11n
125
(K11n
125
)
1
Arc Sequences
7 1 9 10 11 2 10 6 4 8 9
Solving Sequence
4,10 5,8
11 6 7 9 1 3 2
c
4
c
10
c
5
c
7
c
9
c
11
c
3
c
2
c
1
, c
6
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
10
u
9
8u
8
+ 7u
7
+ 12u
6
+ u
5
+ 25u
4
72u
3
27u
2
+ 50u + 25,
14671u
9
4984u
8
+ ··· + 126255b + 192780, 93472u
9
11682u
8
+ ··· + 631275a 852305i
I
u
2
= hu
17
+ u
16
+ ··· + 4u + 3,
33670903502348u
16
+ 54049246568203u
15
+ ··· + 10871069781311b 164468318079608,
205038271164289u
16
+ 320209520593021u
15
+ ··· + 32613209343933a 915506075920007i
I
u
3
= hu
24
+ u
23
+ ··· + 190u + 59,
4.61494 × 10
37
u
23
5.64295 × 10
37
u
22
+ ··· + 4.98789 × 10
39
a 1.52900 × 10
40
,
2.74666 × 10
45
u
23
1.95727 × 10
45
u
22
+ ··· + 7.67583 × 10
47
b 2.51776 × 10
47
i
There are 3 irreducible components with 51 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
10
u
9
+ · · · + 50u + 25, 14671u
9
4984u
8
+ · · · + 126255b +
192780, 93472u
9
11682u
8
+ · · · + 631275a 852305i
(i) Arc colorings
a
4
=
1
0
a
10
=
0.148069u
9
+ 0.0185054u
8
+ ··· + 2.66872u + 1.35013
0.116201u
9
+ 0.0394757u
8
+ ··· 3.21937u 1.52691
a
5
=
0.0550188u
9
0.156418u
8
+ ··· + 7.47723u + 4.92551
0.0261138u
9
0.00196428u
8
+ ··· 1.41659u 0.796681
a
8
=
0
u
a
11
=
0.148069u
9
+ 0.0185054u
8
+ ··· + 2.66872u + 1.35013
0.156461u
9
0.0280464u
8
+ ··· + 6.96050u + 1.71217
a
6
=
0.0610764u
9
0.0551249u
8
+ ··· + 3.51229u + 0.165554
0.0647895u
9
+ 0.0114451u
8
+ ··· 4.61035u 1.38090
a
7
=
0.399506u
9
0.0494761u
8
+ ··· 10.4252u 0.725603
0.211437u
9
0.00902935u
8
+ ··· + 7.67645u + 1.37547
a
9
=
0.0318673u
9
+ 0.0579811u
8
+ ··· 0.550649u 0.176777
0.116201u
9
+ 0.0394757u
8
+ ··· 3.21937u 1.52691
a
1
=
0.252823u
9
+ 0.182600u
8
+ ··· + 3.53136u 1.80114
0.178290u
9
+ 0.0465724u
8
+ ··· 3.68203u 0.164706
a
3
=
0.0811326u
9
+ 0.158383u
8
+ ··· 6.06065u 2.12883
0.0261138u
9
+ 0.00196428u
8
+ ··· + 1.41659u + 0.796681
a
2
=
0.338689u
9
+ 0.0760303u
8
+ ··· + 10.5740u + 3.74531
0.0726229u
9
0.0680448u
8
+ ··· + 8.11202u + 3.58889
a
2
=
0.338689u
9
+ 0.0760303u
8
+ ··· + 10.5740u + 3.74531
0.0726229u
9
0.0680448u
8
+ ··· + 8.11202u + 3.58889
(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 = 2.13111 0.20078I
a = 0.514559 0.106772I
b = 1.363393 0.095887I
7.15848 3.23765I 10.59891 + 4.72266I
u = 2.13111 + 0.20078I
a = 0.514559 + 0.106772I
b = 1.363393 + 0.095887I
7.15848 + 3.23765I 10.59891 4.72266I
u = 0.550303 0.163219I
a = 1.85661 + 1.14324I
b = 0.782055 0.380490I
2.47371 2.31326I 10.56078 + 6.69278I
u = 0.550303 + 0.163219I
a = 1.85661 1.14324I
b = 0.782055 + 0.380490I
2.47371 + 2.31326I 10.56078 6.69278I
u = 0.18682 1.46367I
a = 0.043537 0.437632I
b = 0.231765 + 0.745305I
1.41924 + 3.41496I 4.16112 7.56429I
u = 0.18682 + 1.46367I
a = 0.043537 + 0.437632I
b = 0.231765 0.745305I
1.41924 3.41496I 4.16112 + 7.56429I
u = 1.397910 0.061295I
a = 0.961127 + 0.434676I
b = 0.669161 0.228612I
4.21796 + 4.66670I 8.97137 7.61170I
u = 1.397910 + 0.061295I
a = 0.961127 0.434676I
b = 0.669161 + 0.228612I
4.21796 4.66670I 8.97137 + 7.61170I
u = 1.97032 0.05147I
a = 0.362612 0.100369I
b = 1.51873 0.12956I
5.66336 0.80372I 4.03005 2.76686I
u = 1.97032 + 0.05147I
a = 0.362612 + 0.100369I
b = 1.51873 + 0.12956I
5.66336 + 0.80372I 4.03005 + 2.76686I
3
II.
I
u
2
= hu
17
+u
16
+· · ·+4u+3, 3.37×10
13
u
16
+5.40×10
13
u
15
+· · ·+1.09×10
13
b
1.64 × 10
14
, 2.05 × 10
14
u
16
+ 3.20 × 10
14
u
15
+ · · · + 3.26 × 10
13
a 9.16 × 10
14
i
(i) Arc colorings
a
4
=
1
0
a
10
=
6.28697u
16
9.81840u
15
+ ··· + 85.0571u + 28.0716
3.09729u
16
4.97184u
15
+ ··· + 42.5351u + 15.1290
a
5
=
1.96925u
16
+ 3.38991u
15
+ ··· 35.6333u 13.5767
5.40598u
16
+ 8.91572u
15
+ ··· 80.7377u 28.1528
a
8
=
0
u
a
11
=
6.28697u
16
9.81840u
15
+ ··· + 85.0571u + 28.0716
5.40109u
16
8.66954u
15
+ ··· + 75.5217u + 25.7233
a
6
=
5.04300u
16
8.14029u
15
+ ··· + 71.3598u + 22.3631
0.935822u
16
+ 1.49836u
15
+ ··· 15.7705u 6.49137
a
7
=
7.22048u
16
11.6906u
15
+ ··· + 97.9146u + 36.0853
1.42066u
16
+ 2.13137u
15
+ ··· 21.4537u 5.90775
a
9
=
9.38426u
16
14.7902u
15
+ ··· + 127.592u + 43.2006
3.09729u
16
4.97184u
15
+ ··· + 42.5351u + 15.1290
a
1
=
12.0223u
16
+ 19.1997u
15
+ ··· 164.706u 56.9494
0.993997u
16
1.54491u
15
+ ··· + 18.3395u + 4.93960
a
3
=
7.37523u
16
12.3056u
15
+ ··· + 116.371u + 43.7295
5.40598u
16
8.91572u
15
+ ··· + 80.7377u + 28.1528
a
2
=
11.1815u
16
+ 18.0623u
15
+ ··· 165.012u 55.9365
3.64175u
16
+ 5.91408u
15
+ ··· 49.5643u 16.7132
a
2
=
11.1815u
16
+ 18.0623u
15
+ ··· 165.012u 55.9365
3.64175u
16
+ 5.91408u
15
+ ··· 49.5643u 16.7132
(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 = 2.12614 0.91206I
a = 0.601238 0.118985I
b = 1.22392 + 0.97901I
3.4739 14.0835I 4.55379 + 8.21992I
u = 2.12614 + 0.91206I
a = 0.601238 + 0.118985I
b = 1.22392 0.97901I
3.4739 + 14.0835I 4.55379 8.21992I
u = 2.10853 0.16979I
a = 0.588660 0.182489I
b = 0.724405 + 0.548318I
4.29085 6.01779I 7.18678 + 9.97359I
u = 2.10853 + 0.16979I
a = 0.588660 + 0.182489I
b = 0.724405 0.548318I
4.29085 + 6.01779I 7.18678 9.97359I
u = 0.982078
a = 1.06028
b = 0.641391
1.52652 7.18721
u = 0.383029 0.194414I
a = 0.21378 + 2.76362I
b = 0.746984 + 1.046147I
6.33629 0.97017I 0.559057 + 0.284542I
u = 0.383029 + 0.194414I
a = 0.21378 2.76362I
b = 0.746984 1.046147I
6.33629 + 0.97017I 0.559057 0.284542I
u = 0.226748 0.422497I
a = 1.38743 + 0.67738I
b = 0.512854 0.417660I
0.81159 1.48098I 4.11600 + 4.97481I
u = 0.226748 + 0.422497I
a = 1.38743 0.67738I
b = 0.512854 + 0.417660I
0.81159 + 1.48098I 4.11600 4.97481I
u = 0.615229 0.014283I
a = 0.94755 + 1.84905I
b = 0.909916 + 0.943922I
5.27663 + 7.20759I 2.52796 5.51575I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.615229 + 0.014283I
a = 0.94755 1.84905I
b = 0.909916 0.943922I
5.27663 7.20759I 2.52796 + 5.51575I
u = 0.930360 0.216532I
a = 0.684239 + 0.654110I
b = 0.126925 0.721173I
1.34232 1.94872I 2.56083 + 3.14210I
u = 0.930360 + 0.216532I
a = 0.684239 0.654110I
b = 0.126925 + 0.721173I
1.34232 + 1.94872I 2.56083 3.14210I
u = 1.332853 0.350987I
a = 1.106663 + 0.014416I
b = 0.725128 0.055021I
4.56765 + 3.93288I 12.22005 0.69203I
u = 1.332853 + 0.350987I
a = 1.106663 0.014416I
b = 0.725128 + 0.055021I
4.56765 3.93288I 12.22005 + 0.69203I
u = 1.95705 0.86607I
a = 0.627651 0.110975I
b = 1.05004 + 1.03914I
4.69624 + 7.43319I 2.80359 4.35460I
u = 1.95705 + 0.86607I
a = 0.627651 + 0.110975I
b = 1.05004 1.03914I
4.69624 7.43319I 2.80359 + 4.35460I
6
III. I
u
3
=
hu
24
+u
23
+· · ·+190u+59, 4.61×10
37
u
23
5.64×10
37
u
22
+· · ·+4.99×10
39
a
1.53 × 10
40
, 2.75 × 10
45
u
23
1.96 × 10
45
u
22
+ · · · + 7.68 × 10
47
b 2.52 × 10
47
i
(i) Arc colorings
a
4
=
1
0
a
10
=
0.00925230u
23
+ 0.0113133u
22
+ ··· 1.79943u + 3.06543
0.00357833u
23
+ 0.00254992u
22
+ ··· + 2.97228u + 0.328011
a
5
=
0.0118596u
23
+ 0.00982491u
22
+ ··· + 7.51417u + 1.15459
0.00457664u
23
+ 0.000626340u
22
+ ··· 1.90379u 1.11500
a
8
=
0
u
a
11
=
0.00925230u
23
+ 0.0113133u
22
+ ··· 1.79943u + 3.06543
0.00347074u
23
+ 0.00124126u
22
+ ··· + 2.03480u + 0.206411
a
6
=
0.00651253u
23
0.0102293u
22
+ ··· 1.37827u 3.08564
0.00467414u
23
0.00682862u
22
+ ··· 1.51399u + 0.0428833
a
7
=
0.00697002u
23
+ 0.000234866u
22
+ ··· + 6.71331u 1.49719
0.00203470u
23
0.00162716u
22
+ ··· 1.09874u 0.699717
a
9
=
0.0128306u
23
+ 0.0138632u
22
+ ··· + 1.17285u + 3.39344
0.00357833u
23
+ 0.00254992u
22
+ ··· + 2.97228u + 0.328011
a
1
=
0.0253792u
23
0.0116079u
22
+ ··· 16.6750u + 1.05331
0.00833291u
23
+ 0.00520755u
22
+ ··· + 1.99411u + 1.86520
a
3
=
0.00728297u
23
0.0104513u
22
+ ··· 5.61039u + 1.96041
0.00457664u
23
0.000626340u
22
+ ··· + 1.90379u + 1.11500
a
2
=
0.0338144u
23
0.0192687u
22
+ ··· 18.8172u 2.56210
0.00378331u
23
+ 0.00279273u
22
+ ··· 0.149786u + 1.69171
a
2
=
0.0338144u
23
0.0192687u
22
+ ··· 18.8172u 2.56210
0.00378331u
23
+ 0.00279273u
22
+ ··· 0.149786u + 1.69171
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 2.35550 0.67191I
a = 0.296360 + 0.084537I
b = 1.382991 + 0.000366I
6.25926 1.41510I 12.84625 + 4.90874I
u = 2.35550 + 0.67191I
a = 0.296360 0.084537I
b = 1.382991 0.000366I
6.25926 + 1.41510I 12.84625 4.90874I
u = 2.00666 1.10419I
a = 0.492418 + 0.100236I
b = 1.043135 0.911052I
4.88007 + 0.33584I 2.66351 + 0.41465I
u = 2.00666 + 1.10419I
a = 0.492418 0.100236I
b = 1.043135 + 0.911052I
4.88007 0.33584I 2.66351 0.41465I
u = 0.942897 0.293371I
a = 1.072539 + 0.456265I
b = 1.003937 0.452899I
2.12168 1.41302I 6.31698 1.92930I
u = 0.942897 + 0.293371I
a = 1.072539 0.456265I
b = 1.003937 + 0.452899I
2.12168 + 1.41302I 6.31698 + 1.92930I
u = 0.848687 0.213731I
a = 0.764371 1.070161I
b = 1.00875 1.10787I
4.88007 0.33584I 2.66351 0.41465I
u = 0.848687 + 0.213731I
a = 0.764371 + 1.070161I
b = 1.00875 + 1.10787I
4.88007 + 0.33584I 2.66351 + 0.41465I
u = 0.49005 2.26134I
a = 0.069096 + 0.318845I
b = 0.265048 + 0.154544I
0.74248 3.16396I 9.19277 + 2.56480I
u = 0.49005 + 2.26134I
a = 0.069096 0.318845I
b = 0.265048 0.154544I
0.74248 + 3.16396I 9.19277 2.56480I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.220109 0.211342I
a = 3.76479 0.23078I
b = 0.430835 0.856235I
2.12168 4.24323I 6.31698 + 7.88819I
u = 0.220109 + 0.211342I
a = 3.76479 + 0.23078I
b = 0.430835 + 0.856235I
2.12168 + 4.24323I 6.31698 7.88819I
u = 0.172123 0.671982I
a = 0.72634 1.49179I
b = 0.256438 + 0.045429I
2.12168 1.41302I 6.31698 1.92930I
u = 0.172123 + 0.671982I
a = 0.72634 + 1.49179I
b = 0.256438 0.045429I
2.12168 + 1.41302I 6.31698 + 1.92930I
u = 0.291351 0.679945I
a = 0.401919 + 0.937984I
b = 0.324811 1.039216I
0.74248 3.16396I 9.19277 + 2.56480I
u = 0.291351 + 0.679945I
a = 0.401919 0.937984I
b = 0.324811 + 1.039216I
0.74248 + 3.16396I 9.19277 2.56480I
u = 0.743789 0.351478I
a = 0.577498 1.274339I
b = 0.80956 1.30687I
4.88007 5.99209I 2.66351 + 5.54425I
u = 0.743789 + 0.351478I
a = 0.577498 + 1.274339I
b = 0.80956 + 1.30687I
4.88007 + 5.99209I 2.66351 5.54425I
u = 1.344344 0.300609I
a = 0.503612 0.666684I
b = 1.008369 + 0.230741I
2.12168 4.24323I 6.31698 + 7.88819I
u = 1.344344 + 0.300609I
a = 0.503612 + 0.666684I
b = 1.008369 0.230741I
2.12168 + 4.24323I 6.31698 7.88819I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.60692 0.28833I
a = 0.455115 + 0.081663I
b = 1.60815 0.28919I
6.25926 1.41510I 12.84625 + 4.90874I
u = 1.60692 + 0.28833I
a = 0.455115 0.081663I
b = 1.60815 + 0.28919I
6.25926 + 1.41510I 12.84625 4.90874I
u = 2.20538 0.84982I
a = 0.459744 + 0.160591I
b = 1.19721 0.83587I
4.88007 + 5.99209I 2.66351 5.54425I
u = 2.20538 + 0.84982I
a = 0.459744 0.160591I
b = 1.19721 + 0.83587I
4.88007 5.99209I 2.66351 + 5.54425I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
4
u
3
+ u
2
+ 1)
6
(u
10
+ u
9
+ 3u
8
+ 2u
7
+ 5u
6
+ 2u
5
+ 5u
4
+ 3u
2
+ 1)
(u
17
+ 6u
16
+ ··· + 32u + 8)
c
2
(1 + 2u + 3u
2
+ u
3
+ u
4
)
6
(u
10
+ 5u
9
+ ··· + 6u + 1)
(u
17
+ 6u
16
+ ··· + 32u 64)
c
3
, c
4
, c
8
(u
10
+ u
9
4u
8
4u
7
+ 4u
6
+ 5u
5
+ u
4
2u
3
2u
2
+ 1)
(u
17
+ u
16
+ ··· + 2u
2
1)(u
24
u
23
+ ··· + 12u + 1)
c
5
(u
10
+ u
9
+ ··· 2u
2
+ 1)(u
17
+ u
16
+ ··· 22u
2
+ 1)
(u
24
+ 3u
23
+ ··· + 54u + 107)
c
6
(u
4
u
3
+ u
2
+ 1)
6
(u
10
u
9
+ 3u
8
2u
7
+ 5u
6
2u
5
+ 5u
4
+ 3u
2
+ 1)
(u
17
+ 6u
16
+ ··· + 32u + 8)
c
7
(u
3
+ u
2
1)
8
(u
10
2u
9
u
8
+ 6u
7
2u
6
7u
5
+ 6u
4
+ 4u
3
4u
2
u + 1)
(u
17
7u
16
+ ··· 80u + 16)
c
9
(u
10
u
9
4u
8
+ 4u
7
+ 4u
6
5u
5
+ u
4
+ 2u
3
2u
2
+ 1)
(u
17
+ u
16
+ ··· + 2u
2
1)(u
24
u
23
+ ··· + 12u + 1)
c
10
(u
3
+ u
2
1)
8
(u
10
+ 2u
9
u
8
6u
7
2u
6
+ 7u
5
+ 6u
4
4u
3
4u
2
+ u + 1)
(u
17
7u
16
+ ··· 80u + 16)
c
11
(u
10
+ u
9
+ 3u
8
+ u
6
2u
5
+ 3u
4
+ u
3
+ 2u
2
+ 1)
(u
17
+ 3u
16
+ ··· + 4u + 1)(u
24
+ 3u
23
+ ··· + 846u + 347)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(1 + 2y + 3y
2
+ y
3
+ y
4
)
6
(y
10
+ 5y
9
+ ··· + 6y + 1)
(y
17
+ 6y
16
+ ··· + 32y 64)
c
2
(1 + 2y + 7y
2
+ 5y
3
+ y
4
)
6
(y
10
+ 5y
9
+ ··· + 2y + 1)
(y
17
+ 10y
16
+ ··· + 33280y 4096)
c
3
, c
4
, c
8
c
9
(y
10
9y
9
+ 32y
8
56y
7
+ 48y
6
15y
5
3y
4
+ 6y
2
4y + 1)
(y
17
y
16
+ ··· + 4y 1)(y
24
9y
23
+ ··· 40y + 1)
c
5
(y
10
+ 7y
9
+ 10y
8
9y
7
+ 2y
6
4y
5
+ 3y
3
+ 4y
2
4y + 1)
(y
17
13y
16
+ ··· + 44y 1)(y
24
+ 3y
23
+ ··· + 23192y + 11449)
c
7
, c
10
(y
3
y
2
+ 2y 1)
8
(y
10
6y
9
+ ··· 9y + 1)
(y
17
7y
16
+ ··· 128y 256)
c
11
(y
10
+ 5y
9
+ ··· + 4y + 1)(y
17
27y
16
+ ··· 24y 1)
(y
24
9y
23
+ ··· + 602884y + 120409)
12