10
97
(K10a
12
)
1
Arc Sequences
6 9 7 3 10 4 1 5 2 8
Solving Sequence
1,7 4,8
3 5 6 2 10 9
c
7
c
3
c
4
c
6
c
1
c
10
c
9
c
2
, c
5
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= h4u
2
2u + 7, b + 1, 21a + 8u + 10i
I
u
2
= h4u
17
+ 10u
16
+ ··· + 8u + 1, 6.30878 × 10
21
u
16
+ 1.29423 × 10
22
u
15
+ ··· + 2.07124 × 10
21
b + 3.73078 × 10
21
,
8.35403 × 10
21
u
16
+ 1.68001 × 10
22
u
15
+ ··· + 2.07124 × 10
21
a + 3.93060 × 10
20
i
I
u
3
= hu
28
7u
27
+ ··· 1536u + 181,
1.82876 × 10
68
u
27
1.21857 × 10
69
u
26
+ ··· + 3.01713 × 10
69
b 8.69349 × 10
70
,
5.30558 × 10
70
u
27
+ 3.52385 × 10
71
u
26
+ ··· + 5.46100 × 10
71
a + 2.69444 × 10
73
i
There are 3 irreducible components with 47 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
= h4u
2
2u + 7, b + 1, 21a + 8u + 10i
(i) Arc colorings
a
1
=
1
0
a
7
=
8
21
u
10
21
1
a
4
=
0
u
a
8
=
8
21
u +
11
21
1
a
3
=
4
21
u +
16
21
1
3
u +
2
3
a
5
=
8
21
u
10
21
1
3
u +
1
6
a
6
=
8
21
u
10
21
1
3
u +
1
6
a
2
=
2
7
u +
8
7
1
6
u
1
6
a
10
=
8
21
u +
32
21
1
a
9
=
2
21
u +
8
21
1
6
u
5
6
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1
6
u
61
6
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.250000 1.299038I
a = 0.571429 + 0.494872I
b = 1.00000
1.64493 + 2.02988I 10.12500 0.21651I
u = 0.250000 + 1.299038I
a = 0.571429 0.494872I
b = 1.00000
1.64493 2.02988I 10.12500 + 0.21651I
3
II. I
u
2
=
h4u
17
+10u
16
+· · ·+8u+1, 6.31×10
21
u
16
+1.29×10
22
u
15
+· · ·+2.07×10
21
b+
3.73 × 10
21
, 8.35 × 10
21
u
16
+ 1.68 × 10
22
u
15
+ · · · + 2.07 × 10
21
a + 3.93 × 10
20
i
(i) Arc colorings
a
1
=
1
0
a
7
=
4.03334u
16
8.11109u
15
+ ··· 20.7218u 0.189770
3.04589u
16
6.24854u
15
+ ··· 13.3678u 1.80123
a
4
=
0
u
a
8
=
0.987453u
16
1.86255u
15
+ ··· 7.35404u + 1.61146
3.04589u
16
6.24854u
15
+ ··· 13.3678u 1.80123
a
3
=
1.79623u
16
+ 6.42424u
15
+ ··· + 23.2916u + 6.11933
1.88904u
16
+ 4.86116u
15
+ ··· + 16.0695u + 3.43817
a
5
=
23.6377u
16
+ 54.6701u
15
+ ··· + 94.6543u + 18.6343
10.4065u
16
+ 23.3976u
15
+ ··· + 47.8604u + 8.11492
a
6
=
4.03334u
16
8.11109u
15
+ ··· 20.7218u 0.189770
4.11183u
16
8.52313u
15
+ ··· 16.3040u 2.29429
a
2
=
11.8190u
16
+ 28.5106u
15
+ ··· + 51.0010u + 13.8850
8.93179u
16
+ 20.5994u
15
+ ··· + 39.6256u + 8.98563
a
10
=
4.19505u
16
10.7455u
15
+ ··· 13.3615u 3.18180
9.55765u
16
21.7472u
15
+ ··· 40.1665u 8.25412
a
9
=
30.8652u
16
+ 66.9842u
15
+ ··· + 133.258u + 23.7988
5.01280u
16
+ 9.77738u
15
+ ··· + 18.1671u + 1.67252
(ii) Obstruction class = 1
(iii) Cusp Shapes =
100770932482863059622864
2071244780912044787317
u
16
223891062239633335685016
2071244780912044787317
u
15
+ ···
406918332284594290206608
2071244780912044787317
u
84787820149178858531002
2071244780912044787317
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.18564 2.16518I
a = 0.540925 0.048152I
b = 0.53774 1.38258I
7.7059 + 14.8527I 2.01529 8.44038I
u = 1.18564 + 2.16518I
a = 0.540925 + 0.048152I
b = 0.53774 + 1.38258I
7.7059 14.8527I 2.01529 + 8.44038I
u = 0.778762 1.045725I
a = 1.130820 0.003261I
b = 0.161092 + 1.190926I
5.81019 + 5.32225I 2.45956 7.34338I
u = 0.778762 + 1.045725I
a = 1.130820 + 0.003261I
b = 0.161092 1.190926I
5.81019 5.32225I 2.45956 + 7.34338I
u = 0.34831 1.52104I
a = 0.457588 0.497688I
b = 1.335871 + 0.125893I
1.15632 2.35456I 2.48228 + 6.50501I
u = 0.34831 + 1.52104I
a = 0.457588 + 0.497688I
b = 1.335871 0.125893I
1.15632 + 2.35456I 2.48228 6.50501I
u = 0.293900 0.090615I
a = 2.47276 2.78652I
b = 0.44756 1.37873I
9.32990 + 8.56729I 0.17143 4.34513I
u = 0.293900 + 0.090615I
a = 2.47276 + 2.78652I
b = 0.44756 + 1.37873I
9.32990 8.56729I 0.17143 + 4.34513I
u = 0.271190
a = 1.01411
b = 0.531228
0.869406 11.1450
u = 0.02898 1.67997I
a = 0.531083 0.150112I
b = 0.840094 0.523489I
2.94308 1.91475I 12.50863 + 1.23884I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.02898 + 1.67997I
a = 0.531083 + 0.150112I
b = 0.840094 + 0.523489I
2.94308 + 1.91475I 12.50863 1.23884I
u = 0.102562 0.302615I
a = 2.64218 + 3.06507I
b = 0.047500 + 1.229644I
7.90214 1.97950I 6.13742 + 2.92595I
u = 0.102562 + 0.302615I
a = 2.64218 3.06507I
b = 0.047500 1.229644I
7.90214 + 1.97950I 6.13742 2.92595I
u = 0.137893 0.691342I
a = 1.347081 + 0.334810I
b = 0.222604 0.163997I
0.34103 1.75255I 2.16634 + 2.85736I
u = 0.137893 + 0.691342I
a = 1.347081 0.334810I
b = 0.222604 + 0.163997I
0.34103 + 1.75255I 2.16634 2.85736I
u = 1.28073 1.02818I
a = 0.609497 0.561008I
b = 0.347263 + 1.122362I
1.26847 6.54787I 3.86293 + 7.90993I
u = 1.28073 + 1.02818I
a = 0.609497 + 0.561008I
b = 0.347263 1.122362I
1.26847 + 6.54787I 3.86293 7.90993I
6
III. I
u
3
= hu
28
7u
27
+ · · · 1536u + 181, 1.83 × 10
68
u
27
1.22 × 10
69
u
26
+
· · · + 3.02 × 10
69
b 8.69 × 10
70
, 5.31 × 10
70
u
27
+ 3.52 × 10
71
u
26
+ · · · +
5.46 × 10
71
a + 2.69 × 10
73
i
(i) Arc colorings
a
1
=
1
0
a
7
=
0.0971540u
27
0.645276u
26
+ ··· + 293.200u 49.3396
0.0606125u
27
+ 0.403883u
26
+ ··· 170.263u + 28.8138
a
4
=
0
u
a
8
=
0.157766u
27
1.04916u
26
+ ··· + 463.463u 78.1534
0.0606125u
27
+ 0.403883u
26
+ ··· 170.263u + 28.8138
a
3
=
0.145588u
27
0.962635u
26
+ ··· + 405.369u 65.4714
0.0817127u
27
+ 0.535691u
26
+ ··· 201.061u + 32.7618
a
5
=
0.0587091u
27
+ 0.392847u
26
+ ··· 192.785u + 35.6675
0.0369442u
27
0.247182u
26
+ ··· + 115.799u 19.7141
a
6
=
0.0971540u
27
0.645276u
26
+ ··· + 293.200u 49.3396
0.0474153u
27
+ 0.316626u
26
+ ··· 134.392u + 22.5146
a
2
=
0.124522u
27
0.811945u
26
+ ··· + 301.462u 48.6103
0.00810095u
27
+ 0.0566235u
26
+ ··· 37.6228u + 5.61253
a
10
=
0.253437u
27
+ 1.66276u
26
+ ··· 659.161u + 108.172
0.0724330u
27
0.477447u
26
+ ··· + 199.547u 31.2100
a
9
=
0.0139725u
27
0.103776u
26
+ ··· + 73.7744u 16.1551
0.0162406u
27
+ 0.111152u
26
+ ··· 57.3670u + 11.1272
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.153239u
27
1.01293u
26
+ ··· + 399.774u 66.2933
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.32770 0.73882I
a = 0.373478 0.603187I
b = 0.426047 + 1.000289I
1.35286 + 2.76747I 9.41762 3.21377I
u = 1.32770 + 0.73882I
a = 0.373478 + 0.603187I
b = 0.426047 1.000289I
1.35286 2.76747I 9.41762 + 3.21377I
u = 1.01050 1.86446I
a = 0.520349 0.108187I
b = 0.64484 1.35997I
7.93259 + 2.76747I 1.41762 3.21377I
u = 1.01050 + 1.86446I
a = 0.520349 + 0.108187I
b = 0.64484 + 1.35997I
7.93259 2.76747I 1.41762 + 3.21377I
u = 0.985395 0.376228I
a = 0.889621 0.676242I
b = 0.340282 + 0.137082I
2.09958 + 3.41271I 6.10600 2.62516I
u = 0.985395 + 0.376228I
a = 0.889621 + 0.676242I
b = 0.340282 0.137082I
2.09958 3.41271I 6.10600 + 2.62516I
u = 0.049737 0.918300I
a = 0.725699 0.801705I
b = 1.027093 + 0.175615I
4.48016 3.41271I 1.89400 + 2.62516I
u = 0.049737 + 0.918300I
a = 0.725699 + 0.801705I
b = 1.027093 0.175615I
4.48016 + 3.41271I 1.89400 2.62516I
u = 0.005849 0.295784I
a = 2.06578 2.65187I
b = 0.41210 1.42136I
4.48016 3.41271I 1.89400 + 2.62516I
u = 0.005849 + 0.295784I
a = 2.06578 + 2.65187I
b = 0.41210 + 1.42136I
4.48016 + 3.41271I 1.89400 2.62516I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.00900 1.69198I
a = 0.603983 0.407304I
b = 1.148813 0.016311I
3.28987 + 8.93586I 4.00000 7.26077I
u = 0.00900 + 1.69198I
a = 0.603983 + 0.407304I
b = 1.148813 + 0.016311I
3.28987 8.93586I 4.00000 + 7.26077I
u = 0.34747 1.48466I
a = 0.695117 0.128859I
b = 0.637817 0.252286I
1.35286 2.76747I 9.41762 + 3.21377I
u = 0.34747 + 1.48466I
a = 0.695117 + 0.128859I
b = 0.637817 + 0.252286I
1.35286 + 2.76747I 9.41762 3.21377I
u = 0.401323 0.038356I
a = 1.59325 2.29730I
b = 0.51211 + 1.46812I
7.93259 + 2.76747I 1.41762 3.21377I
u = 0.401323 + 0.038356I
a = 1.59325 + 2.29730I
b = 0.51211 1.46812I
7.93259 2.76747I 1.41762 + 3.21377I
u = 0.695185 0.086492I
a = 0.812530 0.552667I
b = 0.015745 1.176089I
3.31269 1.37770I 3.11410 + 4.12207I
u = 0.695185 + 0.086492I
a = 0.812530 + 0.552667I
b = 0.015745 + 1.176089I
3.31269 + 1.37770I 3.11410 4.12207I
u = 0.76062 1.91607I
a = 0.562695 + 0.101358I
b = 0.136381 + 1.104827I
2.09958 3.41271I 6.10600 + 2.62516I
u = 0.76062 + 1.91607I
a = 0.562695 0.101358I
b = 0.136381 1.104827I
2.09958 + 3.41271I 6.10600 2.62516I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.030601 0.780526I
a = 0.106586 + 0.521711I
b = 0.212363 0.520130I
3.31269 + 1.37770I 3.11410 4.12207I
u = 1.030601 + 0.780526I
a = 0.106586 0.521711I
b = 0.212363 + 0.520130I
3.31269 1.37770I 3.11410 + 4.12207I
u = 1.05218 1.83263I
a = 0.156030 + 0.359169I
b = 0.053811 0.680241I
3.26705 + 1.37770I 4.88590 4.12207I
u = 1.05218 + 1.83263I
a = 0.156030 0.359169I
b = 0.053811 + 0.680241I
3.26705 1.37770I 4.88590 + 4.12207I
u = 1.06267 2.08438I
a = 0.523262 0.061247I
b = 0.56444 1.41873I
3.28987 8.93586I 4.00000 + 7.26077I
u = 1.06267 + 2.08438I
a = 0.523262 + 0.061247I
b = 0.56444 + 1.41873I
3.28987 + 8.93586I 4.00000 7.26077I
u = 1.50844 0.14479I
a = 0.282676 + 0.467229I
b = 0.006983 + 1.150228I
3.26705 + 1.37770I 4.88590 4.12207I
u = 1.50844 + 0.14479I
a = 0.282676 0.467229I
b = 0.006983 1.150228I
3.26705 1.37770I 4.88590 + 4.12207I
10
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(4u
2
2u + 1)(4u
17
2u
16
+ ··· u
2
1)
(u
28
+ 3u
27
+ ··· + 1254u + 653)
c
2
, c
10
(u + 1)
2
(u
17
2u
16
+ ··· 2u 1)(u
28
+ 5u
27
+ ··· + 2u + 1)
c
3
(u
2
+ u + 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
17
+ 3u
15
+ ··· + 17u + 4)
c
4
(u
2
+ u + 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
17
+ 6u
16
+ ··· + 145u 16)
c
5
u
2
(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
17
+ 3u
16
+ ··· 24u 32)
c
6
(u
2
u + 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
17
+ 3u
15
+ ··· + 17u + 4)
c
7
, c
9
(u 1)
2
(u
17
2u
16
+ ··· 2u 1)(u
28
+ 5u
27
+ ··· + 2u + 1)
c
8
(4u
2
+ 2u + 1)(4u
17
2u
16
+ ··· u
2
1)
(u
28
+ 3u
27
+ ··· + 1254u + 653)
11
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
8
(16y
2
+ 4y + 1)(16y
17
+ 132y
16
+ ··· 2y 1)
(y
28
+ 15y
27
+ ··· + 3659320y + 426409)
c
2
, c
7
, c
9
c
10
(y 1)
2
(y
17
+ 10y
16
+ ··· + 8y 1)(y
28
+ 19y
27
+ ··· 10y
2
+ 1)
c
3
, c
6
(y
2
+ y + 1)
(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
17
+ 6y
16
+ ··· + 145y 16)
c
4
(y
2
+ y + 1)
(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
17
+ 10y
16
+ ··· + 44449y 256)
c
5
1y
2
(1.00 + 3.00y + 12.0y
2
+ 28.0y
3
+ 58.0y
4
+ 94.0y
5
+ 126.y
6
+ 144.y
7
+ 137.y
8
+ 111.y
9
+ 74.0y
10
+ 40.0y
11
+ 17.0y
12
+ 5.00y
13
+ 1.00y
14
)
2
(1y
17
+ 5.00000000y
16
+ ··· 6976.00000y 1024.000000)
12