11a
353
(K11a
353
)
1
Arc Sequences
8 6 9 1 10 2 11 3 4 7 5
Solving Sequence
7,10
11
2,8
1 6 3 9 5 4
c
10
c
7
c
1
c
6
c
2
c
8
c
5
c
4
c
3
, c
9
, c
11
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu 1, a + 1, 2b + 1i
I
u
2
= h4b
2
+ 4b 1, a + 1, u + 1i
I
u
3
= ha
4
a
2
+ 1, a
3
+ u, a
3
+ a
2
+ 3b 5a 2i
I
u
4
= hu
28
+ u
27
+ ··· 2u 1,
221044616913u
27
+ 290155714074u
26
+ ··· + 952258493696a 2321054567783,
792700470455u
27
225798479966u
26
+ ··· + 1904516987392b + 1298199346297i
I
u
5
= hu
40
3u
39
+ ··· 16u + 13,
3.76258 × 10
32
u
39
+ 8.77507 × 10
31
u
38
+ ··· + 1.02068 × 10
33
b 1.25131 × 10
34
,
2.55200 × 10
33
u
39
6.60767 × 10
33
u
38
+ ··· + 1.32688 × 10
34
a 3.80580 × 10
34
i
There are 5 irreducible components with 75 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 1, a + 1, 2b + 1i
(i) Arc colorings
a
7
=
0
1
a
10
=
1
1
2
a
11
=
1
3
2
a
2
=
1
0
a
8
=
1
1
2
a
1
=
1
2
1
4
a
6
=
1
1
a
3
=
2
1
a
9
=
1
1
2
a
5
=
3
2
5
4
a
4
=
2
1
a
4
=
2
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 = 1.00000
a = 1.00000
b = 0.500000
3.28987 7.50000
3
II. I
u
2
= h4b
2
+ 4b 1, a + 1, u + 1i
(i) Arc colorings
a
7
=
0
1
a
10
=
1
b
a
11
=
1
b 1
a
2
=
1
0
a
8
=
1
b
a
1
=
b + 1
b
1
4
a
6
=
1
1
a
3
=
2
1
a
9
=
4b + 3
b + 1
a
5
=
b 2
0.75
a
4
=
2b 3
b
1
2
a
4
=
2b 3
b
1
2
(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 = 1.00000
a = 1.00000
b = 1.20711
8.22467 20.0000
u = 1.00000
a = 1.00000
b = 0.207107
8.22467 20.0000
5
III. I
u
3
= ha
4
a
2
+ 1, a
3
+ u, a
3
+ a
2
+ 3b 5a 2i
(i) Arc colorings
a
7
=
0
a
3
a
10
=
a
1
3
a
3
1
3
a
2
+
5
3
a +
2
3
a
11
=
a
1
3
a
3
1
3
a
2
+
2
3
a +
2
3
a
2
=
1
0
a
8
=
a
3
+ a
1
3
a
3
2
3
a
2
+
1
3
a +
1
3
a
1
=
1
3
a
3
1
3
a
2
+
1
3
a +
2
3
2
3
a
3
1
3
a
2
2
3
a +
2
3
a
6
=
a
3
a
3
a
3
=
0
1
a
9
=
a
3
+ a
2
3
a
3
2
3
a
2
+
4
3
a +
1
3
a
5
=
1
3
a
3
2
3
a
2
+
1
3
a +
1
3
4
3
a
2
+
1
3
a
4
=
a
2
+ 1
1
3
a
3
2
3
a
2
1
3
a +
1
3
a
4
=
a
2
+ 1
1
3
a
3
2
3
a
2
1
3
a +
1
3
(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 = 1.00000I
a = 0.866025 0.500000I
b = 0.943376 0.788675I
3.28987 + 2.02988I 6.00000 3.46410I
u = 1.00000I
a = 0.866025 + 0.500000I
b = 0.943376 + 0.788675I
3.28987 2.02988I 6.00000 + 3.46410I
u = 1.00000I
a = 0.866025 0.500000I
b = 1.94338 0.21132I
3.28987 2.02988I 6.00000 + 3.46410I
u = 1.00000I
a = 0.866025 + 0.500000I
b = 1.94338 + 0.21132I
3.28987 + 2.02988I 6.00000 3.46410I
7
IV.
I
u
4
= hu
28
+u
27
+· · ·2u1, 2.21×10
11
u
27
+2.90×10
11
u
26
+· · ·+9.52×10
11
a
2.32 × 10
12
, 7.93× 10
11
u
27
2.26 × 10
11
u
26
+ · · · + 1.90 × 10
12
b + 1.30 × 10
12
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.232127u
27
0.304703u
26
+ ··· 3.30997u + 2.43742
0.416221u
27
+ 0.118559u
26
+ ··· + 1.53325u 0.681642
a
11
=
0.232127u
27
0.304703u
26
+ ··· 3.30997u + 2.43742
0.231896u
27
+ 0.203248u
26
+ ··· + 1.15597u 0.754218
a
2
=
1
0
a
8
=
0.722887u
27
0.240821u
26
+ ··· 4.85080u + 1.53399
0.225086u
27
+ 0.236383u
26
+ ··· + 1.82237u 0.648348
a
1
=
0.464023u
27
0.507951u
26
+ ··· 4.46594u + 3.19164
0.172306u
27
+ 0.100247u
26
+ ··· + 1.72388u 0.972307
a
6
=
u
u
a
3
=
u
2
+ 1
u
2
a
9
=
0.954582u
27
0.561026u
26
+ ··· 7.04483u + 2.50931
0.0710282u
27
+ 0.137612u
26
+ ··· + 1.45714u 0.792144
a
5
=
0.754218u
27
+ 0.522322u
26
+ ··· + 6.43192u 2.66441
0.218089u
27
0.277679u
26
+ ··· 0.866064u + 1.00409
a
4
=
0.710290u
27
+ 0.342533u
26
+ ··· + 8.69552u 3.12843
0.290148u
27
0.350011u
26
+ ··· 1.49376u + 1.17639
a
4
=
0.710290u
27
+ 0.342533u
26
+ ··· + 8.69552u 3.12843
0.290148u
27
0.350011u
26
+ ··· 1.49376u + 1.17639
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.33082
a = 0.501080
b = 0.703758
6.93788 10.0440
u = 1.27260 0.62076I
a = 0.566121 + 0.549832I
b = 0.280520 0.219653I
8.94292 + 2.37104I 16.4575 4.6309I
u = 1.27260 + 0.62076I
a = 0.566121 0.549832I
b = 0.280520 + 0.219653I
8.94292 2.37104I 16.4575 + 4.6309I
u = 0.739750
a = 0.308737
b = 0.924542
6.49503 15.0761
u = 0.576628 1.229497I
a = 0.849293 0.625389I
b = 2.00726 1.06121I
2.41139 11.80855I 9.11501 + 8.63273I
u = 0.576628 + 1.229497I
a = 0.849293 + 0.625389I
b = 2.00726 + 1.06121I
2.41139 + 11.80855I 9.11501 8.63273I
u = 0.565343 0.834957I
a = 1.050140 0.437151I
b = 1.59065 1.49259I
7.22525 3.79352I 14.6297 + 6.8044I
u = 0.565343 + 0.834957I
a = 1.050140 + 0.437151I
b = 1.59065 + 1.49259I
7.22525 + 3.79352I 14.6297 6.8044I
u = 0.443360 0.614385I
a = 1.00472 + 1.43117I
b = 0.409452 + 0.398810I
2.17233 1.46597I 11.44165 + 4.82002I
u = 0.443360 + 0.614385I
a = 1.00472 1.43117I
b = 0.409452 0.398810I
2.17233 + 1.46597I 11.44165 4.82002I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.413566 1.107252I
a = 0.798773 0.659393I
b = 1.74410 0.24157I
5.21251 5.27743I 6.65035 + 6.45030I
u = 0.413566 + 1.107252I
a = 0.798773 + 0.659393I
b = 1.74410 + 0.24157I
5.21251 + 5.27743I 6.65035 6.45030I
u = 0.250785
a = 3.43847
b = 1.30494
6.66293 13.6174
u = 0.125014 0.759051I
a = 1.47338 + 0.23743I
b = 1.44810 + 0.44829I
2.21675 + 2.65344I 14.3737 6.7757I
u = 0.125014 + 0.759051I
a = 1.47338 0.23743I
b = 1.44810 0.44829I
2.21675 2.65344I 14.3737 + 6.7757I
u = 0.274821
a = 0.917979
b = 0.224037
0.508062 19.5922
u = 0.296911 0.949247I
a = 0.872821 + 1.014864I
b = 1.67813 + 0.48220I
2.99004 0.45030I 10.69982 4.65118I
u = 0.296911 + 0.949247I
a = 0.872821 1.014864I
b = 1.67813 0.48220I
2.99004 + 0.45030I 10.69982 + 4.65118I
u = 0.487268 1.106835I
a = 0.968574 + 0.641704I
b = 1.85380 + 1.13963I
1.23650 + 6.41971I 10.13891 5.40596I
u = 0.487268 + 1.106835I
a = 0.968574 0.641704I
b = 1.85380 1.13963I
1.23650 6.41971I 10.13891 + 5.40596I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.555730 1.199456I
a = 0.671335 + 0.521804I
b = 1.71336 + 0.10860I
0.48195 + 10.01536I 10.05091 6.46543I
u = 0.555730 + 1.199456I
a = 0.671335 0.521804I
b = 1.71336 0.10860I
0.48195 10.01536I 10.05091 + 6.46543I
u = 0.569801 0.943146I
a = 0.490194 1.232726I
b = 0.033629 0.390422I
6.49386 + 5.21032I 14.7449 6.2628I
u = 0.569801 + 0.943146I
a = 0.490194 + 1.232726I
b = 0.033629 + 0.390422I
6.49386 5.21032I 14.7449 + 6.2628I
u = 0.67592 1.27886I
a = 0.797288 + 0.577860I
b = 2.14615 + 1.03761I
3.7021 + 16.1510I 12.8937 8.8766I
u = 0.67592 + 1.27886I
a = 0.797288 0.577860I
b = 2.14615 1.03761I
3.7021 16.1510I 12.8937 + 8.8766I
u = 1.084116 0.332892I
a = 0.816759 0.413766I
b = 0.425394 + 0.068144I
3.53318 0.46316I 13.8889 + 10.1726I
u = 1.084116 + 0.332892I
a = 0.816759 + 0.413766I
b = 0.425394 0.068144I
3.53318 + 0.46316I 13.8889 10.1726I
11
V. I
u
5
=
hu
40
3u
39
+· · ·16u+13, 3.76×10
32
u
39
+8.78×10
31
u
38
+· · ·+1.02×10
33
b
1.25 × 10
34
, 2.55 × 10
33
u
39
6.61 × 10
33
u
38
+ · · · + 1.33 × 10
34
a 3.81 × 10
34
i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.192331u
39
+ 0.497986u
38
+ ··· 8.51738u + 2.86824
0.368636u
39
0.0859731u
38
+ ··· 8.69230u + 12.2597
a
11
=
0.192331u
39
+ 0.497986u
38
+ ··· 8.51738u + 2.86824
0.551011u
39
0.585572u
38
+ ··· 7.45612u + 13.2868
a
2
=
1
0
a
8
=
0.0570880u
39
0.210178u
38
+ ··· + 1.31801u 2.25705
0.682063u
39
1.47618u
38
+ ··· 0.398077u + 6.27477
a
1
=
0.641501u
39
+ 2.61722u
38
+ ··· 20.9827u + 11.6808
12.5542u
39
34.9248u
38
+ ··· + 143.372u + 12.7987
a
6
=
u
u
a
3
=
u
2
+ 1
u
2
a
9
=
0.0859023u
39
0.251901u
38
+ ··· + 3.62263u 2.89947
0.765800u
39
1.85834u
38
+ ··· + 3.12660u + 3.99109
a
5
=
0.631022u
39
1.33792u
38
+ ··· 0.351315u + 8.02593
1.66472u
39
16.7390u
38
+ ··· + 189.659u 152.107
a
4
=
0.124411u
39
+ 0.398166u
38
+ ··· 2.48210u + 0.766635
0.684909u
39
0.915620u
38
+ ··· 3.65756u + 13.0927
a
4
=
0.124411u
39
+ 0.398166u
38
+ ··· 2.48210u + 0.766635
0.684909u
39
0.915620u
38
+ ··· 3.65756u + 13.0927
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.966472 0.204987I
a = 1.062267 0.722994I
b = 0.221672 + 0.177889I
0.72067 + 6.27316I 12.10015 6.54347I
u = 0.966472 + 0.204987I
a = 1.062267 + 0.722994I
b = 0.221672 0.177889I
0.72067 6.27316I 12.10015 + 6.54347I
u = 0.78685 1.24207I
a = 0.729482 + 0.508679I
b = 1.68543 + 0.80883I
6.73027 9.64430I 15.6547 + 6.2054I
u = 0.78685 + 1.24207I
a = 0.729482 0.508679I
b = 1.68543 0.80883I
6.73027 + 9.64430I 15.6547 6.2054I
u = 0.539846 1.156749I
a = 0.545893 + 0.352549I
b = 1.174275 0.319453I
3.49387 4.79919I 12.69810 + 3.09464I
u = 0.539846 + 1.156749I
a = 0.545893 0.352549I
b = 1.174275 + 0.319453I
3.49387 + 4.79919I 12.69810 3.09464I
u = 0.536847 1.066783I
a = 0.635978 0.406316I
b = 1.61232 + 0.12503I
4.54605 1.94645I 5.05320 + 4.81876I
u = 0.536847 + 1.066783I
a = 0.635978 + 0.406316I
b = 1.61232 0.12503I
4.54605 + 1.94645I 5.05320 4.81876I
u = 0.531514 0.736461I
a = 0.43783 1.34728I
b = 0.656171 + 0.173872I
7.52808 0.63661I 16.9604 0.1699I
u = 0.531514 + 0.736461I
a = 0.43783 + 1.34728I
b = 0.656171 0.173872I
7.52808 + 0.63661I 16.9604 + 0.1699I
13
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.457374 0.019438I
a = 0.13605 1.57227I
b = 0.641245 + 0.159917I
2.37392 1.80448I 8.82463 + 3.70058I
u = 0.457374 + 0.019438I
a = 0.13605 + 1.57227I
b = 0.641245 0.159917I
2.37392 + 1.80448I 8.82463 3.70058I
u = 0.402724 0.973230I
a = 0.795000 + 0.874208I
b = 1.47973 + 0.78488I
1.14846 2.14390I 13.45592 + 0.24308I
u = 0.402724 + 0.973230I
a = 0.795000 0.874208I
b = 1.47973 0.78488I
1.14846 + 2.14390I 13.45592 0.24308I
u = 0.265345 1.338092I
a = 0.220507 0.622815I
b = 0.615569 0.777760I
4.54605 + 1.94645I 5.05320 4.81876I
u = 0.265345 + 1.338092I
a = 0.220507 + 0.622815I
b = 0.615569 + 0.777760I
4.54605 1.94645I 5.05320 + 4.81876I
u = 0.067663 1.006844I
a = 0.070554 1.049855I
b = 8.41003 + 8.38834I
3.24334 17.8998
u = 0.067663 + 1.006844I
a = 0.070554 + 1.049855I
b = 8.41003 8.38834I
3.24334 17.8998
u = 0.08686 1.50116I
a = 0.306866 + 0.625312I
b = 1.102656 + 0.665990I
0.30488 4.84109I 11.63163 + 6.37981I
u = 0.08686 + 1.50116I
a = 0.306866 0.625312I
b = 1.102656 0.665990I
0.30488 + 4.84109I 11.63163 6.37981I
14
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.169382 1.042147I
a = 0.111475 + 0.685868I
b = 1.89656 + 1.65584I
2.31303 14.9388
u = 0.169382 + 1.042147I
a = 0.111475 0.685868I
b = 1.89656 1.65584I
2.31303 14.9388
u = 0.222487 1.085195I
a = 0.616770 0.211956I
b = 1.182329 + 0.143493I
2.37392 + 1.80448I 8.82463 3.70058I
u = 0.222487 + 1.085195I
a = 0.616770 + 0.211956I
b = 1.182329 0.143493I
2.37392 1.80448I 8.82463 + 3.70058I
u = 0.296099 1.343353I
a = 0.463287 + 0.292122I
b = 1.54793 0.00842I
1.34713 0.58469I 9.20205 + 0.00910I
u = 0.296099 + 1.343353I
a = 0.463287 0.292122I
b = 1.54793 + 0.00842I
1.34713 + 0.58469I 9.20205 0.00910I
u = 0.541181 1.104513I
a = 0.201778 + 0.578355I
b = 0.033752 + 0.273970I
1.34713 + 0.58469I 9.20205 0.00910I
u = 0.541181 + 1.104513I
a = 0.201778 0.578355I
b = 0.033752 0.273970I
1.34713 0.58469I 9.20205 + 0.00910I
u = 0.575083 0.580865I
a = 1.39659 0.72612I
b = 1.34777 1.04742I
7.52808 0.63661I 16.9604 0.1699I
u = 0.575083 + 0.580865I
a = 1.39659 + 0.72612I
b = 1.34777 + 1.04742I
7.52808 + 0.63661I 16.9604 + 0.1699I
15
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.592656 0.264629I
a = 1.38249 + 1.32877I
b = 0.095630 + 0.178172I
1.14846 2.14390I 13.45592 + 0.24308I
u = 0.592656 + 0.264629I
a = 1.38249 1.32877I
b = 0.095630 0.178172I
1.14846 + 2.14390I 13.45592 0.24308I
u = 0.597799 1.194488I
a = 0.715673 0.625396I
b = 1.59978 0.78704I
0.72067 + 6.27316I 12.10015 6.54347I
u = 0.597799 + 1.194488I
a = 0.715673 + 0.625396I
b = 1.59978 + 0.78704I
0.72067 6.27316I 12.10015 + 6.54347I
u = 0.792130 0.867615I
a = 0.847143 + 0.277766I
b = 1.345935 0.150156I
0.30488 + 4.84109I 11.63163 6.37981I
u = 0.792130 + 0.867615I
a = 0.847143 0.277766I
b = 1.345935 + 0.150156I
0.30488 4.84109I 11.63163 + 6.37981I
u = 0.880675 0.188465I
a = 0.313760 0.865984I
b = 0.676385 + 0.372502I
3.49387 4.79919I 12.69810 + 3.09464I
u = 0.880675 + 0.188465I
a = 0.313760 + 0.865984I
b = 0.676385 0.372502I
3.49387 + 4.79919I 12.69810 3.09464I
u = 1.164958 0.329003I
a = 0.845040 + 0.672841I
b = 0.227872 0.431194I
6.73027 9.64430I 15.6547 + 6.2054I
u = 1.164958 + 0.329003I
a = 0.845040 0.672841I
b = 0.227872 + 0.431194I
6.73027 + 9.64430I 15.6547 6.2054I
16
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(2u + 1)(4u
2
4u 1)(9u
4
+ 9u
2
6u + 1)(8u
28
+ 4u
27
+ ··· 13u 1)
(u
40
+ 21u
39
+ ··· 1158u + 199)
c
2
, c
11
(u 1)(u + 1)
2
(u
2
+ 1)
2
(u
28
u
27
+ ··· + 2u 1)
(u
40
+ 3u
39
+ ··· + 16u + 13)
c
3
, c
8
, c
9
u(u
2
2)(u
4
u
2
+ 1)
(1 + 2u u
2
+ 3u
3
+ u
4
+ 6u
5
6u
6
4u
7
+ 13u
8
10u
9
33u
10
7u
11
+ 62u
12
+ 30u
13
62u
14
24u
15
+ 33u
16
+ 8u
17
9u
18
u
19
+ u
20
)
2
(u
28
+ 3u
27
+ ··· + 18u
2
8)
c
4
, c
6
(u 1)
2
(u + 1)(u
2
+ 1)
2
(u
28
u
27
+ ··· + 2u 1)
(u
40
+ 3u
39
+ ··· + 16u + 13)
c
5
(2u 1)(4u
2
+ 4u 1)(9u
4
+ 9u
2
+ 6u + 1)(8u
28
+ 4u
27
+ ··· 13u 1)
(u
40
+ 21u
39
+ ··· 1158u + 199)
c
7
(u 1)(u + 1)
2
(u
2
u + 1)
2
(1 2u u
2
+ u
3
+ 3u
4
+ 4u
5
10u
7
9u
8
+ 4u
9
+ 13u
10
+ 13u
11
2u
12
22u
13
10u
14
+ 16u
15
+ 11u
16
6u
17
5u
18
+ u
19
+ u
20
)
2
(u
28
+ 2u
27
+ ··· + 7u + 8)
c
10
(u 1)
2
(u + 1)(u
2
+ u + 1)
2
(1 2u u
2
+ u
3
+ 3u
4
+ 4u
5
10u
7
9u
8
+ 4u
9
+ 13u
10
+ 13u
11
2u
12
22u
13
10u
14
+ 16u
15
+ 11u
16
6u
17
5u
18
+ u
19
+ u
20
)
2
(u
28
+ 2u
27
+ ··· + 7u + 8)
17
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(4y 1)(16y
2
24y + 1)(81y
4
+ 162y
3
+ 99y
2
18y + 1)
(64y
28
752y
27
+ ··· 97y + 1)
(y
40
+ 11y
39
+ ··· + 1102756y + 39601)
c
2
, c
4
, c
6
c
11
(y 1)
3
(y + 1)
4
(y
28
+ 9y
27
+ ··· 22y + 1)
(y
40
+ 23y
39
+ ··· 724y + 169)
c
3
, c
8
, c
9
y(y 2)
2
(y
2
y + 1)
2
(1 2y 13y
2
23y
3
33y
4
+ 56y
5
+ 140y
6
196y
7
+ 667y
8
2178y
9
+ 3871y
10
5851y
11
+ 8920y
12
1.12 × 10
4
y
13
+ 1.01 × 10
4
y
14
6344y
15
+ 2773y
16
830y
17
+ 163y
18
19y
19
+ y
20
)
2
(y
28
25y
27
+ ··· 288y + 64)
c
7
, c
10
(y 1)
3
(y
2
+ y + 1)
2
(1 2y y
2
+ 9y
3
21y
4
+ 12y
5
+ 48y
6
144y
7
+ 151y
8
+ 58y
9
377y
10
+ 469y
11
132y
12
416y
13
+ 760y
14
720y
15
+ 453y
16
198y
17
+ 59y
18
11y
19
+ y
20
)
2
(y
28
12y
27
+ ··· 2657y + 64)
18