.515505I 2.57655 + 6.02720I
u = 0.958701 0.411863I
a = 0.890834 0.020631I
b = 0.59268 + 1.58586I
2.50599I 3.68111I
u = 0.958701 + 0 .411863I
a = 0.890834 + 0 .020631I
b = 0.59268 1.58586I
2.50599I 3.68111I
u = 1.035418 0.586063I
a = 0.042843 + 1.084994I
b = 2.07306 + 1.29053I
2.34804 + 7.85171I 1.49492 6.74189I
u = 1.035418 + 0 .586063I
a = 0.042843 1.084994I
b = 2.07306 1.29053I
2.34804 7.85171I 1.49492 + 6.74189I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.079988 0.482069I
a = 0.102791 0.988019I
b = 1.69726 1.18884I
4.33954 + 2.65713I 5.08315 1.96927I
u = 1.079988 + 0 .482069I
a = 0.102791 + 0 .988019I
b = 1.69726 + 1 .18884I
4.33954 2.65713I 5.08315 + 1.96927I
u = 1.116725 0.138646I
a = 0.596241 0.275408I
b = 1.08450 1.35720I
0.675636 0.515505I 2.57655 + 6.02720I
u = 1.116725 + 0 .138646I
a = 0.596241 + 0.275408I
b = 1.08450 + 1 .35720I
0.675636 + 0.515505I 2.57655 6.02720I
u = 1.249239 0.262371I
a = 0.358392 0.817084I
b = 0.502811 0.767159I
4.87326 0.03227I 4.84666 + 0.67896I
u = 1.249239 + 0 .262371I
a = 0.358392 + 0 .817084I
b = 0.502811 + 0 .767159I
4.87326 + 0.03227I 4.84666 0.67896I
u = 1.332735 0.180321I
a = 0.488274 + 0.784071I
b = 0.035466 + 0 .492429I
3.29646 5.08791I 1.19601 + 5.66025I
u = 1.332735 + 0 .180321I
a = 0.488274 0.784071I
b = 0.035466 0.492429I
3.29646 + 5.08791I 1.19601 5.66025I
11
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
9
(u 1)
3
(u
3
+ u
2
1)(u
48
+ 5u
47
+ ··· 10u + 1)
c
2
(u + 1)
3
(u
3
+ u
2
+ 2u + 1)(u
48
+ 23u
47
+ ··· + 180u + 1)
c
3
, c
8
u
3
(u
3
u
2
+ 2u 1)(u
48
+ 2u
47
+ ··· 28u 8)
c
4
, c
7
(u + 1)
3
(u
3
u
2
+ 1)(u
48
+ 5u
47
+ ··· 10u + 1)
c
5
, c
6
u
3
(u
3
+ u
2
+ 2u + 1)(u
48
+ 2u
47
+ ··· 28u 8)
c
10
(u 1)
3
(u
3
u
2
+ 2u 1)(u
48
+ 23u
47
+ ··· + 180u + 1)
12
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
, c
7
c
9
(y 1)
3
(y
3
y
2
+ 2y 1)(y
48
23y
47
+ ··· 180y + 1)
c
2
, c
10
(y 1)
3
(y
3
+ 3y
2
+ 2y 1)(y
48
+ 9y
47
+ ··· 29816y + 1)
c
3
, c
5
, c
6
c
8
y
3
(y
3
+ 3y
2
+ 2y 1)(y
48
+ 24y
47
+ ··· 464y + 64)
13