11a
49
(K11a
49
)
1
Arc Sequences
5 1 7 2 4 10 3 11 6 8 9
Solving Sequence
1,5
2 3 4
6,9
11 8 7 10
c
1
c
2
c
4
c
5
c
11
c
8
c
7
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= ha
4
3a
3
+ 5a
2
6a + 4, a
3
+ a
2
a + 2u + 2, a
3
+ 3a
2
+ 2b 3a + 2i
I
u
2
= hu
4
u
3
+ u
2
+ 1, b + 1, u
3
u
2
+ a + ui
I
u
3
= hu
60
4u
59
+ ··· 6u + 1, 2.82387 × 10
17
u
59
1.26103 × 10
18
u
58
+ ··· + 8.05298 × 10
17
b 3.80595 × 10
16
,
1.38918 × 10
18
u
59
+ 5.27349 × 10
18
u
58
+ ··· + 8.05298 × 10
17
a 1.11954 × 10
18
i
There are 3 irreducible components with 68 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
= ha
4
3a
3
+ 5a
2
6a + 4, a
3
+ a
2
a + 2u + 2, a
3
+ 3a
2
+ 2b 3a + 2i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
2
a
3
1
2
a
2
+
1
2
a 1
a
2
=
1
1
2
a
3
1
2
a
2
+
1
2
a
a
3
=
1
2
a
3
+
1
2
a
2
1
2
a + 1
1
2
a
3
1
2
a
2
+
1
2
a
a
4
=
1
2
a
3
+
1
2
a
2
1
2
a + 1
1
2
a
3
1
2
a
2
+
1
2
a
a
6
=
1
0
a
9
=
a
1
2
a
3
3
2
a
2
+
3
2
a 1
a
11
=
a
2
2a + 3
1
2
a
3
+
3
2
a
2
3
2
a
a
8
=
1
2
a
3
1
2
a
2
1
2
a + 1
1
2
a
3
+
3
2
a
2
3
2
a
a
7
=
1
2
a
3
1
2
a
2
1
2
a + 1
1
2
a
3
+
3
2
a
2
3
2
a
a
10
=
1
2
a
3
+
3
2
a
2
1
2
a + 1
1
2
a
3
3
2
a
2
+
3
2
a 1
a
10
=
1
2
a
3
+
3
2
a
2
1
2
a + 1
1
2
a
3
3
2
a
2
+
3
2
a 1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 + 0.866025I
a = 0.19098 1.40126I
b = 1.61803
8.88264 2.02988I 15.5000 2.3454I
u = 0.500000 0.866025I
a = 0.19098 + 1.40126I
b = 1.61803
8.88264 + 2.02988I 15.5000 + 2.3454I
u = 0.500000 0.866025I
a = 1.30902 0.53523I
b = 0.618034
0.98696 + 2.02988I 15.5000 9.2736I
u = 0.500000 + 0.866025I
a = 1.30902 + 0.53523I
b = 0.618034
0.98696 2.02988I 15.5000 + 9.2736I
3
II. I
u
2
= hu
4
u
3
+ u
2
+ 1, b + 1, u
3
u
2
+ a + ui
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
6
=
u
3
u
3
u
2
1
a
9
=
u
3
+ u
2
u
1
a
11
=
u
3
+ u
2
u + 1
1
a
8
=
1
0
a
7
=
u
3
u
3
u
2
1
a
10
=
u
3
+ u
2
u
1
a
10
=
u
3
+ u
2
u
1
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.351808 0.720342I
a = 0.547424 + 1.120873I
b = 1.00000
1.85594 + 1.41510I 15.1414 7.6022I
u = 0.351808 + 0.720342I
a = 0.547424 1.120873I
b = 1.00000
1.85594 1.41510I 15.1414 + 7.6022I
u = 0.851808 0.911292I
a = 0.547424 + 0.585652I
b = 1.00000
5.14581 3.16396I 0.358581 + 1.047693I
u = 0.851808 + 0.911292I
a = 0.547424 0.585652I
b = 1.00000
5.14581 + 3.16396I 0.358581 1.047693I
5
III. I
u
3
=
hu
60
4u
59
+ · · · 6u + 1, 2.82 ×10
17
u
59
1.26 × 10
18
u
58
+ · · · + 8.05× 10
17
b
3.81× 10
16
, 1.39× 10
18
u
59
+5.27 ×10
18
u
58
+· · · +8.05 ×10
17
a 1.12× 10
18
i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
6
=
u
3
u
5
+ u
3
+ u
a
9
=
1.72505u
59
6.54850u
58
+ ··· + 26.8373u + 1.39022
0.350661u
59
+ 1.56592u
58
+ ··· 2.55053u + 0.0472614
a
11
=
3.45045u
59
11.8719u
58
+ ··· + 24.7014u 0.108151
2.09736u
59
+ 8.23633u
58
+ ··· 11.2979u + 1.81095
a
8
=
2.69854u
59
+ 9.78721u
58
+ ··· 2.72116u + 4.45599
1.00661u
59
4.15918u
58
+ ··· + 6.00527u 1.47261
a
7
=
1.69377u
59
5.26308u
58
+ ··· + 15.2624u + 0.956700
1.51199u
59
+ 6.56263u
58
+ ··· 11.1193u + 1.69377
a
10
=
0.152462u
59
+ 0.553640u
58
+ ··· + 17.1437u + 2.97986
0.110576u
59
+ 0.554686u
58
+ ··· 0.308427u 0.243818
a
10
=
0.152462u
59
+ 0.553640u
58
+ ··· + 17.1437u + 2.97986
0.110576u
59
+ 0.554686u
58
+ ··· 0.308427u 0.243818
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.846021 0.263574I
a = 0.730426 + 0.642423I
b = 1.372607 0.240045I
3.38098 + 5.24726I 7.94356 5.01660I
u = 0.846021 + 0.263574I
a = 0.730426 0.642423I
b = 1.372607 + 0.240045I
3.38098 5.24726I 7.94356 + 5.01660I
u = 0.834410 0.714804I
a = 1.294419 0.462921I
b = 1.370482 + 0.211215I
3.80109 2.07837I 9.80025 + 1.22105I
u = 0.834410 + 0.714804I
a = 1.294419 + 0.462921I
b = 1.370482 0.211215I
3.80109 + 2.07837I 9.80025 1.22105I
u = 0.738082 1.010081I
a = 0.43546 + 2.16500I
b = 1.40212 0.25943I
4.71228 + 7.96352I 10.63500 5.90852I
u = 0.738082 + 1.010081I
a = 0.43546 2.16500I
b = 1.40212 + 0.25943I
4.71228 7.96352I 10.63500 + 5.90852I
u = 0.692724 0.774214I
a = 0.83410 + 1.45179I
b = 0.153691 0.545787I
1.086085 + 0.692045I 4.81030 + 0.29508I
u = 0.692724 + 0.774214I
a = 0.83410 1.45179I
b = 0.153691 + 0.545787I
1.086085 0.692045I 4.81030 0.29508I
u = 0.680719 0.940890I
a = 0.31149 1.70748I
b = 0.250540 + 0.658416I
0.56981 + 4.60985I 6.69053 5.91571I
u = 0.680719 + 0.940890I
a = 0.31149 + 1.70748I
b = 0.250540 0.658416I
0.56981 4.60985I 6.69053 + 5.91571I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.649242 0.172450I
a = 0.505415 0.972953I
b = 0.166593 + 0.624695I
1.53389 + 2.11161I 1.83843 4.55656I
u = 0.649242 + 0.172450I
a = 0.505415 + 0.972953I
b = 0.166593 0.624695I
1.53389 2.11161I 1.83843 + 4.55656I
u = 0.608501 0.942456I
a = 0.57515 2.85515I
b = 1.216357 + 0.116183I
1.99576 + 3.02877I 9.18914 0.37480I
u = 0.608501 + 0.942456I
a = 0.57515 + 2.85515I
b = 1.216357 0.116183I
1.99576 3.02877I 9.18914 + 0.37480I
u = 0.550539 0.791247I
a = 1.05956 + 2.01758I
b = 1.050236 0.110830I
1.44417 + 1.61127I 11.6485 15.8847I
u = 0.550539 + 0.791247I
a = 1.05956 2.01758I
b = 1.050236 + 0.110830I
1.44417 1.61127I 11.6485 + 15.8847I
u = 0.508484 1.131971I
a = 0.356209 + 0.908446I
b = 1.401417 + 0.163779I
6.11056 0.33405I 11.54717 + 0.63818I
u = 0.508484 + 1.131971I
a = 0.356209 0.908446I
b = 1.401417 0.163779I
6.11056 + 0.33405I 11.54717 0.63818I
u = 0.474455 0.963893I
a = 1.013159 0.103342I
b = 0.320933 0.297553I
0.66653 + 1.65828I 3.28323 + 3.22527I
u = 0.474455 + 0.963893I
a = 1.013159 + 0.103342I
b = 0.320933 + 0.297553I
0.66653 1.65828I 3.28323 3.22527I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.399913 0.190398I
a = 1.205214 0.371567I
b = 1.119037 + 0.184637I
1.23369 + 0.89939I 5.06416 + 0.58389I
u = 0.399913 + 0.190398I
a = 1.205214 + 0.371567I
b = 1.119037 0.184637I
1.23369 0.89939I 5.06416 0.58389I
u = 0.311462 0.787706I
a = 0.856946 0.064626I
b = 0.1037943 0.0707771I
0.33181 + 1.48905I 3.19515 4.46795I
u = 0.311462 + 0.787706I
a = 0.856946 + 0.064626I
b = 0.1037943 + 0.0707771I
0.33181 1.48905I 3.19515 + 4.46795I
u = 0.153304 1.063440I
a = 0.317563 0.201072I
b = 0.387799 + 0.734072I
2.46678 + 4.55995I 10.63324 6.84099I
u = 0.153304 + 1.063440I
a = 0.317563 + 0.201072I
b = 0.387799 0.734072I
2.46678 4.55995I 10.63324 + 6.84099I
u = 0.147390 1.174605I
a = 1.30719 + 0.74994I
b = 1.45691 0.27854I
8.38340 + 8.24100I 13.2725 6.1295I
u = 0.147390 + 1.174605I
a = 1.30719 0.74994I
b = 1.45691 + 0.27854I
8.38340 8.24100I 13.2725 + 6.1295I
u = 0.089235 1.015874I
a = 2.45653 0.04547I
b = 1.360304 + 0.086702I
4.90942 + 2.39733I 12.95918 3.22106I
u = 0.089235 + 1.015874I
a = 2.45653 + 0.04547I
b = 1.360304 0.086702I
4.90942 2.39733I 12.95918 + 3.22106I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.030561 0.936305I
a = 0.789457 + 0.651261I
b = 0.636405 0.600822I
3.26453 + 0.03248I 13.91224 + 0.15923I
u = 0.030561 + 0.936305I
a = 0.789457 0.651261I
b = 0.636405 + 0.600822I
3.26453 0.03248I 13.91224 0.15923I
u = 0.109251
a = 5.52438
b = 0.489956
0.859418 11.8178
u = 0.171612 1.008352I
a = 1.22301 1.35653I
b = 1.53424 + 0.14331I
10.47485 2.55878I 16.6852 + 0.9369I
u = 0.171612 + 1.008352I
a = 1.22301 + 1.35653I
b = 1.53424 0.14331I
10.47485 + 2.55878I 16.6852 0.9369I
u = 0.598878 0.877781I
a = 0.062044 1.267175I
b = 1.68096 0.03245I
8.49817 2.35434I 3.03721 + 7.33944I
u = 0.598878 + 0.877781I
a = 0.062044 + 1.267175I
b = 1.68096 + 0.03245I
8.49817 + 2.35434I 3.03721 7.33944I
u = 0.615422
a = 1.72722
b = 1.45673
7.21613 11.9743
u = 0.704927 0.962970I
a = 0.864289 0.177634I
b = 0.960690 + 0.700258I
1.22350 4.74489I 7.08737 + 4.52380I
u = 0.704927 + 0.962970I
a = 0.864289 + 0.177634I
b = 0.960690 0.700258I
1.22350 + 4.74489I 7.08737 4.52380I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.706871 0.997331I
a = 0.55158 + 1.99008I
b = 1.46184 0.24029I
0.04969 7.86068I 7.49688 + 6.26874I
u = 0.706871 + 0.997331I
a = 0.55158 1.99008I
b = 1.46184 + 0.24029I
0.04969 + 7.86068I 7.49688 6.26874I
u = 0.727256 1.019210I
a = 0.76545 + 1.39316I
b = 0.318501 0.946527I
3.14485 10.38846I 5.72246 + 8.19302I
u = 0.727256 + 1.019210I
a = 0.76545 1.39316I
b = 0.318501 + 0.946527I
3.14485 + 10.38846I 5.72246 8.19302I
u = 0.730510 1.064478I
a = 0.19549 2.18323I
b = 1.46811 + 0.38180I
2.5536 15.1714I 9.11132 + 8.93146I
u = 0.730510 + 1.064478I
a = 0.19549 + 2.18323I
b = 1.46811 0.38180I
2.5536 + 15.1714I 9.11132 8.93146I
u = 0.746749 0.750172I
a = 0.21661 + 1.66501I
b = 1.022216 0.626579I
1.87463 0.78688I 5.48587 + 0.89304I
u = 0.746749 + 0.750172I
a = 0.21661 1.66501I
b = 1.022216 + 0.626579I
1.87463 + 0.78688I 5.48587 0.89304I
u = 0.773804 0.697114I
a = 0.269799 0.363113I
b = 1.40946 + 0.27587I
0.95728 + 2.24717I 5.61761 1.12563I
u = 0.773804 + 0.697114I
a = 0.269799 + 0.363113I
b = 1.40946 0.27587I
0.95728 2.24717I 5.61761 + 1.12563I
11
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.775127 0.938595I
a = 0.582419 1.068642I
b = 0.397265 + 0.581288I
6.05175 4.76483I 1.11123 + 4.56468I
u = 0.775127 + 0.938595I
a = 0.582419 + 1.068642I
b = 0.397265 0.581288I
6.05175 + 4.76483I 1.11123 4.56468I
u = 0.810322 0.816113I
a = 0.161021 + 0.807105I
b = 0.307636 0.644850I
6.42634 1.17254I 0.566200 + 1.294911I
u = 0.810322 + 0.816113I
a = 0.161021 0.807105I
b = 0.307636 + 0.644850I
6.42634 + 1.17254I 0.566200 1.294911I
u = 0.826891 0.686761I
a = 0.46435 1.34579I
b = 0.260347 + 0.914658I
4.15844 + 4.56410I 3.69697 3.37945I
u = 0.826891 + 0.686761I
a = 0.46435 + 1.34579I
b = 0.260347 0.914658I
4.15844 4.56410I 3.69697 + 3.37945I
u = 0.886564 0.634360I
a = 0.830791 + 0.709189I
b = 1.43351 0.37170I
1.23187 + 9.17979I 7.33609 4.63134I
u = 0.886564 + 0.634360I
a = 0.830791 0.709189I
b = 1.43351 + 0.37170I
1.23187 9.17979I 7.33609 + 4.63134I
u = 0.893193 0.920215I
a = 0.784615 0.685257I
b = 1.182512 + 0.025488I
4.58752 3.28588I 14.2270 + 3.8304I
u = 0.893193 + 0.920215I
a = 0.784615 + 0.685257I
b = 1.182512 0.025488I
4.58752 + 3.28588I 14.2270 3.8304I
12
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
4
u
3
+ u
2
+ 1)(u
60
+ 4u
59
+ ··· + 6u + 1)
c
2
, c
5
(u
2
+ u + 1)
2
(u
4
+ u
3
+ ··· + 2u + 1)(u
60
+ 20u
59
+ ··· 82u + 1)
c
3
u
4
(u
4
u
3
+ ··· 2u + 1)(u
60
+ 2u
59
+ ··· + 16u + 16)
c
4
(u
2
u + 1)
2
(u
4
+ u
3
+ u
2
+ 1)(u
60
+ 4u
59
+ ··· + 6u + 1)
c
6
u
4
(u
2
+ u 1)
2
(u
60
+ 3u
59
+ ··· 24u 16)
c
7
u
4
(u
4
+ u
3
+ ··· + 2u + 1)(u
60
+ 2u
59
+ ··· + 16u + 16)
c
8
(u 1)
4
(u
2
+ u 1)
2
(u
60
+ 7u
59
+ ··· 6u 1)
c
9
u
4
(u
2
u 1)
2
(u
60
+ 3u
59
+ ··· 24u 16)
c
10
, c
11
(u + 1)
4
(u
2
u 1)
2
(u
60
+ 7u
59
+ ··· 6u 1)
13
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
2
(y
4
+ y
3
+ ··· + 2y + 1)(y
60
+ 20y
59
+ ··· 82y + 1)
c
2
, c
5
(y
2
+ y + 1)
2
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
60
+ 44y
59
+ ··· 7010y + 1)
c
3
, c
7
y
4
(y
4
+ 5y
3
+ ··· + 2y + 1)(y
60
+ 30y
59
+ ··· + 1408y + 256)
c
6
, c
9
y
4
(y
2
3y + 1)
2
(y
60
33y
59
+ ··· 576y + 256)
c
8
1.0000000000000(1y 1.0000000000000)
4
(1y
2
3.0000000000000y + 1.0000000000000)
2
(1.00y
60
57.0y
59
+ ··· 48.0y + 1.00)
c
10
1.0000000000000(1y 1.00000000000000)
4
(1y
2
3.0000000000000y + 1.00000000000000)
2
(1.00y
60
57.0y
59
+ ··· 48.0y + 1.00)
c
11
1.0000000000000(1y 1.0000000000000)
4
(1y
2
3.0000000000000y + 1.0000000000000)
2
(1.00y
60
57.0y
59
+ ··· 48.0y + 1.00)
14