11n
7
(K11n
7
)
1
Arc Sequences
5 1 8 2 3 9 3 11 6 8 10
Solving Sequence
3,8 4,11
9 7 6 5 10 1 2
c
3
c
8
c
7
c
6
c
5
c
10
c
11
c
2
c
1
, c
4
, c
9
Representation Ideals
I = I
u
1
\
I
v
1
I
u
1
= hu
39
3u
38
+ ··· + 160u 64,
1.35871 × 10
74
u
38
+ 4.22296 × 10
74
u
37
+ ··· + 6.78965 × 10
75
b 7.58203 × 10
75
,
1.97322 × 10
74
u
38
5.74566 × 10
74
u
37
+ ··· + 1.35793 × 10
76
a + 2.27616 × 10
76
i
I
v
1
= hb
6
+ 3b
5
+ 7b
4
+ 4b
3
+ b
2
+ 2b + 1, 19b
5
45b
4
105b
3
11b
2
15b + v 30, ai
There are 2 irreducible components with 45 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
39
3u
38
+· · ·+160u64, 1.36×10
74
u
38
+4.22×10
74
u
37
+· · ·+6.79×10
75
b
7.58 × 10
75
, 1.97 × 10
74
u
38
5.75 × 10
74
u
37
+ · · · + 1.36 × 10
76
a + 2.28 × 10
76
i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
11
=
0.0145311u
38
+ 0.0423119u
37
+ ··· + 4.20910u 1.67620
0.0200115u
38
0.0621970u
37
+ ··· 9.61102u + 1.11671
a
9
=
0.0207767u
38
+ 0.0602462u
37
+ ··· + 2.06543u 0.671114
0.0162503u
38
0.0497470u
37
+ ··· 7.62454u + 0.958859
a
7
=
u
u
a
6
=
0.00471591u
38
0.00996442u
37
+ ··· + 5.08469u 0.806754
0.00561330u
38
0.0187349u
37
+ ··· 2.86181u + 0.381508
a
5
=
0.0103292u
38
0.0286993u
37
+ ··· + 2.22288u 0.425246
0.00561330u
38
0.0187349u
37
+ ··· 2.86181u + 0.381508
a
10
=
0.0145311u
38
+ 0.0423119u
37
+ ··· + 4.20910u 1.67620
0.0200871u
38
0.0633748u
37
+ ··· 8.88605u + 1.19871
a
1
=
0.0103292u
38
+ 0.0286993u
37
+ ··· 2.22288u + 0.425246
0.00178031u
38
0.00566997u
37
+ ··· 2.56686u + 0.527957
a
2
=
0.000822752u
38
+ 0.00351950u
37
+ ··· + 2.47747u + 0.115491
0.00249633u
38
+ 0.00737375u
37
+ ··· + 1.50215u + 0.140681
a
2
=
0.000822752u
38
+ 0.00351950u
37
+ ··· + 2.47747u + 0.115491
0.00249633u
38
+ 0.00737375u
37
+ ··· + 1.50215u + 0.140681
(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 = 1.015771 0.563299I
a = 0.400629 + 0.467419I
b = 0.589409 0.852841I
1.33536 + 1.46808I 4.04275 4.37387I
u = 1.015771 + 0.563299I
a = 0.400629 0.467419I
b = 0.589409 + 0.852841I
1.33536 1.46808I 4.04275 + 4.37387I
u = 0.764144 0.288695I
a = 1.26262 0.65906I
b = 0.15031 + 1.56666I
2.82585 + 1.96097I 4.21009 0.40138I
u = 0.764144 + 0.288695I
a = 1.26262 + 0.65906I
b = 0.15031 1.56666I
2.82585 1.96097I 4.21009 + 0.40138I
u = 0.58641 1.60233I
a = 0.021261 + 0.795333I
b = 0.05289 2.62347I
5.09332 + 8.17367I 3.75441 5.17214I
u = 0.58641 + 1.60233I
a = 0.021261 0.795333I
b = 0.05289 + 2.62347I
5.09332 8.17367I 3.75441 + 5.17214I
u = 0.525627 0.805732I
a = 0.429621 + 0.235494I
b = 0.035751 0.891969I
1.15178 + 1.50599I 2.56109 2.72315I
u = 0.525627 + 0.805732I
a = 0.429621 0.235494I
b = 0.035751 + 0.891969I
1.15178 1.50599I 2.56109 + 2.72315I
u = 0.391025 0.528783I
a = 0.25451 + 1.68510I
b = 0.000923 0.452010I
2.88333 + 1.52566I 2.95623 6.42875I
u = 0.391025 + 0.528783I
a = 0.25451 1.68510I
b = 0.000923 + 0.452010I
2.88333 1.52566I 2.95623 + 6.42875I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.362636 1.289161I
a = 0.653617 0.135764I
b = 0.1117599 0.0736303I
1.45248 + 3.13609I 0.74981 2.49452I
u = 0.362636 + 1.289161I
a = 0.653617 + 0.135764I
b = 0.1117599 + 0.0736303I
1.45248 3.13609I 0.74981 + 2.49452I
u = 0.32507 1.55745I
a = 0.191541 0.850101I
b = 0.68404 + 2.14659I
8.99304 + 6.36082I 7.31007 4.20770I
u = 0.32507 + 1.55745I
a = 0.191541 + 0.850101I
b = 0.68404 2.14659I
8.99304 6.36082I 7.31007 + 4.20770I
u = 0.277331 0.646400I
a = 1.58033 + 0.79557I
b = 0.375450 0.246768I
2.42328 4.44150I 7.41017 + 1.05267I
u = 0.277331 + 0.646400I
a = 1.58033 0.79557I
b = 0.375450 + 0.246768I
2.42328 + 4.44150I 7.41017 1.05267I
u = 0.023739 0.376403I
a = 0.00176 2.42806I
b = 0.45391 + 3.98939I
1.32570 2.15384I 38.3073 + 0.3658I
u = 0.023739 + 0.376403I
a = 0.00176 + 2.42806I
b = 0.45391 3.98939I
1.32570 + 2.15384I 38.3073 0.3658I
u = 0.08279 1.46418I
a = 0.642531 0.155659I
b = 0.115784 0.127405I
5.68977 + 0.80789I 5.63151 0.39749I
u = 0.08279 + 1.46418I
a = 0.642531 + 0.155659I
b = 0.115784 + 0.127405I
5.68977 0.80789I 5.63151 + 0.39749I
4
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.101788 1.392135I
a = 0.258295 0.919920I
b = 0.65375 + 2.18067I
5.43179 1.52389I 4.49323 + 0.71316I
u = 0.101788 + 1.392135I
a = 0.258295 + 0.919920I
b = 0.65375 2.18067I
5.43179 + 1.52389I 4.49323 0.71316I
u = 0.14680 1.57664I
a = 0.340464 0.843420I
b = 0.62348 + 2.12772I
9.33600 3.03011I 7.58951 + 2.63816I
u = 0.14680 + 1.57664I
a = 0.340464 + 0.843420I
b = 0.62348 2.12772I
9.33600 + 3.03011I 7.58951 2.63816I
u = 0.274747
a = 3.03443
b = 0.503610
1.20362 8.91674
u = 0.40659 1.39442I
a = 0.345713 + 0.549146I
b = 0.00006 2.05405I
0.05191 + 4.16636I 0.57665 9.00427I
u = 0.40659 + 1.39442I
a = 0.345713 0.549146I
b = 0.00006 + 2.05405I
0.05191 4.16636I 0.57665 + 9.00427I
u = 0.45680 1.86686I
a = 0.031347 + 0.713289I
b = 0.03721 2.58839I
10.41507 3.75787I 8.66055 + 3.06635I
u = 0.45680 + 1.86686I
a = 0.031347 0.713289I
b = 0.03721 + 2.58839I
10.41507 + 3.75787I 8.66055 3.06635I
u = 0.51700 1.43416I
a = 0.666754 0.161021I
b = 0.0860983 + 0.1075653I
4.49350 8.17612I 3.77298 + 5.44747I
5
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.51700 + 1.43416I
a = 0.666754 + 0.161021I
b = 0.0860983 0.1075653I
4.49350 + 8.17612I 3.77298 5.44747I
u = 0.612697 0.489080I
a = 1.115050 0.268415I
b = 0.003550 0.690228I
1.92828 0.12066I 7.15424 + 0.12690I
u = 0.612697 + 0.489080I
a = 1.115050 + 0.268415I
b = 0.003550 + 0.690228I
1.92828 + 0.12066I 7.15424 0.12690I
u = 0.77088 1.59345I
a = 0.035485 + 0.801642I
b = 0.02106 2.63014I
8.1954 13.8902I 5.93452 + 8.00354I
u = 0.77088 + 1.59345I
a = 0.035485 0.801642I
b = 0.02106 + 2.63014I
8.1954 + 13.8902I 5.93452 8.00354I
u = 0.792668 0.169516I
a = 0.128021 + 0.417461I
b = 0.713084 0.803498I
0.39510 + 2.82136I 0.57403 4.29661I
u = 0.792668 + 0.169516I
a = 0.128021 0.417461I
b = 0.713084 + 0.803498I
0.39510 2.82136I 0.57403 + 4.29661I
u = 1.69889 0.01850I
a = 0.648911 + 0.174171I
b = 1.163139 0.325427I
3.35360 5.46941I 6.70822 + 8.69559I
u = 1.69889 + 0.01850I
a = 0.648911 0.174171I
b = 1.163139 + 0.325427I
3.35360 + 5.46941I 6.70822 8.69559I
6
II. I
v
1
= hb
6
+ 3b
5
+ 7b
4
+ 4b
3
+ b
2
+ 2b + 1, 19b
5
+ v + · · · 15b 30, ai
(i) Arc colorings
a
3
=
1
0
a
8
=
19b
5
+ 45b
4
+ 105b
3
+ 11b
2
+ 15b + 30
0
a
4
=
1
0
a
11
=
0
b
a
9
=
19b
5
+ 45b
4
+ 105b
3
+ 11b
2
+ 15b + 30
8b
5
+ 19b
4
+ 44b
3
+ 4b
2
+ 5b + 12
a
7
=
19b
5
+ 45b
4
+ 105b
3
+ 11b
2
+ 15b + 30
0
a
6
=
12b
5
27b
4
63b
3
+ b
2
9b 18
14b
5
33b
4
77b
3
7b
2
11b 21
a
5
=
26b
5
60b
4
140b
3
6b
2
20b 39
14b
5
33b
4
77b
3
7b
2
11b 21
a
10
=
32b
5
75b
4
175b
3
14b
2
25b 50
b
a
1
=
12b
5
+ 27b
4
+ 63b
3
b
2
+ 9b + 18
14b
5
+ 33b
4
+ 77b
3
+ 7b
2
+ 11b + 21
a
2
=
42b
5
+ 98b
4
+ 228b
3
+ 14b
2
+ 28b + 64
14b
5
+ 33b
4
+ 77b
3
+ 7b
2
+ 11b + 22
a
2
=
42b
5
+ 98b
4
+ 228b
3
+ 14b
2
+ 28b + 64
14b
5
+ 33b
4
+ 77b
3
+ 7b
2
+ 11b + 22
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 0.162359 + 0.281214I
a = 0
b = 1.16236 2.01326I
1.11345 2.02988I 2.22484 4.65789I
v = 0.162359 0.281214I
a = 0
b = 1.16236 + 2.01326I
1.11345 + 2.02988I 2.22484 + 4.65789I
v = 1.31813 1.15851I
a = 0
b = 0.655769 0.011266I
3.02413 + 0.79824I 0.92725 + 3.21674I
v = 1.31813 + 1.15851I
a = 0
b = 0.655769 + 0.011266I
3.02413 0.79824I 0.92725 3.21674I
v = 0.34423 1.72078I
a = 0
b = 0.318128 0.573545I
3.02413 4.85801I 2.65209 + 7.50333I
v = 0.34423 + 1.72078I
a = 0
b = 0.318128 + 0.573545I
3.02413 + 4.85801I 2.65209 7.50333I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
39
+ 4u
38
+ ··· + 10u 1)
c
2
(u
2
+ u + 1)
3
(u
39
+ 22u
38
+ ··· + 170u 1)
c
3
, c
7
u
6
(u
39
+ 3u
38
+ ··· + 160u + 64)
c
4
(u
2
u + 1)
3
(u
39
+ 4u
38
+ ··· + 10u 1)
c
5
(u
2
+ u + 1)
3
(u
39
+ 4u
38
+ ··· + 602u + 49)
c
6
(u
3
u
2
+ 2u 1)
2
(u
39
+ 3u
38
+ ··· 3u 1)
c
8
(u
3
+ u
2
1)
2
(u
39
+ 3u
38
+ ··· 5u 1)
c
9
(u
3
+ u
2
+ 2u + 1)
2
(u
39
+ 3u
38
+ ··· 3u 1)
c
10
(u
3
u
2
+ 1)
2
(u
39
+ 3u
38
+ ··· 5u 1)
c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
39
+ 23u
38
+ ··· + u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
3
(y
39
+ 22y
38
+ ··· + 170y 1)
c
2
(y
2
+ y + 1)
3
(y
39
6y
38
+ ··· + 31510y 1)
c
3
, c
7
y
6
(y
39
+ 35y
38
+ ··· 23552y 4096)
c
5
(y
2
+ y + 1)
3
(y
39
34y
38
+ ··· + 391706y 2401)
c
6
, c
9
(y
3
+ 3y
2
+ 2y 1)
2
(y
39
+ 9y
38
+ ··· + y 1)
c
8
, c
10
(y
3
y
2
+ 2y 1)
2
(y
39
23y
38
+ ··· + y 1)
c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
39
11y
38
+ ··· + 117y 1)
10