11a
167
(K11a
167
)
1
Arc Sequences
6 1 11 7 2 5 10 4 3 8 9
Solving Sequence
2,6
1
3,9
10 5 7 4 8 11
c
1
c
2
c
9
c
5
c
6
c
4
c
8
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b u, a + u + 1i
I
u
2
= hu
58
+ 2u
57
+ ··· u + 1, 9.20331 × 10
18
u
57
1.37199 × 10
19
u
56
+ ··· + 3.39537 × 10
18
b 4.21028 × 10
18
,
7.03039 × 10
18
u
57
+ 1.85492 × 10
19
u
56
+ ··· + 3.39537 × 10
18
a + 2.49612 × 10
19
i
There are 2 irreducible components with 60 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
2
+ u + 1, b u, a + u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u 1
a
3
=
u
u
a
9
=
u 1
u
a
10
=
0
1
a
5
=
u
u
a
7
=
1
u + 1
a
4
=
1
0
a
8
=
1
u
a
11
=
1
u 1
a
11
=
1
u 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 = 0.500000 0.866025I
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 0.866025I
b = 0.500000 + 0.866025I
1.64493 2.02988I 3.00000 + 3.46410I
3
II. I
u
2
=
hu
58
+2u
57
+· · ·u +1, 9.20×10
18
u
57
1.37× 10
19
u
56
+· · ·+3.40 ×10
18
b
4.21 × 10
18
, 7.03 × 10
18
u
57
+ 1.85 × 10
19
u
56
+ · · · + 3.40 × 10
18
a + 2.50 × 10
19
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
1
=
1
u
2
a
3
=
u
2
+ 1
u
4
a
9
=
2.07058u
57
5.46309u
56
+ ··· + 8.71905u 7.35155
2.71055u
57
+ 4.04078u
56
+ ··· + 2.87153u + 1.24001
a
10
=
0.0678518u
57
1.33180u
56
+ ··· + 7.13402u 4.56590
2.17215u
57
+ 3.03451u
56
+ ··· + 4.72602u + 0.0399380
a
5
=
u
u
a
7
=
u
3
u
3
+ u
a
4
=
u
5
u
u
5
+ u
3
+ u
a
8
=
0.236838u
57
1.43795u
56
+ ··· + 5.04652u 4.51898
2.07684u
57
+ 2.44455u
56
+ ··· + 6.07889u 0.359956
a
11
=
1.42519u
57
1.02253u
56
+ ··· 3.40758u 0.311267
0.425182u
57
1.40266u
56
+ ··· + 2.31128u 1.00001
a
11
=
1.42519u
57
1.02253u
56
+ ··· 3.40758u 0.311267
0.425182u
57
1.40266u
56
+ ··· + 2.31128u 1.00001
(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 = 0.910971 0.834834I
a = 1.39520 + 0.83245I
b = 0.16453 2.92280I
5.95099 9.03935I 1.43078 + 4.42094I
u = 0.910971 + 0.834834I
a = 1.39520 0.83245I
b = 0.16453 + 2.92280I
5.95099 + 9.03935I 1.43078 4.42094I
u = 0.885584 0.866640I
a = 1.50974 1.18822I
b = 0.37841 + 3.18031I
9.19750 2.53947I 3.93592 + 2.26065I
u = 0.885584 + 0.866640I
a = 1.50974 + 1.18822I
b = 0.37841 3.18031I
9.19750 + 2.53947I 3.93592 2.26065I
u = 0.845331 0.960143I
a = 2.46844 0.68770I
b = 0.88375 + 3.17281I
8.90043 + 8.94791I 3.25999 7.24575I
u = 0.845331 + 0.960143I
a = 2.46844 + 0.68770I
b = 0.88375 3.17281I
8.90043 8.94791I 3.25999 + 7.24575I
u = 0.843359 0.871287I
a = 1.58053 0.99032I
b = 2.04212 0.61200I
3.51042 0.65934I 1.64525 + 3.03608I
u = 0.843359 + 0.871287I
a = 1.58053 + 0.99032I
b = 2.04212 + 0.61200I
3.51042 + 0.65934I 1.64525 3.03608I
u = 0.839643 0.992004I
a = 2.25810 + 0.80518I
b = 0.57202 3.02731I
5.4499 + 15.5032I 0.56907 9.02044I
u = 0.839643 + 0.992004I
a = 2.25810 0.80518I
b = 0.57202 + 3.02731I
5.4499 15.5032I 0.56907 + 9.02044I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.823065 0.934173I
a = 0.54799 + 1.61489I
b = 1.96451 1.14977I
3.31523 + 6.86856I 2.26425 8.06853I
u = 0.823065 + 0.934173I
a = 0.54799 1.61489I
b = 1.96451 + 1.14977I
3.31523 6.86856I 2.26425 + 8.06853I
u = 0.817614 0.900250I
a = 2.55651 + 1.15874I
b = 0.38714 3.85396I
1.70109 + 3.05742I 4.60638 2.59890I
u = 0.817614 + 0.900250I
a = 2.55651 1.15874I
b = 0.38714 + 3.85396I
1.70109 3.05742I 4.60638 + 2.59890I
u = 0.769063 0.455452I
a = 0.930791 0.294716I
b = 0.667321 0.337766I
0.20150 + 3.82733I 2.45119 7.09206I
u = 0.769063 + 0.455452I
a = 0.930791 + 0.294716I
b = 0.667321 + 0.337766I
0.20150 3.82733I 2.45119 + 7.09206I
u = 0.587529 0.195087I
a = 0.575520 + 0.301118I
b = 0.685426 + 0.097381I
1.59378 + 0.29446I 6.92249 + 0.09350I
u = 0.587529 + 0.195087I
a = 0.575520 0.301118I
b = 0.685426 0.097381I
1.59378 0.29446I 6.92249 0.09350I
u = 0.568190 1.020443I
a = 0.267371 + 0.661740I
b = 0.379130 0.546655I
1.55389 + 1.10217I 0.55808 + 5.36603I
u = 0.568190 + 1.020443I
a = 0.267371 0.661740I
b = 0.379130 + 0.546655I
1.55389 1.10217I 0.55808 5.36603I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.440330 0.759700I
a = 1.066631 + 0.436456I
b = 0.384922 0.669695I
0.01176 + 1.74270I 0.45658 3.63727I
u = 0.440330 + 0.759700I
a = 1.066631 0.436456I
b = 0.384922 + 0.669695I
0.01176 1.74270I 0.45658 + 3.63727I
u = 0.340458 0.770916I
a = 2.47350 + 3.28718I
b = 3.16149 0.03944I
1.98107 + 1.64703I 6.1662 + 29.1207I
u = 0.340458 + 0.770916I
a = 2.47350 3.28718I
b = 3.16149 + 0.03944I
1.98107 1.64703I 6.1662 29.1207I
u = 0.336157 1.024864I
a = 0.225338 0.792429I
b = 0.112078 + 0.426502I
1.05261 + 3.13615I 2.50844 8.50381I
u = 0.336157 + 1.024864I
a = 0.225338 + 0.792429I
b = 0.112078 0.426502I
1.05261 3.13615I 2.50844 + 8.50381I
u = 0.310122 0.582175I
a = 1.49433 2.38673I
b = 1.265493 0.474592I
1.39991 + 1.11754I 8.74885 3.65436I
u = 0.310122 + 0.582175I
a = 1.49433 + 2.38673I
b = 1.265493 + 0.474592I
1.39991 1.11754I 8.74885 + 3.65436I
u = 0.010811 0.780206I
a = 1.13083 1.06637I
b = 0.336641 0.147588I
1.43717 + 1.52858I 5.32481 4.46151I
u = 0.010811 + 0.780206I
a = 1.13083 + 1.06637I
b = 0.336641 + 0.147588I
1.43717 1.52858I 5.32481 + 4.46151I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.051144 1.040465I
a = 0.694306 + 0.600130I
b = 0.0941538 + 0.0407924I
5.22394 + 4.97791I 7.03726 5.31540I
u = 0.051144 + 1.040465I
a = 0.694306 0.600130I
b = 0.0941538 0.0407924I
5.22394 4.97791I 7.03726 + 5.31540I
u = 0.225458 0.825902I
a = 1.06368 2.01364I
b = 1.09293 + 1.05099I
4.22808 0.48675I 11.22027 + 2.03951I
u = 0.225458 + 0.825902I
a = 1.06368 + 2.01364I
b = 1.09293 1.05099I
4.22808 + 0.48675I 11.22027 2.03951I
u = 0.332519 0.864557I
a = 1.019626 0.147782I
b = 0.137567 + 0.696892I
3.63287 3.82995I 8.64205 + 8.99135I
u = 0.332519 + 0.864557I
a = 1.019626 + 0.147782I
b = 0.137567 0.696892I
3.63287 + 3.82995I 8.64205 8.99135I
u = 0.341034 0.197878I
a = 2.63966 0.02210I
b = 0.304941 0.449084I
1.92464 + 1.08312I 1.72371 1.84781I
u = 0.341034 + 0.197878I
a = 2.63966 + 0.02210I
b = 0.304941 + 0.449084I
1.92464 1.08312I 1.72371 + 1.84781I
u = 0.434758 1.011699I
a = 1.05107 1.46627I
b = 0.635292 + 1.236512I
2.95190 11.16427I 3.60588 + 9.61832I
u = 0.434758 + 1.011699I
a = 1.05107 + 1.46627I
b = 0.635292 1.236512I
2.95190 + 11.16427I 3.60588 9.61832I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.440912 0.897881I
a = 1.28479 + 1.51068I
b = 0.655511 0.976043I
0.81208 5.75126I 0.14596 + 9.28589I
u = 0.440912 + 0.897881I
a = 1.28479 1.51068I
b = 0.655511 + 0.976043I
0.81208 + 5.75126I 0.14596 9.28589I
u = 0.590155 0.416456I
a = 0.964716 + 0.975316I
b = 0.411626 0.943907I
2.32246 + 1.88820I 5.25997 2.06026I
u = 0.590155 + 0.416456I
a = 0.964716 0.975316I
b = 0.411626 + 0.943907I
2.32246 1.88820I 5.25997 + 2.06026I
u = 0.741876 0.285613I
a = 1.143887 0.651964I
b = 0.222922 + 0.926797I
0.59272 + 6.92978I 1.28521 5.28654I
u = 0.741876 + 0.285613I
a = 1.143887 + 0.651964I
b = 0.222922 0.926797I
0.59272 6.92978I 1.28521 + 5.28654I
u = 0.823019 0.992908I
a = 1.17187 + 0.87639I
b = 0.13038 1.73093I
6.63609 7.79931I 2.27644 + 7.29353I
u = 0.823019 + 0.992908I
a = 1.17187 0.87639I
b = 0.13038 + 1.73093I
6.63609 + 7.79931I 2.27644 7.29353I
u = 0.838722 0.920465I
a = 0.561004 0.957597I
b = 1.93336 1.29288I
4.97538 3.90988I 4.24532 6.23565I
u = 0.838722 + 0.920465I
a = 0.561004 + 0.957597I
b = 1.93336 + 1.29288I
4.97538 + 3.90988I 4.24532 + 6.23565I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.848099 0.894531I
a = 1.07319 + 1.36400I
b = 1.65226 1.13672I
5.05665 2.36415I 3.35828 + 10.69478I
u = 0.848099 + 0.894531I
a = 1.07319 1.36400I
b = 1.65226 + 1.13672I
5.05665 + 2.36415I 3.35828 10.69478I
u = 0.867514 0.951168I
a = 1.31254 0.69248I
b = 0.09282 + 2.05203I
8.26855 3.73272I 4.21505 + 0.37316I
u = 0.867514 + 0.951168I
a = 1.31254 + 0.69248I
b = 0.09282 2.05203I
8.26855 + 3.73272I 4.21505 0.37316I
u = 0.895121 0.818579I
a = 0.892761 + 0.255381I
b = 0.50150 1.65644I
7.18623 + 1.43763I 3.61939 2.17602I
u = 0.895121 + 0.818579I
a = 0.892761 0.255381I
b = 0.50150 + 1.65644I
7.18623 1.43763I 3.61939 + 2.17602I
u = 0.897894 0.891847I
a = 1.155737 0.290111I
b = 0.41707 + 1.99877I
8.46147 2.78521I 4.56237 + 4.61833I
u = 0.897894 + 0.891847I
a = 1.155737 + 0.290111I
b = 0.41707 1.99877I
8.46147 + 2.78521I 4.56237 4.61833I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
58
+ 2u
57
+ ··· u + 1)
c
2
, c
6
(u
2
+ u + 1)(u
58
+ 14u
57
+ ··· + 5u + 1)
c
3
(u
2
+ u + 1)(u
58
+ 4u
57
+ ··· + u + 1)
c
4
(u
2
u + 1)(u
58
+ 14u
57
+ ··· + 5u + 1)
c
5
(u
2
u + 1)(u
58
+ 2u
57
+ ··· u + 1)
c
7
(u 1)
2
(u
58
+ 3u
57
+ ··· + 10u + 1)
c
8
(u
2
u + 1)(u
58
+ 4u
57
+ ··· + 21u + 1)
c
9
(u
2
u + 1)(u
58
+ 2u
57
+ ··· 403u + 77)
c
10
(u + 1)
2
(u
58
+ 3u
57
+ ··· + 10u + 1)
c
11
u
2
(u
58
+ 9u
57
+ ··· + 12u + 4)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
2
+ y + 1)(y
58
+ 14y
57
+ ··· + 5y + 1)
c
2
, c
4
, c
6
(y
2
+ y + 1)(y
58
+ 62y
57
+ ··· + 53y + 1)
c
3
(y
2
+ y + 1)(y
58
+ 10y
57
+ ··· + 5y + 1)
c
7
, c
10
(y 1)
2
(y
58
33y
57
+ ··· + 122y + 1)
c
8
(y
2
+ y + 1)(y
58
46y
57
+ ··· + 117y + 1)
c
9
(y
2
+ y + 1)(y
58
58y
57
+ ··· 89259y + 5929)
c
11
y
2
(y
58
15y
57
+ ··· 168y + 16)
12