11a
1
(K11a
1
)
1
Arc Sequences
5 1 8 2 3 10 4 6 11 7 9
Solving Sequence
1,5
2 3 6
4,9
8 7 11 10
c
1
c
2
c
5
c
4
c
8
c
7
c
11
c
9
c
3
, c
6
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= ha
6
3a
5
+ 9a
4
18a
3
+ 45a
2
54a + 27, 3a
5
+ 7a
4
15a
3
+ 33a
2
+ 99b 36a + 72,
8a
5
+ 15a
4
51a
3
+ 99a
2
261a + 99u + 225i
I
u
2
= hu
69
4u
68
+ ··· 3u + 1, 6u
68
39u
67
+ ··· + 4b 17, 19u
68
+ 81u
67
+ ··· + 4a + 30i
There are 2 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
= ha
6
3a
5
+ 9a
4
18a
3
+ 45a
2
54a + 27, 3a
5
+ 99b + · · · 36a +
72, 8a
5
+ 99u + · · · 261a + 225i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
0.0808081a
5
0.151515a
4
+ ··· + 2.63636a 2.27273
a
2
=
1
0.0808081a
5
0.151515a
4
+ ··· + 2.63636a 1.27273
a
3
=
0.0808081a
5
+ 0.151515a
4
+ ··· 2.63636a + 2.27273
0.0808081a
5
0.151515a
4
+ ··· + 2.63636a 1.27273
a
6
=
1
0
a
4
=
0.0808081a
5
+ 0.151515a
4
+ ··· 2.63636a + 2.27273
0.0808081a
5
0.151515a
4
+ ··· + 2.63636a 1.27273
a
9
=
a
0.0303030a
5
0.0707071a
4
+ ··· + 0.363636a 0.727273
a
8
=
0.0303030a
5
0.0707071a
4
+ ··· + 1.36364a 0.727273
0.0303030a
5
0.0707071a
4
+ ··· + 0.363636a 0.727273
a
7
=
0.0303030a
5
0.0707071a
4
+ ··· + 1.36364a 0.727273
0.0303030a
5
0.0707071a
4
+ ··· + 0.363636a 0.727273
a
11
=
0.0202020a
5
+ 0.121212a
4
+ ··· 0.909091a + 1.81818
0.0202020a
5
0.121212a
4
+ ··· + 0.909091a 0.818182
a
10
=
0.0303030a
5
+ 0.0404040a
4
+ ··· + 0.363636a + 1.27273
0.0101010a
5
0.0505051a
4
+ ··· + 0.545455a 1.09091
a
10
=
0.0303030a
5
+ 0.0404040a
4
+ ··· + 0.363636a + 1.27273
0.0101010a
5
0.0505051a
4
+ ··· + 0.545455a 1.09091
(ii) Obstruction class = 1
(iii) Cusp Shapes =
2
33
a
5
25
99
a
4
+
7
11
a
3
2a
2
+
52
11
a
93
11
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.80940 2.14698I
b = 0.215080 + 1.307141I
3.02413 + 4.85801I 2.23639 5.66123I
u = 0.500000 + 0.866025I
a = 0.80940 + 2.14698I
b = 0.215080 1.307141I
3.02413 4.85801I 2.23639 + 5.66123I
u = 0.500000 0.866025I
a = 0.854760 0.493496I
b = 0.569840
1.11345 + 2.02988I 5.31735 1.07831I
u = 0.500000 + 0.866025I
a = 0.854760 + 0.493496I
b = 0.569840
1.11345 2.02988I 5.31735 + 1.07831I
u = 0.500000 + 0.866025I
a = 1.45464 1.77445I
b = 0.215080 + 1.307141I
3.02413 + 0.79824I 0.946254 0.677361I
u = 0.500000 0.866025I
a = 1.45464 + 1.77445I
b = 0.215080 1.307141I
3.02413 0.79824I 0.946254 + 0.677361I
3
II. I
u
2
= hu
69
4u
68
+ · · · 3u + 1, 6u
68
39u
67
+ · · · + 4b 17, 19u
68
+
81u
67
+ · · · + 4a + 30i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
4
=
u
u
3
+ u
a
9
=
19
4
u
68
81
4
u
67
+ ··· +
71
4
u
15
2
3
2
u
68
+
39
4
u
67
+ ···
19
2
u +
17
4
a
8
=
21
4
u
68
37
2
u
67
+ ··· +
37
4
u
13
4
5
2
u
68
+
63
4
u
67
+ ···
25
2
u +
21
4
a
7
=
19
4
u
68
41
2
u
67
+ ··· +
67
4
u
27
4
1
2
u
68
+
25
4
u
67
+ ···
17
2
u +
19
4
a
11
=
1
2
u
68
7
4
u
67
+ ···
7
4
u
2
1
4
1
4
u
68
+ u
67
+ ··· +
1
4
u +
1
4
a
10
=
15
4
u
68
+
57
4
u
67
+ ···
29
4
u +
7
2
3
4
u
68
23
4
u
67
+ ··· +
25
4
u
7
2
a
10
=
15
4
u
68
+
57
4
u
67
+ ···
29
4
u +
7
2
3
4
u
68
23
4
u
67
+ ··· +
25
4
u
7
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
68
+
45
2
u
67
+ ··· 25u + 8
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.733440 0.695585I
a = 0.32108 + 2.76125I
b = 0.46759 1.36494I
2.53315 + 7.57441I 2.31688 8.51844I
u = 0.733440 + 0.695585I
a = 0.32108 2.76125I
b = 0.46759 + 1.36494I
2.53315 7.57441I 2.31688 + 8.51844I
u = 0.711006 0.663567I
a = 0.98352 2.14293I
b = 0.037928 + 1.078992I
3.30960 + 2.06237I 0.16192 3.38913I
u = 0.711006 + 0.663567I
a = 0.98352 + 2.14293I
b = 0.037928 1.078992I
3.30960 2.06237I 0.16192 + 3.38913I
u = 0.671797 0.919890I
a = 1.34580 1.77652I
b = 0.444041 + 1.303308I
1.87091 2.24800I 4.33887 + 3.21036I
u = 0.671797 + 0.919890I
a = 1.34580 + 1.77652I
b = 0.444041 1.303308I
1.87091 + 2.24800I 4.33887 3.21036I
u = 0.666104 0.796156I
a = 1.070086 + 0.710756I
b = 0.914669 0.073074I
2.00723 + 2.55767I 10.61202 4.63475I
u = 0.666104 + 0.796156I
a = 1.070086 0.710756I
b = 0.914669 + 0.073074I
2.00723 2.55767I 10.61202 + 4.63475I
u = 0.643133 0.942473I
a = 0.42434 + 1.85341I
b = 0.061202 1.072385I
2.48833 + 3.11204I 2.48193 1.92377I
u = 0.643133 + 0.942473I
a = 0.42434 1.85341I
b = 0.061202 + 1.072385I
2.48833 3.11204I 2.48193 + 1.92377I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.639153 0.277410I
a = 0.88351 1.90881I
b = 0.44295 + 1.35230I
2.84817 4.66103I 0.13544 + 3.76116I
u = 0.639153 + 0.277410I
a = 0.88351 + 1.90881I
b = 0.44295 1.35230I
2.84817 + 4.66103I 0.13544 3.76116I
u = 0.626215 0.346171I
a = 1.42777 + 1.55739I
b = 0.007523 1.100475I
3.53880 + 0.84616I 1.80381 1.98615I
u = 0.626215 + 0.346171I
a = 1.42777 1.55739I
b = 0.007523 + 1.100475I
3.53880 0.84616I 1.80381 + 1.98615I
u = 0.527223 1.086462I
a = 2.27892 0.60251I
b = 0.091941 + 1.023900I
1.42174 + 3.69530I 1.96425 2.66569I
u = 0.527223 + 1.086462I
a = 2.27892 + 0.60251I
b = 0.091941 1.023900I
1.42174 3.69530I 1.96425 + 2.66569I
u = 0.519317 1.110641I
a = 2.39007 + 1.20449I
b = 0.51412 1.37725I
0.48440 + 9.18841I 4.07732 7.78667I
u = 0.519317 + 1.110641I
a = 2.39007 1.20449I
b = 0.51412 + 1.37725I
0.48440 9.18841I 4.07732 + 7.78667I
u = 0.489928 0.621160I
a = 0.667738 + 0.145242I
b = 0.154354 0.007742I
0.74701 + 1.37700I 2.48134 4.28508I
u = 0.489928 + 0.621160I
a = 0.667738 0.145242I
b = 0.154354 + 0.007742I
0.74701 1.37700I 2.48134 + 4.28508I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.471607 0.968231I
a = 0.643850 0.089048I
b = 0.021337 0.157694I
0.30141 + 2.59969I 1.01042 4.25911I
u = 0.471607 + 0.968231I
a = 0.643850 + 0.089048I
b = 0.021337 + 0.157694I
0.30141 2.59969I 1.01042 + 4.25911I
u = 0.449639 1.095021I
a = 1.03463 + 1.19600I
b = 1.035073 0.110219I
4.13213 + 3.64791I 10.05983 4.17616I
u = 0.449639 + 1.095021I
a = 1.03463 1.19600I
b = 1.035073 + 0.110219I
4.13213 3.64791I 10.05983 + 4.17616I
u = 0.411327
a = 0.640170
b = 0.829237
1.54956 5.79308
u = 0.343205 1.083996I
a = 0.206360 + 0.776401I
b = 0.511101 + 1.222992I
0.73989 1.80119I 6.05149 + 1.58800I
u = 0.343205 + 1.083996I
a = 0.206360 0.776401I
b = 0.511101 1.222992I
0.73989 + 1.80119I 6.05149 1.58800I
u = 0.298471 1.014101I
a = 0.286357 0.551459I
b = 0.204156 0.919601I
0.06717 + 3.13357I 4.80388 4.92855I
u = 0.298471 + 1.014101I
a = 0.286357 + 0.551459I
b = 0.204156 + 0.919601I
0.06717 3.13357I 4.80388 + 4.92855I
u = 0.016008 0.604528I
a = 0.099442 1.386711I
b = 0.670638 + 0.275762I
1.57618 0.35138I 8.53058 + 0.76832I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.016008 + 0.604528I
a = 0.099442 + 1.386711I
b = 0.670638 0.275762I
1.57618 + 0.35138I 8.53058 0.76832I
u = 0.220475 1.197467I
a = 0.373221 + 0.265803I
b = 0.58748 1.42496I
4.53517 + 7.27554I 8.25882 4.98616I
u = 0.220475 + 1.197467I
a = 0.373221 0.265803I
b = 0.58748 + 1.42496I
4.53517 7.27554I 8.25882 + 4.98616I
u = 0.226126 1.167245I
a = 0.590926 0.232388I
b = 0.243884 + 0.983091I
3.30752 + 1.68121I 6.48774 0.23677I
u = 0.226126 + 1.167245I
a = 0.590926 + 0.232388I
b = 0.243884 0.983091I
3.30752 1.68121I 6.48774 + 0.23677I
u = 0.289382 1.197349I
a = 0.523502 0.299785I
b = 1.195541 0.170515I
9.42165 + 0.97729I 13.02252 0.67092I
u = 0.289382 + 1.197349I
a = 0.523502 + 0.299785I
b = 1.195541 + 0.170515I
9.42165 0.97729I 13.02252 + 0.67092I
u = 0.331275 1.142270I
a = 1.003824 + 0.301325I
b = 0.335164 0.497496I
4.64351 0.95645I 6.95143 + 0.40009I
u = 0.331275 + 1.142270I
a = 1.003824 0.301325I
b = 0.335164 + 0.497496I
4.64351 + 0.95645I 6.95143 0.40009I
u = 0.365690 1.185439I
a = 1.037630 0.590896I
b = 0.723201 + 1.156097I
6.45549 5.60193I 9.88794 + 5.72624I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.365690 + 1.185439I
a = 1.037630 + 0.590896I
b = 0.723201 1.156097I
6.45549 + 5.60193I 9.88794 5.72624I
u = 0.466517 0.537675I
a = 1.10137 3.09841I
b = 0.29431 + 1.40224I
3.75509 3.78687I 0.204318 + 0.628842I
u = 0.466517 + 0.537675I
a = 1.10137 + 3.09841I
b = 0.29431 1.40224I
3.75509 + 3.78687I 0.204318 0.628842I
u = 0.469960 1.031002I
a = 1.33009 + 1.65429I
b = 0.20461 1.45685I
2.24452 0.13399I 5.97208 + 2.96612I
u = 0.469960 + 1.031002I
a = 1.33009 1.65429I
b = 0.20461 + 1.45685I
2.24452 + 0.13399I 5.97208 2.96612I
u = 0.491725 1.042926I
a = 0.79382 1.80473I
b = 0.056888 + 1.390471I
2.44272 6.15643I 4.67340 + 8.38565I
u = 0.491725 + 1.042926I
a = 0.79382 + 1.80473I
b = 0.056888 1.390471I
2.44272 + 6.15643I 4.67340 8.38565I
u = 0.492791 1.160646I
a = 0.767436 0.283309I
b = 0.790973 0.979369I
5.59193 2.78605I 9.11319 + 1.16015I
u = 0.492791 + 1.160646I
a = 0.767436 + 0.283309I
b = 0.790973 + 0.979369I
5.59193 + 2.78605I 9.11319 1.16015I
u = 0.516637 0.488301I
a = 1.59115 + 2.67785I
b = 0.052494 1.282723I
4.09856 + 2.01146I 1.16053 4.35116I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.516637 + 0.488301I
a = 1.59115 2.67785I
b = 0.052494 + 1.282723I
4.09856 2.01146I 1.16053 + 4.35116I
u = 0.527078 1.135363I
a = 0.248874 + 0.241584I
b = 0.441249 + 0.311910I
3.29307 6.95572I 4.27136 + 6.38165I
u = 0.527078 + 1.135363I
a = 0.248874 0.241584I
b = 0.441249 0.311910I
3.29307 + 6.95572I 4.27136 6.38165I
u = 0.549001 1.165560I
a = 0.34679 1.49283I
b = 1.181786 + 0.308658I
7.64943 9.45868I 10.40135 + 7.32694I
u = 0.549001 + 1.165560I
a = 0.34679 + 1.49283I
b = 1.181786 0.308658I
7.64943 + 9.45868I 10.40135 7.32694I
u = 0.577394 1.144402I
a = 1.87964 + 1.29373I
b = 0.262889 1.112564I
0.95244 9.80543I 3.41645 + 5.83408I
u = 0.577394 + 1.144402I
a = 1.87964 1.29373I
b = 0.262889 + 1.112564I
0.95244 + 9.80543I 3.41645 5.83408I
u = 0.583785 1.154610I
a = 1.96927 1.89338I
b = 0.54586 + 1.49451I
2.0811 15.6439I 5.11812 + 10.33480I
u = 0.583785 + 1.154610I
a = 1.96927 + 1.89338I
b = 0.54586 1.49451I
2.0811 + 15.6439I 5.11812 10.33480I
u = 0.702651 0.239503I
a = 0.314791 + 0.466421I
b = 0.327437 0.294115I
0.72685 + 2.26761I 0.99875 3.18261I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.702651 + 0.239503I
a = 0.314791 0.466421I
b = 0.327437 + 0.294115I
0.72685 2.26761I 0.99875 + 3.18261I
u = 0.755004 0.112601I
a = 0.56540 1.33572I
b = 0.662786 + 1.026456I
2.57806 1.76748I 5.73344 + 2.63178I
u = 0.755004 + 0.112601I
a = 0.56540 + 1.33572I
b = 0.662786 1.026456I
2.57806 + 1.76748I 5.73344 2.63178I
u = 0.806399 0.324578I
a = 0.69907 1.86919I
b = 0.214984 + 1.090892I
1.48274 + 4.64133I 0.50996 2.25276I
u = 0.806399 + 0.324578I
a = 0.69907 + 1.86919I
b = 0.214984 1.090892I
1.48274 4.64133I 0.50996 + 2.25276I
u = 0.807663 0.242666I
a = 0.831962 + 0.642579I
b = 1.129337 0.260107I
4.92005 + 4.43213I 7.83379 3.89811I
u = 0.807663 + 0.242666I
a = 0.831962 0.642579I
b = 1.129337 + 0.260107I
4.92005 4.43213I 7.83379 + 3.89811I
u = 0.832356 0.319128I
a = 0.20208 + 2.46516I
b = 0.53327 1.46283I
0.41311 + 10.39027I 2.35000 6.90796I
u = 0.832356 + 0.319128I
a = 0.20208 2.46516I
b = 0.53327 + 1.46283I
0.41311 10.39027I 2.35000 + 6.90796I
11
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
69
+ 4u
68
+ ··· 3u 1)
c
2
(u
2
+ u + 1)
3
(u
69
+ 34u
68
+ ··· 3u 1)
c
3
, c
7
u
6
(u
69
+ u
68
+ ··· + 224u 64)
c
4
(u
2
u + 1)
3
(u
69
+ 4u
68
+ ··· 3u 1)
c
5
(u
2
+ u + 1)
3
(u
69
+ 4u
68
+ ··· + 5265u + 1153)
c
6
(u
3
+ u
2
1)
2
(u
69
+ 3u
68
+ ··· + 2u + 1)
c
8
(u
3
+ u
2
+ 2u + 1)
2
(u
69
+ 3u
68
+ ··· 7540u 937)
c
9
(u
3
u
2
+ 2u 1)
2
(u
69
+ 23u
68
+ ··· 4u + 1)
c
10
(u
3
u
2
+ 1)
2
(u
69
+ 3u
68
+ ··· + 2u + 1)
c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
69
+ 23u
68
+ ··· 4u + 1)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
3
(y
69
+ 34y
68
+ ··· 3y 1)
c
2
(y
2
+ y + 1)
3
(y
69
+ 6y
68
+ ··· + 29y 1)
c
3
, c
7
y
6
(y
69
+ 35y
68
+ ··· 44032y 4096)
c
5
(y
2
+ y + 1)
3
(y
69
22y
68
+ ··· + 5956197y 1329409)
c
6
(y
3
y
2
+ 2y 1)
2
(y
69
23y
68
+ ··· 4y 1)
c
8
(y
3
+ 3y
2
+ 2y 1)
2
(y
69
11y
68
+ ··· + 1.05001 × 10
7
y 877969)
c
9
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
69
+ 49y
68
+ ··· 4y 1)
c
10
(y
3
y
2
+ 2y 1)
2
(y
69
23y
68
+ ··· 4y 1)
13