10
138
(K10n
1
)
1
Arc Sequences
8 4 10 6 8 9 2 5 2 3
Solving Sequence
2,9 5,10
8 6 1 4 3 7
c
9
c
8
c
5
c
1
c
4
c
3
c
7
c
2
, c
6
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hb
2
+ b + 1, b + a, u 1i
I
u
2
= hu
2
+ u + 1, a + 1, b + u + 1i
I
u
3
= hu
7
u
6
+ 9u
5
14u
4
+ 23u
3
14u
2
+ 8u 1,
115u
6
+ 86u
5
982u
4
+ 1372u
3
2128u
2
+ 277b + 859u 542,
285u
6
177u
5
+ 2530u
4
3075u
3
+ 5623u
2
+ 277a 2454u + 1837i
I
u
4
= hu
14
+ u
13
+ ··· + 2u + 1,
1205138480001u
13
+ 3533853013940u
12
+ ··· + 51132550709402b + 88002408176841,
40546748429631u
13
51692838289772u
12
+ ··· + 51132550709402a 37861651496541i
There are 4 irreducible components with 25 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
+ b + 1, b + a, u 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
b
b
a
5
=
0
1
a
10
=
2b
b
a
8
=
b
0
a
6
=
b 1
1
a
1
=
1
0
a
4
=
b
b
a
3
=
b + 2
b 1
a
7
=
b
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.00000
a = 0.500000 + 0.866025I
b = 0.500000 0.866025I
0 3.00000
u = 1.00000
a = 0.500000 0.866025I
b = 0.500000 + 0.866025I
0 3.00000
3
II. I
u
2
= hu
2
+ u + 1, a + 1, b + u + 1i
(i) Arc colorings
a
2
=
1
0
a
9
=
1
u 1
a
5
=
0
u
a
10
=
u
u 1
a
8
=
1
0
a
6
=
u
u
a
1
=
1
0
a
4
=
1
u + 1
a
3
=
u + 2
u
a
7
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u + 4
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 1.00000
b = 0.500000 + 0.866025I
4.05977I 6.92820I
u = 0.500000 + 0.866025I
a = 1.00000
b = 0.500000 0.866025I
4.05977I 6.92820I
5
III. I
u
3
= hu
7
u
6
+ 9u
5
14u
4
+ 23u
3
14u
2
+ 8u 1, 115u
6
+ 86u
5
+
· · · + 277b 542, 285u
6
177u
5
+ · · · + 277a + 1837i
(i) Arc colorings
a
2
=
1
0
a
9
=
1.02888u
6
+ 0.638989u
5
+ ··· + 8.85921u 6.63177
0.415162u
6
0.310469u
5
+ ··· 3.10108u + 1.95668
a
5
=
0
u
a
10
=
1.44404u
6
+ 0.949458u
5
+ ··· + 11.9603u 8.58845
0.415162u
6
0.310469u
5
+ ··· 3.10108u + 1.95668
a
8
=
1.02888u
6
+ 0.638989u
5
+ ··· + 8.85921u 6.63177
0.151625u
6
0.104693u
5
+ ··· 1.01083u + 1.56679
a
6
=
1.07942u
6
1.00722u
5
+ ··· 11.8628u + 6.48736
0.440433u
6
+ 0.494585u
5
+ ··· + 5.60289u 1.88448
a
1
=
1.88448u
6
1.44404u
5
+ ··· 17.5632u + 11.4729
0.541516u
6
+ 0.231047u
5
+ ··· + 3.61011u 2.59567
a
4
=
0.877256u
6
0.534296u
5
+ ··· 7.84838u + 5.06498
0.0722022u
6
+ 0.0974729u
5
+ ··· + 2.14801u 1.07942
a
3
=
0.617329u
6
+ 0.783394u
5
+ ··· + 8.11552u 3.37906
0.342960u
6
0.212996u
5
+ ··· 1.95307u + 0.877256
a
7
=
0.877256u
6
+ 0.534296u
5
+ ··· + 7.84838u 5.06498
0.151625u
6
0.104693u
5
+ ··· 1.01083u + 1.56679
(ii) Obstruction class = 1
(iii) Cusp Shapes =
218
277
u
6
+
232
277
u
5
+
1770
277
u
4
+
622
277
u
3
+
946
277
u
2
+
2240
277
u
702
277
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.46632 2.74126I
a = 0.422265 0.273313I
b = 2.01513 0.33917I
11.0685 10.4672I 6.45679 + 5.97165I
u = 0.46632 + 2.74126I
a = 0.422265 + 0.273313I
b = 2.01513 + 0.33917I
11.0685 + 10.4672I 6.45679 5.97165I
u = 0.158515
a = 5.69902
b = 1.64063
3.61413 1.15355
u = 0.260920 0.655876I
a = 0.614097 + 0.651143I
b = 0.023311 0.507189I
0.184850 + 1.357363I 2.08591 4.58406I
u = 0.260920 + 0.655876I
a = 0.614097 0.651143I
b = 0.023311 + 0.507189I
0.184850 1.357363I 2.08591 + 4.58406I
u = 0.626141 1.116013I
a = 0.885874 0.284512I
b = 0.718123 0.224799I
4.21141 3.35522I 7.88053 + 3.75965I
u = 0.626141 + 1.116013I
a = 0.885874 + 0.284512I
b = 0.718123 + 0.224799I
4.21141 + 3.35522I 7.88053 3.75965I
7
IV. I
u
4
=
hu
14
+u
13
+· · ·+2u +1, 1.21×10
12
u
13
+3.53× 10
12
u
12
+· · ·+5.11 ×10
13
b +
8.80× 10
13
, 4.05×10
13
u
13
5.17 × 10
13
u
12
+· · · +5.11 × 10
13
a 3.79 × 10
13
i
(i) Arc colorings
a
2
=
1
0
a
9
=
0.792973u
13
+ 1.01096u
12
+ ··· + 26.6378u + 0.740461
0.0235689u
13
0.0691116u
12
+ ··· 0.915315u 1.72106
a
5
=
0
u
a
10
=
0.769404u
13
+ 1.08007u
12
+ ··· + 27.5531u + 2.46153
0.0235689u
13
0.0691116u
12
+ ··· 0.915315u 1.72106
a
8
=
0.792973u
13
+ 1.01096u
12
+ ··· + 26.6378u + 0.740461
0.166089u
13
0.178205u
12
+ ··· 2.14426u 1.93905
a
6
=
1.31222u
13
1.03317u
12
+ ··· 26.4352u 0.994374
0.128536u
13
+ 0.0552761u
12
+ ··· + 3.66944u + 1.44202
a
1
=
1.44202u
13
+ 1.57055u
12
+ ··· + 44.8024u + 1.21459
0.274408u
13
0.288127u
12
+ ··· 6.04501u 3.23784
a
4
=
0.626884u
13
0.832752u
12
+ ··· 24.4935u + 1.19859
0.184867u
13
+ 0.262200u
12
+ ··· + 6.69113u + 1.58682
a
3
=
0.829049u
13
0.665834u
12
+ ··· 22.3564u + 8.44585
0.585885u
13
+ 0.660074u
12
+ ··· + 17.9742u + 1.02851
a
7
=
0.626884u
13
+ 0.832752u
12
+ ··· + 24.4935u 1.19859
0.166089u
13
0.178205u
12
+ ··· 2.14426u 1.93905
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
3838300796474
25566275354701
u
13
+
3851477719173
25566275354701
u
12
+ ··· +
391630963996580
25566275354701
u
108530281491461
25566275354701
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.684697 0.025265I
a = 0.86141 + 1.42487I
b = 0.306521 0.195372I
1.11654 3.28492I 6.60141 + 2.44171I
u = 0.684697 + 0.025265I
a = 0.86141 1.42487I
b = 0.306521 + 0.195372I
1.11654 + 3.28492I 6.60141 2.44171I
u = 0.51639 2.58562I
a = 0.468436 0.254302I
b = 1.75993 0.38141I
12.1121 7.77053
u = 0.51639 + 2.58562I
a = 0.468436 + 0.254302I
b = 1.75993 + 0.38141I
12.1121 7.77053
u = 0.46039 1.77594I
a = 0.524911 + 0.031807I
b = 0.83720 1.20669I
1.11654 + 3.28492I 6.60141 2.44171I
u = 0.46039 + 1.77594I
a = 0.524911 0.031807I
b = 0.83720 + 1.20669I
1.11654 3.28492I 6.60141 + 2.44171I
u = 0.026394 0.197164I
a = 0.09255 5.01567I
b = 1.73502 + 0.09983I
7.46645 + 4.93043I 4.23989 2.98386I
u = 0.026394 + 0.197164I
a = 0.09255 + 5.01567I
b = 1.73502 0.09983I
7.46645 4.93043I 4.23989 + 2.98386I
u = 0.251357 0.560891I
a = 0.670116 + 1.116089I
b = 0.000704 0.384496I
0.165382 + 1.372838I 2.77344 4.48022I
u = 0.251357 + 0.560891I
a = 0.670116 1.116089I
b = 0.000704 + 0.384496I
0.165382 1.372838I 2.77344 + 4.48022I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.43046 2.68133I
a = 0.433855 0.248841I
b = 1.92690 0.44916I
7.46645 + 4.93043I 4.23989 2.98386I
u = 0.43046 + 2.68133I
a = 0.433855 + 0.248841I
b = 1.92690 + 0.44916I
7.46645 4.93043I 4.23989 + 2.98386I
u = 0.506054 0.754738I
a = 0.551357 + 0.355463I
b = 0.038076 0.785867I
0.165382 + 1.372838I 2.77344 4.48022I
u = 0.506054 + 0.754738I
a = 0.551357 0.355463I
b = 0.038076 + 0.785867I
0.165382 1.372838I 2.77344 + 4.48022I
10
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
7
u
4
(u
7
2u
6
3u
5
+ 8u
4
2u
3
2u
2
u + 2)
2
(u
7
+ 5u
6
+ 10u
5
+ 13u
4
+ 18u
3
+ 20u
2
+ 12u + 4)
c
2
, c
4
(u
2
u + 1)
2
(u
7
+ 5u
6
+ 11u
5
+ 10u
4
u
3
8u
2
4u 1)
(u
14
+ 8u
13
+ ··· + 9u + 1)
c
3
, c
8
(u
2
u + 1)
2
(u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ 1)
(u
14
+ 2u
13
+ ··· + u + 1)
c
5
, c
10
(u
2
+ u + 1)
2
(u
7
+ u
6
+ 3u
5
+ 2u
4
+ 3u
3
+ 2u
2
+ 1)
(u
14
+ 2u
13
+ ··· + u + 1)
c
6
, c
9
(u
2
u + 1)
2
(u
7
+ u
6
5u
5
2u
4
+ 7u
3
4u
2
+ 2u 1)
(u
14
+ 2u
13
+ ··· + 5u + 1)
11
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
y
4
(y
7
10y
6
+ 37y
5
62y
4
+ 50y
3
32y
2
+ 9y 4)
2
(y
7
5y
6
+ 6y
5
+ 15y
4
+ 4y
3
72y
2
16y 16)
c
2
, c
4
(y
2
+ y + 1)
2
(y
7
3y
6
+ 19y
5
50y
4
+ 83y
3
36y
2
1)
(y
14
4y
13
+ ··· 15y + 1)
c
3
, c
5
, c
8
c
10
(y
2
+ y + 1)
2
(y
7
+ 5y
6
+ 11y
5
+ 10y
4
y
3
8y
2
4y 1)
(y
14
+ 8y
13
+ ··· + 9y + 1)
c
6
, c
9
(y
2
+ y + 1)
2
(y
7
11y
6
+ 43y
5
62y
4
+ 15y
3
+ 8y
2
4y 1)
(y
14
16y
13
+ ··· + 9y + 1)
12