11n
123
(K11n
123
)
1
Arc Sequences
6 1 9 8 2 5 10 11 1 5 4
Solving Sequence
2,5
6 7 1
3,11
10 8 4 9
c
5
c
6
c
1
c
2
c
10
c
7
c
4
c
9
c
3
, c
8
, c
11
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= ha
4
a
3
+ 2a + 1, a
3
+ 2a
2
a + u 1, a
3
+ 2a
2
+ b 2a 2i
I
u
2
= hu
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1, u
3
2u
2
+ b + 2u 1, u
5
2u
4
+ 4u
3
4u
2
+ a + 3ui
I
u
3
= ha
22
+ 4a
21
+ ··· + 97a + 47, 1.11015 × 10
30
u 1.65327 × 10
29
a
21
+ ··· 1.03794 × 10
31
a 2.73663 × 10
30
,
1.11015 × 10
30
b + 3.42462 × 10
29
a
21
+ ··· + 1.93051 × 10
31
a + 1.61394 × 10
31
i
I
u
4
= hu
16
+ 5u
15
+ ··· + 25u + 4,
u
13
2u
12
6u
11
8u
10
13u
9
17u
8
20u
7
26u
6
22u
5
21u
4
14u
3
9u
2
+ b 5u 1,
11u
15
+ 43u
14
+ ··· + 4a + 39i
There are 4 irreducible components with 48 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
a
3
+ 2a + 1, a
3
+ 2a
2
a + u 1, a
3
+ 2a
2
+ b 2a 2i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
a
3
2a
2
+ a + 1
a
6
=
a
3
2a
2
+ a + 1
a
3
2a
2
+ a + 1
a
7
=
a
3
2a
2
+ a + 1
a
3
2a
2
+ a + 2
a
1
=
a
3
+ 2a
2
a 1
a
3
+ 2a
2
a 2
a
3
=
0
a
3
2a
2
+ a + 1
a
11
=
a
a
3
2a
2
+ 2a + 2
a
10
=
a
2a
3
3a
2
+ 2a + 3
a
8
=
a
3
2a
2
+ 2a + 1
3a
3
5a
2
+ 3a + 5
a
4
=
2a
3
3a
2
+ a + 4
a
3
+ 2a
2
3a
a
9
=
a
3
2a
2
+ 2a + 1
3a
3
5a
2
+ 3a + 5
a
9
=
a
3
2a
2
+ 2a + 1
3a
3
5a
2
+ 3a + 5
(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 = 0.621964 0.187730I
b = 0.121964 1.053756I
1.64493 + 2.02988I 7.50000 + 0.86603I
u = 0.500000 + 0.866025I
a = 0.621964 + 0.187730I
b = 0.121964 + 1.053756I
1.64493 2.02988I 7.50000 0.86603I
u = 0.500000 + 0.866025I
a = 1.12196 1.05376I
b = 1.62196 0.18773I
1.64493 2.02988I 7.50000 0.86603I
u = 0.500000 0.866025I
a = 1.12196 + 1.05376I
b = 1.62196 + 0.18773I
1.64493 + 2.02988I 7.50000 + 0.86603I
3
II. I
u
2
= hu
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1, u
3
2u
2
+ b + 2u 1, u
5
2u
4
+ 4u
3
4u
2
+ a + 3ui
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
11
=
u
5
+ 2u
4
4u
3
+ 4u
2
3u
u
3
+ 2u
2
2u + 1
a
10
=
u
5
+ 2u
4
4u
3
+ 4u
2
3u
u
2
u + 1
a
8
=
u
4
+ 2u
3
3u
2
+ 3u 2
u
5
+ 2u
4
2u
3
+ 2u
2
u
a
4
=
u
5
u
4
+ 2u
3
+ 2
u
5
2u
4
+ 4u
3
4u
2
+ 3u 1
a
9
=
u
5
+ 2u
4
4u
3
+ 4u
2
2u
u
4
2u
3
+ 4u
2
2u + 2
a
9
=
u
5
+ 2u
4
4u
3
+ 4u
2
2u
u
4
2u
3
+ 4u
2
2u + 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 = 0.034417 0.580231I
a = 1.13629 + 1.13718I
b = 0.363148 + 1.047058I
1.27956 + 3.69612I 2.32375 5.61497I
u = 0.034417 + 0.580231I
a = 1.13629 1.13718I
b = 0.363148 1.047058I
1.27956 3.69612I 2.32375 + 5.61497I
u = 0.096993 1.308888I
a = 0.846700 0.549053I
b = 2.10396 0.09547I
4.36362 4.05299I 2.48265 + 9.09326I
u = 0.096993 + 1.308888I
a = 0.846700 + 0.549053I
b = 2.10396 + 0.09547I
4.36362 + 4.05299I 2.48265 9.09326I
u = 0.937424 0.916243I
a = 0.482988 0.383006I
b = 0.740809 + 0.043150I
3.99825 3.41127I 0.80640 + 5.19600I
u = 0.937424 + 0.916243I
a = 0.482988 + 0.383006I
b = 0.740809 0.043150I
3.99825 + 3.41127I 0.80640 5.19600I
5
III.
I
u
3
= ha
22
+ 4a
21
+ · · · + 97a + 47, 1.11 × 10
30
u 1.65 × 10
29
a
21
+ · · · 1.04 ×
10
31
a2.74×10
30
, 1.11×10
30
b+3.42×10
29
a
21
+· · ·+1.93×10
31
a+1.61×10
31
i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
0.148923a
21
+ 0.339449a
20
+ ··· + 9.34953a + 2.46510
a
6
=
0.148923a
21
+ 0.339449a
20
+ ··· + 9.34953a + 2.46510
0.148923a
21
+ 0.339449a
20
+ ··· + 9.34953a + 2.46510
a
7
=
0.148923a
21
+ 0.339449a
20
+ ··· + 9.34953a + 2.46510
0.398518a
21
+ 1.20653a
20
+ ··· + 18.7370a + 18.8088
a
1
=
0.197214a
21
+ 0.473950a
20
+ ··· + 8.89023a + 3.07089
0.197214a
21
+ 0.473950a
20
+ ··· + 8.89023a + 2.07089
a
3
=
0.156327a
21
+ 0.675487a
20
+ ··· + 10.6308a + 17.5760
0.0408872a
21
+ 0.201537a
20
+ ··· + 1.74061a + 14.5051
a
11
=
a
0.308482a
21
0.935338a
20
+ ··· 17.3897a 14.5381
a
10
=
a
0.623390a
21
1.77061a
20
+ ··· 34.4486a 23.8072
a
8
=
0.531579a
21
1.50040a
20
+ ··· 25.3250a 19.8230
0.948107a
21
2.58160a
20
+ ··· 44.4383a 31.8347
a
4
=
1.00685a
21
3.02184a
20
+ ··· 53.2147a 45.5981
2.07713a
21
6.21708a
20
+ ··· 105.701a 93.7398
a
9
=
0.813886a
21
2.42211a
20
+ ··· 39.0830a 36.0694
1.82073a
21
5.44395a
20
+ ··· 92.2976a 81.6674
a
9
=
0.813886a
21
2.42211a
20
+ ··· 39.0830a 36.0694
1.82073a
21
5.44395a
20
+ ··· 92.2976a 81.6674
(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 = 0.371033 0.270161I
a = 1.56375 1.12340I
b = 0.235267 1.218702I
0.25878 + 3.13682I 5.62912 1.87495I
u = 0.371033 + 0.270161I
a = 1.56375 + 1.12340I
b = 0.235267 + 1.218702I
0.25878 3.13682I 5.62912 + 1.87495I
u = 0.221199 1.131889I
a = 1.015248 0.145269I
b = 2.92424 0.05703I
2.74251 5.63735I 0.48609 + 8.17754I
u = 0.221199 + 1.131889I
a = 1.015248 + 0.145269I
b = 2.92424 + 0.05703I
2.74251 + 5.63735I 0.48609 8.17754I
u = 0.52365 + 1.35993I
a = 1.011995 0.185578I
b = 2.23805 0.64099I
10.59453 + 5.64581I 3.10897 3.66343I
u = 0.52365 1.35993I
a = 1.011995 + 0.185578I
b = 2.23805 + 0.64099I
10.59453 5.64581I 3.10897 + 3.66343I
u = 0.617799 0.778228I
a = 0.772875 1.064530I
b = 1.356090 0.300734I
1.20928 + 2.43685I 2.45208 7.14380I
u = 0.617799 + 0.778228I
a = 0.772875 + 1.064530I
b = 1.356090 + 0.300734I
1.20928 2.43685I 2.45208 + 7.14380I
u = 0.617799 0.778228I
a = 0.734637 0.378412I
b = 0.177438 0.637479I
1.20928 + 2.43685I 2.45208 7.14380I
u = 0.617799 + 0.778228I
a = 0.734637 + 0.378412I
b = 0.177438 + 0.637479I
1.20928 2.43685I 2.45208 + 7.14380I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.371033 + 0.270161I
a = 0.26489 2.44265I
b = 0.411141 + 0.216736I
0.25878 3.13682I 5.62912 + 1.87495I
u = 0.371033 0.270161I
a = 0.26489 + 2.44265I
b = 0.411141 0.216736I
0.25878 + 3.13682I 5.62912 1.87495I
u = 1.06351
a = 0.116699 1.384196I
b = 0.256101 + 0.093486I
6.33840 1.16744
u = 1.06351
a = 0.116699 + 1.384196I
b = 0.256101 0.093486I
6.33840 1.16744
u = 0.029838 1.264782I
a = 0.127099 0.316107I
b = 0.596986 + 1.102979I
5.00595 + 2.60776I 5.49826 2.04245I
u = 0.029838 + 1.264782I
a = 0.127099 + 0.316107I
b = 0.596986 1.102979I
5.00595 2.60776I 5.49826 + 2.04245I
u = 0.221199 + 1.131889I
a = 0.82678 1.38094I
b = 0.716292 0.998646I
2.74251 + 5.63735I 0.48609 8.17754I
u = 0.221199 1.131889I
a = 0.82678 + 1.38094I
b = 0.716292 + 0.998646I
2.74251 5.63735I 0.48609 + 8.17754I
u = 0.52365 1.35993I
a = 1.239903 0.198277I
b = 2.51307 0.01980I
10.59453 5.64581I 3.10897 + 3.66343I
u = 0.52365 + 1.35993I
a = 1.239903 + 0.198277I
b = 2.51307 + 0.01980I
10.59453 + 5.64581I 3.10897 3.66343I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.029838 1.264782I
a = 1.286314 0.299862I
b = 2.43750 0.42518I
5.00595 + 2.60776I 5.49826 2.04245I
u = 0.029838 + 1.264782I
a = 1.286314 + 0.299862I
b = 2.43750 + 0.42518I
5.00595 2.60776I 5.49826 + 2.04245I
8
IV. I
u
4
=
hu
16
+5u
15
+· · ·+25u+4, u
13
2u
12
+· · ·+b1, 11u
15
+43u
14
+· · ·+4a+39i
(i) Arc colorings
a
2
=
1
0
a
5
=
0
u
a
6
=
u
u
a
7
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
11
=
11
4
u
15
43
4
u
14
+ ··· 48u
39
4
u
13
+ 2u
12
+ ··· + 5u + 1
a
10
=
11
4
u
15
43
4
u
14
+ ··· 48u
39
4
3u
15
12u
14
+ ··· 59u 11
a
8
=
11
4
u
15
+
43
4
u
14
+ ··· + 58u +
43
4
3u
15
+ 12u
14
+ ··· + 59u + 11
a
4
=
13
4
u
15
49
4
u
14
+ ··· 69u
49
4
u
14
+ 2u
13
+ ··· 4u 1
a
9
=
25
4
u
15
+
93
4
u
14
+ ··· + 98u +
69
4
3u
15
+ 11u
14
+ ··· + 29u + 5
a
9
=
25
4
u
15
+
93
4
u
14
+ ··· + 98u +
69
4
3u
15
+ 11u
14
+ ··· + 29u + 5
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.160333 0.003661I
a = 0.303683 + 1.229775I
b = 0.050547 0.230239I
5.07602 + 7.48555I 0.37702 4.74920I
u = 1.160333 + 0.003661I
a = 0.303683 1.229775I
b = 0.050547 + 0.230239I
5.07602 7.48555I 0.37702 + 4.74920I
u = 0.54868 1.39870I
a = 1.174395 0.035505I
b = 2.59319 + 0.08166I
9.5135 + 13.5240I 1.40716 7.09485I
u = 0.54868 + 1.39870I
a = 1.174395 + 0.035505I
b = 2.59319 0.08166I
9.5135 13.5240I 1.40716 + 7.09485I
u = 0.54609 1.44233I
a = 0.872029 + 0.296610I
b = 2.06967 + 0.57462I
9.63409 1.32861I 2.22452 + 1.49647I
u = 0.54609 + 1.44233I
a = 0.872029 0.296610I
b = 2.06967 0.57462I
9.63409 + 1.32861I 2.22452 1.49647I
u = 0.432484 0.256526I
a = 1.40295 0.58537I
b = 0.416387 0.230792I
0.807041 + 1.112271I 4.93277 4.22512I
u = 0.432484 + 0.256526I
a = 1.40295 + 0.58537I
b = 0.416387 + 0.230792I
0.807041 1.112271I 4.93277 + 4.22512I
u = 0.369944 0.821753I
a = 0.034260 0.715157I
b = 0.183984 0.642300I
0.30256 + 1.83448I 0.83977 3.70409I
u = 0.369944 + 0.821753I
a = 0.034260 + 0.715157I
b = 0.183984 + 0.642300I
0.30256 1.83448I 0.83977 + 3.70409I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.181272 1.345516I
a = 0.842177 0.427246I
b = 2.01613 0.11456I
4.12337 + 3.39887I 1.84182 + 0.78536I
u = 0.181272 + 1.345516I
a = 0.842177 + 0.427246I
b = 2.01613 + 0.11456I
4.12337 3.39887I 1.84182 0.78536I
u = 0.053151 0.986056I
a = 1.237597 + 0.475690I
b = 1.68480 0.60416I
0.087550 + 0.115204I 2.02768 + 0.43913I
u = 0.053151 + 0.986056I
a = 1.237597 0.475690I
b = 1.68480 + 0.60416I
0.087550 0.115204I 2.02768 0.43913I
u = 0.791950 0.933769I
a = 0.112206 + 0.242982I
b = 0.524858 0.403689I
4.72589 3.00353I 9.61260 + 1.22526I
u = 0.791950 + 0.933769I
a = 0.112206 0.242982I
b = 0.524858 + 0.403689I
4.72589 + 3.00353I 9.61260 1.22526I
11
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
6
2u
5
+ 4u
4
4u
3
+ 4u
2
u + 1)
(1 + 4u 8u
2
+ 9u
3
13u
4
+ 12u
5
13u
6
+ 12u
7
8u
8
+ 6u
9
2u
10
+ u
11
)
2
(u
16
+ 5u
15
+ ··· + 25u + 4)
c
2
, c
6
(u
2
+ u + 1)
2
(u
6
+ 4u
5
+ 8u
4
+ 14u
3
+ 16u
2
+ 7u + 1)
(1 18u
2
57u
3
81u
4
62u
5
5u
6
+ 46u
7
+ 52u
8
+ 28u
9
+ 8u
10
+ u
11
)
2
(u
16
+ 11u
15
+ ··· + 15u + 16)
c
3
, c
10
(u
4
u
3
+ 3u
2
u + 1)(u
6
+ u
4
u
3
+ 2u
2
u + 1)
(u
16
+ 8u
14
+ ··· + u + 1)(u
22
+ 10u
20
+ ··· 265u + 47)
c
4
, c
11
(u
4
u
3
+ 3u
2
u + 1)(u
6
u
5
+ 2u
4
u
3
+ u
2
+ 1)
(u
16
+ u
15
+ ··· + 8u
2
+ 1)(u
22
+ 2u
21
+ ··· + 3u + 1)
c
5
(u
2
u + 1)
2
(u
6
+ 2u
5
+ 4u
4
+ 4u
3
+ 4u
2
+ u + 1)
(1 + 4u 8u
2
+ 9u
3
13u
4
+ 12u
5
13u
6
+ 12u
7
8u
8
+ 6u
9
2u
10
+ u
11
)
2
(u
16
+ 5u
15
+ ··· + 25u + 4)
c
7
, c
9
(u + 1)
4
(u
6
2u
5
+ 5u
4
5u
3
+ 4u
2
3u + 1)
(u
16
2u
15
+ ··· 5u + 1)(u
22
+ 5u
21
+ ··· 94u + 53)
c
8
u
4
(u
6
8u
5
+ 30u
4
65u
3
+ 84u
2
62u + 21)
(4 10u u
2
+ 25u
3
22u
4
10u
5
+ 29u
6
15u
7
5u
8
+ 10u
9
5u
10
+ u
11
)
2
(u
16
+ 11u
15
+ ··· + 9u + 2)
12
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)
2
(y
6
+ 4y
5
+ 8y
4
+ 14y
3
+ 16y
2
+ 7y + 1)
(1 18y
2
57y
3
81y
4
62y
5
5y
6
+ 46y
7
+ 52y
8
+ 28y
9
+ 8y
10
+ y
11
)
2
(y
16
+ 11y
15
+ ··· + 15y + 16)
c
2
, c
6
(y
2
+ y + 1)
2
(y
6
16y
4
+ 6y
3
+ 76y
2
17y + 1)
(1 36y 486y
2
+ 323y
3
+ 431y
4
322y
5
209y
6
+ 346y
7
172y
8
+ 44y
9
8y
10
+ y
11
)
2
(y
16
9y
15
+ ··· + 3679y + 256)
c
3
, c
10
(y
4
+ 5y
3
+ 9y
2
+ 5y + 1)(y
6
+ 2y
5
+ 5y
4
+ 5y
3
+ 4y
2
+ 3y + 1)
(y
16
+ 16y
15
+ ··· + 13y + 1)(y
22
+ 20y
21
+ ··· 5647y + 2209)
c
4
, c
11
(y
4
+ 5y
3
+ 9y
2
+ 5y + 1)(y
6
+ 3y
5
+ 4y
4
+ 5y
3
+ 5y
2
+ 2y + 1)
(y
16
+ 9y
15
+ ··· + 16y + 1)(y
22
+ 12y
20
+ ··· + 21y + 1)
c
7
, c
9
(y 1)
4
(y
6
+ 6y
5
+ 13y
4
+ 5y
3
4y
2
y + 1)
(y
16
+ 20y
15
+ ··· + 89y + 1)(y
22
+ 23y
21
+ ··· 34700y + 2809)
c
8
y
4
(y
6
4y
5
+ 28y
4
135y
3
+ 256y
2
316y + 441)
(16 + 108y 325y
2
+ 549y
3
586y
4
+ 456y
5
291y
6
+ 145y
7
55y
8
+ 20y
9
5y
10
+ y
11
)
2
(y
16
9y
15
+ ··· + 67y + 4)
13