11n
43
(K11n
43
)
1
Arc Sequences
4 1 6 2 8 4 10 11 5 6 9
Solving Sequence
9,11 1,4
2 5 10 8 6 3 7
c
11
c
1
c
4
c
9
c
8
c
5
c
3
c
7
c
2
, c
6
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= ha
5
+ a
4
2a
3
a
2
+ a 1, u + 1, b + a + 1i
I
u
2
= hu
2
+ 6u + 4, 4a u 6, 2b 3u 4i
I
u
3
= hu
51
10u
50
+ ··· 1299926u + 166228,
2.30096 × 10
247
u
50
+ 2.14840 × 10
248
u
49
+ ··· + 8.58527 × 10
249
b + 5.98416 × 10
252
,
7.20161 × 10
251
u
50
6.71919 × 10
252
u
49
+ ··· + 3.56778 × 10
254
a 1.91794 × 10
257
i
There are 3 irreducible components with 58 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
5
+ a
4
2a
3
a
2
+ a 1, u + 1, b + a + 1i
(i) Arc colorings
a
9
=
0
1
a
11
=
a
a 1
a
1
=
a
1
a
4
=
1
0
a
2
=
a + 1
1
a
5
=
a
1
a
10
=
a
2
a 1
a
8
=
a
2
a
2
+ a 1
a
6
=
a
4
+ a
3
+ a
2
2a + 1
0
a
3
=
1
0
a
7
=
a
4
+ a
3
+ a
2
2a + 1
0
a
7
=
a
4
+ a
3
+ a
2
2a + 1
0
(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 = 1.00000
a = 1.41878 0.21917I
b = 0.418784 + 0.219165I
7.51750 4.40083I 22.0438 + 5.2094I
u = 1.00000
a = 1.41878 + 0.21917I
b = 0.418784 0.219165I
7.51750 + 4.40083I 22.0438 5.2094I
u = 1.00000
a = 0.309916 0.549911I
b = 1.30992 + 0.54991I
1.97403 + 1.53058I 13.4575 4.4032I
u = 1.00000
a = 0.309916 + 0.549911I
b = 1.30992 0.54991I
1.97403 1.53058I 13.4575 + 4.4032I
u = 1.00000
a = 1.21774
b = 2.21774
4.04602 2.99735
3
II. I
u
2
= hu
2
+ 6u + 4, 4a u 6, 2b 3u 4i
(i) Arc colorings
a
9
=
0
u
a
11
=
1
4
u +
3
2
3
2
u + 2
a
1
=
1
4
u +
3
2
1
2
u + 2
a
4
=
1
0
a
2
=
1
4
u
1
2
1
2
u + 2
a
5
=
0.5
1
2
u + 3
a
10
=
1
4
u
u + 1
a
8
=
1
4
u
3
2
1
2
u 2
a
6
=
3
4
u
7
2
1
2
u 2
a
3
=
u
9
2
1
2
u 3
a
7
=
1
4
u
3
2
1
2
u 2
a
7
=
1
4
u
3
2
1
2
u 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 = 5.23607
a = 0.190983
b = 5.85410
2.63189 61.0000
u = 0.763932
a = 1.30902
b = 0.854102
10.5276 61.0000
5
III. I
u
3
= hu
51
10u
50
+ · · · 1299926u + 166228, 2.30 × 10
247
u
50
+
2.15 × 10
248
u
49
+ · · · + 8.59 × 10
249
b + 5.98 × 10
252
, 7.20 × 10
251
u
50
6.72 ×
10
252
u
49
+ · · · + 3.57 × 10
254
a 1.92 × 10
257
i
(i) Arc colorings
a
9
=
0
u
a
11
=
0.00201851u
50
+ 0.0188330u
49
+ ··· 3295.21u + 537.572
0.00268013u
50
0.0250243u
49
+ ··· + 4333.21u 697.027
a
1
=
0.00201851u
50
+ 0.0188330u
49
+ ··· 3295.21u + 537.572
0.00170221u
50
0.0160030u
49
+ ··· + 2911.05u 472.262
a
4
=
1
0
a
2
=
0.00372072u
50
+ 0.0348360u
49
+ ··· 6206.26u + 1009.83
0.00170221u
50
0.0160030u
49
+ ··· + 2911.05u 472.262
a
5
=
0.0000154275u
50
+ 0.000525031u
49
+ ··· 543.989u + 97.7439
0.000532763u
50
+ 0.00507315u
49
+ ··· 1100.08u + 193.914
a
10
=
0.000821253u
50
0.00850985u
49
+ ··· + 2652.73u 476.131
0.00489093u
50
+ 0.0450196u
49
+ ··· 6756.96u + 1034.01
a
8
=
0.00188618u
50
0.0172771u
49
+ ··· + 2602.41u 410.593
0.00128608u
50
+ 0.0120461u
49
+ ··· 2167.31u + 354.534
a
6
=
0.00686743u
50
+ 0.0638716u
49
+ ··· 10603.9u + 1676.32
0.00242860u
50
+ 0.0223262u
49
+ ··· 3434.21u + 542.395
a
3
=
0.000685616u
50
+ 0.00703395u
49
+ ··· 2308.84u + 435.713
0.00230835u
50
0.0219226u
49
+ ··· + 4244.18u 696.333
a
7
=
0.00443882u
50
+ 0.0415454u
49
+ ··· 7169.70u + 1133.93
0.00242860u
50
+ 0.0223262u
49
+ ··· 3434.21u + 542.395
a
7
=
0.00443882u
50
+ 0.0415454u
49
+ ··· 7169.70u + 1133.93
0.00242860u
50
+ 0.0223262u
49
+ ··· 3434.21u + 542.395
(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 = 1.80161 1.46261I
a = 0.276410 0.592485I
b = 2.21486 1.51948I
0.8642 + 15.7945I 8.70527 8.85601I
u = 1.80161 + 1.46261I
a = 0.276410 + 0.592485I
b = 2.21486 + 1.51948I
0.8642 15.7945I 8.70527 + 8.85601I
u = 1.56097 0.89821I
a = 0.407933 0.530935I
b = 0.926330 0.946559I
1.87901 + 7.98783I 8.12954 7.19035I
u = 1.56097 + 0.89821I
a = 0.407933 + 0.530935I
b = 0.926330 + 0.946559I
1.87901 7.98783I 8.12954 + 7.19035I
u = 1.44652 0.39904I
a = 0.022304 0.815827I
b = 0.381586 + 0.631553I
6.12236 2.89222I 3.53220 + 2.32311I
u = 1.44652 + 0.39904I
a = 0.022304 + 0.815827I
b = 0.381586 0.631553I
6.12236 + 2.89222I 3.53220 2.32311I
u = 1.037452 0.842943I
a = 0.651884 + 0.241242I
b = 1.44851 + 0.34423I
1.43007 + 1.84298I 5.28738 8.98031I
u = 1.037452 + 0.842943I
a = 0.651884 0.241242I
b = 1.44851 0.34423I
1.43007 1.84298I 5.28738 + 8.98031I
u = 1.036358 0.030953I
a = 1.41997 + 0.16512I
b = 0.678961 0.291256I
7.16136 4.34566I 0.37647 + 1.57270I
u = 1.036358 + 0.030953I
a = 1.41997 0.16512I
b = 0.678961 + 0.291256I
7.16136 + 4.34566I 0.37647 1.57270I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.988752 0.796474I
a = 0.552353 + 0.430698I
b = 0.256174 0.459940I
0.62734 + 3.24727I 5.87868 5.45997I
u = 0.988752 + 0.796474I
a = 0.552353 0.430698I
b = 0.256174 + 0.459940I
0.62734 3.24727I 5.87868 + 5.45997I
u = 0.844649 0.486018I
a = 0.253196 0.842569I
b = 0.093126 0.808944I
5.28316 + 2.70789I 3.72797 4.62977I
u = 0.844649 + 0.486018I
a = 0.253196 + 0.842569I
b = 0.093126 + 0.808944I
5.28316 2.70789I 3.72797 + 4.62977I
u = 0.813306 0.333594I
a = 0.57173 + 1.44411I
b = 1.76359 + 0.14903I
0.08535 + 1.42859I 8.20291 2.86015I
u = 0.813306 + 0.333594I
a = 0.57173 1.44411I
b = 1.76359 0.14903I
0.08535 1.42859I 8.20291 + 2.86015I
u = 0.785342
a = 1.22565
b = 0.886433
10.4502 42.5646
u = 0.660965 0.501008I
a = 0.634697 1.116633I
b = 1.56703 1.07565I
1.46329 + 2.48395I 7.42637 4.33344I
u = 0.660965 + 0.501008I
a = 0.634697 + 1.116633I
b = 1.56703 + 1.07565I
1.46329 2.48395I 7.42637 + 4.33344I
u = 0.27550 2.57946I
a = 0.146051 + 0.445382I
b = 0.67013 + 3.02777I
3.69039 + 2.13393I 15.8264 4.5625I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.27550 + 2.57946I
a = 0.146051 0.445382I
b = 0.67013 3.02777I
3.69039 2.13393I 15.8264 + 4.5625I
u = 0.062767 0.424232I
a = 1.36666 + 0.97822I
b = 0.407189 + 0.301069I
0.61038 + 1.48999I 4.46560 4.54978I
u = 0.062767 + 0.424232I
a = 1.36666 0.97822I
b = 0.407189 0.301069I
0.61038 1.48999I 4.46560 + 4.54978I
u = 0.20421 1.46615I
a = 0.058655 + 0.821689I
b = 1.68897 + 1.82716I
4.21875 0.45905I 12.7704 + 7.0072I
u = 0.20421 + 1.46615I
a = 0.058655 0.821689I
b = 1.68897 1.82716I
4.21875 + 0.45905I 12.7704 7.0072I
u = 0.437043 0.267925I
a = 1.054485 + 0.668518I
b = 1.004931 + 0.482365I
4.27954 3.84215I 4.79842 + 0.74687I
u = 0.437043 + 0.267925I
a = 1.054485 0.668518I
b = 1.004931 0.482365I
4.27954 + 3.84215I 4.79842 0.74687I
u = 0.558386
a = 0.0219166
b = 0.643615
1.00288 10.0668
u = 0.593389 0.094339I
a = 1.61131 + 0.74377I
b = 0.828996 + 0.546872I
4.54085 + 1.95941I 3.52747 2.51429I
u = 0.593389 + 0.094339I
a = 1.61131 0.74377I
b = 0.828996 0.546872I
4.54085 1.95941I 3.52747 + 2.51429I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.675060 0.634705I
a = 1.49986 + 1.15386I
b = 0.19543 + 1.80529I
2.26568 2.29719I 12.40272 + 3.03914I
u = 0.675060 + 0.634705I
a = 1.49986 1.15386I
b = 0.19543 1.80529I
2.26568 + 2.29719I 12.40272 3.03914I
u = 0.879883 0.018476I
a = 1.262509 0.553591I
b = 0.473863 + 0.492979I
3.74009 5.32281I 9.19635 + 5.98834I
u = 0.879883 + 0.018476I
a = 1.262509 + 0.553591I
b = 0.473863 0.492979I
3.74009 + 5.32281I 9.19635 5.98834I
u = 1.005875 0.283256I
a = 0.972494 0.946838I
b = 1.88300 0.74357I
3.27749 9.36362I 6.17125 + 5.41405I
u = 1.005875 + 0.283256I
a = 0.972494 + 0.946838I
b = 1.88300 + 0.74357I
3.27749 + 9.36362I 6.17125 5.41405I
u = 1.092616 0.151383I
a = 0.223936 + 0.454070I
b = 1.110942 0.690529I
1.13007 1.28368I 3.18177 + 0.90248I
u = 1.092616 + 0.151383I
a = 0.223936 0.454070I
b = 1.110942 + 0.690529I
1.13007 + 1.28368I 3.18177 0.90248I
u = 1.11835 1.54296I
a = 0.133468 0.675000I
b = 1.18811 1.96050I
2.32039 8.39966I 9.43233 + 7.03660I
u = 1.11835 + 1.54296I
a = 0.133468 + 0.675000I
b = 1.18811 + 1.96050I
2.32039 + 8.39966I 9.43233 7.03660I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.351318 0.257254I
a = 0.304164 + 0.912622I
b = 0.98715 1.08343I
4.63878 8.97661I 6.03126 + 7.51919I
u = 1.351318 + 0.257254I
a = 0.304164 0.912622I
b = 0.98715 + 1.08343I
4.63878 + 8.97661I 6.03126 7.51919I
u = 1.42266 0.55927I
a = 0.268768 0.377077I
b = 0.008939 0.301567I
0.294650 1.061436I 4.84581 + 2.42805I
u = 1.42266 + 0.55927I
a = 0.268768 + 0.377077I
b = 0.008939 + 0.301567I
0.294650 + 1.061436I 4.84581 2.42805I
u = 1.47863
a = 0.773712
b = 1.78643
2.29513 1.15087
u = 1.51078 1.23807I
a = 0.210973 + 0.793232I
b = 2.13231 + 0.83382I
0.91339 6.46505I 11.09691 + 8.15111I
u = 1.51078 + 1.23807I
a = 0.210973 0.793232I
b = 2.13231 0.83382I
0.91339 + 6.46505I 11.09691 8.15111I
u = 1.98819 5.76383I
a = 0.050744 + 0.149118I
b = 2.67487 + 5.65169I
2.64938 + 0.11132I 58.9394 3.8883I
u = 1.98819 + 5.76383I
a = 0.050744 0.149118I
b = 2.67487 5.65169I
2.64938 0.11132I 58.9394 + 3.8883I
u = 2.49810 1.34713I
a = 0.291593 + 0.246256I
b = 2.02428 + 1.56194I
1.01385 3.52246I 1.47352 5.19905I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 2.49810 + 1.34713I
a = 0.291593 0.246256I
b = 2.02428 1.56194I
1.01385 + 3.52246I 1.47352 + 5.19905I
11
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
5
(u
2
+ u 1)(u
51
+ 7u
50
+ ··· 81u
2
+ 1)
c
2
(u + 1)
5
(u
2
+ 3u + 1)(u
51
+ 23u
50
+ ··· + 162u + 1)
c
3
u
5
(u
2
+ u 1)(u
51
+ 2u
50
+ ··· 96u + 32)
c
4
(u + 1)
5
(u
2
u 1)(u
51
+ 7u
50
+ ··· 81u
2
+ 1)
c
5
(u
2
3u + 1)(u
5
3u
4
+ ··· u + 1)(u
51
+ 3u
50
+ ··· + 2u + 1)
c
6
u
5
(u
2
u 1)(u
51
+ 2u
50
+ ··· 96u + 32)
c
7
u
2
(u
5
u
4
+ ··· + u 1)(u
51
+ 8u
50
+ ··· + 64u + 4)
c
8
(u 1)
2
(u
5
+ u
4
+ ··· + u 1)(u
51
+ 4u
50
+ ··· 87u 1)
c
9
(u
2
3u + 1)(u
5
u
4
+ ··· + u + 1)(u
51
+ 5u
50
+ ··· 402u 137)
c
10
(u
2
3u + 1)(u
5
+ u
4
+ ··· + u + 1)(u
51
+ u
50
+ ··· 4u + 31)
c
11
(u + 1)
2
(u
5
u
4
+ ··· + u + 1)(u
51
+ 4u
50
+ ··· 87u 1)
12
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y 1)
5
(y
2
3y + 1)(y
51
23y
50
+ ··· + 162y 1)
c
2
(y 1)
5
(y
2
7y + 1)(y
51
+ 17y
50
+ ··· + 12790y 1)
c
3
y
5
(y
2
3y + 1)(y
51
+ 30y
50
+ ··· 8704y 1024)
c
5
(y
2
7y + 1)(y
5
y
4
+ ··· + 3y 1)(y
51
15y
50
+ ··· + 20y 1)
c
6
y
5
(y
2
3y + 1)(y
51
+ 30y
50
+ ··· 8704y 1024)
c
7
y
2
(y
5
+ 3y
4
+ ··· y 1)(y
51
12y
50
+ ··· + 1272y 16)
c
8
(y 1)
2
(y
5
5y
4
+ ··· y 1)(y
51
30y
50
+ ··· + 6683y 1)
c
9
(y
2
7y + 1)(y
5
5y
4
+ 8y
3
3y
2
y 1)
(y
51
+ 37y
50
+ ··· + 211472y 18769)
c
10
(y
2
7y + 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
51
+ 29y
50
+ ··· + 22708y 961)
c
11
(y 1)
2
(y
5
5y
4
+ ··· y 1)(y
51
30y
50
+ ··· + 6683y 1)
13