10
118
(K10a
88
)
1
Arc Sequences
8 6 1 2 9 10 4 5 3 7
Solving Sequence
5,8
9
2,6
1 4 3 7 10
c
8
c
5
c
1
c
4
c
3
c
7
c
10
c
2
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
8
+ 4u
7
+ 10u
6
+ 16u
5
+ 15u
4
+ 8u
3
+ u
2
u 1,
7u
7
+ 22u
6
+ 49u
5
+ 65u
4
+ 40u
3
+ 16u
2
+ 5a + 4u + 6,
4u
7
14u
6
33u
5
50u
4
40u
3
22u
2
+ 5b 3u + 3i
I
u
2
= hu
56
+ 3u
55
+ ··· 23u + 1,
6.82989 × 10
168
u
55
2.30077 × 10
169
u
54
+ ··· + 3.18715 × 10
168
b 2.74292 × 10
167
,
3.95296 × 10
169
u
55
+ 1.15025 × 10
170
u
54
+ ··· + 3.18715 × 10
168
a 1.57535 × 10
170
i
There are 2 irreducible components with 64 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
= hu
8
+4u
7
+· · ·u1, 7u
7
+22u
6
+· · ·+5a+6, 4u
7
14u
6
+· · ·+5b+3i
(i) Arc colorings
a
5
=
0
u
a
8
=
7
5
u
7
22
5
u
6
+ ···
4
5
u
6
5
4
5
u
7
+
14
5
u
6
+ ··· +
3
5
u
3
5
a
9
=
7
5
u
7
22
5
u
6
+ ···
4
5
u
6
5
1
5
u
7
+
1
5
u
6
+ ··· +
2
5
u +
3
5
a
2
=
1
0
a
6
=
1
5
u
7
+
6
5
u
6
+ ···
23
5
u
7
5
1
5
u
7
+
1
5
u
6
+ ··· +
7
5
u +
3
5
a
1
=
1
5
u
7
+
11
5
u
6
+ ··· +
17
5
u +
3
5
4
5
u
7
+
9
5
u
6
+ ···
12
5
u
3
5
a
4
=
u
u
a
3
=
8
5
u
7
+
33
5
u
6
+ ··· +
16
5
u +
4
5
3
5
u
7
13
5
u
6
+ ···
6
5
u +
1
5
a
7
=
u
7
3u
6
6u
5
7u
4
2u
3
u 2
2
5
u
7
+
7
5
u
6
+ ··· +
4
5
u +
1
5
a
10
=
3
5
u
7
3
5
u
6
+ ··· +
24
5
u
4
5
1
5
u
7
6
5
u
6
+ ···
7
5
u +
2
5
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
7
28u
6
66u
5
95u
4
74u
3
30u
2
+ 3u + 4
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.41225
a = 0.200595
b = 0.747139
4.69721 8.72999
u = 0.767995 0.666287I
a = 0.508133 1.088906I
b = 1.13851 + 1.06522I
1.62267 + 2.99663I 2.80411 6.12718I
u = 0.767995 + 0.666287I
a = 0.508133 + 1.088906I
b = 1.13851 1.06522I
1.62267 2.99663I 2.80411 + 6.12718I
u = 0.50000 1.75726I
a = 0.044875 + 0.652463I
b = 0.564069 0.825728I
6.39156I 8.17644I
u = 0.50000 + 1.75726I
a = 0.044875 0.652463I
b = 0.564069 + 0.825728I
6.39156I 8.17644I
u = 0.232005 0.666287I
a = 1.48817 + 0.58927I
b = 0.468348 + 0.438200I
1.62267 + 2.99663I 2.80411 6.12718I
u = 0.232005 + 0.666287I
a = 1.48817 0.58927I
b = 0.468348 0.438200I
1.62267 2.99663I 2.80411 + 6.12718I
u = 0.412253
a = 3.15077
b = 1.33844
4.69721 8.72999
3
II. I
u
2
= hu
56
+ 3u
55
+ · · · 23u + 1, 6.83 × 10
168
u
55
2.30 × 10
169
u
54
+
· · · + 3.19 × 10
168
b 2.74 × 10
167
, 3.95 × 10
169
u
55
+ 1.15 × 10
170
u
54
+ · · · +
3.19 × 10
168
a 1.58 × 10
170
i
(i) Arc colorings
a
5
=
0
u
a
8
=
12.4028u
55
36.0902u
54
+ ··· 1342.02u + 49.4282
2.14295u
55
+ 7.21890u
54
+ ··· + 47.7260u + 0.0860620
a
9
=
12.4028u
55
36.0902u
54
+ ··· 1342.02u + 49.4282
4.03769u
55
+ 13.7281u
54
+ ··· + 85.8492u 1.03221
a
2
=
1
0
a
6
=
33.4160u
55
+ 112.827u
54
+ ··· + 1195.99u 72.1648
4.02724u
55
13.0159u
54
+ ··· 252.557u + 13.7718
a
1
=
26.3503u
55
86.2454u
54
+ ··· 1554.87u + 97.6109
0.433894u
55
0.367947u
54
+ ··· 203.451u + 8.98716
a
4
=
u
u
a
3
=
8.50724u
55
+ 28.3882u
54
+ ··· 94.8859u + 59.0027
2.41988u
55
7.10554u
54
+ ··· 245.324u + 8.60202
a
7
=
14.4351u
55
42.9938u
54
+ ··· 1396.17u + 51.3365
4.17529u
55
+ 14.1225u
54
+ ··· + 101.877u 1.82226
a
10
=
20.4131u
55
58.9377u
54
+ ··· 2959.14u + 198.353
0.245067u
55
0.109465u
54
+ ··· 119.627u + 3.16003
(ii) Obstruction class = 1
(iii) Cusp Shapes = 32.4574u
55
+ 97.4359u
54
+ ··· + 3415.56u 174.213
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.83724
a = 0.539542
b = 1.28361
3.28334 1.95804
u = 1.234157 0.567369I
a = 0.159065 0.457452I
b = 0.330263 + 0.747920I
5.38278 + 1.94709I 9.07863 3.78322I
u = 1.234157 + 0.567369I
a = 0.159065 + 0.457452I
b = 0.330263 0.747920I
5.38278 1.94709I 9.07863 + 3.78322I
u = 1.20127
a = 1.24340
b = 0.299877
1.54290 6.38462
u = 1.16823 1.56942I
a = 0.155570 + 0.525956I
b = 0.987700 0.648175I
2.03150 + 6.02280I 3.73094 6.75893I
u = 1.16823 + 1.56942I
a = 0.155570 0.525956I
b = 0.987700 + 0.648175I
2.03150 6.02280I 3.73094 + 6.75893I
u = 1.07432 1.36291I
a = 0.327597 + 0.689372I
b = 0.495818 0.985480I
1.18098 + 5.45507I 4.29936 4.75401I
u = 1.07432 + 1.36291I
a = 0.327597 0.689372I
b = 0.495818 + 0.985480I
1.18098 5.45507I 4.29936 + 4.75401I
u = 0.97319 1.32442I
a = 0.282435 0.696343I
b = 0.874453 + 0.768327I
1.91385 + 4.83850I 2.53419 6.95729I
u = 0.97319 + 1.32442I
a = 0.282435 + 0.696343I
b = 0.874453 0.768327I
1.91385 4.83850I 2.53419 + 6.95729I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.962342 0.676541I
a = 0.689895 0.262727I
b = 0.867481 + 0.465833I
3.60092 + 1.79783I 4.19052 2.40869I
u = 0.962342 + 0.676541I
a = 0.689895 + 0.262727I
b = 0.867481 0.465833I
3.60092 1.79783I 4.19052 + 2.40869I
u = 0.856742 0.694261I
a = 0.758989 + 0.691101I
b = 1.63614 0.85817I
3.60092 + 1.79783I 4.19052 2.40869I
u = 0.856742 + 0.694261I
a = 0.758989 0.691101I
b = 1.63614 + 0.85817I
3.60092 1.79783I 4.19052 + 2.40869I
u = 0.735334 1.174977I
a = 0.389510 0.807225I
b = 1.16393 + 1.16318I
2.03150 + 6.02280I 3.73094 6.75893I
u = 0.735334 + 1.174977I
a = 0.389510 + 0.807225I
b = 1.16393 1.16318I
2.03150 6.02280I 3.73094 + 6.75893I
u = 0.730117 0.548940I
a = 0.566769 + 0.452477I
b = 0.531582 0.418877I
1.26494 + 1.26950I 4.50876 0.89106I
u = 0.730117 + 0.548940I
a = 0.566769 0.452477I
b = 0.531582 + 0.418877I
1.26494 1.26950I 4.50876 + 0.89106I
u = 0.717229 0.283512I
a = 0.840307 0.676792I
b = 0.614585 + 0.660088I
3.06725 + 0.91106I 2.78612 2.04256I
u = 0.717229 + 0.283512I
a = 0.840307 + 0.676792I
b = 0.614585 0.660088I
3.06725 0.91106I 2.78612 + 2.04256I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.498094 0.903498I
a = 0.28993 + 1.46329I
b = 0.695004 0.172107I
5.06898 + 2.94565I 6.46008 3.65784I
u = 0.498094 + 0.903498I
a = 0.28993 1.46329I
b = 0.695004 + 0.172107I
5.06898 2.94565I 6.46008 + 3.65784I
u = 0.494241 1.313962I
a = 0.019080 1.012746I
b = 0.907848 + 0.325017I
2.56248 + 7.32114I 3.13433 7.29187I
u = 0.494241 + 1.313962I
a = 0.019080 + 1.012746I
b = 0.907848 0.325017I
2.56248 7.32114I 3.13433 + 7.29187I
u = 0.268110 0.634954I
a = 1.310042 + 0.329790I
b = 0.606940 + 0.092167I
1.17763 + 1.90833I 2.60907 2.19068I
u = 0.268110 + 0.634954I
a = 1.310042 0.329790I
b = 0.606940 0.092167I
1.17763 1.90833I 2.60907 + 2.19068I
u = 0.162599 1.215955I
a = 0.0437655 + 0.0905220I
b = 0.671481 0.714028I
1.91385 + 4.83850I 2.53419 6.95729I
u = 0.162599 + 1.215955I
a = 0.0437655 0.0905220I
b = 0.671481 + 0.714028I
1.91385 4.83850I 2.53419 + 6.95729I
u = 0.050804 0.543574I
a = 0.110165 + 0.413409I
b = 0.783480 + 0.627461I
1.26494 + 1.26950I 4.50876 0.89106I
u = 0.050804 + 0.543574I
a = 0.110165 0.413409I
b = 0.783480 0.627461I
1.26494 1.26950I 4.50876 + 0.89106I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.0825027
a = 16.9254
b = 1.36704
3.28334 1.95804
u = 0.234535 0.184663I
a = 5.09042 + 0.29576I
b = 0.417667 0.587372I
2.12732 2.99186I 13.6584 + 6.9170I
u = 0.234535 + 0.184663I
a = 5.09042 0.29576I
b = 0.417667 + 0.587372I
2.12732 + 2.99186I 13.6584 6.9170I
u = 0.241391 0.195151I
a = 1.57991 + 2.79050I
b = 0.246039 1.330326I
1.17763 1.90833I 2.60907 + 2.19068I
u = 0.241391 + 0.195151I
a = 1.57991 2.79050I
b = 0.246039 + 1.330326I
1.17763 + 1.90833I 2.60907 2.19068I
u = 0.337369 0.654550I
a = 0.63381 + 1.55304I
b = 1.048226 0.400499I
3.06725 0.91106I 2.78612 + 2.04256I
u = 0.337369 + 0.654550I
a = 0.63381 1.55304I
b = 1.048226 + 0.400499I
3.06725 + 0.91106I 2.78612 2.04256I
u = 0.392381 0.245133I
a = 1.59086 2.00173I
b = 0.44552 + 1.55785I
2.56248 7.32114I 3.13433 + 7.29187I
u = 0.392381 + 0.245133I
a = 1.59086 + 2.00173I
b = 0.44552 1.55785I
2.56248 + 7.32114I 3.13433 7.29187I
u = 0.711761 0.476270I
a = 0.57001 + 1.29958I
b = 0.768372 0.912758I
3.96282I 12.0335I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.711761 + 0.476270I
a = 0.57001 1.29958I
b = 0.768372 + 0.912758I
3.96282I 12.0335I
u = 0.87217 1.98277I
a = 0.260010 + 0.481550I
b = 0.584292 0.088844I
1.18098 + 5.45507I 4.29936 4.75401I
u = 0.87217 + 1.98277I
a = 0.260010 0.481550I
b = 0.584292 + 0.088844I
1.18098 5.45507I 4.29936 + 4.75401I
u = 0.884782 0.578822I
a = 0.710976 + 1.197379I
b = 1.34569 0.94875I
2.12732 2.99186I 13.6584 + 6.9170I
u = 0.884782 + 0.578822I
a = 0.710976 1.197379I
b = 1.34569 + 0.94875I
2.12732 + 2.99186I 13.6584 6.9170I
u = 0.963651 0.662726I
a = 0.416599 0.875936I
b = 0.951068 + 0.899353I
5.00346 9.83371I 3.71372 + 7.72363I
u = 0.963651 + 0.662726I
a = 0.416599 + 0.875936I
b = 0.951068 0.899353I
5.00346 + 9.83371I 3.71372 7.72363I
u = 0.994968 0.036107I
a = 0.273483 0.811050I
b = 0.964815 + 1.009115I
5.06898 + 2.94565I 6.46008 3.65784I
u = 0.994968 + 0.036107I
a = 0.273483 + 0.811050I
b = 0.964815 1.009115I
5.06898 2.94565I 6.46008 + 3.65784I
u = 1.04075 1.16900I
a = 0.552115 0.556202I
b = 0.768710 + 0.080853I
5.38278 + 1.94709I 9.07863 3.78322I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.04075 + 1.16900I
a = 0.552115 + 0.556202I
b = 0.768710 0.080853I
5.38278 1.94709I 9.07863 + 3.78322I
u = 1.09944 1.02141I
a = 0.461267 0.891526I
b = 1.33347 + 0.88382I
5.00346 9.83371I 3.71372 + 7.72363I
u = 1.09944 + 1.02141I
a = 0.461267 + 0.891526I
b = 1.33347 0.88382I
5.00346 + 9.83371I 3.71372 7.72363I
u = 1.25812 1.19692I
a = 0.395794 + 0.774371I
b = 1.29911 0.91025I
15.5452I 8.47183I
u = 1.25812 + 1.19692I
a = 0.395794 0.774371I
b = 1.29911 + 0.91025I
15.5452I 8.47183I
u = 1.54115
a = 0.938846
b = 1.13343
1.54290 6.38462
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
8
u
6
+ ··· u
2
+ 1)(u
56
+ 3u
55
+ ··· 86u 29)
c
2
(u
8
u
7
+ ··· 2u 1)(u
56
u
54
+ ··· 12u + 1)
c
3
(u
8
+ 4u
7
+ 10u
6
+ 16u
5
+ 15u
4
+ 8u
3
+ u
2
u 1)
(u
56
+ 3u
55
+ ··· 23u + 1)
c
4
(u
8
4u
7
+ 10u
6
16u
5
+ 15u
4
8u
3
+ u
2
+ u 1)
(u
56
+ 3u
55
+ ··· 23u + 1)
c
5
, c
10
(u
8
3u
6
+ ··· 3u + 1)(u
56
+ u
55
+ ··· 21u 1)
c
6
, c
8
(u
8
3u
6
+ ··· + 3u + 1)(u
56
+ u
55
+ ··· 21u 1)
c
7
(u
8
u
6
+ ··· u
2
+ 1)(u
56
+ 3u
55
+ ··· 86u 29)
c
9
(u
8
+ u
7
+ ··· + 2u 1)(u
56
u
54
+ ··· 12u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
(y
8
2y
7
+ 7y
6
12y
5
+ 5y
4
12y
3
+ 7y
2
2y + 1)
(y
56
13y
55
+ ··· 26246y + 841)
c
2
, c
9
(y
8
+ y
7
y
6
13y
5
6y
4
+ 13y
3
+ 9y
2
+ 2y + 1)
(y
56
2y
55
+ ··· 526y + 1)
c
3
, c
4
(y
8
+ 4y
7
+ 2y
6
18y
5
5y
4
22y
3
13y
2
3y + 1)
(y
56
+ 5y
55
+ ··· 155y + 1)
c
5
, c
8
(y
8
6y
7
+ 17y
6
31y
5
+ 42y
4
45y
3
+ 35y
2
15y + 1)
(y
56
41y
55
+ ··· 91y + 1)
c
6
(1.68718 + 1.00000y)(0.0802538 + 1.00000y)(1.00000y
2
2.61067y + 2.03839)(1.00000y
2
1.99444y + 2.22802)(1.00000y
2
+ 0.372539y + 1.62618)
(1.00y
56
41.0y
55
+ ··· 91.0y + 1.00)
c
10
(1.68718 + 1.00000y)(0.0802538 + 1.00000y)(1.00000y
2
2.61067y + 2.03839)(1.00000y
2
1.99444y + 2.22802)(1.00000y
2
+ 0.372539y + 1.62618)
(1.00y
56
41.0y
55
+ ··· 91.0y + 1.00)
12