10
114
(K10a
77
)
1
Arc Sequences
5 6 8 7 9 10 4 1 2 3
Solving Sequence
3,8 4,10
1 9 7 5 6 2
c
3
c
10
c
8
c
7
c
4
c
6
c
2
c
1
, c
5
, c
9
Representation Ideals
I =
3
\
i=1
I
u
i
\
I
v
1
I
u
1
= hu
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 11u
4
+ 9u
3
+ 7u
2
+ 2u + 1, u
5
+ 2u
4
+ 4u
3
+ 4u
2
+ b + 3u + 1,
u
7
2u
6
5u
5
5u
4
5u
3
u
2
+ a + 2i
I
u
2
= hb
30
b
29
+ ··· 6b 11, 4.83325 × 10
39
u + 4.53585 × 10
39
b
29
+ ··· + 1.22874 × 10
41
b + 6.97154 × 10
40
,
1.12547 × 10
40
b
29
+ 3.10447 × 10
39
b
28
+ ··· + 4.83325 × 10
39
a 1.74419 × 10
41
i
I
u
3
= hu
23
5u
22
+ ··· 46u + 7, 149u
22
622u
21
+ ··· + 293b 2738,
1695u
22
+ 8293u
21
+ ··· + 2051a + 795i
I
v
1
= hb + 1, v 1, ai
There are 4 irreducible components with 62 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
8
+ 2u
7
+ · · · + 2u + 1, u
5
+ 2u
4
+ 4u
3
+ 4u
2
+ b + 3u + 1, u
7
2u
6
5u
5
5u
4
5u
3
u
2
+ a + 2i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
10
=
u
7
+ 2u
6
+ 5u
5
+ 5u
4
+ 5u
3
+ u
2
2
u
5
2u
4
4u
3
4u
2
3u 1
a
1
=
u
7
+ 2u
6
+ 6u
5
+ 7u
4
+ 9u
3
+ 5u
2
+ 3u 1
u
5
2u
4
4u
3
4u
2
3u 1
a
9
=
u
7
2u
6
5u
5
7u
4
8u
3
7u
2
5u 2
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u + 1
a
7
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
2u
2
a
6
=
u
7
2u
6
6u
5
8u
4
11u
3
8u
2
6u
u
3
+ u
2
+ 2u + 1
a
2
=
u
7
+ 2u
6
+ 5u
5
+ 6u
4
+ 6u
3
+ 3u
2
+ u 1
u
6
2u
5
5u
4
6u
3
6u
2
4u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
7
u
6
3u
5
+ 8u
4
+ 17u
3
+ 19u
2
+ 18u + 5
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.768546 0.720795I
a = 0.346208 0.283354I
b = 0.562759 + 0.266496I
0.48271 + 2.83701I 5.21159 10.60912I
u = 0.768546 + 0.720795I
a = 0.346208 + 0.283354I
b = 0.562759 0.266496I
0.48271 2.83701I 5.21159 + 10.60912I
u = 0.19859 1.50044I
a = 1.58392 + 0.07117I
b = 1.108336 + 0.872786I
6.67501 + 5.79166I 2.06170 5.06823I
u = 0.19859 + 1.50044I
a = 1.58392 0.07117I
b = 1.108336 0.872786I
6.67501 5.79166I 2.06170 + 5.06823I
u = 0.057100 0.488588I
a = 1.86606 + 0.38637I
b = 0.138522 + 0.871772I
0.43885 + 3.70343I 0.65225 5.99436I
u = 0.057100 + 0.488588I
a = 1.86606 0.38637I
b = 0.138522 0.871772I
0.43885 3.70343I 0.65225 + 5.99436I
u = 0.024235 1.274499I
a = 1.29619 1.02778I
b = 0.309617 1.251957I
2.47121 3.78237I 0.87896 + 6.92362I
u = 0.024235 + 1.274499I
a = 1.29619 + 1.02778I
b = 0.309617 + 1.251957I
2.47121 + 3.78237I 0.87896 6.92362I
3
II.
I
u
2
= hb
30
b
29
+· · ·6b11, 4.83×10
39
u+4.54×10
39
b
29
+· · ·+1.23×10
41
b+
6.97 × 10
40
, 1.13× 10
40
b
29
+ 3.10 × 10
39
b
28
+ · · · + 4.83 × 10
39
a 1.74 × 10
41
i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
0.938468b
29
+ 0.231798b
28
+ ··· 25.4227b 14.4241
a
4
=
1
0.621946b
29
+ 0.113749b
28
+ ··· 16.0520b 9.16115
a
10
=
2.32859b
29
0.642314b
28
+ ··· + 67.5031b + 36.0872
b
a
1
=
2.32859b
29
0.642314b
28
+ ··· + 66.5031b + 36.0872
b
a
9
=
1.63516b
29
+ 0.526941b
28
+ ··· 52.3036b 25.0122
1
a
7
=
0.938468b
29
0.231798b
28
+ ··· + 25.4227b + 14.4241
0.720162b
29
+ 0.145766b
28
+ ··· 21.7325b 12.3770
a
5
=
0.621946b
29
0.113749b
28
+ ··· + 16.0520b + 10.1611
0.716268b
29
+ 0.161571b
28
+ ··· 22.7645b 13.1804
a
6
=
0.822296b
29
0.105908b
28
+ ··· + 19.3467b + 13.0881
0.410201b
29
0.0990500b
28
+ ··· + 9.87931b + 3.93058
a
2
=
1.18892b
29
0.274975b
28
+ ··· + 31.4928b + 20.5819
0.289932b
29
0.122102b
28
+ ··· + 7.51160b + 1.82876
(ii) Obstruction class = 1
(iii) Cusp Shapes = 11.7075b
29
+ 3.83458b
28
+ ··· 389.096b 186.775
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.15235 + 1.51729I
a = 1.88912 0.40929I
b = 1.63678 0.95520I
8.32063 5.47678I 8.29813 + 5.38780I
u = 0.15235 1.51729I
a = 1.88912 + 0.40929I
b = 1.63678 + 0.95520I
8.32063 + 5.47678I 8.29813 5.38780I
u = 0.29798 + 1.53037I
a = 1.137816 0.082271I
b = 0.968761 0.886910I
5.55973 6.84757I 1.00546 + 10.27446I
u = 0.29798 1.53037I
a = 1.137816 + 0.082271I
b = 0.968761 + 0.886910I
5.55973 + 6.84757I 1.00546 10.27446I
u = 0.374558 + 0.641779I
a = 0.406606 + 0.029021I
b = 0.877609 0.842947I
1.31377 3.39671I 3.52800 + 8.19673I
u = 0.374558 0.641779I
a = 0.406606 0.029021I
b = 0.877609 + 0.842947I
1.31377 + 3.39671I 3.52800 8.19673I
u = 0.100337 1.375655I
a = 2.26778 0.50606I
b = 0.689826 0.421097I
3.32174 5.59550I 0.66951 + 7.79345I
u = 0.100337 + 1.375655I
a = 2.26778 + 0.50606I
b = 0.689826 + 0.421097I
3.32174 + 5.59550I 0.66951 7.79345I
u = 0.825834 + 0.538674I
a = 0.048366 + 0.223957I
b = 0.476814 0.494120I
1.13071 2.72262I 11.6934 + 8.2204I
u = 0.825834 0.538674I
a = 0.048366 0.223957I
b = 0.476814 + 0.494120I
1.13071 + 2.72262I 11.6934 8.2204I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.388845 0.104061I
a = 2.50710 + 2.08358I
b = 0.400846 0.866321I
1.42898 3.92960I 9.71569 + 7.98755I
u = 0.388845 + 0.104061I
a = 2.50710 2.08358I
b = 0.400846 + 0.866321I
1.42898 + 3.92960I 9.71569 7.98755I
u = 0.678314
a = 1.12014
b = 0.327578
1.01641 9.27192
u = 0.000696 + 1.255425I
a = 0.217717 0.136538I
b = 0.259040 1.130626I
1.82383 2.53738I 2.44510 + 1.72215I
u = 0.000696 1.255425I
a = 0.217717 + 0.136538I
b = 0.259040 + 1.130626I
1.82383 + 2.53738I 2.44510 1.72215I
u = 0.100337 1.375655I
a = 0.00710 1.64119I
b = 0.27520 2.16220I
3.32174 5.59550I 0.66951 + 7.79345I
u = 0.100337 + 1.375655I
a = 0.00710 + 1.64119I
b = 0.27520 + 2.16220I
3.32174 + 5.59550I 0.66951 7.79345I
u = 0.388845 0.104061I
a = 0.917626 + 0.970239I
b = 0.51204 1.36623I
1.42898 3.92960I 9.71569 + 7.98755I
u = 0.388845 + 0.104061I
a = 0.917626 0.970239I
b = 0.51204 + 1.36623I
1.42898 + 3.92960I 9.71569 7.98755I
u = 0.374558 0.641779I
a = 1.12046 + 1.56750I
b = 0.736028 0.024323I
1.31377 + 3.39671I 3.52800 8.19673I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.374558 + 0.641779I
a = 1.12046 1.56750I
b = 0.736028 + 0.024323I
1.31377 3.39671I 3.52800 + 8.19673I
u = 0.825834 0.538674I
a = 0.609385 + 0.321722I
b = 0.740636 0.362219I
1.13071 + 2.72262I 11.6934 8.2204I
u = 0.825834 + 0.538674I
a = 0.609385 0.321722I
b = 0.740636 + 0.362219I
1.13071 2.72262I 11.6934 + 8.2204I
u = 0.15235 1.51729I
a = 1.30588 + 0.66451I
b = 0.789858 0.466052I
8.32063 + 5.47678I 8.29813 5.38780I
u = 0.15235 + 1.51729I
a = 1.30588 0.66451I
b = 0.789858 + 0.466052I
8.32063 5.47678I 8.29813 + 5.38780I
u = 0.678314
a = 0.499081
b = 0.982011
1.01641 9.27192
u = 0.29798 1.53037I
a = 1.62064 + 0.22014I
b = 1.181025 0.498484I
5.55973 + 6.84757I 1.00546 10.27446I
u = 0.29798 + 1.53037I
a = 1.62064 0.22014I
b = 1.181025 + 0.498484I
5.55973 6.84757I 1.00546 + 10.27446I
u = 0.000696 + 1.255425I
a = 2.14838 0.80506I
b = 1.247670 0.599225I
1.82383 2.53738I 2.44510 + 1.72215I
u = 0.000696 1.255425I
a = 2.14838 + 0.80506I
b = 1.247670 + 0.599225I
1.82383 + 2.53738I 2.44510 1.72215I
7
III. I
u
3
= hu
23
5u
22
+ · · · 46u + 7, 149u
22
622u
21
+ · · · + 293b
2738, 1695u
22
+ 8293u
21
+ · · · + 2051a + 795i
(i) Arc colorings
a
3
=
1
0
a
8
=
0
u
a
4
=
1
u
2
a
10
=
0.826426u
22
4.04339u
21
+ ··· + 22.5958u 0.387616
0.508532u
22
+ 2.12287u
21
+ ··· 51.6758u + 9.34471
a
1
=
1.33496u
22
6.16626u
21
+ ··· + 74.2716u 9.73233
0.508532u
22
+ 2.12287u
21
+ ··· 51.6758u + 9.34471
a
9
=
3.13701u
22
+ 14.7157u
21
+ ··· 146.794u + 20.1351
0.969283u
22
4.75768u
21
+ ··· + 125.167u 21.9590
a
7
=
u
u
3
+ u
a
5
=
u
2
+ 1
u
4
2u
2
a
6
=
1.11555u
22
4.72111u
21
+ ··· + 61.6090u 12.8684
0.436860u
22
0.890785u
21
+ ··· 12.6007u + 4.75085
a
2
=
0.689907u
22
3.07752u
21
+ ··· + 27.7835u 2.87226
0.136519u
22
+ 0.965870u
21
+ ··· 24.8123u + 3.51536
(ii) Obstruction class = 1
(iii) Cusp Shapes =
1041
293
u
22
5380
293
u
21
+ ··· +
116025
293
u
22191
293
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.638103 0.842766I
a = 0.316038 0.073048I
b = 0.181627 + 0.040696I
0.67100 + 2.44356I 2.46207 + 5.34596I
u = 0.638103 + 0.842766I
a = 0.316038 + 0.073048I
b = 0.181627 0.040696I
0.67100 2.44356I 2.46207 5.34596I
u = 0.541870
a = 0.868184
b = 0.305112
1.26878 7.95589
u = 0.134358 1.265936I
a = 0.846718 + 0.501284I
b = 0.294667 + 0.675459I
2.55142 + 2.44221I 0.25016 2.15872I
u = 0.134358 + 1.265936I
a = 0.846718 0.501284I
b = 0.294667 0.675459I
2.55142 2.44221I 0.25016 + 2.15872I
u = 0.024762 1.413529I
a = 1.67002 + 0.21095I
b = 1.206411 0.634398I
6.65304 + 0.20600I 5.87376 + 0.49624I
u = 0.024762 + 1.413529I
a = 1.67002 0.21095I
b = 1.206411 + 0.634398I
6.65304 0.20600I 5.87376 0.49624I
u = 0.15014 1.58348I
a = 1.058526 0.429300I
b = 0.990467 + 0.202654I
9.33057 + 2.66158I 5.53368 3.29637I
u = 0.15014 + 1.58348I
a = 1.058526 + 0.429300I
b = 0.990467 0.202654I
9.33057 2.66158I 5.53368 + 3.29637I
u = 0.21309 1.44798I
a = 2.13202 + 0.04331I
b = 1.53737 1.10594I
7.80750 6.31614I 8.73055 + 7.98600I
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.21309 + 1.44798I
a = 2.13202 0.04331I
b = 1.53737 + 1.10594I
7.80750 + 6.31614I 8.73055 7.98600I
u = 0.26611 1.40784I
a = 1.28504 + 0.84124I
b = 1.380755 0.121423I
7.16412 2.89840I 6.40584 + 0.61240I
u = 0.26611 + 1.40784I
a = 1.28504 0.84124I
b = 1.380755 + 0.121423I
7.16412 + 2.89840I 6.40584 0.61240I
u = 0.30585 1.51255I
a = 1.89673 0.13828I
b = 1.49687 + 0.93220I
7.0586 15.3049I 1.91417 + 8.23545I
u = 0.30585 + 1.51255I
a = 1.89673 + 0.13828I
b = 1.49687 0.93220I
7.0586 + 15.3049I 1.91417 8.23545I
u = 0.476919 0.256901I
a = 0.785964 + 0.743449I
b = 0.986987 + 0.192913I
1.97196 + 0.18097I 4.00287 + 0.43243I
u = 0.476919 + 0.256901I
a = 0.785964 0.743449I
b = 0.986987 0.192913I
1.97196 0.18097I 4.00287 0.43243I
u = 0.571973 0.376783I
a = 0.443055 + 1.084283I
b = 1.116235 0.856342I
1.92126 3.42239I 4.06899 + 7.96024I
u = 0.571973 + 0.376783I
a = 0.443055 1.084283I
b = 1.116235 + 0.856342I
1.92126 + 3.42239I 4.06899 7.96024I
u = 0.746057 0.716204I
a = 0.018474 0.755049I
b = 0.926152 0.445909I
1.41145 + 5.79407I 0.88331 5.20349I
10
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.746057 + 0.716204I
a = 0.018474 + 0.755049I
b = 0.926152 + 0.445909I
1.41145 5.79407I 0.88331 + 5.20349I
u = 0.838014 0.461206I
a = 0.496263 0.531351I
b = 1.18542 + 0.78671I
0.67120 11.14214I 1.22299 + 8.55675I
u = 0.838014 + 0.461206I
a = 0.496263 + 0.531351I
b = 1.18542 0.78671I
0.67120 + 11.14214I 1.22299 8.55675I
11
IV. I
v
1
= hb + 1, v 1, ai
(i) Arc colorings
a
3
=
1
0
a
8
=
1
0
a
4
=
1
0
a
10
=
0
1
a
1
=
1
1
a
9
=
0
1
a
7
=
1
0
a
5
=
1
0
a
6
=
1
1
a
2
=
0
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 6
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
v
1
1(vol +
1CS) Cusp shape
v = 1.00000
a = 0
b = 1.00000
1.64493 6.00000
13
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
6
(u 1)(u
8
u
7
+ ··· + 2u
2
+ 1)(u
23
+ 2u
22
+ ··· 2u + 1)
(u
30
u
29
+ ··· 16u + 1)
c
2
, c
5
(u + 1)(u
8
+ 2u
6
+ ··· + u + 1)(u
23
u
22
+ ··· u 1)
(u
30
+ u
29
+ ··· 6u 1)
c
3
, c
4
u(u
8
+ 2u
7
+ 6u
6
+ 8u
5
+ 11u
4
+ 9u
3
+ 7u
2
+ 2u + 1)
(1 4u
2
8u
3
2u
4
+ 24u
5
+ 58u
6
+ 94u
7
+ 109u
8
+ 104u
9
+ 78u
10
+ 52u
11
+ 25u
12
+ 12u
13
+ 3u
14
+ u
15
)
2
(u
23
5u
22
+ ··· 46u + 7)
c
7
u(u
8
2u
7
+ 6u
6
8u
5
+ 11u
4
9u
3
+ 7u
2
2u + 1)
(1 4u
2
8u
3
2u
4
+ 24u
5
+ 58u
6
+ 94u
7
+ 109u
8
+ 104u
9
+ 78u
10
+ 52u
11
+ 25u
12
+ 12u
13
+ 3u
14
+ u
15
)
2
(u
23
5u
22
+ ··· 46u + 7)
c
8
, c
10
(u + 1)(u
8
3u
7
+ 6u
6
9u
5
+ 12u
4
11u
3
+ 8u
2
4u + 1)
(u
23
+ 2u
22
+ ··· + 14u 1)(u
30
u
29
+ ··· 6u 11)
c
9
u(u
8
+ 5u
7
+ 13u
6
+ 20u
5
+ 22u
4
+ 18u
3
+ 12u
2
+ 5u + 1)
(1 + 4u
2
4u
3
6u
4
+ 16u
5
6u
6
18u
7
+ 27u
8
42u
10
+ 62u
11
49u
12
+ 24u
13
7u
14
+ u
15
)
2
(u
23
+ 14u
22
+ ··· 43u 7)
14
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y 1)(y
8
+ 3y
7
+ 6y
6
+ 9y
5
+ 12y
4
+ 11y
3
+ 8y
2
+ 4y + 1)
(y
23
+ 2y
22
+ ··· 12y 1)(y
30
+ 3y
29
+ ··· 92y + 1)
c
2
, c
5
(y 1)(y
8
+ 4y
7
+ 8y
6
+ 11y
5
+ 12y
4
+ 9y
3
+ 6y
2
+ 3y + 1)
(y
23
9y
22
+ ··· + 25y 1)(y
30
+ 7y
29
+ ··· 48y + 1)
c
3
, c
4
, c
7
y(y
8
+ 8y
7
+ 26y
6
+ 46y
5
+ 55y
4
+ 53y
3
+ 35y
2
+ 10y + 1)
(1 + 8y 12y
2
68y
3
142y
4
+ 20y
5
+ 494y
6
+ 858y
7
+ 1051y
8
+ 1260y
9
+ 1238y
10
+ 834y
11
+ 363y
12
+ 98y
13
+ 15y
14
+ y
15
)
2
(y
23
+ 23y
22
+ ··· + 198y 49)
c
8
, c
10
(y 1)(y
8
+ 3y
7
+ 6y
6
+ 13y
5
+ 20y
4
+ 11y
3
+ 1)
(y
23
18y
22
+ ··· + 68y 1)(y
30
+ 3y
29
+ ··· 1972y + 121)
c
9
y(y
8
+ y
7
+ 13y
6
+ 16y
5
+ 28y
4
+ 30y
3
+ 8y
2
y + 1)
(1 + 8y 28y
2
+ 52y
3
62y
4
+ 28y
5
50y
6
+ 26y
7
+ 27y
8
+ 124y
9
34y
10
+ 70y
11
13y
12
+ 14y
13
y
14
+ y
15
)
2
(y
23
+ 26y
21
+ ··· + 225y 49)
15