11a
81
(K11a
81
)
1
Arc Sequences
6 1 10 9 2 3 4 11 7 8 5
Solving Sequence
2,5
6 1 3
7,9
4 11 8 10
c
5
c
1
c
2
c
6
c
4
c
11
c
8
c
10
c
3
, c
7
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b + u, a 1i
I
u
2
= hu
65
2u
64
+ ··· + u + 1, 1.19217 × 10
21
u
64
+ 1.28632 × 10
21
u
63
+ ··· + 3.92161 × 10
20
a 9.38507 × 10
20
,
1.12437 × 10
21
u
64
+ 2.08104 × 10
21
u
63
+ ··· + 3.92161 × 10
20
b + 9.56908 × 10
20
i
There are 2 irreducible components with 67 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
2
+ u + 1, b + u, a 1i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
u 1
a
3
=
0
u
a
7
=
u
u + 1
a
9
=
1
u
a
4
=
u
1
a
11
=
u
u
a
8
=
u + 1
0
a
10
=
1
u
a
10
=
1
u
(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 = 1.00000
b = 0.500000 + 0.866025I
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 1.00000
b = 0.500000 0.866025I
1.64493 2.02988I 3.00000 + 3.46410I
3
II. I
u
2
=
hu
65
2u
64
+· · · +u+1, 1.19×10
21
u
64
+1.29×10
21
u
63
+· · · +3.92×10
20
a
9.39 × 10
20
, 1.12× 10
21
u
64
+ 2.08 × 10
21
u
63
+ · · · + 3.92 × 10
20
b + 9.57 × 10
20
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
7
=
u
7
2u
5
2u
3
u
7
u
5
+ u
a
9
=
3.04000u
64
3.28008u
63
+ ··· + 4.27119u + 2.39317
2.86712u
64
5.30659u
63
+ ··· + 1.95308u 2.44009
a
4
=
5.41183u
64
+ 10.2518u
63
+ ··· 0.187153u + 4.72588
0.371835u
64
+ 4.36917u
63
+ ··· + 11.7089u + 4.79741
a
11
=
u
2
+ 1
u
4
a
8
=
3.24000u
64
3.47980u
63
+ ··· + 5.21094u + 1.57707
3.03387u
64
5.47353u
63
+ ··· + 2.13727u 2.43980
a
10
=
2.84000u
64
2.88110u
63
+ ··· + 4.21647u + 2.50783
2.69775u
64
4.93896u
63
+ ··· + 2.26672u 2.24110
a
10
=
2.84000u
64
2.88110u
63
+ ··· + 4.21647u + 2.50783
2.69775u
64
4.93896u
63
+ ··· + 2.26672u 2.24110
(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 = 0.895488 0.125785I
a = 0.471660 1.144842I
b = 0.376735 + 0.287159I
7.23534 2.53854I 7.87268 + 2.81634I
u = 0.895488 + 0.125785I
a = 0.471660 + 1.144842I
b = 0.376735 0.287159I
7.23534 + 2.53854I 7.87268 2.81634I
u = 0.792642
a = 4.50659
b = 0.401921
4.52444 15.6457
u = 0.775998 0.055269I
a = 0.78184 + 1.25281I
b = 0.272202 0.513943I
2.68758 0.98018I 2.13142 1.09675I
u = 0.775998 + 0.055269I
a = 0.78184 1.25281I
b = 0.272202 + 0.513943I
2.68758 + 0.98018I 2.13142 + 1.09675I
u = 0.681051 0.492285I
a = 0.055199 1.411262I
b = 0.444176 0.382574I
1.66713 5.95884I 1.87973 + 5.22029I
u = 0.681051 + 0.492285I
a = 0.055199 + 1.411262I
b = 0.444176 + 0.382574I
1.66713 + 5.95884I 1.87973 5.22029I
u = 0.571139 0.977969I
a = 1.213615 + 0.140366I
b = 1.62844 0.58978I
3.06278 + 10.75517I 3.84780 9.80074I
u = 0.571139 + 0.977969I
a = 1.213615 0.140366I
b = 1.62844 + 0.58978I
3.06278 10.75517I 3.84780 + 9.80074I
u = 0.526288 1.234516I
a = 0.873939 + 0.185569I
b = 2.29059 0.00902I
10.58887 + 7.68258I 9.86968 6.20420I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.526288 + 1.234516I
a = 0.873939 0.185569I
b = 2.29059 + 0.00902I
10.58887 7.68258I 9.86968 + 6.20420I
u = 0.487308 0.547787I
a = 0.211960 + 1.272069I
b = 0.231211 0.350100I
1.87042 1.68269I 3.98101 + 2.38110I
u = 0.487308 + 0.547787I
a = 0.211960 1.272069I
b = 0.231211 + 0.350100I
1.87042 + 1.68269I 3.98101 2.38110I
u = 0.484262 0.914397I
a = 0.921884 0.333475I
b = 2.04011 + 0.02854I
0.85073 + 5.73380I 0.10369 9.30698I
u = 0.484262 + 0.914397I
a = 0.921884 + 0.333475I
b = 2.04011 0.02854I
0.85073 5.73380I 0.10369 + 9.30698I
u = 0.477979 1.199781I
a = 0.716881 + 0.641319I
b = 2.26994 + 1.41320I
6.01852 + 5.55484I 4.97164 2.03636I
u = 0.477979 + 1.199781I
a = 0.716881 0.641319I
b = 2.26994 1.41320I
6.01852 5.55484I 4.97164 + 2.03636I
u = 0.457248 1.211201I
a = 0.94032 2.63123I
b = 1.90067 5.72471I
8.07040 + 4.48064I 9.41196 1.03498I
u = 0.457248 + 1.211201I
a = 0.94032 + 2.63123I
b = 1.90067 + 5.72471I
8.07040 4.48064I 9.41196 + 1.03498I
u = 0.432502 1.204003I
a = 1.029901 + 0.236107I
b = 2.78335 + 1.12697I
6.34559 + 3.27358I 5.96586 5.01774I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.432502 + 1.204003I
a = 1.029901 0.236107I
b = 2.78335 1.12697I
6.34559 3.27358I 5.96586 + 5.01774I
u = 0.384245 1.272527I
a = 0.694634 + 0.601416I
b = 1.61318 + 0.48242I
11.61322 + 1.86120I 11.51231 1.02799I
u = 0.384245 + 1.272527I
a = 0.694634 0.601416I
b = 1.61318 0.48242I
11.61322 1.86120I 11.51231 + 1.02799I
u = 0.379172 0.954880I
a = 0.501065 + 0.020288I
b = 0.95093 + 2.01516I
3.54308 + 4.03035I 8.72949 8.88482I
u = 0.379172 + 0.954880I
a = 0.501065 0.020288I
b = 0.95093 2.01516I
3.54308 4.03035I 8.72949 + 8.88482I
u = 0.287539 0.230477I
a = 2.03295 + 0.85038I
b = 0.980634 0.158120I
1.91160 0.92204I 3.30845 + 0.88312I
u = 0.287539 + 0.230477I
a = 2.03295 0.85038I
b = 0.980634 + 0.158120I
1.91160 + 0.92204I 3.30845 0.88312I
u = 0.255209 0.948908I
a = 1.042638 0.374322I
b = 2.04192 + 0.64047I
4.38791 + 0.83519I 11.90532 1.80779I
u = 0.255209 + 0.948908I
a = 1.042638 + 0.374322I
b = 2.04192 0.64047I
4.38791 0.83519I 11.90532 + 1.80779I
u = 0.042517 0.896108I
a = 0.268781 0.514883I
b = 1.50275 1.00485I
1.88737 1.49983I 7.09909 + 4.37897I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.042517 + 0.896108I
a = 0.268781 + 0.514883I
b = 1.50275 + 1.00485I
1.88737 + 1.49983I 7.09909 4.37897I
u = 0.026760 1.174523I
a = 1.029311 + 0.303686I
b = 1.65520 + 0.65340I
7.09632 4.52850I 10.34033 + 4.25760I
u = 0.026760 + 1.174523I
a = 1.029311 0.303686I
b = 1.65520 0.65340I
7.09632 + 4.52850I 10.34033 4.25760I
u = 0.268889 0.735800I
a = 0.25691 1.97444I
b = 2.05899 1.16628I
1.59108 1.22204I 11.25340 + 3.73985I
u = 0.268889 + 0.735800I
a = 0.25691 + 1.97444I
b = 2.05899 + 1.16628I
1.59108 + 1.22204I 11.25340 3.73985I
u = 0.350482 0.867729I
a = 1.52415 + 2.82114I
b = 2.39364 + 0.40176I
2.05273 1.79214I 3.9608 27.3479I
u = 0.350482 + 0.867729I
a = 1.52415 2.82114I
b = 2.39364 0.40176I
2.05273 + 1.79214I 3.9608 + 27.3479I
u = 0.393745 1.260008I
a = 0.955717 0.999785I
b = 2.67784 1.24472I
12.22785 + 6.64210I 8.44808 3.04159I
u = 0.393745 + 1.260008I
a = 0.955717 + 0.999785I
b = 2.67784 + 1.24472I
12.22785 6.64210I 8.44808 + 3.04159I
u = 0.416907 0.814224I
a = 0.477464 0.365364I
b = 0.394803 0.423441I
0.06086 1.78141I 0.16593 + 3.65886I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.416907 + 0.814224I
a = 0.477464 + 0.365364I
b = 0.394803 + 0.423441I
0.06086 + 1.78141I 0.16593 3.65886I
u = 0.420103 1.222354I
a = 0.592326 + 0.479793I
b = 2.61282 + 0.57494I
6.68706 + 0.82473I 6.76197 2.72355I
u = 0.420103 + 1.222354I
a = 0.592326 0.479793I
b = 2.61282 0.57494I
6.68706 0.82473I 6.76197 + 2.72355I
u = 0.448174 1.226545I
a = 0.558710 + 0.688441I
b = 1.70773 + 0.09090I
10.71551 2.40472I 12.03987 + 0.33417I
u = 0.448174 + 1.226545I
a = 0.558710 0.688441I
b = 1.70773 0.09090I
10.71551 + 2.40472I 12.03987 0.33417I
u = 0.457366 1.034949I
a = 0.241762 0.730168I
b = 0.747461 0.669285I
0.33001 3.05900I 1.51979 + 5.07455I
u = 0.457366 + 1.034949I
a = 0.241762 + 0.730168I
b = 0.747461 + 0.669285I
0.33001 + 3.05900I 1.51979 5.07455I
u = 0.468861 1.223143I
a = 0.789355 + 0.234003I
b = 2.74601 0.33365I
10.56670 6.75210I 11.57161 + 6.48191I
u = 0.468861 + 1.223143I
a = 0.789355 0.234003I
b = 2.74601 + 0.33365I
10.56670 + 6.75210I 11.57161 6.48191I
u = 0.488804 0.385911I
a = 1.289379 + 0.459288I
b = 0.403529 0.183900I
1.49244 0.86359I 5.03880 + 1.53624I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.488804 + 0.385911I
a = 1.289379 0.459288I
b = 0.403529 + 0.183900I
1.49244 + 0.86359I 5.03880 1.53624I
u = 0.490148 1.212657I
a = 0.852756 + 0.043707I
b = 3.07486 + 0.48626I
6.18604 9.87507I 5.65212 + 8.90207I
u = 0.490148 + 1.212657I
a = 0.852756 0.043707I
b = 3.07486 0.48626I
6.18604 + 9.87507I 5.65212 8.90207I
u = 0.517285 1.227495I
a = 1.385000 0.258737I
b = 3.77402 0.28861I
11.3347 16.0510I 7.14104 + 8.98990I
u = 0.517285 + 1.227495I
a = 1.385000 + 0.258737I
b = 3.77402 + 0.28861I
11.3347 + 16.0510I 7.14104 8.98990I
u = 0.637047 0.936500I
a = 0.491753 + 0.351693I
b = 0.317828 + 0.930103I
2.16059 1.39700I 10.38048 4.03093I
u = 0.637047 + 0.936500I
a = 0.491753 0.351693I
b = 0.317828 0.930103I
2.16059 + 1.39700I 10.38048 + 4.03093I
u = 0.688274 0.597046I
a = 0.760075 + 0.431895I
b = 0.177796 + 0.445216I
1.20524 3.66120I 3.96415 + 8.51812I
u = 0.688274 + 0.597046I
a = 0.760075 0.431895I
b = 0.177796 0.445216I
1.20524 + 3.66120I 3.96415 8.51812I
u = 0.816612 0.075250I
a = 0.445787 1.172264I
b = 0.036633 + 0.982310I
2.82170 + 5.12577I 2.55858 6.00397I
10
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.816612 + 0.075250I
a = 0.445787 + 1.172264I
b = 0.036633 0.982310I
2.82170 5.12577I 2.55858 + 6.00397I
u = 0.824553 0.021951I
a = 0.660739 1.105182I
b = 0.732613 + 0.631471I
7.00163 + 2.10905I 8.59887 3.22074I
u = 0.824553 + 0.021951I
a = 0.660739 + 1.105182I
b = 0.732613 0.631471I
7.00163 2.10905I 8.59887 + 3.22074I
u = 0.873776 0.112840I
a = 0.86301 + 1.85824I
b = 0.325555 0.658517I
7.98517 + 11.01329I 4.27925 5.99396I
u = 0.873776 + 0.112840I
a = 0.86301 1.85824I
b = 0.325555 + 0.658517I
7.98517 11.01329I 4.27925 + 5.99396I
11
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
65
+ 2u
64
+ ··· + u 1)
c
2
(u
2
+ u + 1)(u
65
+ 36u
64
+ ··· 5u 1)
c
3
(u
2
u + 1)(u
65
+ 2u
64
+ ··· 77u 209)
c
4
(u
2
u + 1)(u
65
14u
63
+ ··· 51079u 9713)
c
5
(u
2
u + 1)(u
65
+ 2u
64
+ ··· + u 1)
c
6
(u
2
+ u + 1)(u
65
+ 2u
64
+ ··· 27u + 17)
c
7
(u
2
+ u + 1)(u
65
+ 2u
64
+ ··· + u + 1)
c
8
(u 1)
2
(u
65
+ 3u
64
+ ··· 6u 1)
c
9
u
2
(u
65
+ 11u
64
+ ··· 12u 4)
c
10
(u + 1)
2
(u
65
+ 3u
64
+ ··· 6u 1)
c
11
(u
2
u + 1)(u
65
+ 2u
64
+ ··· 27u + 17)
12
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)(y
65
+ 36y
64
+ ··· 5y 1)
c
2
(y
2
+ y + 1)(y
65
12y
64
+ ··· 97y 1)
c
3
(y
2
+ y + 1)(y
65
80y
64
+ ··· + 680999y 43681)
c
4
(y
2
+ y + 1)(y
65
28y
64
+ ··· 2.28328 × 10
9
y 9.43424 × 10
7
)
c
6
, c
11
(y
2
+ y + 1)(y
65
60y
64
+ ··· 15013y 289)
c
7
(y
2
+ y + 1)(y
65
+ 12y
64
+ ··· 5y 1)
c
8
(y 1)
2
(y
65
51y
64
+ ··· 94y 1)
c
9
y
2
(y
65
+ 15y
64
+ ··· 152y 16)
c
10
(y 1)
2
(y
65
51y
64
+ ··· 94y 1)
13