11n
2
(K11n
2
)
1
Arc Sequences
5 1 7 2 3 10 3 6 11 7 9
Solving Sequence
6,10 3,7
4 11 5 9 1 2 8
c
6
c
3
c
10
c
5
c
9
c
11
c
2
c
8
c
1
, c
4
, c
7
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hb
6
+ b
4
2b
3
+ b
2
b + 1, b
5
b
3
+ b
2
b + u, b
5
+ b
3
2b
2
+ b + a 1i
I
u
2
= hu
34
4u
33
+ ··· + 19u + 2, 2.02040 × 10
47
u
33
8.89539 × 10
47
u
32
+ ··· + 3.56653 × 10
48
a 1.32472 × 10
49
,
6.93199 × 10
47
u
33
2.97737 × 10
48
u
32
+ ··· + 3.56653 × 10
48
b + 3.45419 × 10
48
i
There are 2 irreducible components with 40 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
6
+ b
4
2b
3
+ b
2
b + 1, b
5
b
3
+ b
2
b + u, b
5
+ b
3
2b
2
+ b + a 1i
(i) Arc colorings
a
6
=
0
b
5
+ b
3
b
2
+ b
a
10
=
b
5
b
3
+ 2b
2
b + 1
b
a
3
=
1
0
a
7
=
b
5
2b
2
0
a
4
=
1
0
a
11
=
b
5
2b
3
+ b
2
2b + 2
b
a
5
=
b
5
+ b
3
b
2
+ b
b
5
+ b
3
b
2
+ b
a
9
=
b
5
2b
2
b
5
b
4
b
3
+ b
2
+ b
a
1
=
0
b
5
b
3
+ b
2
b
a
2
=
1
b
5
b
3
+ b
2
b + 1
a
8
=
b
5
2b
2
0
a
8
=
b
5
2b
2
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 = 0.500000 + 0.866025I
a = 0.377439 + 0.653743I
b = 0.662359 1.147239I
1.11345 2.02988I 12.72167 + 5.84990I
u = 0.500000 0.866025I
a = 0.377439 0.653743I
b = 0.662359 + 1.147239I
1.11345 + 2.02988I 12.72167 5.84990I
u = 0.500000 0.866025I
a = 0.206350 + 1.132315I
b = 0.155769 0.854759I
3.02413 0.79824I 4.03424 1.64667I
u = 0.500000 + 0.866025I
a = 0.206350 1.132315I
b = 0.155769 + 0.854759I
3.02413 + 0.79824I 4.03424 + 1.64667I
u = 0.500000 0.866025I
a = 1.083789 + 0.387453I
b = 0.818128 0.292480I
3.02413 + 4.85801I 2.74410 7.22587I
u = 0.500000 + 0.866025I
a = 1.083789 0.387453I
b = 0.818128 + 0.292480I
3.02413 4.85801I 2.74410 + 7.22587I
3
II. I
u
2
=
hu
34
4u
33
+· · ·+19u+2, 2.02×10
47
u
33
8.90×10
47
u
32
+· · ·+3.57×10
48
a
1.32 × 10
49
, 6.93 × 10
47
u
33
2.98 × 10
48
u
32
+ · · · + 3.57 × 10
48
b + 3.45 × 10
48
i
(i) Arc colorings
a
6
=
0
u
a
10
=
0.0566489u
33
+ 0.249413u
32
+ ··· 5.11924u + 3.71431
0.194363u
33
+ 0.834808u
32
+ ··· 6.94430u 0.968503
a
3
=
1
0
a
7
=
0.262175u
33
+ 1.16952u
32
+ ··· 18.1862u 0.106750
0.188672u
33
+ 0.803320u
32
+ ··· 8.39530u 1.16451
a
4
=
0.565009u
33
+ 2.47063u
32
+ ··· 24.6283u 1.48149
0.103580u
33
0.417380u
32
+ ··· + 6.42017u + 1.72226
a
11
=
0.459219u
33
1.91262u
32
+ ··· + 22.1283u + 6.42053
0.00359309u
33
+ 0.0151139u
32
+ ··· + 1.52077u + 0.151496
a
5
=
u
u
a
9
=
0.0735031u
33
+ 0.366199u
32
+ ··· 9.79089u + 1.05776
0.184383u
33
+ 0.779503u
32
+ ··· 9.61985u 1.30888
a
1
=
0.861132u
33
3.54811u
32
+ ··· + 49.0150u + 9.94134
0.213655u
33
0.901649u
32
+ ··· + 9.49945u + 1.33718
a
2
=
0.865650u
33
3.57299u
32
+ ··· + 49.2355u + 9.82824
0.218173u
33
0.926527u
32
+ ··· + 9.71994u + 1.22408
a
8
=
0.0735031u
33
+ 0.366199u
32
+ ··· 9.79089u + 1.05776
0.188672u
33
+ 0.803320u
32
+ ··· 8.39530u 1.16451
a
8
=
0.0735031u
33
+ 0.366199u
32
+ ··· 9.79089u + 1.05776
0.188672u
33
+ 0.803320u
32
+ ··· 8.39530u 1.16451
(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 = 1.85741
a = 0.624508
b = 3.01445
8.01260 11.0104
u = 1.71014 0.58956I
a = 0.679658 + 0 .152204I
b = 2.60659 0. 69691I
3.72457 7.16368I 7.25453 + 4.71165I
u = 1.71014 + 0 .58956I
a = 0.679658 0 .152204I
b = 2.60659 + 0. 69691I
3.72457 + 7.16368I 7.25453 4.71165I
u = 1.50124 0.41047I
a = 0.188374 0 .589925I
b = 0.068931 + 0.160639I
2.09915 1.64840I 5.13395 + 0.24192I
u = 1.50124 + 0 .41047I
a = 0.188374 + 0 .589925I
b = 0.068931 0.160639I
2.09915 + 1.64840I 5.13395 0.24192I
u = 0.984070 0.059591I
a = 1.031790 0.503137I
b = 1.40385 0.50545I
1.53468 5.70085I 9.72834 + 6.45202I
u = 0.984070 + 0 .059591I
a = 1.031790 + 0.503137I
b = 1.40385 + 0.50545I
1.53468 + 5.70085I 9.72834 6.45202I
u = 0.906064 0.616536I
a = 0.302256 + 0.678123I
b = 1.307523 0.508561I
0.45796 1.44409I 6.46359 0.67387I
u = 0.906064 + 0 .616536I
a = 0.302256 0.678123I
b = 1.307523 + 0.508561I
0.45796 + 1.44409I 6.46359 + 0.67387I
u = 0.56021 1.45979I
a = 0.555134 0 .398716I
b = 1.88194 + 0. 27022I
1.13021 4.95087I 8.36503 + 5.99635I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.56021 + 1 .45979I
a = 0.555134 + 0 .398716I
b = 1.88194 0.27022I
1.13021 + 4.95087I 8.36503 5.99635I
u = 0.451045 0.554820I
a = 0.103250 + 0 .483276I
b = 0.506649 0.510193I
0.37760 1.65869I 3.04964 + 3 .10072I
u = 0.451045 + 0 .554820I
a = 0.103250 0.483276I
b = 0.506649 + 0 .510193I
0.37760 + 1.65869I 3.04964 3.10072I
u = 0.207441 0.108198I
a = 4.15395 1 .20468I
b = 0.684159 0.910758I
3.59663 0.10881I 14.08590 + 0.63240I
u = 0.207441 + 0 .108198I
a = 4.15395 + 1 .20468I
b = 0.684159 + 0.910758I
3.59663 + 0.10881I 14.08590 0.63240I
u = 0.202628 1.380975I
a = 0.329694 0.623936I
b = 1.256161 + 0 .544934I
1.44216 0.29929I 6.95462 + 0 .76731I
u = 0.202628 + 1 .380975I
a = 0.329694 + 0.623936I
b = 1.256161 0.544934I
1.44216 + 0.29929I 6.95462 0.76731I
u = 0.175058
a = 3.67600
b = 0.378410
0.881314 11.5071
u = 0.168899 0.431096I
a = 0.90559 + 2.29022I
b = 0.469617 0.460736I
2.63328 1.69138I 7.83080 + 4 .78233I
u = 0.168899 + 0.431096I
a = 0.90559 2.29022I
b = 0.469617 + 0.460736I
2.63328 + 1.69138I 7.83080 4.78233I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.280735 0.372811I
a = 2.50415 + 0.58715I
b = 0.165027 0.676362I
2.37182 + 4.13713I 9.61954 1.32790I
u = 0.280735 + 0. 372811I
a = 2.50415 0.58715I
b = 0.165027 + 0.676362I
2.37182 4.13713I 9.61954 + 1.32790I
u = 0.690909 0.226743I
a = 0.83431 + 1.16273I
b = 1.020510 + 0.010217I
1.86820 2.41838I 3.21586 + 3.79872I
u = 0.690909 + 0. 226743I
a = 0.83431 1.16273I
b = 1.020510 0.010217I
1.86820 + 2.41838I 3.21586 3.79872I
u = 1.45290 0.11438I
a = 0.236928 0.600726I
b = 0.199355 0.085492I
6.32362 2.72594I 8.84636 + 2.76466I
u = 1.45290 + 0 .11438I
a = 0.236928 + 0.600726I
b = 0.199355 + 0.085492I
6.32362 + 2.72594I 8.84636 2.76466I
u = 1.47963 0.39496I
a = 0.804051 + 0.201390I
b = 2.43803 0.86862I
8.42620 + 2.66430I 11.45013 1.91985I
u = 1.47963 + 0 .39496I
a = 0.804051 0.201390I
b = 2.43803 + 0.86862I
8.42620 2.66430I 11.45013 + 1 .91985I
u = 1.73421 0.38352I
a = 0.201277 0.551551I
b = 0.171783 + 0.093820I
5.42766 + 6.75489I 7.92215 3.41714I
u = 1.73421 + 0 .38352I
a = 0.201277 + 0.551551I
b = 0.171783 0.093820I
5.42766 6.75489I 7.92215 + 3.41714I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.84786 0.00711I
a = 0.648133 0 .028688I
b = 3.05274 0. 09619I
12.04691 + 5.13421I 13.68860 3.30024I
u = 1.84786 + 0.00711I
a = 0.648133 + 0 .028688I
b = 3.05274 + 0. 09619I
12.04691 5.13421I 13.68860 + 3.30024I
u = 1.88393 0.46444I
a = 0.631127 + 0.213635I
b = 2.73058 0.76264I
7.0419 + 12 .5932I 9.63219 7.64177I
u = 1.88393 + 0.46444I
a = 0.631127 0.213635I
b = 2.73058 + 0.76264I
7.0419 12.5932I 9.63219 + 7.64177I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
3
(u
34
+ 4u
33
+ ··· + 7u + 1)
c
2
(u
2
+ u + 1)
3
(u
34
+ 20u
33
+ ··· 31u + 1)
c
3
, c
7
u
6
(u
34
+ u
33
+ ··· 160u 64)
c
4
(u
2
u + 1)
3
(u
34
+ 4u
33
+ ··· + 7u + 1)
c
5
(u
2
+ u + 1)
3
(u
34
+ 4u
33
+ ··· 19u + 2)
c
6
(u
3
+ u
2
1)
2
(u
34
+ 3u
33
+ ··· + 3u
2
1)
c
8
(u
3
+ u
2
+ 2u + 1)
2
(u
34
+ 3u
33
+ ··· 4u 1)
c
9
(u
3
u
2
+ 2u 1)
2
(u
34
+ 13u
33
+ ··· + 6u + 1)
c
10
(u
3
u
2
+ 1)
2
(u
34
+ 3u
33
+ ··· + 3u
2
1)
c
11
(u
3
+ u
2
+ 2u + 1)
2
(u
34
+ 13u
33
+ ··· + 6u + 1)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)
3
(y
34
+ 20y
33
+ ··· 31y + 1)
c
2
(y
2
+ y + 1)
3
(y
34
8y
33
+ ··· 1215y + 1)
c
3
, c
7
y
6
(y
34
35y
33
+ ··· 29696y + 4096)
c
5
(y
2
+ y + 1)
3
(y
34
36y
33
+ ··· 209y + 4)
c
6
, c
10
(y
3
y
2
+ 2y 1)
2
(y
34
13y
33
+ ··· 6y + 1)
c
8
(y
3
+ 3y
2
+ 2y 1)
2
(y
34
41y
33
+ ··· 6y + 1)
c
9
, c
11
(y
3
+ 3y
2
+ 2y 1)
2
(y
34
+ 19y
33
+ ··· + 122y + 1)
10