10
107
(K10a
66
)
1
Arc Sequences
6 9 1 8 7 2 10 3 5 4
Solving Sequence
2,9 3,7
6 1 5 10 8 4
c
2
c
6
c
1
c
5
c
9
c
8
c
4
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
8
2u
6
u
5
+ 3u
4
+ 2u
3
2u
2
u + 1, u
7
+ u
5
+ u
4
2u
3
u
2
+ b,
u
7
u
6
+ u
5
+ 2u
4
2u
2
+ a u 1i
I
u
2
= hu
54
+ u
53
+ ··· + 14u + 7, 4.47089 × 10
41
u
53
+ 7.36898 × 10
41
u
52
+ ··· + 1.12922 × 10
42
b 5.30063 × 10
41
,
2.11123 × 10
43
u
53
7.77800 × 10
42
u
52
+ ··· + 7.90452 × 10
42
a 1.62649 × 10
44
i
There are 2 irreducible components with 62 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
2u
6
u
5
+ 3u
4
+ 2u
3
2u
2
u + 1, u
7
+ u
5
+ u
4
2u
3
u
2
+ b, u
7
u
6
+ u
5
+ 2u
4
2u
2
+ a u 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
u
7
+ u
6
u
5
2u
4
+ 2u
2
+ u + 1
u
7
u
5
u
4
+ 2u
3
+ u
2
a
3
=
2u
7
+ 3u
5
+ 3u
4
4u
3
4u
2
+ 2
u
6
+ 2u
4
+ u
3
2u
2
2u + 2
a
7
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
10
=
u
7
u
5
u
4
+ u
3
+ u
2
+ 1
u
7
u
5
2u
4
+ 2u
3
+ 2u
2
1
a
8
=
2u
6
3u
4
2u
3
+ 4u
2
+ 3u 2
u
6
u
5
u
4
+ 3u
2
1
a
4
=
u
7
+ u
5
+ 2u
4
2u
3
2u
2
+ 1
u
6
+ 2u
4
2u
2
u + 2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6u
7
u
6
+ 13u
5
+ 9u
4
15u
3
14u
2
+ 9u + 3
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.929887 0.300978I
a = 1.283114 + 0.114850I
b = 0.11414 1.61519I
3.90365 1.24143I 8.13667 + 0.29040I
u = 0.929887 + 0.300978I
a = 1.283114 0.114850I
b = 0.11414 + 1.61519I
3.90365 + 1.24143I 8.13667 0.29040I
u = 0.819536 0.880313I
a = 0.193792 0.566802I
b = 0.110933 0.652805I
3.61840 3.26075I 5.09230 + 4.26286I
u = 0.819536 + 0.880313I
a = 0.193792 + 0.566802I
b = 0.110933 + 0.652805I
3.61840 + 3.26075I 5.09230 4.26286I
u = 0.612814 0.310228I
a = 2.24044 0.56042I
b = 0.487513 0.687654I
1.19791 2.24783I 2.34914 + 3.96490I
u = 0.612814 + 0.310228I
a = 2.24044 + 0.56042I
b = 0.487513 + 0.687654I
1.19791 + 2.24783I 2.34914 3.96490I
u = 1.136609 0.491905I
a = 1.21734 + 1.01828I
b = 0.484309 0.994840I
0.91267 + 5.73534I 1.12017 7.06636I
u = 1.136609 + 0.491905I
a = 1.21734 1.01828I
b = 0.484309 + 0.994840I
0.91267 5.73534I 1.12017 + 7.06636I
3
II. I
u
2
=
hu
54
+u
53
+· · ·+14u+7, 4.47×10
41
u
53
+7.37×10
41
u
52
+· · ·+1.13×10
42
b
5.30× 10
41
, 2.11×10
43
u
53
7.78 ×10
42
u
52
+· · · +7.90 ×10
42
a 1.63× 10
44
i
(i) Arc colorings
a
2
=
1
0
a
9
=
2.67092u
53
+ 0.983995u
52
+ ··· + 19.4586u + 20.5767
0.395928u
53
0.652574u
52
+ ··· + 5.72839u + 0.469408
a
3
=
1.45501u
53
+ 0.271643u
52
+ ··· + 10.9328u + 13.2661
1.50088u
53
+ 0.182843u
52
+ ··· + 14.4896u + 11.7498
a
7
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
5
=
u
u
3
+ u
a
10
=
2.27605u
53
+ 0.352308u
52
+ ··· + 18.9949u + 16.2195
0.400034u
53
1.00625u
52
+ ··· 0.814728u 5.54554
a
8
=
1.03118u
53
+ 0.626767u
52
+ ··· + 9.32897u + 16.7897
0.938421u
53
+ 0.522019u
52
+ ··· + 9.97411u + 13.9236
a
4
=
1.48598u
53
+ 0.728886u
52
+ ··· + 11.2675u + 14.1439
2.97609u
53
+ 0.415054u
52
+ ··· + 27.7498u + 23.4347
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4.88150u
53
+ 2.21402u
52
+ ··· + 30.7782u + 41.8665
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.180667 0.487156I
a = 1.039446 + 0.669296I
b = 0.130666 0.651090I
0.77663 4.41165I 0.11668 + 2.93869I
u = 1.180667 + 0.487156I
a = 1.039446 0.669296I
b = 0.130666 + 0.651090I
0.77663 + 4.41165I 0.11668 2.93869I
u = 1.151001 0.681898I
a = 1.67980 0.45500I
b = 0.68491 + 1.33958I
0.7043 15.3529I 0.13234 + 9.01820I
u = 1.151001 + 0.681898I
a = 1.67980 + 0.45500I
b = 0.68491 1.33958I
0.7043 + 15.3529I 0.13234 9.01820I
u = 1.132699 0.087147I
a = 0.174872 0.906117I
b = 0.186915 + 1.303552I
6.71595 + 1.65367I 8.08588 1.23488I
u = 1.132699 + 0.087147I
a = 0.174872 + 0.906117I
b = 0.186915 1.303552I
6.71595 1.65367I 8.08588 + 1.23488I
u = 1.068485 0.499039I
a = 1.74680 + 0.84184I
b = 0.91484 1.16843I
1.89398 6.72384I 4.13785 + 10.41753I
u = 1.068485 + 0.499039I
a = 1.74680 0.84184I
b = 0.91484 + 1.16843I
1.89398 + 6.72384I 4.13785 10.41753I
u = 0.989157 0.495426I
a = 1.067117 + 0.767421I
b = 0.548827 + 0.151414I
0.20915 4.60279I 0.54557 + 5.84363I
u = 0.989157 + 0.495426I
a = 1.067117 0.767421I
b = 0.548827 0.151414I
0.20915 + 4.60279I 0.54557 5.84363I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.899764 0.742029I
a = 0.724782 + 0.345508I
b = 0.450150 + 0.291052I
4.02222 2.85318I 6.13116 + 2.92462I
u = 0.899764 + 0.742029I
a = 0.724782 0.345508I
b = 0.450150 0.291052I
4.02222 + 2.85318I 6.13116 2.92462I
u = 0.879867 0.454315I
a = 1.100277 + 0.147502I
b = 0.14202 1.78593I
3.14719 1.85744I 0.22547 + 4.53165I
u = 0.879867 + 0.454315I
a = 1.100277 0.147502I
b = 0.14202 + 1.78593I
3.14719 + 1.85744I 0.22547 4.53165I
u = 0.874847 0.775331I
a = 0.154685 + 0.827917I
b = 0.108880 + 0.350450I
4.11998 2.94109I 7.18387 1.28923I
u = 0.874847 + 0.775331I
a = 0.154685 0.827917I
b = 0.108880 0.350450I
4.11998 + 2.94109I 7.18387 + 1.28923I
u = 0.854958 0.391145I
a = 2.58305 0.83382I
b = 0.060105 + 0.744957I
0.562845 + 1.252740I 1.233700 + 0.528973I
u = 0.854958 + 0.391145I
a = 2.58305 + 0.83382I
b = 0.060105 0.744957I
0.562845 1.252740I 1.233700 0.528973I
u = 0.777980 0.072248I
a = 0.72166 1.26050I
b = 0.860654 0.510820I
0.47962 + 3.24903I 2.21240 6.16822I
u = 0.777980 + 0.072248I
a = 0.72166 + 1.26050I
b = 0.860654 + 0.510820I
0.47962 3.24903I 2.21240 + 6.16822I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.765445 1.034508I
a = 0.449043 0.111414I
b = 0.289712 0.661831I
4.15647 3.52492I 9.67282 + 9.11035I
u = 0.765445 + 1.034508I
a = 0.449043 + 0.111414I
b = 0.289712 + 0.661831I
4.15647 + 3.52492I 9.67282 9.11035I
u = 0.461230 0.959704I
a = 0.715768 + 0.788735I
b = 0.655812 + 1.216395I
2.82560 + 9.37445I 2.26303 5.61235I
u = 0.461230 + 0.959704I
a = 0.715768 0.788735I
b = 0.655812 1.216395I
2.82560 9.37445I 2.26303 + 5.61235I
u = 0.407541 0.536964I
a = 1.43833 + 0.05831I
b = 0.494572 0.085251I
1.46648 + 0.54178I 5.93628 0.17488I
u = 0.407541 + 0.536964I
a = 1.43833 0.05831I
b = 0.494572 + 0.085251I
1.46648 0.54178I 5.93628 + 0.17488I
u = 0.164204 0.530511I
a = 1.30727 1.15076I
b = 0.695616 0.960177I
0.35569 + 2.65328I 0.15342 4.21264I
u = 0.164204 + 0.530511I
a = 1.30727 + 1.15076I
b = 0.695616 + 0.960177I
0.35569 2.65328I 0.15342 + 4.21264I
u = 0.250165 0.961789I
a = 0.281115 0.024481I
b = 0.436774 0.867594I
3.39532 + 0.06056I 6.47502 0.56421I
u = 0.250165 + 0.961789I
a = 0.281115 + 0.024481I
b = 0.436774 + 0.867594I
3.39532 0.06056I 6.47502 + 0.56421I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.495862 0.516272I
a = 0.490181 1.234826I
b = 0.127283 0.982994I
1.44225 + 1.32993I 2.25893 3.81749I
u = 0.495862 + 0.516272I
a = 0.490181 + 1.234826I
b = 0.127283 + 0.982994I
1.44225 1.32993I 2.25893 + 3.81749I
u = 0.503300 0.772891I
a = 0.852963 + 1.003000I
b = 0.355394 + 1.070242I
1.38784 3.43862I 1.22590 + 4.16430I
u = 0.503300 + 0.772891I
a = 0.852963 1.003000I
b = 0.355394 1.070242I
1.38784 + 3.43862I 1.22590 4.16430I
u = 0.602787 0.758609I
a = 1.51785 0.57529I
b = 1.118211 + 0.388876I
5.49306 3.12677I 5.13785 + 1.91103I
u = 0.602787 + 0.758609I
a = 1.51785 + 0.57529I
b = 1.118211 0.388876I
5.49306 + 3.12677I 5.13785 1.91103I
u = 0.752778 0.535726I
a = 1.083709 + 0.022994I
b = 0.541794 + 1.014990I
2.22290 1.39898I 4.85913 0.38785I
u = 0.752778 + 0.535726I
a = 1.083709 0.022994I
b = 0.541794 1.014990I
2.22290 + 1.39898I 4.85913 + 0.38785I
u = 0.873475 0.311207I
a = 0.116159 0.327387I
b = 0.513290 0.460675I
1.49546 + 0.86217I 4.23255 0.90919I
u = 0.873475 + 0.311207I
a = 0.116159 + 0.327387I
b = 0.513290 + 0.460675I
1.49546 0.86217I 4.23255 + 0.90919I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.928292 0.548530I
a = 1.71930 1.45525I
b = 0.356434 + 1.125184I
1.65482 + 5.78575I 2.51560 7.01331I
u = 0.928292 + 0.548530I
a = 1.71930 + 1.45525I
b = 0.356434 1.125184I
1.65482 5.78575I 2.51560 + 7.01331I
u = 0.946417 0.512780I
a = 1.85789 + 0.01886I
b = 0.499584 1.293067I
2.54046 + 2.78962I 3.14255 2.96255I
u = 0.946417 + 0.512780I
a = 1.85789 0.01886I
b = 0.499584 + 1.293067I
2.54046 2.78962I 3.14255 + 2.96255I
u = 1.024259 0.645350I
a = 0.842657 + 0.863048I
b = 1.315114 + 0.239226I
4.21764 + 8.45863I 2.40338 7.53990I
u = 1.024259 + 0.645350I
a = 0.842657 0.863048I
b = 1.315114 0.239226I
4.21764 8.45863I 2.40338 + 7.53990I
u = 1.075856 0.638642I
a = 1.93223 0.18933I
b = 0.472599 + 1.159996I
3.08313 + 8.79179I 2.77233 8.25769I
u = 1.075856 + 0.638642I
a = 1.93223 + 0.18933I
b = 0.472599 1.159996I
3.08313 8.79179I 2.77233 + 8.25769I
u = 1.077881 0.335260I
a = 0.668992 + 0.027582I
b = 0.472068 1.265797I
2.94679 + 0.31182I 2.63431 + 1.82332I
u = 1.077881 + 0.335260I
a = 0.668992 0.027582I
b = 0.472068 + 1.265797I
2.94679 0.31182I 2.63431 1.82332I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.227392 0.690056I
a = 0.989561 + 0.495861I
b = 0.365421 1.174400I
0.50528 + 5.97519I 4.12477 7.22399I
u = 1.227392 + 0.690056I
a = 0.989561 0.495861I
b = 0.365421 + 1.174400I
0.50528 5.97519I 4.12477 + 7.22399I
u = 1.349382 0.002358I
a = 0.035556 0.593174I
b = 0.366982 + 1.181522I
3.96372 6.42189I 3.89989 + 6.62120I
u = 1.349382 + 0.002358I
a = 0.035556 + 0.593174I
b = 0.366982 1.181522I
3.96372 + 6.42189I 3.89989 6.62120I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
8
2u
6
+ ··· u + 1)(u
54
+ u
53
+ ··· + 14u + 7)
c
2
(u
8
+ 4u
6
+ ··· u + 1)(u
54
+ u
53
+ ··· + 9u
2
+ 1)
c
3
(u
8
+ u
7
+ ··· + 4u
2
+ 1)(u
54
+ 2u
53
+ ··· + 75u + 19)
c
4
(u
8
u
5
u
4
+ u + 1)(u
54
+ 5u
53
+ ··· + u + 2)
c
5
(u
8
4u
7
+ 10u
6
17u
5
+ 23u
4
22u
3
+ 14u
2
5u + 1)
(u
54
+ 23u
53
+ ··· + 406u + 49)
c
6
(u
8
2u
6
+ ··· + u + 1)(u
54
+ u
53
+ ··· + 14u + 7)
c
7
(u
8
2u
6
+ ··· + 4u + 1)(u
54
+ 7u
53
+ ··· + 123u + 49)
c
8
(u
8
+ 4u
6
+ ··· + u + 1)(u
54
+ u
53
+ ··· + 9u
2
+ 1)
c
9
(u
8
u
7
u
4
+ u
3
+ 1)(u
54
+ 3u
51
+ ··· 19u + 1)
c
10
(u
8
u
7
+ ··· + 4u
2
+ 1)(u
54
+ 2u
53
+ ··· + 75u + 19)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
8
4y
7
+ 10y
6
17y
5
+ 23y
4
22y
3
+ 14y
2
5y + 1)
(y
54
23y
53
+ ··· 406y + 49)
c
2
, c
8
(y
8
+ 8y
7
+ 26y
6
+ 47y
5
+ 55y
4
+ 42y
3
+ 22y
2
+ 7y + 1)
(y
54
+ 33y
53
+ ··· + 18y + 1)
c
3
, c
10
(y
8
+ 7y
7
+ 22y
6
+ 42y
5
+ 55y
4
+ 47y
3
+ 26y
2
+ 8y + 1)
(y
54
+ 36y
53
+ ··· + 911y + 361)
c
4
(y
8
2y
6
+ ··· y + 1)(y
54
3y
53
+ ··· + 43y + 4)
c
5
(y
8
+ 4y
7
+ 10y
6
+ 23y
5
+ 23y
4
+ 10y
3
+ 22y
2
+ 3y + 1)
(y
54
+ 21y
53
+ ··· + 31458y + 2401)
c
7
(y
8
4y
7
+ 4y
6
+ 3y
5
+ 2y
4
+ 4y
3
+ 4y
2
4y + 1)
(y
54
15y
53
+ ··· 60601y + 2401)
c
9
(y
8
y
7
+ ··· 2y
2
+ 1)(y
54
+ 42y
52
+ ··· 17y + 1)
12