11a
123
(K11a
123
)
1
Arc Sequences
5 1 9 6 2 10 3 11 7 4 8
Solving Sequence
1,5
2 3 6
4,8
7 11 9 10
c
1
c
2
c
5
c
4
c
7
c
11
c
8
c
10
c
3
, c
6
, c
9
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hb
2
+ 1, u 1, b + a + 1i
I
u
2
= hu
3
+ u
2
1, b + 1, 2a + 2u + 1i
I
u
3
= hb
38
+ 7b
37
+ ··· + 19b + 2,
3.87628 × 10
24
b
37
2.32129 × 10
25
b
36
+ ··· + 6.25748 × 10
24
a + 2.72523 × 10
25
,
6.25748 × 10
24
u + 3.46920 × 10
24
b
37
+ ··· + 1.08438 × 10
26
b + 8.56490 × 10
24
i
I
u
4
= hu
25
2u
24
+ ··· + 17u 4,
31496267249u
24
1954119851u
23
+ ··· + 353576605306b 294716823050,
258242463245u
24
+ 140426199118u
23
+ ··· + 707153210612a + 177949498093i
There are 4 irreducible components with 68 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
2
+ 1, u 1, b + a + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
1
a
2
=
1
1
a
3
=
0
1
a
6
=
1
0
a
4
=
1
1
a
8
=
b 1
b
a
7
=
b 1
1
a
11
=
b + 2
1
a
9
=
b
0
a
10
=
1
b
a
10
=
1
b
(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 = 1.00000 1.00000I
b = 1.00000I
1.64493 8.00000
u = 1.00000
a = 1.00000 + 1.00000I
b = 1.00000I
1.64493 8.00000
3
II. I
u
2
= hu
3
+ u
2
1, b + 1, 2a + 2u + 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
2
+ u 1
a
4
=
u
2
+ 1
u
2
a
8
=
u
1
2
1
a
7
=
3
2
u
1
2
u
2
+
1
2
u
3
2
a
11
=
u +
1
2
1
a
9
=
2u
2
a
10
=
1
2
u
1
2
u
2
1
2
u
1
2
a
10
=
1
2
u
1
2
u
2
1
2
u
1
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 = 0.877439 0.744862I
a = 0.377439 + 0.744862I
b = 1.00000
1.37919 2.82812I 13.06503 + 2.38969I
u = 0.877439 + 0.744862I
a = 0.377439 0.744862I
b = 1.00000
1.37919 + 2.82812I 13.06503 2.38969I
u = 0.754878
a = 1.25488
b = 1.00000
2.75839 4.61995
5
III.
I
u
3
= hb
38
+ 7b
37
+ · · · + 19b + 2, 3.88 × 10
24
b
37
2.32 × 10
25
b
36
+ · · · + 6.26 ×
10
24
a+2.73×10
25
, 6.26×10
24
u+3.47×10
24
b
37
+· · ·+1.08×10
26
b+8.56×10
24
i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
0.554408b
37
3.54083b
36
+ ··· 17.3294b 1.36874
a
2
=
1
0.227736b
37
1.66018b
36
+ ··· 10.1333b 0.702067
a
3
=
0.227736b
37
+ 1.66018b
36
+ ··· + 10 .1333b + 1.70207
0.227736b
37
1.66018b
36
+ ··· 10.1333b 0.702067
a
6
=
0.554408b
37
+ 3.54083b
36
+ ··· + 17.3294b + 1.36874
0.0118647b
37
+ 0.0994000b
36
+ ··· + 3 .72968b + 0.899144
a
4
=
0.566272b
37
3.64023b
36
+ ··· 21.0590b 2.26789
0.827976b
37
5.35236b
36
+ ··· 22.0105b 2.41100
a
8
=
0.619464b
37
+ 3.70962b
36
+ ··· 26.9083b 4.35515
b
a
7
=
0.697027b
37
+ 4.34130b
36
+ ··· 8.65905b 2.11472
0.159401b
37
1.21306b
36
+ ··· + 2 .57938b + 1.23294
a
11
=
0.626621b
37
+ 5.13921b
36
+ ··· + 16 .1250b + 2.23893
b
2
a
9
=
0.133406b
37
1.12349b
36
+ ··· 17.2415b 3.10191
b
3
+ b
a
10
=
0.441205b
37
+ 3.72282b
36
+ ··· + 21 .9821b + 3.51328
0.114250b
37
+ 0.642591b
36
+ ··· + 6.20657b + 1.22215
a
10
=
0.441205b
37
+ 3.72282b
36
+ ··· + 21 .9821b + 3.51328
0.114250b
37
+ 0.642591b
36
+ ··· + 6.20657b + 1.22215
(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 = 0.975971 + 0 .799116I
a = 0.229088 + 0 .972721I
b = 1.211894 0.090804I
7.18622 9.88550I 6.86128 + 7.31129I
u = 0.975971 0.799116I
a = 0.229088 0.972721I
b = 1.211894 + 0.090804I
7.18622 + 9.88550I 6.86128 7.31129I
u = 0.809650 0.858173I
a = 0.565390 0.425293I
b = 1.179350 0.174669I
7.70394 3.71612I 5.80100 + 2.45937I
u = 0.809650 + 0 .858173I
a = 0.565390 + 0 .425293I
b = 1.179350 + 0.174669I
7.70394 + 3.71612I 5.80100 2.45937I
u = 0.978202 + 0.313897I
a = 0.849902 0.223654I
b = 0.852454 0.070284I
0.16029 + 5.52702I 12.4279 7.0025I
u = 0.978202 0.313897I
a = 0.849902 + 0.223654I
b = 0.852454 + 0.070284I
0.16029 5.52702I 12.4279 + 7.0025I
u = 0.903405 + 0 .838368I
a = 0.98440 + 2.02627I
b = 0.66723 1.46348I
11.59749 3.11880I 2.41376 + 2.69239I
u = 0.903405 0.838368I
a = 0.98440 2.02627I
b = 0.66723 + 1.46348I
11.59749 + 3.11880I 2.41376 2.69239I
u = 0.103765 + 0.589022I
a = 0.054068 0.769378I
b = 0.625152 0.214266I
2.82151 2.32534I 6.27174 + 3.09456I
u = 0.103765 0.589022I
a = 0.054068 + 0 .769378I
b = 0.625152 + 0.214266I
2.82151 + 2.32534I 6.27174 3.09456I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.903405 0.838368I
a = 0.86034 + 1 .32154I
b = 0.60508 1.51193I
11.59749 + 3.11880I 2.41376 2.69239I
u = 0.903405 + 0 .838368I
a = 0.86034 1.32154I
b = 0.60508 + 1.51193I
11.59749 3.11880I 2.41376 + 2.69239I
u = 0.635698 0.450549I
a = 1.94079 + 0.02901I
b = 0.463625 0.999486I
4.70093 1.72326I 4.18035 + 5.18112I
u = 0.635698 + 0.450549I
a = 1.94079 0.02901I
b = 0.463625 + 0.999486I
4.70093 + 1.72326I 4.18035 5.18112I
u = 0.964317 + 0 .230449I
a = 0.491180 + 0.989591I
b = 0.217752 0.156279I
0.332249 0.168160I 14.1683 + 0.9143I
u = 0.964317 0.230449I
a = 0.491180 0.989591I
b = 0.217752 + 0.156279I
0.332249 + 0.168160I 14.1683 0.9143I
u = 0.635698 + 0.450549I
a = 0.230067 + 0.608858I
b = 0.213170 1.280762I
4.70093 + 1.72326I 4.18035 5.18112I
u = 0.635698 0.450549I
a = 0.230067 0.608858I
b = 0.213170 + 1.280762I
4.70093 1.72326I 4.18035 + 5.18112I
u = 0.949254 0.773576I
a = 1.61357 + 1.95000I
b = 0.116946 1.190011I
5.72757 4.39903I 8.93348 + 2.80289I
u = 0.949254 + 0.773576I
a = 1.61357 1.95000I
b = 0.116946 + 1.190011I
5.72757 + 4.39903I 8.93348 2.80289I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.820272 + 0.802988I
a = 0.09753 + 2.28421I
b = 0.052144 1.206600I
6.12368 + 1.53005I 8.20605 2.54963I
u = 0.820272 0.802988I
a = 0.09753 2.28421I
b = 0.052144 + 1.206600I
6.12368 1.53005I 8.20605 + 2.54963I
u = 0.667698
a = 6.45400 + 5 .52977I
b = 0.072948 1.007950I
2.38250 15.4722
u = 0.667698
a = 6.45400 5.52977I
b = 0.072948 + 1 .007950I
2.38250 15.4722
u = 0.964317 + 0 .230449I
a = 0.985016 0.274220I
b = 0.134173 0.930763I
0.332249 0.168160I 14.1683 + 0.9143I
u = 0.964317 0.230449I
a = 0.985016 + 0.274220I
b = 0.134173 + 0 .930763I
0.332249 + 0.168160I 14.1683 0.9143I
u = 0.103765 0.589022I
a = 0.56231 + 1.56828I
b = 0.223313 1.232587I
2.82151 + 2.32534I 6.27174 3.09456I
u = 0.103765 + 0.589022I
a = 0.56231 1.56828I
b = 0.223313 + 1 .232587I
2.82151 2.32534I 6.27174 + 3.09456I
u = 0.820272 0.802988I
a = 0.377406 0.541506I
b = 0.357882 0.461087I
6.12368 1.53005I 8.20605 + 2.54963I
u = 0.820272 + 0.802988I
a = 0.377406 + 0 .541506I
b = 0.357882 + 0 .461087I
6.12368 + 1.53005I 8.20605 2.54963I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.949254 + 0.773576I
a = 0.620189 0.094867I
b = 0.423801 0.309302I
5.72757 + 4.39903I 8.93348 2.80289I
u = 0.949254 0.773576I
a = 0.620189 + 0 .094867I
b = 0.423801 + 0 .309302I
5.72757 4.39903I 8.93348 + 2.80289I
u = 0.978202 + 0.313897I
a = 1.41152 + 0 .19798I
b = 0.462406 1.206878I
0.16029 + 5.52702I 12.4279 7.0025I
u = 0.978202 0.313897I
a = 1.41152 0.19798I
b = 0.462406 + 1 .206878I
0.16029 5.52702I 12.4279 + 7.0025I
u = 0.809650 + 0 .858173I
a = 0.62274 + 1.72512I
b = 0.48129 1.51282I
7.70394 + 3.71612I 5.80100 2.45937I
u = 0.809650 0.858173I
a = 0.62274 1.72512I
b = 0.48129 + 1 .51282I
7.70394 3.71612I 5.80100 + 2.45937I
u = 0.975971 0.799116I
a = 1.30550 + 2 .02733I
b = 0.54897 1.49405I
7.18622 + 9.88550I 6.86128 7.31129I
u = 0.975971 + 0 .799116I
a = 1.30550 2.02733I
b = 0.54897 + 1 .49405I
7.18622 9.88550I 6.86128 + 7.31129I
10
IV. I
u
4
=
hu
25
2u
24
+· · · +17u 4, 3.15× 10
10
u
24
1.95 × 10
9
u
23
+· · · +3.54 × 10
11
b
2.95× 10
11
, 2.58× 10
11
u
24
+1.40 × 10
11
u
23
+· · · +7.07 × 10
11
a + 1.78 ×10
11
i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
4
=
u
3
u
5
u
3
+ u
a
8
=
0.365186u
24
0.198580u
23
+ ··· 5.28891u 0.251642
0.0890790u
24
+ 0.00552672u
23
+ ··· 1.23519u + 0.833530
a
7
=
0.267603u
24
0.170382u
23
+ ··· 4.06703u 0.0336839
0.227826u
24
+ 0.244721u
23
+ ··· + 1 .30904u + 0.0124582
a
11
=
0.361710u
24
+ 0.276963u
23
+ ··· + 5 .53501u + 0.291576
0.135603u
24
0.0416754u
23
+ ··· + 1.55999u 0.715415
a
9
=
0.690139u
24
0.455342u
23
+ ··· 9.24059u 0.0115544
0.314729u
24
+ 0.224657u
23
+ ··· 1.21944u + 0.939187
a
10
=
0.323410u
24
+ 0.155971u
23
+ ··· + 3 .99738u + 0.846565
0.110684u
24
0.103445u
23
+ ··· + 0 .590987u 0.502652
a
10
=
0.323410u
24
+ 0.155971u
23
+ ··· + 3 .99738u + 0.846565
0.110684u
24
0.103445u
23
+ ··· + 0 .590987u 0.502652
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.142430 0.382078I
a = 1.268034 + 0.422167I
b = 0.409233 1.264710I
3.90197 10.08687I 8.02551 + 8.45672I
u = 1.142430 + 0.382078I
a = 1.268034 0.422167I
b = 0.409233 + 1.264710I
3.90197 + 10.08687I 8.02551 8.45672I
u = 1.076915 0.848675I
a = 1.04834 1.34169I
b = 0.153566 + 1 .310872I
10.62702 6.42707I 2.25103 + 5.26300I
u = 1.076915 + 0.848675I
a = 1.04834 + 1 .34169I
b = 0.153566 1.310872I
10.62702 + 6.42707I 2.25103 5.26300I
u = 0.873066 0.713586I
a = 0.343451 + 0.174556I
b = 0.321236 + 0.053568I
2.48771 2.73173I 3.19620 + 2.74281I
u = 0.873066 + 0.713586I
a = 0.343451 0.174556I
b = 0.321236 0.053568I
2.48771 + 2.73173I 3.19620 2.74281I
u = 0.791605 0.124227I
a = 1.178096 0.337562I
b = 1.040975 0.275015I
2.97368 0.31797I 15.9411 + 13.1085I
u = 0.791605 + 0.124227I
a = 1.178096 + 0 .337562I
b = 1.040975 + 0 .275015I
2.97368 + 0.31797I 15.9411 13.1085I
u = 0.749482 1.006981I
a = 0.24489 + 1.53795I
b = 0.065690 1.341366I
11.66026 0.32743I 0.684836 + 0.302346I
u = 0.749482 + 1.006981I
a = 0.24489 1.53795I
b = 0.065690 + 1 .341366I
11.66026 + 0.32743I 0.684836 0.302346I
12
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.096629 0.930099I
a = 0.10164 1.51799I
b = 0.269095 + 1.312202I
7.47156 + 5.57189I 2.30391 4.65826I
u = 0.096629 + 0.930099I
a = 0.10164 + 1 .51799I
b = 0.269095 1.312202I
7.47156 5.57189I 2.30391 + 4.65826I
u = 0.341201
a = 1.17598
b = 0.331472
0.684794 14.4595
u = 0.746418 0.296773I
a = 0.848177 0.452473I
b = 0.188697 + 0 .408041I
0.644117 + 0.222126I 12.29988 + 0.69416I
u = 0.746418 + 0 .296773I
a = 0.848177 + 0.452473I
b = 0.188697 0.408041I
0.644117 0.222126I 12.29988 0.69416I
u = 0.770950 0.937113I
a = 0.30856 + 1 .63900I
b = 0.48655 1.44328I
12.8491 9.5213I 4.07416 + 4.01278I
u = 0.770950 + 0 .937113I
a = 0.30856 1.63900I
b = 0.48655 + 1.44328I
12.8491 + 9.5213I 4.07416 4.01278I
u = 0.831869 0.519995I
a = 0.60247 + 1 .53377I
b = 0.604789 0.633838I
0.94821 + 3.43572I 13.2032 6.7534I
u = 0.831869 + 0 .519995I
a = 0.60247 1.53377I
b = 0.604789 + 0 .633838I
0.94821 3.43572I 13.2032 + 6.7534I
u = 0.890108 0.788825I
a = 0.718513 + 0.942420I
b = 1.43833 + 0 .04819I
2.16301 + 2.96631I 1.51503 3.55668I
13
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.890108 + 0 .788825I
a = 0.718513 0.942420I
b = 1.43833 0.04819I
2.16301 2.96631I 1.51503 + 3.55668I
u = 1.033464 0.815877I
a = 1.39808 1.83834I
b = 0.53103 + 1.43134I
12.0149 + 15.9713I 5.35626 8.57789I
u = 1.033464 + 0 .815877I
a = 1.39808 + 1.83834I
b = 0.53103 1.43134I
12.0149 15.9713I 5.35626 + 8.57789I
u = 1.286718 0.219140I
a = 0.295022 + 0 .018174I
b = 0.140637 1.182506I
2.59539 1.53840I 3.54408 + 4.89308I
u = 1.286718 + 0 .219140I
a = 0.295022 0.018174I
b = 0.140637 + 1.182506I
2.59539 + 1.53840I 3.54408 4.89308I
14
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u 1)
2
(u
3
+ u
2
1)
(1 u
2
+ 3u
3
2u
4
4u
5
+ 10u
6
+ 8u
7
15u
8
10u
9
+ 17u
10
+ 11u
11
13u
12
8u
13
+ 8u
14
+ 6u
15
3u
16
2u
17
+ u
18
+ u
19
)
2
(u
25
+ 2u
24
+ ··· + 17 u + 4)
c
2
(u + 1)
2
(u
3
+ u
2
+ 2u + 1)
(1 + 2u 3u
2
15u
3
22u
4
+ 40u
5
+ 224u
6
+ 536u
7
+ 891u
8
+ 1170u
9
+ 1257u
10
+ 1145u
11
+ 887u
12
+ 592u
13
+ 336u
14
+ 164u
15
+ 65u
16
+ 22u
17
+ 5u
18
+ u
19
)
2
(u
25
+ 8u
24
+ ··· + 241 u + 16)
c
3
u
3
(u
2
+ 1)
(1 + 2u + u
2
+ 5u
3
+ 10u
5
+ 14u
7
3u
8
+ 18u
9
5u
10
+ 19u
11
7u
12
+ 16u
13
6u
14
+ 10u
15
3u
16
+ 4u
17
u
18
+ u
19
)
2
(u
25
+ 3u
24
+ ··· 96u 128)
c
4
(u 1)
2
(u
3
u
2
+ 2u 1)
(1 + 2u 3u
2
15u
3
22u
4
+ 40u
5
+ 224u
6
+ 536u
7
+ 891u
8
+ 1170u
9
+ 1257u
10
+ 1145u
11
+ 887u
12
+ 592u
13
+ 336u
14
+ 164u
15
+ 65u
16
+ 22u
17
+ 5u
18
+ u
19
)
2
(u
25
+ 8u
24
+ ··· + 241 u + 16)
c
5
(u + 1)
2
(u
3
u
2
+ 1)
(1 u
2
+ 3u
3
2u
4
4u
5
+ 10u
6
+ 8u
7
15u
8
10u
9
+ 17u
10
+ 11u
11
13u
12
8u
13
+ 8u
14
+ 6u
15
3u
16
2u
17
+ u
18
+ u
19
)
2
(u
25
+ 2u
24
+ ··· + 17 u + 4)
c
6
, c
8
(u 1)
3
(u
2
+ 1)(u
25
3u
24
+ ··· + 2u 1)(u
38
+ 7u
37
+ ··· + 19u + 2)
c
7
(u
2
+ 2u + 2)(8u
3
+ 4u
2
+ 4u + 1)(8u
25
4u
24
+ ··· + 2u 2)
(u
38
+ 5u
37
+ ··· 8230u + 15341)
c
9
, c
11
(u + 1)
3
(u
2
+ 1)(u
25
3u
24
+ ··· + 2u 1)(u
38
+ 7u
37
+ ··· + 19u + 2)
c
10
(u
2
2u + 2)(8u
3
4u
2
+ 4u 1)(8u
25
4u
24
+ ··· + 2u 2)
(u
38
+ 5u
37
+ ··· 8230u + 15341)
15
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y 1)
2
(y
3
y
2
+ 2y 1)
(1 + 2y + 3y
2
15y
3
+ 22y
4
+ 40y
5
224y
6
+ 536y
7
891y
8
+ 1170y
9
1257y
10
+ 1145y
11
887y
12
+ 592y
13
336y
14
+ 164y
15
65y
16
+ 22y
17
5y
18
+ y
19
)
2
(y
25
8y
24
+ ··· + 241y 16)
c
2
, c
4
(y 1)
2
(y
3
+ 3y
2
+ 2y 1)
(1 + 10y 25y
2
195y
3
+ 22y
4
+ 2888y
5
+ 7156y
6
+ 9704y
7
+ 1.26 × 10
4
y
8
+ 1.67 × 10
4
y
9
+ 1.81 × 10
4
y
10
+ 1.63 × 10
4
y
11
+ 1.36 × 10
4
y
12
+ 1.02 × 10
4
y
13
+ 6120y
14
+ 2684y
15
+ 815y
16
+ 162y
17
+ 19y
18
+ y
19
)
2
(y
25
+ 20y
24
+ ··· + 19233 y 256)
c
3
y
3
(y + 1)
2
(1 + 2y + 19y
2
+ 65y
3
+ 162y
4
+ 328y
5
+ 560y
6
+ 836y
7
+ 1093y
8
+ 1270y
9
+ 1311y
10
+ 1201y
11
+ 973y
12
+ 688y
13
+ 420y
14
+ 216y
15
+ 91y
16
+ 30y
17
+ 7y
18
+ y
19
)
2
(y
25
+ 9y
24
+ ··· 72704y 16384)
c
6
, c
8
, c
9
c
11
(y 1)
3
(y + 1)
2
(y
25
+ 17y
24
+ ··· + 16y 1)
(y
38
+ 27y
37
+ ··· 21y + 4)
c
7
, c
10
(y
2
+ 4)(64y
3
+ 48y
2
+ 8y 1)(64y
25
+ 1072y
24
+ ··· + 20 y 4)
(y
38
+ 23y
37
+ ··· + 2526338154 y + 235346281)
16