10
164
(K10n
38
)
1
Arc Sequences
6 8 7 8 3 1 10 6 3 4
Solving Sequence
7,10 4,8
1 3 6 2 5 9
c
7
c
10
c
3
c
6
c
1
c
5
c
9
c
2
, c
4
, c
8
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hu
6
+ u
4
u
3
+ 3u
2
u + 1, b + u, 3u
5
+ u
4
+ 2u
3
3u
2
+ 2a + 8u + 1i
I
u
2
= hu
12
+ u
9
+ 6u
8
+ u
7
u
6
+ 2u
5
+ 5u
4
2u
3
+ 2u + 1, b u, 138u
11
105u
10
+ ··· + 142a + 155i
I
u
3
= hu
16
3u
15
+ ··· + 4u + 1, 1670834153u
15
+ 5539576302u
14
+ ··· + 864094501a + 498705292,
22976741298u
15
+ 77906464811u
14
+ ··· + 11233228513b 21004036137i
There are 3 irreducible components with 34 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
u
3
+ 3u
2
u + 1, b + u, 3u
5
+ u
4
+ 2u
3
3u
2
+ 2a + 8u + 1i
(i) Arc colorings
a
7
=
0
u
a
10
=
3
2
u
5
1
2
u
4
+ ··· 4u
1
2
u
a
4
=
1
0
a
8
=
u
5
2u
4
u
3
u 4
1
2
u
5
1
2
u
4
+
1
2
u
2
1
2
a
1
=
3
2
u
5
1
2
u
4
+ ··· 3u
1
2
u
a
3
=
1
u
2
a
6
=
u
4
u
2
+ u 3
1
2
u
5
1
2
u
4
u
3
+
1
2
u
2
1
2
a
2
=
1
2
u
5
3
2
u
4
3
2
u
2
+ 3u
7
2
1
2
u
5
+
1
2
u
4
+ ··· + u
1
2
a
5
=
1
2
u
5
3
2
u
4
u
3
1
2
u
2
5
2
u
4
u
3
u
2
1
a
9
=
2u
5
u
4
u
3
+ 2u
2
4u 1
1
2
u
5
1
2
u
4
+ ··· u
1
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 7u
5
+ 5u
4
+ 6u
3
3u
2
+ 11u + 10
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.87236 1.13524I
a = 0.102208 + 0.744112I
b = 0.87236 + 1.13524I
4.06966 4.74338I 5.65982 + 6.07362I
u = 0.87236 + 1.13524I
a = 0.102208 0.744112I
b = 0.87236 1.13524I
4.06966 + 4.74338I 5.65982 6.07362I
u = 0.125253 0.619808I
a = 1.59453 + 2.06251I
b = 0.125253 + 0.619808I
2.50509 + 1.44331I 12.78155 4.91052I
u = 0.125253 + 0.619808I
a = 1.59453 2.06251I
b = 0.125253 0.619808I
2.50509 1.44331I 12.78155 + 4.91052I
u = 0.747107 0.813589I
a = 0.507683 + 0.055394I
b = 0.747107 + 0.813589I
4.92982 2.38212I 1.44137 + 0.69060I
u = 0.747107 + 0.813589I
a = 0.507683 0.055394I
b = 0.747107 0.813589I
4.92982 + 2.38212I 1.44137 0.69060I
3
II. I
u
2
= hu
12
+ u
9
+ · · · + 2u + 1, b u, 138u
11
105u
10
+ · · · + 142a + 155i
(i) Arc colorings
a
7
=
0
u
a
10
=
0.971831u
11
+ 0.739437u
10
+ ··· 2.60563u 1.09155
u
a
4
=
1
0
a
8
=
0.517606u
11
+ 0.0246479u
10
+ ··· 0.00352113u + 2.13028
0.464789u
11
+ 0.0492958u
10
+ ··· + 0.492958u 0.739437
a
1
=
0.971831u
11
+ 0.739437u
10
+ ··· 3.60563u 1.09155
u
a
3
=
1
u
2
a
6
=
1.44718u
11
0.0739437u
10
+ ··· + 1.01056u + 3.60915
0.464789u
11
+ 0.0492958u
10
+ ··· + 0.492958u 0.739437
a
2
=
0.0422535u
11
0.359155u
10
+ ··· + 0.408451u + 0.387324
0.468310u
11
+ 0.144366u
10
+ ··· 0.806338u 0.665493
a
5
=
0.978873u
11
+ 0.0704225u
10
+ ··· + 0.204225u + 2.94366
0.0669014u
11
0.193662u
10
+ ··· + 0.313380u 0.595070
a
9
=
0.507042u
11
+ 0.690141u
10
+ ··· 3.09859u 0.352113
0.169014u
11
0.0633803u
10
+ ··· + 1.36620u 0.0492958
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
72
71
u
11
+
169
71
u
10
19
71
u
9
56
71
u
8
340
71
u
7
+
943
71
u
6
+
100
71
u
5
288
71
u
4
394
71
u
3
+
984
71
u
2
369
71
u
192
71
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.925706 1.050553I
a = 0.084968 0.559096I
b = 0.925706 1.050553I
3.38867 4.08003I 1.46265 + 0.78652I
u = 0.925706 + 1.050553I
a = 0.084968 + 0.559096I
b = 0.925706 + 1.050553I
3.38867 + 4.08003I 1.46265 0.78652I
u = 0.894529 0.606911I
a = 0.05396 1.57242I
b = 0.894529 0.606911I
7.98844 5.04592I 3.50212 + 4.93530I
u = 0.894529 + 0.606911I
a = 0.05396 + 1.57242I
b = 0.894529 + 0.606911I
7.98844 + 5.04592I 3.50212 4.93530I
u = 0.444254 0.260304I
a = 0.66941 + 1.85880I
b = 0.444254 0.260304I
1.58084 1.46904I 1.29817 + 5.01402I
u = 0.444254 + 0.260304I
a = 0.66941 1.85880I
b = 0.444254 + 0.260304I
1.58084 + 1.46904I 1.29817 5.01402I
u = 0.433167 0.820343I
a = 1.34256 1.09793I
b = 0.433167 0.820343I
2.63922 + 4.58392I 1.89423 6.22117I
u = 0.433167 + 0.820343I
a = 1.34256 + 1.09793I
b = 0.433167 + 0.820343I
2.63922 4.58392I 1.89423 + 6.22117I
u = 0.727666 0.459131I
a = 0.041129 0.695889I
b = 0.727666 0.459131I
1.29616 + 0.86105I 4.70470 1.78151I
u = 0.727666 + 0.459131I
a = 0.041129 + 0.695889I
b = 0.727666 + 0.459131I
1.29616 0.86105I 4.70470 + 1.78151I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.10366 1.16882I
a = 0.163031 0.926828I
b = 1.10366 1.16882I
4.56023 + 12.50669I 0.52291 6.78913I
u = 1.10366 + 1.16882I
a = 0.163031 + 0.926828I
b = 1.10366 + 1.16882I
4.56023 12.50669I 0.52291 + 6.78913I
6
III.
I
u
3
= hu
16
3u
15
+· · ·+4u+1, 1.67 ×10
9
u
15
+5.54×10
9
u
14
+· · ·+8.64×10
8
a+
4.99 × 10
8
, 2.30 × 10
10
u
15
+ 7.79 × 10
10
u
14
+ · · · + 1.12 × 10
10
b 2.10 × 10
10
i
(i) Arc colorings
a
7
=
0
u
a
10
=
1.93362u
15
6.41085u
14
+ ··· + 11.5996u 0.577142
2.04543u
15
6.93536u
14
+ ··· + 10.9045u + 1.86981
a
4
=
1
0
a
8
=
2.50110u
15
6.52849u
14
+ ··· + 28.7759u + 8.29535
0.161484u
15
+ 0.0864436u
14
+ ··· + 4.46296u + 2.07506
a
1
=
0.111802u
15
+ 0.524513u
14
+ ··· + 0.695051u 2.44696
2.04543u
15
6.93536u
14
+ ··· + 10.9045u + 1.86981
a
3
=
1
u
2
a
6
=
0.888198u
15
2.47549u
14
+ ··· + 12.6951u + 1.55304
1.45142u
15
4.13944u
14
+ ··· + 13.6179u + 4.66725
a
2
=
0.888198u
15
2.47549u
14
+ ··· + 12.6951u + 1.55304
1.34644u
15
4.00323u
14
+ ··· + 11.9733u + 4.47814
a
5
=
2.23464u
15
6.47872u
14
+ ··· + 24.6683u + 6.03118
1.68993u
15
4.95565u
14
+ ··· + 15.1088u + 4.70335
a
9
=
0.0729519u
15
+ 0.441341u
14
+ ··· + 0.188785u 3.05693
2.21877u
15
7.30458u
14
+ ··· + 12.2278u + 2.70227
(ii) Obstruction class = 1
(iii) Cusp Shapes
=
62712632292
11233228513
u
15
+
222551627880
11233228513
u
14
+ ···
386527091664
11233228513
u
36877222054
11233228513
7
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.804589 0.808792I
a = 0.039716 + 1.092695I
b = 0.88699 + 1.31736I
1.85594 5.19385I 0.17326 + 6.02890I
u = 0.804589 + 0.808792I
a = 0.039716 1.092695I
b = 0.88699 1.31736I
1.85594 + 5.19385I 0.17326 6.02890I
u = 0.44895 1.60911I
a = 0.471928 0.086965I
b = 1.051694 0.235939I
5.14581 0.61478I 3.82674 1.44464I
u = 0.44895 + 1.60911I
a = 0.471928 + 0.086965I
b = 1.051694 + 0.235939I
5.14581 + 0.61478I 3.82674 + 1.44464I
u = 0.321200 0.647019I
a = 0.60564 + 1.61717I
b = 0.160429 + 0.464095I
1.85594 1.13408I 0.173262 0.899303I
u = 0.321200 + 0.647019I
a = 0.60564 1.61717I
b = 0.160429 0.464095I
1.85594 + 1.13408I 0.173262 + 0.899303I
u = 0.311042 0.310121I
a = 0.59074 1.72691I
b = 1.60753 1.13440I
5.14581 3.44499I 3.82674 + 8.37284I
u = 0.311042 + 0.310121I
a = 0.59074 + 1.72691I
b = 1.60753 + 1.13440I
5.14581 + 3.44499I 3.82674 8.37284I
u = 0.160429 0.464095I
a = 2.32074 + 1.03318I
b = 0.321200 + 0.647019I
1.85594 + 1.13408I 0.173262 + 0.899303I
u = 0.160429 + 0.464095I
a = 2.32074 1.03318I
b = 0.321200 0.647019I
1.85594 1.13408I 0.173262 0.899303I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.88699 1.31736I
a = 0.176417 + 0.765387I
b = 0.804589 + 0.808792I
1.85594 + 5.19385I 0.17326 6.02890I
u = 0.88699 + 1.31736I
a = 0.176417 0.765387I
b = 0.804589 0.808792I
1.85594 5.19385I 0.17326 + 6.02890I
u = 1.051694 0.235939I
a = 0.172190 0.723564I
b = 0.44895 1.60911I
5.14581 0.61478I 3.82674 1.44464I
u = 1.051694 + 0.235939I
a = 0.172190 + 0.723564I
b = 0.44895 + 1.60911I
5.14581 + 0.61478I 3.82674 + 1.44464I
u = 1.60753 1.13440I
a = 0.357193 + 0.196042I
b = 0.311042 0.310121I
5.14581 3.44499I 3.82674 + 8.37284I
u = 1.60753 + 1.13440I
a = 0.357193 0.196042I
b = 0.311042 + 0.310121I
5.14581 + 3.44499I 3.82674 8.37284I
9
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
u + 1)
8
(u
6
u
5
+ ··· + 3u
2
+ 2)(u
12
+ 8u
11
+ ··· + 96u + 16)
c
2
, c
8
(u
6
u
5
+ ··· + u + 1)(u
12
+ u
11
+ ··· + 2u + 1)
(u
16
u
15
+ ··· + 48u + 19)
c
3
, c
10
(u
6
+ u
4
u
3
+ 3u
2
u + 1)
(u
12
u
9
+ 6u
8
u
7
u
6
2u
5
+ 5u
4
+ 2u
3
2u + 1)
(u
16
+ 3u
15
+ ··· 4u + 1)
c
4
, c
9
(u
6
u
5
+ ··· + 2u + 2)(u
12
u
11
+ ··· + 4u
2
+ 2)
(u
16
+ u
15
+ ··· 6u + 1)
c
5
(u
4
+ 3u
3
+ u
2
2u + 1)
4
(u
6
+ 4u
5
+ 6u
4
+ 8u
3
+ 10u
2
+ 4u + 1)
(u
12
9u
11
+ ··· 20u + 16)
c
6
(u
2
u + 1)
8
(u
6
+ u
5
+ ··· + 3u
2
+ 2)(u
12
+ 8u
11
+ ··· + 96u + 16)
c
7
(u
4
u
3
+ u
2
+ 1)
4
(u
6
+ 3u
5
+ 5u
4
+ 3u
3
+ u
2
+ 1)
(u
12
+ 8u
11
+ ··· + 22u + 4)
10
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
2
+ y + 1)
8
(y
6
+ 5y
5
+ 13y
4
+ 21y
3
+ 21y
2
+ 12y + 4)
(y
12
+ 6y
11
+ ··· 640y + 256)
c
2
, c
8
(y
6
+ 3y
5
+ ··· 5y + 1)(y
12
+ 17y
11
+ ··· + 6y + 1)
(y
16
+ 15y
15
+ ··· + 2332y + 361)
c
3
, c
10
(y
6
+ 2y
5
+ ··· + 5y + 1)(y
12
+ 12y
10
+ ··· 4y + 1)
(y
16
+ 3y
15
+ ··· + 8y + 1)
c
4
, c
9
(y
6
+ 3y
5
+ ··· 8y + 4)(y
12
+ 5y
11
+ ··· + 16y + 4)
(y
16
+ 7y
15
+ ··· + 134y + 1)
c
5
(1 2y + 15y
2
7y
3
+ y
4
)
4
(y
6
4y
5
+ ··· + 4y + 1)
(y
12
11y
11
+ ··· + 80y + 256)
c
7
(y
4
+ y
3
+ 3y
2
+ 2y + 1)
4
(y
6
+ y
5
+ 9y
4
+ 3y
3
+ 11y
2
+ 2y + 1)
(y
12
+ 2y
11
+ ··· + 84y + 16)
11