11a
219
(K11a
219
)
1
Arc Sequences
7 1 9 11 10 2 6 4 3 5 8
Solving Sequence
2,6
7 8 1
3,10
5 11 4 9
c
6
c
7
c
1
c
2
c
5
c
10
c
4
c
9
c
3
, c
8
, c
11
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
6
+ u
4
+ 2u
2
+ 1, u
4
+ a + 1, u
5
+ u
3
u
2
+ b + 2ui
I
u
2
= ha
28
+ 2a
27
+ ··· 4a + 1, 1.18729 × 10
22
u 7.70275 × 10
20
a
27
+ ··· + 5.89764 × 10
22
a + 3.59677 × 10
21
,
1.18729 × 10
22
b + 5.17592 × 10
21
a
27
+ ··· + 5.45145 × 10
22
a 1.16803 × 10
22
i
I
u
3
= hu
21
3u
20
+ ··· + 3u + 2, u
20
+ u
19
+ ··· + 2a 3, 3u
20
+ 8u
19
+ ··· + b + 5i
There are 3 irreducible components with 55 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
6
+ u
4
+ 2u
2
+ 1, u
4
+ a + 1, u
5
+ u
3
u
2
+ b + 2ui
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
7
=
u
u
a
8
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
10
=
u
4
1
u
5
u
3
+ u
2
2u
a
5
=
u
5
u
3
2u
u
5
+ u
4
u
3
+ 1
a
11
=
0
u
4
u
2
1
a
4
=
u
5
u
3
2u
u
5
+ u
4
u
3
u + 1
a
9
=
u
4
+ u 1
u
5
+ u
2
u
a
9
=
u
4
+ u 1
u
5
+ u
2
u
(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 = 0.744862 0.877439I
a = 0.662359 + 0.562280I
b = 0.77736 + 1.96950I
0.26574 + 2.82812I 3.50976 2.97945I
u = 0.744862 + 0.877439I
a = 0.662359 0.562280I
b = 0.77736 1.96950I
0.26574 2.82812I 3.50976 + 2.97945I
u = 0.754878I
a = 1.32472
b = 0.56984 + 1.32472I
4.40332 3.01951
u = 0.754878I
a = 1.32472
b = 0.56984 1.32472I
4.40332 3.01951
u = 0.744862 0.877439I
a = 0.662359 0.562280I
b = 0.347200 0.644782I
0.26574 2.82812I 3.50976 + 2.97945I
u = 0.744862 + 0.877439I
a = 0.662359 + 0.562280I
b = 0.347200 + 0.644782I
0.26574 + 2.82812I 3.50976 2.97945I
3
II.
I
u
2
= ha
28
+2a
27
+· · ·4a+1, 1.19×10
22
u7.70×10
20
a
27
+· · ·+5.90×10
22
a+
3.60 × 10
21
, 1.19 × 10
22
b + 5.18 × 10
21
a
27
+ · · · + 5.45 × 10
22
a 1.17 × 10
22
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
0.0648767a
27
+ 0.423197a
26
+ ··· 4.96731a 0.302940
a
7
=
0.0648767a
27
+ 0.423197a
26
+ ··· 4.96731a 0.302940
0.0648767a
27
+ 0.423197a
26
+ ··· 4.96731a 0.302940
a
8
=
0.0648767a
27
+ 0.423197a
26
+ ··· 4.96731a 0.302940
0.101611a
27
+ 0.751072a
26
+ ··· 9.44921a + 1.34626
a
1
=
0.0531771a
27
0.164665a
26
+ ··· + 7.71802a + 0.0171482
0.0531771a
27
0.164665a
26
+ ··· + 7.71802a 0.982852
a
3
=
0.141682a
27
+ 0.192432a
26
+ ··· + 4.86511a 1.25498
0.0885052a
27
+ 0.357096a
26
+ ··· 2.85291a 1.27213
a
10
=
a
0.435944a
27
1.03656a
26
+ ··· 4.59150a + 0.983778
a
5
=
0.269325a
27
0.698352a
26
+ ··· 1.10890a + 0.293443
0.698546a
27
1.68593a
26
+ ··· 6.94417a + 0.501838
a
11
=
0.0905420a
27
0.131111a
26
+ ··· + 0.358204a + 0.228204
0.526677a
27
1.29960a
26
+ ··· 4.92459a + 1.29687
a
4
=
0.676533a
27
1.76247a
26
+ ··· 2.91213a + 0.819798
1.20635a
27
3.21336a
26
+ ··· 7.58959a + 0.817334
a
9
=
0.394443a
27
+ 1.10767a
26
+ ··· + 5.79152a + 0.144194
0.0164785a
27
+ 0.275238a
26
+ ··· + 4.22169a + 0.106815
a
9
=
0.394443a
27
+ 1.10767a
26
+ ··· + 5.79152a + 0.144194
0.0164785a
27
+ 0.275238a
26
+ ··· + 4.22169a + 0.106815
(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 = 0.517057 + 0.454483I
a = 1.24793 1.39851I
b = 0.340098 0.254510I
3.26705 + 1.37770I 4.88590 4.12207I
u = 0.517057 0.454483I
a = 1.24793 + 1.39851I
b = 0.340098 + 0.254510I
3.26705 1.37770I 4.88590 + 4.12207I
u = 0.494052 + 0.663856I
a = 1.16133 0.99760I
b = 0.434654 + 0.130908I
3.31269 + 1.37770I 3.11410 4.12207I
u = 0.494052 0.663856I
a = 1.16133 + 0.99760I
b = 0.434654 0.130908I
3.31269 1.37770I 3.11410 + 4.12207I
u = 0.683715 1.025590I
a = 1.077215 0.503914I
b = 1.92424 2.11049I
3.28987 + 8.93586I 4.00000 7.26077I
u = 0.683715 + 1.025590I
a = 1.077215 + 0.503914I
b = 1.92424 + 2.11049I
3.28987 8.93586I 4.00000 + 7.26077I
u = 0.622207 + 1.001065I
a = 1.061821 0.597490I
b = 1.78405 1.27929I
4.48016 + 3.41271I 1.89400 2.62516I
u = 0.622207 1.001065I
a = 1.061821 + 0.597490I
b = 1.78405 + 1.27929I
4.48016 3.41271I 1.89400 + 2.62516I
u = 0.772300 0.626535I
a = 0.72403 1.22509I
b = 0.891898 0.018580I
2.09958 3.41271I 6.10600 + 2.62516I
u = 0.772300 + 0.626535I
a = 0.72403 + 1.22509I
b = 0.891898 + 0.018580I
2.09958 + 3.41271I 6.10600 2.62516I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.050221 1.076786I
a = 0.586708 0.735945I
b = 0.546110 + 0.350420I
7.93259 2.76747I 1.41762 + 3.21377I
u = 0.050221 + 1.076786I
a = 0.586708 + 0.735945I
b = 0.546110 0.350420I
7.93259 + 2.76747I 1.41762 3.21377I
u = 0.772300 + 0.626535I
a = 0.396027 0.837541I
b = 1.57280 + 0.52715I
2.09958 + 3.41271I 6.10600 2.62516I
u = 0.772300 0.626535I
a = 0.396027 + 0.837541I
b = 1.57280 0.52715I
2.09958 3.41271I 6.10600 + 2.62516I
u = 0.727524 + 0.860849I
a = 0.032281 0.180723I
b = 0.335562 + 0.520004I
1.35286 2.76747I 9.41762 + 3.21377I
u = 0.727524 0.860849I
a = 0.032281 + 0.180723I
b = 0.335562 0.520004I
1.35286 + 2.76747I 9.41762 3.21377I
u = 0.622207 1.001065I
a = 0.264474 0.873381I
b = 1.223200 0.438742I
4.48016 3.41271I 1.89400 + 2.62516I
u = 0.622207 + 1.001065I
a = 0.264474 + 0.873381I
b = 1.223200 + 0.438742I
4.48016 + 3.41271I 1.89400 2.62516I
u = 0.727524 + 0.860849I
a = 0.668637 0.469979I
b = 0.004978 1.243892I
1.35286 2.76747I 9.41762 + 3.21377I
u = 0.727524 0.860849I
a = 0.668637 + 0.469979I
b = 0.004978 + 1.243892I
1.35286 + 2.76747I 9.41762 3.21377I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.683715 1.025590I
a = 0.692585 0.419157I
b = 1.91463 + 1.19881I
3.28987 + 8.93586I 4.00000 7.26077I
u = 0.683715 + 1.025590I
a = 0.692585 + 0.419157I
b = 1.91463 1.19881I
3.28987 8.93586I 4.00000 + 7.26077I
u = 0.494052 0.663856I
a = 1.093782 0.168458I
b = 0.51695 1.72422I
3.31269 1.37770I 3.11410 + 4.12207I
u = 0.494052 + 0.663856I
a = 1.093782 + 0.168458I
b = 0.51695 + 1.72422I
3.31269 + 1.37770I 3.11410 4.12207I
u = 0.517057 0.454483I
a = 1.150316 0.200394I
b = 0.484640 1.252975I
3.26705 1.37770I 4.88590 + 4.12207I
u = 0.517057 + 0.454483I
a = 1.150316 + 0.200394I
b = 0.484640 + 1.252975I
3.26705 + 1.37770I 4.88590 4.12207I
u = 0.050221 + 1.076786I
a = 1.352990 0.024562I
b = 2.17648 + 0.45551I
7.93259 + 2.76747I 1.41762 3.21377I
u = 0.050221 1.076786I
a = 1.352990 + 0.024562I
b = 2.17648 0.45551I
7.93259 2.76747I 1.41762 + 3.21377I
7
III.
I
u
3
= hu
21
3u
20
+· · ·+3u+2, u
20
+u
19
+· · ·+2a3, 3u
20
+8u
19
+· · ·+b+5i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
7
=
u
u
a
8
=
u
u
3
+ u
a
1
=
u
2
+ 1
u
2
a
3
=
u
4
+ u
2
+ 1
u
4
a
10
=
1
2
u
20
1
2
u
19
+ ··· +
5
2
u +
3
2
3u
20
8u
19
+ ··· 9u 5
a
5
=
1
2
u
20
+
3
2
u
19
+ ··· +
5
2
u +
3
2
u
19
u
18
+ ··· + 3u + 1
a
11
=
u
6
+ u
4
+ 2u
2
+ 1
u
8
+ 2u
6
+ 2u
4
+ 2u
2
a
4
=
3
2
u
20
+
9
2
u
19
+ ··· +
15
2
u +
9
2
2u
19
3u
18
+ ··· + 8u + 3
a
9
=
1
2
u
20
1
2
u
19
+ ··· +
1
2
u +
1
2
2u
20
5u
19
+ ··· 6u 3
a
9
=
1
2
u
20
1
2
u
19
+ ··· +
1
2
u +
1
2
2u
20
5u
19
+ ··· 6u 3
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.782310 0.279066I
a = 0.82987 1.64858I
b = 0.302112 1.099951I
10.20411 + 4.07649I 2.56533 2.84794I
u = 0.782310 + 0.279066I
a = 0.82987 + 1.64858I
b = 0.302112 + 1.099951I
10.20411 4.07649I 2.56533 + 2.84794I
u = 0.697631 0.876351I
a = 0.330904 + 0.320326I
b = 0.152951 + 0.357896I
1.38918 + 2.68088I 7.84813 2.28119I
u = 0.697631 + 0.876351I
a = 0.330904 0.320326I
b = 0.152951 0.357896I
1.38918 2.68088I 7.84813 + 2.28119I
u = 0.517849 1.076628I
a = 1.141156 0.679586I
b = 0.716866 1.078900I
12.61954 + 0.61749I 1.09619 1.91653I
u = 0.517849 + 1.076628I
a = 1.141156 + 0.679586I
b = 0.716866 + 1.078900I
12.61954 0.61749I 1.09619 + 1.91653I
u = 0.342162
a = 1.09553
b = 0.344773
0.610872 16.2597
u = 0.123120 1.145068I
a = 1.378774 0.078260I
b = 1.54819 + 0.86740I
15.0783 + 6.7163I 2.88981 3.97813I
u = 0.123120 + 1.145068I
a = 1.378774 + 0.078260I
b = 1.54819 0.86740I
15.0783 6.7163I 2.88981 + 3.97813I
u = 0.057404 0.911869I
a = 0.428320 + 0.366701I
b = 0.681197 0.111656I
1.99273 + 1.21629I 2.57418 5.92996I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.057404 + 0.911869I
a = 0.428320 0.366701I
b = 0.681197 + 0.111656I
1.99273 1.21629I 2.57418 + 5.92996I
u = 0.695179 0.960843I
a = 0.537858 + 0.263146I
b = 1.61612 0.78241I
2.49171 5.86529I 9.54952 + 8.01834I
u = 0.695179 + 0.960843I
a = 0.537858 0.263146I
b = 1.61612 + 0.78241I
2.49171 + 5.86529I 9.54952 8.01834I
u = 0.717299 1.057827I
a = 1.132055 + 0.447554I
b = 1.63546 + 2.55764I
9.5708 13.4578I 1.56507 + 7.58317I
u = 0.717299 + 1.057827I
a = 1.132055 0.447554I
b = 1.63546 2.55764I
9.5708 + 13.4578I 1.56507 7.58317I
u = 0.732239 0.738407I
a = 0.256789 0.598200I
b = 1.008546 + 0.717533I
3.16640 + 0.40908I 11.72672 2.09398I
u = 0.732239 + 0.738407I
a = 0.256789 + 0.598200I
b = 1.008546 0.717533I
3.16640 0.40908I 11.72672 + 2.09398I
u = 0.841522 0.895718I
a = 0.833451 0.760554I
b = 0.53773 1.97865I
3.09312 3.11987I 1.81385 + 2.72222I
u = 0.841522 + 0.895718I
a = 0.833451 + 0.760554I
b = 0.53773 + 1.97865I
3.09312 + 3.11987I 1.81385 2.72222I
u = 0.863155 0.627345I
a = 0.59056 + 1.30824I
b = 1.39226 + 0.47933I
8.25449 + 7.57688I 3.21336 3.12167I
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.863155 + 0.627345I
a = 0.59056 1.30824I
b = 1.39226 0.47933I
8.25449 7.57688I 3.21336 + 3.12167I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u
6
+ u
4
+ 2u
2
+ 1)
(1 + u + 2u
2
+ 4u
4
+ 6u
6
2u
7
+ 7u
8
3u
9
+ 6u
10
2u
11
+ 3u
12
u
13
+ u
14
)
2
(u
21
+ 3u
20
+ ··· + 3u 2)
c
2
, c
7
(u
3
+ u
2
+ 2u + 1)
2
(1 + 3u + 12u
2
+ 28u
3
+ 58u
4
+ 94u
5
+ 126u
6
+ 144u
7
+ 137u
8
+ 111u
9
+ 74u
10
+ 40u
11
+ 17u
12
+ 5u
13
+ u
14
)
2
(u
21
+ 7u
20
+ ··· + 21u 4)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(u
2
+ 1)
3
(u
21
+ 13u
19
+ ··· + 12u
3
+ 1)(u
28
+ u
27
+ ··· 6u + 1)
c
11
(u
6
3u
4
+ 2u
2
+ 1)
(1 3u + 12u
2
28u
3
+ 58u
4
94u
5
+ 126u
6
144u
7
+ 137u
8
111u
9
+ 74u
10
40u
11
+ 17u
12
5u
13
+ u
14
)
2
(u
21
+ 15u
20
+ ··· + 2103u + 266)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
3
+ y
2
+ 2y + 1)
2
(1 + 3y + 12y
2
+ 28y
3
+ 58y
4
+ 94y
5
+ 126y
6
+ 144y
7
+ 137y
8
+ 111y
9
+ 74y
10
+ 40y
11
+ 17y
12
+ 5y
13
+ y
14
)
2
(y
21
+ 7y
20
+ ··· + 21y 4)
c
2
, c
7
(y
3
+ 3y
2
+ 2y 1)
2
(1 + 15y + 92y
2
+ 296y
3
+ 534y
4
+ 486y
5
+ 50y
6
356y
7
355y
8
105y
9
+ 66y
10
+ 80y
11
+ 37y
12
+ 9y
13
+ y
14
)
2
(y
21
+ 15y
20
+ ··· + 1137y 16)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y + 1)
6
(y
21
+ 26y
20
+ ··· + 24y
2
1)(y
28
+ 23y
27
+ ··· + 16y + 1)
c
11
(y
3
3y
2
+ 2y + 1)
2
(1 + 15y + 92y
2
+ 296y
3
+ 534y
4
+ 486y
5
+ 50y
6
356y
7
355y
8
105y
9
+ 66y
10
+ 80y
11
+ 37y
12
+ 9y
13
+ y
14
)
2
(y
21
+ 3y
20
+ ··· + 343765y 70756)
12