11n
134
(K11n
134
)
1
Arc Sequences
7 1 10 1 8 2 11 5 4 7 9
Solving Sequence
7,11 2,8
1 3 6 5 10 4 9
c
7
c
1
c
2
c
6
c
5
c
10
c
3
c
9
c
4
, c
8
, c
11
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
4
2b
3
+ b + 1, b
2
b + u, b
2
+ b + a + 1i
I
u
2
= hu
7
+ 3u
5
+ u
4
+ u
3
+ 3u
2
u + 1, u
5
+ 2u
3
+ u
2
+ a u + 2, 2u
6
u
5
5u
4
4u
3
u
2
+ b 4ui
I
u
3
= hu
34
u
33
+ ··· + 223u + 11,
2.32190 × 10
44
u
33
+ 5.14830 × 10
43
u
32
+ ··· + 8.33489 × 10
45
b 2.24805 × 10
45
,
3.69177 × 10
45
u
33
4.10987 × 10
45
u
32
+ ··· + 4.58419 × 10
46
a + 7.72452 × 10
47
i
There are 3 irreducible components with 45 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
4
2b
3
+ b + 1, b
2
b + u, b
2
+ b + a + 1i
(i) Arc colorings
a
7
=
0
b
2
+ b
a
11
=
b
2
b 1
b
a
2
=
1
0
a
8
=
b
2
b 1
b
2
+ 2b
a
1
=
1
b
2
b 1
a
3
=
b
2
b
b
2
+ b
a
6
=
b
2
b
b
2
+ b
a
5
=
b
2
2b + 1
b
3
+ 2b
2
2
a
10
=
b
2
b 1
b
2
+ 2b
a
4
=
b
2
2b + 1
b
3
+ 2b
2
2
a
9
=
2b
3
4b
2
+ b + 1
b
3
+ 2b
2
b + 1
a
9
=
2b
3
4b
2
+ b + 1
b
3
+ 2b
2
b + 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 = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 0.473561 0.444772I
1.64493 + 2.02988I 5.75416 8.47377I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0.473561 + 0.444772I
1.64493 2.02988I 5.75416 + 8.47377I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 1.47356 0.44477I
1.64493 2.02988I 13.74584 + 2.78456I
u = 0.500000 0.866025I
a = 0.500000 + 0.866025I
b = 1.47356 + 0.44477I
1.64493 + 2.02988I 13.74584 2.78456I
3
II. I
u
2
= hu
7
+ 3u
5
+ u
4
+ u
3
+ 3u
2
u + 1, u
5
+ 2u
3
+ u
2
+ a u +
2, 2u
6
u
5
5u
4
4u
3
u
2
+ b 4ui
(i) Arc colorings
a
7
=
0
u
a
11
=
u
5
2u
3
u
2
+ u 2
2u
6
+ u
5
+ 5u
4
+ 4u
3
+ u
2
+ 4u
a
2
=
1
0
a
8
=
u
6
+ 2u
5
+ 3u
4
+ 5u
3
+ 2u
2
+ u + 2
2u
6
u
5
5u
4
3u
3
2u + 1
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
6
=
u
u
a
5
=
u
5
u
4
3u
3
3u
2
2u 1
u
6
2u
4
u
3
+ u
2
2u
a
10
=
u
5
2u
3
u
2
+ u 2
2u
6
+ 2u
5
+ 5u
4
+ 6u
3
+ 2u
2
+ 3u + 1
a
4
=
u
5
u
4
3u
3
2u
2
u
u
6
u
4
+ 2u
2
2u
a
9
=
2u
6
+ u
5
+ 5u
4
+ 4u
3
+ 4u 1
u
6
2u
4
+ u
2
+ 1
a
9
=
2u
6
+ u
5
+ 5u
4
+ 4u
3
+ 4u 1
u
6
2u
4
+ u
2
+ 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 = 1.05920
a = 0.471275
b = 0.0837912
2.66082 22.3155
u = 0.01853 1.56068I
a = 0.695549 + 0.028024I
b = 1.71879 + 0.41737I
5.24144 + 3.67154I 2.26281 7.80389I
u = 0.01853 + 1.56068I
a = 0.695549 0.028024I
b = 1.71879 0.41737I
5.24144 3.67154I 2.26281 + 7.80389I
u = 0.257656 0.546312I
a = 1.147929 0.420828I
b = 0.03261 1.59526I
0.57687 5.05320I 8.58977 + 8.09248I
u = 0.257656 + 0.546312I
a = 1.147929 + 0.420828I
b = 0.03261 + 1.59526I
0.57687 + 5.05320I 8.58977 8.09248I
u = 0.290472 0.988874I
a = 0.31198 1.68286I
b = 0.793294 + 0.932250I
4.48789 1.14089I 4.48970 0.40326I
u = 0.290472 + 0.988874I
a = 0.31198 + 1.68286I
b = 0.793294 0.932250I
4.48789 + 1.14089I 4.48970 + 0.40326I
5
III. I
u
3
=
hu
34
u
33
+· · ·+223u+11, 2.32×10
44
u
33
+5.15×10
43
u
32
+· · ·+8.33×10
45
b
2.25 × 10
45
, 3.69 × 10
45
u
33
4.11 × 10
45
u
32
+ · · · + 4.58 × 10
46
a + 7.72 × 10
47
i
(i) Arc colorings
a
7
=
0
u
a
11
=
0.0805326u
33
+ 0.0896531u
32
+ ··· 58.8143u 16.8504
0.0278576u
33
0.00617680u
32
+ ··· + 15.2706u + 0.269715
a
2
=
1
0
a
8
=
0.0699058u
33
+ 0.0645796u
32
+ ··· 54.0427u 14.6112
0.0418675u
33
+ 0.0709545u
32
+ ··· + 1.56185u 0.354007
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
6
=
u
u
a
5
=
0.0681426u
33
0.0884559u
32
+ ··· + 43.2625u + 11.3413
0.0327994u
33
0.0141026u
32
+ ··· + 24.7789u + 1.59675
a
10
=
0.0805326u
33
+ 0.0896531u
32
+ ··· 58.8143u 16.8504
0.0253178u
33
0.00597640u
32
+ ··· + 14.1226u + 0.169390
a
4
=
0.0559432u
33
0.101198u
32
+ ··· + 22.2639u + 9.96804
0.0425057u
33
0.0183210u
32
+ ··· + 30.4752u + 1.87111
a
9
=
0.167638u
33
+ 0.192459u
32
+ ··· 103.012u 24.1298
0.00860991u
33
0.00519110u
32
+ ··· + 5.05582u 0.502284
a
9
=
0.167638u
33
+ 0.192459u
32
+ ··· 103.012u 24.1298
0.00860991u
33
0.00519110u
32
+ ··· + 5.05582u 0.502284
(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 = 1.20680
a = 0.0229102
b = 0.309082
2.39902 9.40475
u = 0.563765 1.003299I
a = 0.346581 + 0.213718I
b = 1.061007 0.721232I
0.24666 + 2.01770I 4.06600 3.01899I
u = 0.563765 + 1.003299I
a = 0.346581 0.213718I
b = 1.061007 + 0.721232I
0.24666 2.01770I 4.06600 + 3.01899I
u = 0.507321 0.832042I
a = 0.59647 1.40947I
b = 0.430828 + 0.791426I
5.26138 + 1.82293I 12.08014 4.65351I
u = 0.507321 + 0.832042I
a = 0.59647 + 1.40947I
b = 0.430828 0.791426I
5.26138 1.82293I 12.08014 + 4.65351I
u = 0.424431 0.480697I
a = 1.29745 + 0.87860I
b = 0.360469 0.623058I
1.94989 + 1.48394I 11.56368 + 0.88523I
u = 0.424431 + 0.480697I
a = 1.29745 0.87860I
b = 0.360469 + 0.623058I
1.94989 1.48394I 11.56368 0.88523I
u = 0.321774 0.391240I
a = 0.843663 0.126792I
b = 0.088896 0.631993I
0.417699 + 1.309314I 4.01409 5.53427I
u = 0.321774 + 0.391240I
a = 0.843663 + 0.126792I
b = 0.088896 + 0.631993I
0.417699 1.309314I 4.01409 + 5.53427I
u = 0.25509 1.72919I
a = 0.521499 0.195573I
b = 1.54407 + 0.28780I
4.70800 + 5.46166I 4.56324 8.11055I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.25509 + 1.72919I
a = 0.521499 + 0.195573I
b = 1.54407 0.28780I
4.70800 5.46166I 4.56324 + 8.11055I
u = 0.218497 0.654411I
a = 1.076141 0.331238I
b = 0.63419 + 1.67208I
1.19710 4.22413I 2.51642 + 1.58423I
u = 0.218497 + 0.654411I
a = 1.076141 + 0.331238I
b = 0.63419 1.67208I
1.19710 + 4.22413I 2.51642 1.58423I
u = 0.12009 1.49407I
a = 0.705285 0.090250I
b = 1.99892 0.19745I
5.69136 + 2.93128I 2.16566 + 0.03766I
u = 0.12009 + 1.49407I
a = 0.705285 + 0.090250I
b = 1.99892 + 0.19745I
5.69136 2.93128I 2.16566 0.03766I
u = 0.0612607
a = 14.0023
b = 0.515132
1.42028 5.52282
u = 0.03541 1.72852I
a = 0.857507 + 0.779137I
b = 1.79251 0.47467I
10.20856 2.99636I 2.11867 + 2.22916I
u = 0.03541 + 1.72852I
a = 0.857507 0.779137I
b = 1.79251 + 0.47467I
10.20856 + 2.99636I 2.11867 2.22916I
u = 0.03451 1.58243I
a = 0.856086 + 0.124269I
b = 1.74222 0.39783I
4.73573 + 2.86800I 6.77612 0.77716I
u = 0.03451 + 1.58243I
a = 0.856086 0.124269I
b = 1.74222 + 0.39783I
4.73573 2.86800I 6.77612 + 0.77716I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.00268 1.76509I
a = 0.728771 0.653862I
b = 1.77297 + 0.49209I
11.55113 + 3.90804I 1.18914 2.96330I
u = 0.00268 + 1.76509I
a = 0.728771 + 0.653862I
b = 1.77297 0.49209I
11.55113 3.90804I 1.18914 + 2.96330I
u = 0.106891 0.794920I
a = 0.440312 1.192643I
b = 0.031368 0.880258I
1.76083 + 4.62537I 1.34288 4.38948I
u = 0.106891 + 0.794920I
a = 0.440312 + 1.192643I
b = 0.031368 + 0.880258I
1.76083 4.62537I 1.34288 + 4.38948I
u = 0.42724 1.62646I
a = 0.877338 0.522648I
b = 2.11076 + 0.33915I
11.26852 5.15956I 1.46171 + 2.86631I
u = 0.42724 + 1.62646I
a = 0.877338 + 0.522648I
b = 2.11076 0.33915I
11.26852 + 5.15956I 1.46171 2.86631I
u = 0.45939 1.70921I
a = 0.915268 + 0.479980I
b = 2.07741 0.45415I
10.3471 13.0489I 2.99134 + 6.41926I
u = 0.45939 + 1.70921I
a = 0.915268 0.479980I
b = 2.07741 + 0.45415I
10.3471 + 13.0489I 2.99134 6.41926I
u = 0.515707 0.964004I
a = 0.522904 + 1.019360I
b = 1.379550 + 0.157107I
1.03541 1.94405I 0.403250 + 0.742376I
u = 0.515707 + 0.964004I
a = 0.522904 1.019360I
b = 1.379550 0.157107I
1.03541 + 1.94405I 0.403250 0.742376I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.867297 0.660290I
a = 0.902246 + 0.520939I
b = 0.648010 0.185694I
4.16665 0.08364I 1.19426 + 1.11998I
u = 0.867297 + 0.660290I
a = 0.902246 0.520939I
b = 0.648010 + 0.185694I
4.16665 + 0.08364I 1.19426 1.11998I
u = 1.241064 0.526290I
a = 0.651812 0.637629I
b = 0.508462 + 0.155388I
3.08759 6.63875I 2.63219 + 6.12497I
u = 1.241064 + 0.526290I
a = 0.651812 + 0.637629I
b = 0.508462 0.155388I
3.08759 + 6.63875I 2.63219 6.12497I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)
2
(u
7
+ 3u
5
+ u
4
+ u
3
+ 3u
2
u + 1)
(u
34
+ u
33
+ ··· 223u + 11)
c
2
(u
2
+ u + 1)
2
(u
7
+ 6u
6
+ 11u
5
+ 3u
4
11u
3
13u
2
5u 1)
(u
34
+ 39u
33
+ ··· 30545u + 121)
c
3
(u
4
+ u
3
+ 3u
2
+ u + 1)(u
7
+ 2u
5
+ u
3
+ u
2
+ u + 1)
(u
34
+ 2u
32
+ ··· 30u + 11)
c
4
u
4
(u
7
u
6
+ 2u
5
4u
4
+ 13u
3
28u
2
+ 37u 21)
(u
34
+ 6u
33
+ ··· + 88u 16)
c
5
(u
4
+ u
3
+ 3u
2
+ u + 1)(u
7
+ u
6
+ u
5
+ u
4
+ 2u
2
+ 1)
(u
34
+ 3u
33
+ ··· + 73u + 11)
c
6
(u
2
u + 1)
2
(u
7
+ 3u
5
u
4
+ u
3
3u
2
u 1)
(u
34
+ u
33
+ ··· 223u + 11)
c
7
(u + 1)
4
(u
7
4u
6
+ 6u
5
6u
4
+ 5u
3
u
2
u + 1)
(u
34
+ u
33
+ ··· + 33u 1)
c
8
(u
4
u
3
+ 3u
2
u + 1)(u
7
u
6
+ u
5
u
4
2u
2
1)
(u
34
+ 3u
33
+ ··· + 73u + 11)
c
9
(u
4
u
3
+ 3u
2
u + 1)(u
7
+ 2u
5
+ u
3
u
2
+ u 1)
(u
34
+ 2u
32
+ ··· 30u + 11)
c
10
(u 1)
4
(u
7
+ 4u
6
+ 6u
5
+ 6u
4
+ 5u
3
+ u
2
u 1)
(u
34
+ u
33
+ ··· + 33u 1)
c
11
(u
4
2u
3
+ u + 1)(u
7
u
6
u
5
+ 4u
4
u
3
3u
2
+ 3u 1)
(u
34
4u
32
+ ··· + 8u 1)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
2
+ y + 1)
2
(y
7
+ 6y
6
+ 11y
5
+ 3y
4
11y
3
13y
2
5y 1)
(y
34
+ 39y
33
+ ··· 30545y + 121)
c
2
(y
2
+ y + 1)
2
(y
7
14y
6
+ 63y
5
105y
4
+ 101y
3
53y
2
y 1)
(y
34
89y
33
+ ··· 1039040457y + 14641)
c
3
, c
9
(y
4
+ 5y
3
+ 9y
2
+ 5y + 1)(y
7
+ 4y
6
+ 6y
5
+ 6y
4
+ 5y
3
+ y
2
y 1)
(y
34
+ 4y
33
+ ··· 1054y + 121)
c
4
y
4
(y
7
+ 3y
6
+ 22y
5
+ 54y
4
+ 51y
3
+ 10y
2
+ 193y 441)
(y
34
+ 18y
33
+ ··· 5440y + 256)
c
5
, c
8
(y
4
+ 5y
3
+ 9y
2
+ 5y + 1)(y
7
+ y
6
y
5
5y
4
6y
3
6y
2
4y 1)
(y
34
+ 29y
33
+ ··· + 1513y + 121)
c
7
, c
10
(y 1)
4
(y
7
4y
6
2y
5
+ 14y
4
+ 9y
3
+ y
2
+ 3y 1)
(y
34
+ 3y
33
+ ··· 1291y + 1)
c
11
(y
4
4y
3
+ 6y
2
y + 1)(y
7
3y
6
+ ··· + 3y 1)
(y
34
8y
33
+ ··· 28y + 1)
12