11n
159
(K11n
159
)
1
Arc Sequences
10 9 1 10 11 2 3 11 7 5 8
Solving Sequence
5,11 6,8
9 1 10 2 4 3 7
c
5
c
8
c
11
c
10
c
1
c
4
c
3
c
7
c
2
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
16
3u
15
+ 3u
14
+ 12u
13
48u
12
+ 56u
11
53u
9
+ 273u
8
41u
7
195u
6
+ 4u
5
+ 18u
4
+ 92u
3
56u
2
+ 4,
3.16949 × 10
17
u
15
7.07957 × 10
17
u
14
+ ··· + 1.32086 × 10
18
b 1.55276 × 10
18
,
9.26588 × 10
17
u
15
2.45910 × 10
18
u
14
+ ··· + 1.32086 × 10
18
a 2.29756 × 10
18
i
I
u
2
= hu
51
34u
49
+ ··· 38412u + 7316,
1.45415 × 10
228
u
50
+ 8.60197 × 10
227
u
49
+ ··· + 6.33871 × 10
230
b 1.67155 × 10
232
,
1.21828 × 10
232
u
50
+ 6.24948 × 10
231
u
49
+ ··· + 2.31870 × 10
234
a 1.33856 × 10
236
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
16
3u
15
+· · ·56u
2
+4, 3.17×10
17
u
15
7.08×10
17
u
14
+· · ·+1.32×10
18
b
1.55 × 10
18
, 9.27 × 10
17
u
15
2.46 × 10
18
u
14
+ · · · + 1.32 × 10
18
a 2.30 × 10
18
i
(i) Arc colorings
a
5
=
1
0
a
11
=
0.701502u
15
+ 1.86173u
14
+ ··· + 17.6190u + 1.73944
0.239956u
15
+ 0.535981u
14
+ ··· 0.776797u + 1.17556
a
6
=
0.926114u
15
+ 2.28655u
14
+ ··· + 7.85036u + 9.38981
0.652004u
15
+ 1.54407u
14
+ ··· + 3.97597u + 3.39278
a
8
=
0
u
a
9
=
0.0605993u
15
+ 0.172207u
14
+ ··· + 5.99307u 5.68242
0.491797u
15
1.27320u
14
+ ··· 7.38981u 3.70446
a
1
=
0.701502u
15
+ 1.86173u
14
+ ··· + 17.6190u + 1.73944
0.122329u
15
+ 0.233345u
14
+ ··· 3.58281u + 0.204471
a
10
=
0.461546u
15
+ 1.32575u
14
+ ··· + 18.3958u + 0.563874
0.239956u
15
+ 0.535981u
14
+ ··· 0.776797u + 1.17556
a
2
=
0.308338u
15
+ 0.599709u
14
+ ··· 8.52734u + 5.30417
0.529675u
15
1.31073u
14
+ ··· 7.55877u 4.18831
a
4
=
0.274110u
15
0.742474u
14
+ ··· 3.87439u 3.99703
0.652004u
15
1.54407u
14
+ ··· 3.97597u 3.39278
a
3
=
0.0341537u
15
0.206494u
14
+ ··· 4.65119u 3.82147
1.13480u
15
2.79387u
14
+ ··· 13.3924u 7.91133
a
7
=
0.335168u
15
+ 1.08785u
14
+ ··· + 14.0657u + 4.41868
1.35331u
15
+ 3.31553u
14
+ ··· + 13.6398u + 8.89533
a
7
=
0.335168u
15
+ 1.08785u
14
+ ··· + 14.0657u + 4.41868
1.35331u
15
+ 3.31553u
14
+ ··· + 13.6398u + 8.89533
(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 = 2.36684
a = 0.677361
b = 1.55538
7.57445 38.0720
u = 1.161410 0.095564I
a = 0.346013 + 1.040218I
b = 0.586743 + 0.122702I
1.52177 5.31010I 3.27149 + 8.11960I
u = 1.161410 + 0.095564I
a = 0.346013 1.040218I
b = 0.586743 0.122702I
1.52177 + 5.31010I 3.27149 8.11960I
u = 0.377131 0.656461I
a = 1.215041 + 0.315082I
b = 0.961986 + 0.640032I
3.33249 0.34499I 2.01757 0.37341I
u = 0.377131 + 0.656461I
a = 1.215041 0.315082I
b = 0.961986 0.640032I
3.33249 + 0.34499I 2.01757 + 0.37341I
u = 0.228520
a = 5.57510
b = 0.425060
2.72211 31.3502
u = 0.07571 1.67351I
a = 0.437584 0.030033I
b = 1.255520 + 0.125305I
3.97957 + 3.37369I 2.06218 1.25376I
u = 0.07571 + 1.67351I
a = 0.437584 + 0.030033I
b = 1.255520 0.125305I
3.97957 3.37369I 2.06218 + 1.25376I
u = 0.494902 0.214225I
a = 1.47481 + 1.80275I
b = 1.330788 + 0.053274I
1.66733 5.08939I 7.99749 + 6.39804I
u = 0.494902 + 0.214225I
a = 1.47481 1.80275I
b = 1.330788 0.053274I
1.66733 + 5.08939I 7.99749 6.39804I
3
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.615830 0.101124I
a = 0.656078 + 0.968979I
b = 0.934791 + 0.689406I
3.32910 4.68558I 1.72806 + 9.06253I
u = 0.615830 + 0.101124I
a = 0.656078 0.968979I
b = 0.934791 0.689406I
3.32910 + 4.68558I 1.72806 9.06253I
u = 1.08865 2.23627I
a = 0.238082 0.072838I
b = 0.638027 0.074543I
1.44991 + 2.46756I 14.2433 0.1322I
u = 1.08865 + 2.23627I
a = 0.238082 + 0.072838I
b = 0.638027 + 0.074543I
1.44991 2.46756I 14.2433 + 0.1322I
u = 2.06113 0.76403I
a = 0.159682 + 0.607574I
b = 0.93054 + 1.06757I
7.35161 + 3.81019I 1.49699 3.24326I
u = 2.06113 + 0.76403I
a = 0.159682 0.607574I
b = 0.93054 1.06757I
7.35161 3.81019I 1.49699 + 3.24326I
4
II. I
u
2
= hu
51
34u
49
+ · · · 38412u + 7316, 1.45 × 10
228
u
50
+ 8.60 ×
10
227
u
49
+ · · · + 6.34 × 10
230
b 1.67 × 10
232
, 1.22 × 10
232
u
50
+ 6.25 ×
10
231
u
49
+ · · · + 2.32 × 10
234
a 1.34 × 10
236
i
(i) Arc colorings
a
5
=
1
0
a
11
=
0.00525414u
50
0.00269525u
49
+ ··· 247.225u + 57.7291
0.00229408u
50
0.00135705u
49
+ ··· 111.456u + 26.3705
a
6
=
0.00296957u
50
0.00162766u
49
+ ··· 131.251u + 27.2276
0.00440175u
50
0.00213971u
49
+ ··· 214.319u + 59.2652
a
8
=
0
u
a
9
=
0.00115778u
50
+ 0.000735404u
49
+ ··· 138.952u + 69.7801
0.00162766u
50
+ 0.000739317u
49
+ ··· + 88.8394u 21.7254
a
1
=
0.00525414u
50
0.00269525u
49
+ ··· 247.225u + 57.7291
0.00104626u
50
0.000871739u
49
+ ··· 46.3650u + 6.65203
a
10
=
0.00296006u
50
0.00133820u
49
+ ··· 135.769u + 31.3586
0.00229408u
50
0.00135705u
49
+ ··· 111.456u + 26.3705
a
2
=
0.0184954u
50
0.00843906u
49
+ ··· 980.077u + 269.220
0.00386291u
50
+ 0.00169577u
49
+ ··· + 191.570u 53.8895
a
4
=
0.00143218u
50
0.000512051u
49
+ ··· 83.0680u + 34.0376
0.00440175u
50
+ 0.00213971u
49
+ ··· + 214.319u 59.2652
a
3
=
0.0112004u
50
0.00401831u
49
+ ··· 610.861u + 198.564
0.00450096u
50
+ 0.00137021u
49
+ ··· + 252.703u 82.8963
a
7
=
0.00351161u
50
0.00346346u
49
+ ··· 100.687u 8.08016
0.00129940u
50
+ 0.000481082u
49
+ ··· + 68.4875u 24.4037
a
7
=
0.00351161u
50
0.00346346u
49
+ ··· 100.687u 8.08016
0.00129940u
50
+ 0.000481082u
49
+ ··· + 68.4875u 24.4037
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
5
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 2.38629
a = 0.647409
b = 1.52947
7.66230 53.2495
u = 2.37629 0.46498I
a = 0.127338 + 0.649402I
b = 1.18021 + 0.96200I
9.1940 15.1810I 2.73248 + 7.71054I
u = 2.37629 + 0.46498I
a = 0.127338 0.649402I
b = 1.18021 0.96200I
9.1940 + 15.1810I 2.73248 7.71054I
u = 2.28181 0.57934I
a = 0.110451 0.549019I
b = 1.038830 0.830920I
7.59339 1.64029I 1.00681 1.43004I
u = 2.28181 + 0.57934I
a = 0.110451 + 0.549019I
b = 1.038830 + 0.830920I
7.59339 + 1.64029I 1.00681 + 1.43004I
u = 2.06822 1.02485I
a = 0.221493 0.563667I
b = 0.841737 0.908889I
8.21456 4.85073I 3.48162 + 7.40715I
u = 2.06822 + 1.02485I
a = 0.221493 + 0.563667I
b = 0.841737 + 0.908889I
8.21456 + 4.85073I 3.48162 7.40715I
u = 1.55773 0.55114I
a = 0.571127 + 0.797456I
b = 0.83700 + 1.22014I
10.36515 + 7.36778I 1.14610 3.78229I
u = 1.55773 + 0.55114I
a = 0.571127 0.797456I
b = 0.83700 1.22014I
10.36515 7.36778I 1.14610 + 3.78229I
u = 1.53991 0.85223I
a = 0.520884 0.378590I
b = 0.656206 0.469667I
1.69267 + 6.49416I 2.47045 12.27311I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.53991 + 0.85223I
a = 0.520884 + 0.378590I
b = 0.656206 + 0.469667I
1.69267 6.49416I 2.47045 + 12.27311I
u = 1.24329 0.75083I
a = 0.334468 0.990349I
b = 0.423218 0.667986I
3.16584 5.71287I 1.14428 + 8.06159I
u = 1.24329 + 0.75083I
a = 0.334468 + 0.990349I
b = 0.423218 + 0.667986I
3.16584 + 5.71287I 1.14428 8.06159I
u = 1.02313 1.01983I
a = 0.620456 + 0.528966I
b = 1.064110 + 0.166907I
0.53609 2.28831I 3.93640 + 2.00082I
u = 1.02313 + 1.01983I
a = 0.620456 0.528966I
b = 1.064110 0.166907I
0.53609 + 2.28831I 3.93640 2.00082I
u = 0.592682 0.555401I
a = 1.54610 0.87819I
b = 1.04433 0.98128I
4.12435 + 1.05287I 0.46055 2.90458I
u = 0.592682 + 0.555401I
a = 1.54610 + 0.87819I
b = 1.04433 + 0.98128I
4.12435 1.05287I 0.46055 + 2.90458I
u = 0.510268 0.017817I
a = 1.48241 + 0.70636I
b = 0.590936 + 0.281707I
1.328987 + 0.424799I 9.68382 3.33114I
u = 0.510268 + 0.017817I
a = 1.48241 0.70636I
b = 0.590936 0.281707I
1.328987 0.424799I 9.68382 + 3.33114I
u = 0.392233
a = 3.38822
b = 0.202368
2.55208 16.4856
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.311992
a = 1.70410
b = 0.745090
1.10699 8.71624
u = 0.16119 1.62879I
a = 0.194743 + 0.027405I
b = 1.209941 0.110828I
4.50411 + 3.58227I 16.0667 6.6127I
u = 0.16119 + 1.62879I
a = 0.194743 0.027405I
b = 1.209941 + 0.110828I
4.50411 3.58227I 16.0667 + 6.6127I
u = 0.090556 0.440679I
a = 0.338264 + 0.064793I
b = 0.965350 0.733322I
2.97946 + 4.11273I 4.99561 + 1.01748I
u = 0.090556 + 0.440679I
a = 0.338264 0.064793I
b = 0.965350 + 0.733322I
2.97946 4.11273I 4.99561 1.01748I
u = 0.345885 0.807680I
a = 0.626153 + 0.879066I
b = 0.460423 0.213816I
3.18456 + 2.09749I 1.52387 2.56668I
u = 0.345885 + 0.807680I
a = 0.626153 0.879066I
b = 0.460423 + 0.213816I
3.18456 2.09749I 1.52387 + 2.56668I
u = 0.415759 0.165762I
a = 0.36088 2.71304I
b = 0.788429 0.102113I
0.82395 + 4.71884I 8.89552 4.87376I
u = 0.415759 + 0.165762I
a = 0.36088 + 2.71304I
b = 0.788429 + 0.102113I
0.82395 4.71884I 8.89552 + 4.87376I
u = 0.470782 0.141394I
a = 0.391310 0.101889I
b = 0.752804 + 0.724541I
3.57803 1.46219I 0.29485 + 4.98451I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.470782 + 0.141394I
a = 0.391310 + 0.101889I
b = 0.752804 0.724541I
3.57803 + 1.46219I 0.29485 4.98451I
u = 0.589639 0.285800I
a = 1.050199 0.048592I
b = 0.318964 + 0.525247I
0.18643 + 2.24007I 4.31467 4.82703I
u = 0.589639 + 0.285800I
a = 1.050199 + 0.048592I
b = 0.318964 0.525247I
0.18643 2.24007I 4.31467 + 4.82703I
u = 0.592081 0.474386I
a = 1.39245 + 1.18413I
b = 0.965038 + 0.987508I
4.33426 + 8.28696I 3.05654 6.59545I
u = 0.592081 + 0.474386I
a = 1.39245 1.18413I
b = 0.965038 0.987508I
4.33426 8.28696I 3.05654 + 6.59545I
u = 0.818290 0.099735I
a = 0.06664 + 1.58947I
b = 1.50242 + 0.25890I
0.75736 + 4.17149I 1.94325 1.93943I
u = 0.818290 + 0.099735I
a = 0.06664 1.58947I
b = 1.50242 0.25890I
0.75736 4.17149I 1.94325 + 1.93943I
u = 0.87425 2.17955I
a = 0.299163 0.102570I
b = 0.248978 0.026080I
0.97092 + 2.65596I 1.16196 5.22442I
u = 0.87425 + 2.17955I
a = 0.299163 + 0.102570I
b = 0.248978 + 0.026080I
0.97092 2.65596I 1.16196 + 5.22442I
u = 1.377274 0.174891I
a = 0.221169 0.776602I
b = 0.835661 0.381025I
1.47252 + 3.57882I 5.02464 4.02671I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.377274 + 0.174891I
a = 0.221169 + 0.776602I
b = 0.835661 + 0.381025I
1.47252 3.57882I 5.02464 + 4.02671I
u = 1.42034 0.42459I
a = 0.684601 0.962006I
b = 0.77539 1.18366I
9.56909 + 0.79682I 0.499984 0.266984I
u = 1.42034 + 0.42459I
a = 0.684601 + 0.962006I
b = 0.77539 + 1.18366I
9.56909 0.79682I 0.499984 + 0.266984I
u = 1.50417 0.21769I
a = 0.085193 + 0.865353I
b = 0.119557 + 0.894385I
4.26515 + 0.63551I 0.338676 + 0.991047I
u = 1.50417 + 0.21769I
a = 0.085193 0.865353I
b = 0.119557 0.894385I
4.26515 0.63551I 0.338676 0.991047I
u = 2.01327 0.78642I
a = 0.172310 + 0.575098I
b = 1.21747 + 1.06307I
6.78593 + 5.05905I 0.50964 9.05421I
u = 2.01327 + 0.78642I
a = 0.172310 0.575098I
b = 1.21747 1.06307I
6.78593 5.05905I 0.50964 + 9.05421I
u = 2.08330 0.73020I
a = 0.124708 + 0.588162I
b = 0.78943 + 1.25281I
8.06935 + 3.19037I 7.11325 + 0.50531I
u = 2.08330 + 0.73020I
a = 0.124708 0.588162I
b = 0.78943 1.25281I
8.06935 3.19037I 7.11325 0.50531I
u = 2.30420 0.31137I
a = 0.210667 0.701356I
b = 1.21227 0.95119I
8.17422 + 6.90097I 1.66288 5.67986I
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 2.30420 + 0.31137I
a = 0.210667 + 0.701356I
b = 1.21227 + 0.95119I
8.17422 6.90097I 1.66288 + 5.67986I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
16
u
15
+ ··· + 6u
2
1)(u
51
+ 4u
50
+ ··· + 10304u + 1139)
c
2
(u
16
+ u
14
+ ··· 3u 1)(u
51
+ u
50
+ ··· + 197u + 3)
c
3
(u
16
+ 9u
15
+ ··· + 48u + 4)(u
51
+ 4u
50
+ ··· 668u + 28)
c
4
, c
5
(u
16
6u
14
+ ··· u + 1)(u
51
+ u
50
+ ··· 15u 1)
c
6
(u
16
3u
15
+ ··· 3u 1)(u
51
+ 14u
49
+ ··· 3611u 487)
c
7
(u
16
+ 2u
14
+ ··· 2u 1)(u
51
+ u
50
+ ··· + 2734u + 367)
c
8
(u
16
+ u
15
+ ··· u 1)(u
51
+ 19u
49
+ ··· + 13u + 1)
c
9
(u
16
4u
15
+ ··· + 3u + 1)(u
51
+ 3u
50
+ ··· + 17u + 1)
c
10
(u
16
6u
14
+ ··· + u + 1)(u
51
+ u
50
+ ··· 15u 1)
c
11
(u
16
u
15
+ ··· + u 1)(u
51
+ 19u
49
+ ··· + 13u + 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
16
+ y
15
+ ··· 12y + 1)
(y
51
+ 48y
50
+ ··· + 51493582y 1297321)
c
2
(y
16
+ 2y
15
+ ··· 15y + 1)(y
51
11y
50
+ ··· + 48673y 9)
c
3
(y
16
13y
15
+ ··· 1600y + 16)(y
51
54y
50
+ ··· + 361104y 784)
c
4
, c
5
, c
10
(y
16
12y
15
+ ··· 19y + 1)(y
51
13y
50
+ ··· + 69y 1)
c
6
(y
16
+ 5y
15
+ ··· 11y + 1)
(y
51
+ 28y
50
+ ··· 14382675y 237169)
c
7
(y
16
+ 4y
15
+ ··· 12y + 1)(y
51
13y
50
+ ··· + 4303142y 134689)
c
8
, c
11
(y
16
+ 7y
15
+ ··· + 13y + 1)(y
51
+ 38y
50
+ ··· 63y 1)
c
9
(y
16
4y
15
+ ··· 9y + 1)(y
51
+ 3y
50
+ ··· + 231y 1)
12