11n
181
(K11n
181
)
1
Arc Sequences
6 8 1 10 9 1 4 2 5 4 8
Solving Sequence
2,6
1
7,9
5 10 4 8 3 11
c
1
c
6
c
5
c
9
c
4
c
8
c
2
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
7
+ 2u
5
+ u
4
+ u
3
+ u
2
u 1, b + u, u
6
u
5
+ 3u
4
u
3
+ 2u
2
+ a 1i
I
u
2
= hu
13
+ 5u
11
u
10
+ 15u
9
3u
8
+ 23u
7
4u
6
+ 22u
5
3u
4
+ 10u
3
2u
2
+ 2u 1, b + u,
57u
12
33u
11
+ ··· + 64a + 47i
I
u
3
= hu
16
+ u
15
+ ··· + 22u + 31,
3020100992u
15
+ 17915450451u
14
+ ··· + 339868204009b + 273458732915,
756017073639u
15
1082850030952u
14
+ ··· + 10535914324279a 12506104179146i
There are 3 irreducible components with 36 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
7
+ 2u
5
+ u
4
+ u
3
+ u
2
u 1, b + u, u
6
u
5
+ 3u
4
u
3
+ 2u
2
+ a 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
u
3
+ u
a
9
=
u
6
+ u
5
3u
4
+ u
3
2u
2
+ 1
u
a
5
=
u
6
2u
4
u
3
u + 2
u
6
+ 2u
4
+ u
2
+ u 1
a
10
=
u
6
u
5
+ 3u
4
u
3
+ 2u
2
2
u
6
3u
4
2u
2
+ 1
a
4
=
u
6
+ u
5
2u
4
+ u
3
u
2
+ 1
u
4
+ 2u
2
a
8
=
u
6
+ u
5
3u
4
+ u
3
2u
2
u + 1
u
a
3
=
u
6
+ u
5
2u
4
+ u
3
+ 2
u
2
a
11
=
u
6
u
5
+ 3u
4
2u
3
+ 2u
2
u 1
u
2
1
a
11
=
u
6
u
5
+ 3u
4
2u
3
+ 2u
2
u 1
u
2
1
(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 = 0.684504 0.212754I
a = 0.33756 1.93236I
b = 0.684504 + 0.212754I
0.58425 + 1.95701I 10.82069 1.34837I
u = 0.684504 + 0.212754I
a = 0.33756 + 1.93236I
b = 0.684504 0.212754I
0.58425 1.95701I 10.82069 + 1.34837I
u = 0.192319 1.190842I
a = 0.587751 + 0.132804I
b = 0.192319 + 1.190842I
4.79738 0.94912I 1.21872 + 0.82233I
u = 0.192319 + 1.190842I
a = 0.587751 0.132804I
b = 0.192319 1.190842I
4.79738 + 0.94912I 1.21872 0.82233I
u = 0.511889 1.253215I
a = 1.161310 0.409623I
b = 0.511889 + 1.253215I
13.16474 + 2.34118I 0.965786 0.952471I
u = 0.511889 + 1.253215I
a = 1.161310 + 0.409623I
b = 0.511889 1.253215I
13.16474 2.34118I 0.965786 + 0.952471I
u = 0.729869
a = 0.471993
b = 0.729869
4.19405 9.98962
3
II. I
u
2
= hu
13
+ 5u
11
+ · · · + 2u 1, b + u, 57u
12
33u
11
+ · · · + 64a + 47i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
u
3
+ u
a
9
=
0.890625u
12
+ 0.515625u
11
+ ··· + 1.10938u 0.734375
u
a
5
=
0.406250u
12
+ 0.343750u
11
+ ··· + 4.40625u 1.15625
0.140625u
12
+ 0.234375u
11
+ ··· + 1.14063u 0.515625
a
10
=
0.531250u
12
0.781250u
11
+ ··· + 1.53125u + 0.718750
0.0468750u
12
0.0781250u
11
+ ··· 0.0468750u + 0.171875
a
4
=
3
4
u
12
3
4
u
11
+ ··· +
11
4
u +
1
4
0.0156250u
12
0.640625u
11
+ ··· 0.984375u + 0.609375
a
8
=
0.890625u
12
+ 0.515625u
11
+ ··· + 0.109375u 0.734375
u
a
3
=
0.515625u
12
0.140625u
11
+ ··· + 2.51563u + 0.109375
u
2
a
11
=
0.296875u
12
1.17188u
11
+ ··· 1.70313u + 2.57813
0.234375u
12
0.609375u
11
+ ··· + 0.234375u + 0.140625
a
11
=
0.296875u
12
1.17188u
11
+ ··· 1.70313u + 2.57813
0.234375u
12
0.609375u
11
+ ··· + 0.234375u + 0.140625
(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 = 0.76994 1.47811I
a = 0.931114 0.335072I
b = 0.76994 + 1.47811I
6.50636 11.34500I 6.41522 + 5.59283I
u = 0.76994 + 1.47811I
a = 0.931114 + 0.335072I
b = 0.76994 1.47811I
6.50636 + 11.34500I 6.41522 5.59283I
u = 0.454277 1.006096I
a = 0.093987 + 0.805953I
b = 0.454277 + 1.006096I
3.02361 2.70878I 9.87229 + 2.50117I
u = 0.454277 + 1.006096I
a = 0.093987 0.805953I
b = 0.454277 1.006096I
3.02361 + 2.70878I 9.87229 2.50117I
u = 0.345858 0.592455I
a = 1.35097 0.54415I
b = 0.345858 + 0.592455I
2.39689 1.47210I 6.76905 + 4.68228I
u = 0.345858 + 0.592455I
a = 1.35097 + 0.54415I
b = 0.345858 0.592455I
2.39689 + 1.47210I 6.76905 4.68228I
u = 0.150630 0.774375I
a = 1.54813 + 0.76688I
b = 0.150630 + 0.774375I
1.92199 1.66881I 4.76442 + 0.86409I
u = 0.150630 + 0.774375I
a = 1.54813 0.76688I
b = 0.150630 0.774375I
1.92199 + 1.66881I 4.76442 0.86409I
u = 0.356605
a = 0.881921
b = 0.356605
0.542082 18.2828
u = 0.589560 1.048774I
a = 1.26834 0.65193I
b = 0.589560 + 1.048774I
12.09747 + 2.52656I 9.24277 2.75851I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.589560 + 1.048774I
a = 1.26834 + 0.65193I
b = 0.589560 1.048774I
12.09747 2.52656I 9.24277 + 2.75851I
u = 0.651583 1.236761I
a = 0.714908 + 0.187738I
b = 0.651583 + 1.236761I
1.53379 + 7.84030I 8.79484 6.42108I
u = 0.651583 + 1.236761I
a = 0.714908 0.187738I
b = 0.651583 1.236761I
1.53379 7.84030I 8.79484 + 6.42108I
6
III. I
u
3
=
hu
16
+u
15
+· · ·+22u+31, 3.02×10
9
u
15
+1.79×10
10
u
14
+· · ·+3.40×10
11
b+
2.73× 10
11
, 7.56 ×10
11
u
15
1.08 ×10
12
u
14
+· · · +1.05 ×10
13
a 1.25× 10
13
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
7
=
u
u
3
+ u
a
9
=
0.0717562u
15
+ 0.102777u
14
+ ··· + 2.64959u + 1.18700
0.00888609u
15
0.0527129u
14
+ ··· 0.917073u 0.804602
a
5
=
0.0456383u
15
+ 0.00210365u
14
+ ··· + 1.00937u + 1.43837
0.0466726u
15
0.00819680u
14
+ ··· + 0.200327u 2.32362
a
10
=
0.0311097u
15
0.0880219u
14
+ ··· 2.05895u 0.233069
0.0214961u
15
+ 0.0759945u
14
+ ··· + 1.14894u + 1.31477
a
4
=
0.0372499u
15
+ 0.0153993u
14
+ ··· + 0.787082u 1.68262
0.00152993u
15
0.0806813u
14
+ ··· 0.717701u 2.17728
a
8
=
0.0806423u
15
+ 0.0500641u
14
+ ··· + 1.73252u + 0.382395
0.00888609u
15
0.0527129u
14
+ ··· 0.917073u 0.804602
a
3
=
0.00667175u
15
0.107263u
14
+ ··· 0.604653u 3.18254
0.0682838u
15
0.228891u
14
+ ··· 3.69147u 5.03188
a
11
=
0.0372499u
15
0.0153993u
14
+ ··· 0.787082u + 1.68262
0.0305782u
15
+ 0.122662u
14
+ ··· + 1.39174u + 1.49991
a
11
=
0.0372499u
15
0.0153993u
14
+ ··· 0.787082u + 1.68262
0.0305782u
15
+ 0.122662u
14
+ ··· + 1.39174u + 1.49991
(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 = 1.55524 0.42761I
a = 0.192476 0.945314I
b = 0.176984 + 1.046895I
2.84290 + 3.16396I 6.17326 2.56480I
u = 1.55524 + 0.42761I
a = 0.192476 + 0.945314I
b = 0.176984 1.046895I
2.84290 3.16396I 6.17326 + 2.56480I
u = 0.589780 0.682901I
a = 0.711330 + 0.035736I
b = 1.159017 + 0.482637I
4.15885 1.41510I 9.82674 + 4.90874I
u = 0.589780 + 0.682901I
a = 0.711330 0.035736I
b = 1.159017 0.482637I
4.15885 + 1.41510I 9.82674 4.90874I
u = 0.309677 0.918028I
a = 0.626050 0.219220I
b = 0.092247 1.363224I
3.73684 1.41510I 9.82674 + 4.90874I
u = 0.309677 + 0.918028I
a = 0.626050 + 0.219220I
b = 0.092247 + 1.363224I
3.73684 + 1.41510I 9.82674 4.90874I
u = 0.26048 1.62229I
a = 0.943043 + 0.086771I
b = 0.786926 1.059081I
10.73858 3.16396I 6.17326 + 2.56480I
u = 0.26048 + 1.62229I
a = 0.943043 0.086771I
b = 0.786926 + 1.059081I
10.73858 + 3.16396I 6.17326 2.56480I
u = 0.092247 1.363224I
a = 0.389627 0.263479I
b = 0.309677 0.918028I
3.73684 + 1.41510I 9.82674 4.90874I
u = 0.092247 + 1.363224I
a = 0.389627 + 0.263479I
b = 0.309677 + 0.918028I
3.73684 1.41510I 9.82674 + 4.90874I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.176984 1.046895I
a = 1.45821 + 0.14634I
b = 1.55524 + 0.42761I
2.84290 + 3.16396I 6.17326 2.56480I
u = 0.176984 + 1.046895I
a = 1.45821 0.14634I
b = 1.55524 0.42761I
2.84290 3.16396I 6.17326 + 2.56480I
u = 0.786926 1.059081I
a = 0.991866 + 0.637958I
b = 0.26048 1.62229I
10.73858 + 3.16396I 6.17326 2.56480I
u = 0.786926 + 1.059081I
a = 0.991866 0.637958I
b = 0.26048 + 1.62229I
10.73858 3.16396I 6.17326 + 2.56480I
u = 1.159017 0.482637I
a = 0.135341 0.493664I
b = 0.589780 + 0.682901I
4.15885 1.41510I 9.82674 + 4.90874I
u = 1.159017 + 0.482637I
a = 0.135341 + 0.493664I
b = 0.589780 0.682901I
4.15885 + 1.41510I 9.82674 4.90874I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
8
(u
7
+ 2u
5
+ ··· u 1)(u
13
+ 5u
11
+ ··· + 2u 1)
(u
16
+ u
15
+ ··· + 22u + 31)
c
2
, c
6
(u
7
+ 2u
5
+ ··· u + 1)(u
13
+ 5u
11
+ ··· + 2u 1)
(u
16
+ u
15
+ ··· + 22u + 31)
c
3
(u
2
u 1)
8
(u
7
+ 3u
6
+ 3u
5
+ 4u
4
+ 6u
3
+ u
2
u + 2)
(u
13
+ 12u
12
+ ··· 16u 16)
c
4
, c
5
(u
4
u
3
+ 3u
2
2u + 1)
4
(u
7
+ 5u
5
+ 7u
3
+ 2u 1)
(u
13
+ 5u
12
+ ··· + 30u + 4)
c
7
, c
11
(u
7
u
6
+ ··· + 2u
2
+ 1)(u
13
+ u
12
+ ··· + 3u + 1)
(u
16
+ u
15
+ ··· 48u + 19)
c
9
, c
10
(u
4
u
3
+ 3u
2
2u + 1)
4
(u
7
+ 5u
5
+ 7u
3
+ 2u + 1)
(u
13
+ 5u
12
+ ··· + 30u + 4)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
2
, c
6
c
8
(y
7
+ 4y
6
+ ··· + 3y 1)(y
13
+ 10y
12
+ ··· + 30y
2
1)
(y
16
+ 7y
15
+ ··· + 4104y + 961)
c
3
(y
2
3y + 1)
8
(y
7
3y
6
3y
5
+ 12y
4
+ 10y
3
29y
2
3y 4)
(y
13
6y
12
+ ··· + 4480y 256)
c
4
, c
5
, c
9
c
10
(y
4
+ 5y
3
+ 7y
2
+ 2y + 1)
4
(y
7
+ 10y
6
+ 39y
5
+ 74y
4
+ 69y
3
+ 28y
2
+ 4y 1)
(y
13
+ 15y
12
+ ··· + 124y 16)
c
7
, c
11
(y
7
3y
6
+ ··· 4y 1)(y
13
17y
12
+ ··· + 25y 1)
(y
16
9y
15
+ ··· 4356y + 361)
11