11n
35
(K11n
35
)
1
Arc Sequences
4 1 6 2 11 4 10 5 6 8 9
Solving Sequence
1,4
2
5,9
8 11 6 7 3 10
c
1
c
4
c
8
c
11
c
5
c
6
c
3
c
10
c
2
, c
7
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
5
+ b
4
+ 2b
3
+ b
2
+ b + 1, u 1, b
4
2b
2
+ a 2i
I
u
2
= hu
2
+ u 1, b, a + 3u + 5i
I
u
3
= hu
51
+ 7u
50
+ ··· 81u
2
+ 1, 2.47871 × 10
39
u
50
+ 1.88813 × 10
40
u
49
+ ··· + 2.18939 × 10
39
a + 1.19965 × 10
39
,
7.64232 × 10
39
u
50
+ 4.53219 × 10
40
u
49
+ ··· + 8.75756 × 10
39
b + 1.97436 × 10
40
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
= hb
5
+ b
4
+ 2b
3
+ b
2
+ b + 1, u 1, b
4
2b
2
+ a 2i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
1
a
2
=
1
1
a
5
=
1
0
a
9
=
b
4
+ 2b
2
+ 2
b
a
8
=
b
4
+ 2b
2
b + 2
b
a
11
=
b
4
b
2
+ b
b
2
a
6
=
0
b
4
a
7
=
0
b
4
a
3
=
0
1
a
10
=
b
4
+ 2b
2
+ 2
b
4
a
10
=
b
4
+ 2b
2
+ 2
b
4
(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 = 3.52181
b = 0.766826
4.04602 2.99735
u = 1.00000
a = 0.142272 0.509071I
b = 0.455697 1.200152I
7.51750 + 4.40083I 22.0438 5.2094I
u = 1.00000
a = 0.142272 + 0.509071I
b = 0.455697 + 1.200152I
7.51750 4.40083I 22.0438 + 5.2094I
u = 1.00000
a = 0.881366 0.489365I
b = 0.339110 0.822375I
1.97403 1.53058I 13.4575 + 4.4032I
u = 1.00000
a = 0.881366 + 0.489365I
b = 0.339110 + 0.822375I
1.97403 + 1.53058I 13.4575 4.4032I
3
II. I
u
2
= hu
2
+ u 1, b, a + 3u + 5i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u + 1
a
5
=
u
u + 1
a
9
=
3u 5
0
a
8
=
2u 4
1
a
11
=
1
0
a
6
=
1
u + 1
a
7
=
1
0
a
3
=
u
u + 1
a
10
=
2u 3
1
a
10
=
2u 3
1
(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 = 1.61803
a = 0.145898
b = 0
10.5276 61.0000
u = 0.618034
a = 6.85410
b = 0
2.63189 61.0000
5
III. I
u
3
=
hu
51
+7u
50
+· · ·81u
2
+1, 2.48×10
39
u
50
+1.89×10
40
u
49
+· · ·+2.19×10
39
a+
1.20 × 10
39
, 7.64 × 10
39
u
50
+ 4.53 × 10
40
u
49
+ · · · + 8.76 × 10
39
b + 1.97 × 10
40
i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
u
u
3
+ u
a
9
=
1.13215u
50
8.62401u
49
+ ··· + 75.3242u 0 .547939
0.872654u
50
5.17518u
49
+ ··· + 5.03419u 2.25447
a
8
=
1.36430u
50
9.37394u
49
+ ··· + 72.3435u + 0 .0541095
2.27233u
50
13.7744u
49
+ ··· + 8.24704u 3.73165
a
11
=
2.21621u
50
14.3423u
49
+ ··· + 4.63645u 11 .0109
0.0694094u
50
0.202698u
49
+ ··· + 9.54759u 1.43747
a
6
=
0.109838u
50
+ 0.455800u
49
+ ··· 35.3698u + 2.83830
2.10336u
50
+ 11.3955u
49
+ ··· 4.83182u + 1 .99352
a
7
=
0.109838u
50
+ 0.455800u
49
+ ··· 35.3698u + 2.83830
1.41865u
50
+ 6.00756u
49
+ ··· 4.72198u + 0 .768857
a
3
=
u
2
+ 1
u
2
a
10
=
2.02006u
50
13.1436u
49
+ ··· + 56.6295u 3 .02939
3.40837u
50
22.0852u
49
+ ··· + 10.9393u 5 .67809
a
10
=
2.02006u
50
13.1436u
49
+ ··· + 56.6295u 3 .02939
3.40837u
50
22.0852u
49
+ ··· + 10.9393u 5 .67809
(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 = 1.64237
a = 0.00866580
b = 0.265959
10.4502 42.5646
u = 1.227371 0.667999I
a = 0.663202 0.675600I
b = 0.918945 0.630687I
1.87901 7.98783I 8.12954 + 7.19035I
u = 1.227371 + 0.667999I
a = 0.663202 + 0.675600I
b = 0.918945 + 0 .630687I
1.87901 + 7.98783I 8.12954 7.19035I
u = 1.213469 0.718395I
a = 1.34702 + 0 .86109I
b = 1.31215 + 1.04774I
0.8642 15.7945I 8.70527 + 8.85601I
u = 1.213469 + 0.718395I
a = 1.34702 0.86109I
b = 1.31215 1.04774I
0.8642 + 15.7945I 8.70527 8.85601I
u = 1.086370 0.734156I
a = 1.44695 0.84504I
b = 1.53144 1.12477I
4.63878 8.97661I 6.03126 + 7.51919I
u = 1.086370 + 0.734156I
a = 1.44695 + 0.84504I
b = 1.53144 + 1 .12477I
4.63878 + 8.97661I 6.03126 7.51919I
u = 1.035773 0.809017I
a = 0.932196 + 0 .624404I
b = 0.826302 + 0.379791I
4.27954 3.84215I 4.79842 + 0.74687I
u = 1.035773 + 0.809017I
a = 0.932196 0.624404I
b = 0.826302 0.379791I
4.27954 + 3.84215I 4.79842 0.74687I
u = 0.993259 0.639230I
a = 0.579456 + 0.533405I
b = 0.83461 1.79554I
0.91339 6.46505I 11.09691 + 8.15111I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.993259 + 0.639230I
a = 0.579456 0.533405I
b = 0.83461 + 1.79554I
0.91339 + 6.46505I 11.09691 8.15111I
u = 0.923670 0.689242I
a = 0.789827 + 1.033602I
b = 0.495586 + 0 .521010I
1.01385 3.52246I 1.47352 5.19905I
u = 0.923670 + 0.689242I
a = 0.789827 1.033602I
b = 0.495586 0.521010I
1.01385 + 3.52246I 1.47352 + 5.19905I
u = 0.872616 0.581002I
a = 1.29175 + 1 .14812I
b = 1.88392 + 0.37218I
2.26568 2.29719I 12.40272 + 3.03914I
u = 0.872616 + 0.581002I
a = 1.29175 1.14812I
b = 1.88392 0.37218I
2.26568 + 2.29719I 12.40272 3.03914I
u = 0.834183 0.022964I
a = 0.240873 0.744504I
b = 0.447898 1.297843I
7.16136 + 4.34566I 0.37647 1.57270I
u = 0.834183 + 0.022964I
a = 0.240873 + 0.744504I
b = 0.447898 + 1.297843I
7.16136 4.34566I 0.37647 + 1.57270I
u = 0.788139 0.707702I
a = 0.54082 1.43941I
b = 0.259903 0.709097I
1.43007 1.84298I 5.28738 + 8.98031I
u = 0.788139 + 0.707702I
a = 0.54082 + 1.43941I
b = 0.259903 + 0 .709097I
1.43007 + 1.84298I 5.28738 8.98031I
u = 0.719478 1.003154I
a = 0.872802 + 0 .173176I
b = 1.015781 0.052361I
5.28316 2.70789I 3.72797 + 4.62977I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.719478 + 1.003154I
a = 0.872802 0.173176I
b = 1.015781 + 0.052361I
5.28316 + 2.70789I 3.72797 4.62977I
u = 0.665417 0.633317I
a = 1.341349 + 0 .037908I
b = 0.20863 + 1.70015I
0.08535 + 1.42859I 8.20291 2.86015I
u = 0.665417 + 0.633317I
a = 1.341349 0.037908I
b = 0.20863 1.70015I
0.08535 1.42859I 8.20291 + 2.86015I
u = 0.603362 0.919845I
a = 0.955773 0.890743I
b = 1.53585 + 0 .75787I
6.12236 + 2.89222I 3.53220 2.32311I
u = 0.603362 + 0.919845I
a = 0.955773 + 0.890743I
b = 1.53585 0 .75787I
6.12236 2.89222I 3.53220 + 2.32311I
u = 0.437529 1.071434I
a = 1.073637 + 0 .605444I
b = 1.30783 0.82877I
3.27749 + 9.36362I 6.17125 5.41405I
u = 0.437529 + 1.071434I
a = 1.073637 0.605444I
b = 1.30783 + 0.82877I
3.27749 9.36362I 6.17125 + 5.41405I
u = 0.359247 0.982542I
a = 0.744457 0.170926I
b = 1.095466 + 0 .271570I
4.54085 + 1.95941I 3.52747 2.51429I
u = 0.359247 + 0.982542I
a = 0.744457 + 0.170926I
b = 1.095466 0.271570I
4.54085 1.95941I 3.52747 + 2.51429I
u = 0.218706 0.056088I
a = 2.35076 + 2 .94041I
b = 0.149528 + 0 .895127I
0.61038 + 1.48999I 4.46560 4.54978I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.218706 + 0.056088I
a = 2.35076 2.94041I
b = 0.149528 0.895127I
0.61038 1.48999I 4.46560 + 4.54978I
u = 0.0948151
a = 11.7096
b = 0.594393
2.29513 1.15087
u = 0.650449
a = 1.17006
b = 0.110617
1.00288 10.0668
u = 0.709519 0.862840I
a = 0.751349 0.774960I
b = 0.855985 + 0.416787I
1.46329 2.48395I 7.42637 + 4.33344I
u = 0.709519 + 0. 862840I
a = 0.751349 + 0 .774960I
b = 0.855985 0.416787I
1.46329 + 2.48395I 7.42637 4.33344I
u = 0.742417 0.124155I
a = 2.43511 6.15388I
b = 0.012912 0.273501I
2.64938 + 0.11132I 58.9394 3.8883I
u = 0.742417 + 0. 124155I
a = 2.43511 + 6 .15388I
b = 0.012912 + 0.273501I
2.64938 0.11132I 58.9394 + 3.8883I
u = 0.744937 0.557874I
a = 1.79438 0.06696I
b = 1.049408 + 0 .670680I
0.62734 + 3.24727I 5.87868 5.45997I
u = 0.744937 + 0. 557874I
a = 1.79438 + 0.06696I
b = 1.049408 0.670680I
0.62734 3.24727I 5.87868 + 5.45997I
u = 0.923570 0.305757I
a = 0.416243 1.068126I
b = 0.620880 + 0.980403I
3.69039 + 2.13393I 15.8264 4.5625I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.923570 + 0 .305757I
a = 0.416243 + 1 .068126I
b = 0.620880 0.980403I
3.69039 2.13393I 15.8264 + 4.5625I
u = 0.990585 0.737126I
a = 1.46798 0.33121I
b = 1.072347 0.755687I
2.32039 + 8.39966I 9.43233 7.03660I
u = 0.990585 + 0 .737126I
a = 1.46798 + 0 .33121I
b = 1.072347 + 0.755687I
2.32039 8.39966I 9.43233 + 7.03660I
u = 1.051020 0.519122I
a = 0.212768 + 0.967576I
b = 0.806427 0.022325I
0.294650 + 1.061436I 4.84581 2.42805I
u = 1.051020 + 0 .519122I
a = 0.212768 0.967576I
b = 0.806427 + 0 .022325I
0.294650 1.061436I 4.84581 + 2.42805I
u = 1.073296 0.130259I
a = 2.73037 1.97243I
b = 0.752137 0.540476I
4.21875 0.45905I 12.7704 + 7.0072I
u = 1.073296 + 0 .130259I
a = 2.73037 + 1 .97243I
b = 0.752137 + 0.540476I
4.21875 + 0.45905I 12.7704 7.0072I
u = 1.222938 0.084782I
a = 0.687532 + 0 .411301I
b = 0.715736 + 0 .439898I
1.13007 + 1.28368I 3.18177 0.90248I
u = 1.222938 + 0 .084782I
a = 0.687532 0.411301I
b = 0.715736 0.439898I
1.13007 1.28368I 3.18177 + 0.90248I
u = 1.46886 0.10563I
a = 0.173747 + 0.050346I
b = 0.921422 + 0.643101I
3.74009 5.32281I 9.19635 + 5.98834I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.46886 + 0 .10563I
a = 0.173747 0.050346I
b = 0.921422 0.643101I
3.74009 + 5.32281I 9.19635 5.98834I
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 1)
2
(u
5
+ u
4
+ ··· + u 1)(u
51
+ 4u
50
+ ··· 87u 1)
c
8
(u
2
+ 3u + 1)(u
5
+ u
4
+ ··· + u + 1)(u
51
+ u
50
+ ··· 4u + 31)
c
9
(u
2
+ 3u + 1)(u
5
u
4
+ ··· + u + 1)(u
51
+ 5u
50
+ ··· 402u 137)
c
10
(u + 1)
2
(u
5
u
4
+ ··· + u + 1)(u
51
+ 4u
50
+ ··· 87u 1)
c
11
u
2
(u
5
+ u
4
+ ··· + u + 1)(u
51
+ 8u
50
+ ··· + 64u + 4)
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
, c
6
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
7
, c
10
(y 1)
2
(y
5
5y
4
+ ··· y 1)(y
51
30y
50
+ ··· + 6683y 1)
c
8
(y
2
7y + 1)(y
5
+ 3y
4
+ 4y
3
+ y
2
y 1)
(y
51
+ 29y
50
+ ··· + 22708y 961)
c
9
(y
2
7y + 1)(y
5
5y
4
+ 8y
3
3y
2
y 1)
(y
51
+ 37y
50
+ ··· + 211472y 18769)
c
11
y
2
(y
5
+ 3y
4
+ ··· y 1)(y
51
12y
50
+ ··· + 1272y 16)
13