11n
162
(K11n
162
)
1
Arc Sequences
10 5 1 7 10 3 11 5 3 8 4
Solving Sequence
4,11
1
3,8
7 5 2 10 6 9
c
11
c
3
c
7
c
4
c
2
c
10
c
5
c
9
c
1
, c
6
, c
8
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu
6
7u
5
+ 18u
4
22u
3
+ 15u
2
5u + 1, 2u
5
+ 13u
4
29u
3
+ 27u
2
+ a 12u,
u
5
+ 7u
4
17u
3
+ 18u
2
+ b 10u + 2i
I
u
2
= hb
6
+ 2b
5
+ 4b
4
+ 6b
3
+ 6b
2
+ 5b + 3, b
4
2b
3
3b
2
3b + u 1, b
5
2b
4
4b
3
5b
2
5b + a 4i
I
u
3
= hu
13
24u
12
+ ··· 2016u + 480,
219031775u
12
+ 4687623230u
11
+ ··· + 903641744b + 43146415952,
2696650997u
12
+ 58148670678u
11
+ ··· + 27109252320a + 610906263072i
I
u
4
= hb
26
3b
25
+ ··· 9b + 1,
280010027772943b
25
+ 640188464864309u + ··· 9045506250176506b + 911334568428454,
1931825958972041b
25
5354645722954962b
24
+ ··· + 1920565394592927a 4009504723628393i
There are 4 irreducible components with 51 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
7u
5
+ 18u
4
22u
3
+ 15u
2
5u + 1, 2u
5
+ 13u
4
29u
3
+
27u
2
+ a 12u, u
5
+ 7u
4
17u
3
+ 18u
2
+ b 10u + 2i
(i) Arc colorings
a
4
=
1
0
a
11
=
2u
5
13u
4
+ 29u
3
27u
2
+ 12u
u
5
7u
4
+ 17u
3
18u
2
+ 10u 2
a
1
=
u
5
6u
4
+ 12u
3
9u
2
+ 2u + 2
u
5
7u
4
+ 17u
3
18u
2
+ 10u 2
a
3
=
2u
5
+ 14u
4
35u
3
+ 39u
2
22u + 5
u
3
+ 4u
2
4u + 2
a
8
=
0
u
a
7
=
2u
5
14u
4
+ 36u
3
43u
2
+ 26u 6
u
3
4u
2
+ 5u 2
a
5
=
3u
5
+ 20u
4
47u
3
+ 48u
2
24u + 3
u
5
+ 8u
4
22u
3
+ 26u
2
15u + 3
a
2
=
u
4
+ 7u
3
17u
2
+ 17u 7
u
3
4u
2
+ 5u 3
a
10
=
2u
5
13u
4
+ 29u
3
27u
2
+ 12u
u
5
6u
4
+ 13u
3
13u
2
+ 7u 1
a
6
=
u
4
+ 7u
3
17u
2
+ 17u 7
u
5
+ 7u
4
17u
3
+ 17u
2
7u
a
9
=
u
4
7u
3
+ 17u
2
17u + 7
u
5
6u
4
+ 12u
3
9u
2
+ 2u + 2
a
9
=
u
4
7u
3
+ 17u
2
17u + 7
u
5
6u
4
+ 12u
3
9u
2
+ 2u + 2
(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 = 0.211778 0.330391I
a = 2.74807 0.64577I
b = 0.368622 1.044697I
9.91965 + 2.91185I 0.22592 3.63955I
u = 0.211778 + 0.330391I
a = 2.74807 + 0.64577I
b = 0.368622 + 1.044697I
9.91965 2.91185I 0.22592 + 3.63955I
u = 0.717525 0.672354I
a = 0.033577 0.670495I
b = 0.474902 0.458521I
1.33814 0.90202I 11.17385 + 4.13696I
u = 0.717525 + 0.672354I
a = 0.033577 + 0.670495I
b = 0.474902 + 0.458521I
1.33814 + 0.90202I 11.17385 4.13696I
u = 2.57070 0.32705I
a = 0.214491 + 0.482108I
b = 0.393720 + 1.309502I
4.57797 + 6.62522I 5.60023 6.47362I
u = 2.57070 + 0.32705I
a = 0.214491 0.482108I
b = 0.393720 1.309502I
4.57797 6.62522I 5.60023 + 6.47362I
3
II. I
u
2
= hb
6
+ 2b
5
+ 4b
4
+ 6b
3
+ 6b
2
+ 5b + 3, b
4
2b
3
3b
2
3b + u
1, b
5
2b
4
4b
3
5b
2
5b + a 4i
(i) Arc colorings
a
4
=
1
0
a
11
=
b
5
+ 2b
4
+ 4b
3
+ 5b
2
+ 5b + 4
b
a
1
=
b
5
+ 2b
4
+ 4b
3
+ 5b
2
+ 4b + 4
b
a
3
=
b
3
+ 2b
2
+ b + 4
b
2
a
8
=
0
b
4
+ 2b
3
+ 3b
2
+ 3b + 1
a
7
=
b
5
2b
4
4b
3
4b
2
5b 2
b
3
+ b
2
+ b + 1
a
5
=
b
5
+ b
4
+ 2b
3
+ 3b
2
+ b + 2
b
4
2b
3
3b
2
3b 2
a
2
=
b
3
+ b
2
+ b + 2
b
5
+ 2b
4
+ 2b
3
+ b
2
1
a
10
=
b
5
+ 2b
4
+ 4b
3
+ 5b
2
+ 5b + 4
b
5
+ 2b
4
+ 2b
3
+ 2b
2
+ b 1
a
6
=
b
5
+ 2b
4
+ 2b
3
+ 4b
2
+ 2b + 2
b
5
+ 2b
4
+ 4b
3
+ 5b
2
+ 4b + 1
a
9
=
2b
5
+ 2b
4
+ 6b
3
+ 6b
2
+ 5b + 5
b
4
+ b
3
+ b
2
+ b 1
a
9
=
2b
5
+ 2b
4
+ 6b
3
+ 6b
2
+ 5b + 5
b
4
+ b
3
+ b
2
+ b 1
(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 = 0.426050 + 0.368989I
a = 1.62708 1.36745I
b = 1.044140 0.390425I
1.07850 + 1.58317I 4.02349 3.48462I
u = 0.426050 0.368989I
a = 1.62708 + 1.36745I
b = 1.044140 + 0.390425I
1.07850 1.58317I 4.02349 + 3.48462I
u = 0.426050 0.368989I
a = 0.94687 1.73644I
b = 0.188646 1.182977I
1.07850 1.58317I 4.02349 + 3.48462I
u = 0.426050 + 0.368989I
a = 0.94687 + 1.73644I
b = 0.188646 + 1.182977I
1.07850 + 1.58317I 4.02349 3.48462I
u = 3.14790
a = 0.073950 0.405345I
b = 0.232786 1.275985I
11.0025 0.953017
u = 3.14790
a = 0.073950 + 0.405345I
b = 0.232786 + 1.275985I
11.0025 0.953017
5
III. I
u
3
= hu
13
24u
12
+ · · · 2016u + 480, 2.19 × 10
8
u
12
+ 4.69 ×
10
9
u
11
+ · · · + 9.04 × 10
8
b + 4.31 × 10
10
, 2.70 × 10
9
u
12
+ 5.81 × 10
10
u
11
+
· · · + 2.71 × 10
10
a + 6.11 × 10
11
i
(i) Arc colorings
a
4
=
1
0
a
11
=
0.0994735u
12
2.14498u
11
+ ··· + 88.4038u 22.5350
0.242388u
12
5.18748u
11
+ ··· + 178.004u 47.7473
a
1
=
0.142914u
12
+ 3.04250u
11
+ ··· 89.5997u + 25.2123
0.242388u
12
5.18748u
11
+ ··· + 178.004u 47.7473
a
3
=
0.0565209u
12
+ 1.24812u
11
+ ··· 52.9216u + 15.8826
0.105111u
12
2.30917u
11
+ ··· + 89.5168u 23.3232
a
8
=
0
u
a
7
=
0.0485900u
12
+ 1.06105u
11
+ ··· 36.5952u + 8.44059
0.105111u
12
+ 2.30917u
11
+ ··· 88.5168u + 23.3232
a
5
=
0.0843754u
12
1.82264u
11
+ ··· + 66.1376u 17.2594
0.0111211u
12
+ 0.349662u
11
+ ··· 36.7390u + 9.95305
a
2
=
0.129345u
12
+ 2.74183u
11
+ ··· 75.8747u + 22.8445
0.211793u
12
4.54561u
11
+ ··· + 159.148u 42.2710
a
10
=
0.0994735u
12
2.14498u
11
+ ··· + 88.4038u 22.5350
0.387441u
12
+ 8.27096u
11
+ ··· 262.903u + 68.5989
a
6
=
0.178070u
12
3.82028u
11
+ ··· + 130.508u 33.6483
0.453399u
12
+ 9.71423u
11
+ ··· 325.341u + 85.4737
a
9
=
0.0943531u
12
+ 1.98564u
11
+ ··· 55.2274u + 15.3443
0.504250u
12
10.7447u
11
+ ··· + 352.239u 92.3938
a
9
=
0.0943531u
12
+ 1.98564u
11
+ ··· 55.2274u + 15.3443
0.504250u
12
10.7447u
11
+ ··· + 352.239u 92.3938
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.696637
a = 0.314085
b = 0.218803
0.973187 8.95549
u = 0.199032 0.612878I
a = 0.561529 + 1.138335I
b = 0.809423 0.117584I
5.36911 + 2.95494I 7.16291 2.81901I
u = 0.199032 + 0.612878I
a = 0.561529 1.138335I
b = 0.809423 + 0.117584I
5.36911 2.95494I 7.16291 + 2.81901I
u = 0.263411 0.441744I
a = 0.312290 1.271242I
b = 0.479303 0.472811I
0.52618 1.43256I 3.73519 + 6.28375I
u = 0.263411 + 0.441744I
a = 0.312290 + 1.271242I
b = 0.479303 + 0.472811I
0.52618 + 1.43256I 3.73519 6.28375I
u = 2.34475 0.66404I
a = 0.204341 0.493346I
b = 0.151527 1.292462I
6.35965 0.39707I 2.26484 + 2.21487I
u = 2.34475 + 0.66404I
a = 0.204341 + 0.493346I
b = 0.151527 + 1.292462I
6.35965 + 0.39707I 2.26484 2.21487I
u = 2.53526 0.14222I
a = 0.127215 0.575447I
b = 0.40436 1.44082I
6.52797 5.89125I 0.30193 + 3.39089I
u = 2.53526 + 0.14222I
a = 0.127215 + 0.575447I
b = 0.40436 + 1.44082I
6.52797 + 5.89125I 0.30193 3.39089I
u = 2.93435 0.11492I
a = 0.185374 0.504984I
b = 0.48592 1.50310I
15.6748 + 13.1069I 1.18846 5.87531I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 2.93435 + 0.11492I
a = 0.185374 + 0.504984I
b = 0.48592 + 1.50310I
15.6748 13.1069I 1.18846 + 5.87531I
u = 4.07152 1.55549I
a = 0.093363 + 0.274076I
b = 0.046196 + 1.261130I
12.31696 2.15873I 3.13107 + 3.07256I
u = 4.07152 + 1.55549I
a = 0.093363 0.274076I
b = 0.046196 1.261130I
12.31696 + 2.15873I 3.13107 3.07256I
8
IV.
I
u
4
= hb
26
3b
25
+· · ·9b+1, 6.40×10
14
u2.80×10
14
b
25
+· · ·9.05×10
15
b+
9.11 × 10
14
, 1.93 × 10
15
b
25
5.35 × 10
15
b
24
+ · · · + 1.92 × 10
15
a 4.01 × 10
15
i
(i) Arc colorings
a
4
=
1
0
a
11
=
1.00586b
25
+ 2.78806b
24
+ ··· 18.7038b + 2.08767
b
a
1
=
1.00586b
25
+ 2.78806b
24
+ ··· 19.7038b + 2.08767
b
a
3
=
0.229532b
25
0.636550b
24
+ ··· + 6.96510b 0.00586315
b
2
a
8
=
0
0.437387b
25
1.02758b
24
+ ··· + 14.1294b 1.42354
a
7
=
0.973498b
25
2.54878b
24
+ ··· + 15.1574b 3.22989
0.738314b
25
1.84839b
24
+ ··· + 15.4595b 1.32528
a
5
=
0.353112b
25
+ 1.05017b
24
+ ··· 8.22868b + 0.546894
0.624907b
25
2.12314b
24
+ ··· + 25.2114b 3.59109
a
2
=
0.0982585b
25
0.595703b
24
+ ··· + 11.7692b 2.21440
0.851882b
25
2.18042b
24
+ ··· + 12.8033b + 1.43267
a
10
=
1.00586b
25
+ 2.78806b
24
+ ··· 18.7038b + 2.08767
0.0591345b
25
+ 0.0776802b
24
+ ··· + 2.44901b 1.85007
a
6
=
1.39158b
25
+ 3.70478b
24
+ ··· 30.6015b + 5.15885
0.682959b
25
+ 1.75673b
24
+ ··· 14.2343b + 1.11476
a
9
=
0.585691b
25
+ 1.68463b
24
+ ··· 13.7189b + 2.97508
0.00282387b
25
+ 0.201032b
24
+ ··· + 1.47093b 1.71634
a
9
=
0.585691b
25
+ 1.68463b
24
+ ··· 13.7189b + 2.97508
0.00282387b
25
+ 0.201032b
24
+ ··· + 1.47093b 1.71634
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.212184 0.601220I
a = 1.78123 + 0.51570I
b = 1.207500 0.051879I
1.265646 0.287296I 2.88954 0.89161I
u = 0.212184 + 0.601220I
a = 1.78123 0.51570I
b = 1.207500 + 0.051879I
1.265646 + 0.287296I 2.88954 + 0.89161I
u = 2.55159 0.69397I
a = 0.204696 0.440581I
b = 0.45644 1.36303I
5.76658 5.41588I 0.94009 + 2.54727I
u = 2.55159 + 0.69397I
a = 0.204696 + 0.440581I
b = 0.45644 + 1.36303I
5.76658 + 5.41588I 0.94009 2.54727I
u = 1.02005
a = 0.247472 0.411191I
b = 0.252433 0.419434I
0.889471 5.46516
u = 1.02005
a = 0.247472 + 0.411191I
b = 0.252433 + 0.419434I
0.889471 5.46516
u = 0.131463 0.355843I
a = 0.308504 + 1.360323I
b = 0.247316 1.177338I
2.03958 2.67289I 1.48191 + 5.19592I
u = 0.131463 + 0.355843I
a = 0.308504 1.360323I
b = 0.247316 + 1.177338I
2.03958 + 2.67289I 1.48191 5.19592I
u = 0.212184 0.601220I
a = 0.70704 1.75888I
b = 0.067900 1.180333I
1.265646 0.287296I 2.88954 0.89161I
u = 0.212184 + 0.601220I
a = 0.70704 + 1.75888I
b = 0.067900 + 1.180333I
1.265646 + 0.287296I 2.88954 + 0.89161I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 2.31351 + 0.01822I
a = 0.332532 0.674434I
b = 0.052461 1.357580I
13.70611 0.67900I 1.58151 + 0.47574I
u = 2.31351 0.01822I
a = 0.332532 + 0.674434I
b = 0.052461 + 1.357580I
13.70611 + 0.67900I 1.58151 0.47574I
u = 0.726221 1.104014I
a = 0.623167 + 0.767003I
b = 0.160428 0.147487I
8.78740 1.45996I 3.65650 + 0.27682I
u = 0.726221 + 1.104014I
a = 0.623167 0.767003I
b = 0.160428 + 0.147487I
8.78740 + 1.45996I 3.65650 0.27682I
u = 2.55159 + 0.69397I
a = 0.301845 0.452092I
b = 0.216549 1.266236I
5.76658 + 5.41588I 0.94009 2.54727I
u = 2.55159 0.69397I
a = 0.301845 + 0.452092I
b = 0.216549 + 1.266236I
5.76658 5.41588I 0.94009 + 2.54727I
u = 0.492555 + 0.725715I
a = 0.55639 1.37912I
b = 0.326804 1.347110I
10.00278 + 7.01304I 2.38090 4.98186I
u = 0.492555 0.725715I
a = 0.55639 + 1.37912I
b = 0.326804 + 1.347110I
10.00278 7.01304I 2.38090 + 4.98186I
u = 0.726221 + 1.104014I
a = 0.159963 0.040090I
b = 0.394224 1.244999I
8.78740 + 1.45996I 3.65650 0.27682I
u = 0.726221 1.104014I
a = 0.159963 + 0.040090I
b = 0.394224 + 1.244999I
8.78740 1.45996I 3.65650 + 0.27682I
11
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.131463 0.355843I
a = 3.13717 + 0.46398I
b = 0.524618 0.069053I
2.03958 2.67289I 1.48191 + 5.19592I
u = 0.131463 + 0.355843I
a = 3.13717 0.46398I
b = 0.524618 + 0.069053I
2.03958 + 2.67289I 1.48191 5.19592I
u = 2.31351 0.01822I
a = 0.027296 0.586589I
b = 0.78161 1.55425I
13.70611 + 0.67900I 1.58151 0.47574I
u = 2.31351 + 0.01822I
a = 0.027296 + 0.586589I
b = 0.78161 + 1.55425I
13.70611 0.67900I 1.58151 + 0.47574I
u = 0.492555 + 0.725715I
a = 1.06159 1.17084I
b = 1.274903 0.275508I
10.00278 + 7.01304I 2.38090 4.98186I
u = 0.492555 0.725715I
a = 1.06159 + 1.17084I
b = 1.274903 + 0.275508I
10.00278 7.01304I 2.38090 + 4.98186I
12
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
3
+ u
2
2u + 1)
2
(u
6
u
5
+ 8u
3
+ 3u
2
u + 3)
(1 + 47u 142u
2
+ 100u
3
+ 127u
4
214u
5
+ 26u
6
+ 122u
7
59u
8
26u
9
+ 24u
10
4u
12
+ u
13
)
2
(u
13
+ 16u
12
+ ··· + 384u + 32)
c
2
, c
5
(u
6
+ 2u
4
+ 3u
3
2u
2
2u + 1)(u
6
+ 3u
4
3u
3
u + 3)
(u
13
+ 12u
11
+ ··· 7u
2
1)(u
26
+ u
25
+ ··· 193u + 61)
c
3
, c
10
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 3u
2
+ 2u + 1)
(u
6
+ 2u
5
+ ··· + 5u + 3)(u
13
+ u
12
+ ··· + 4u 1)
(u
26
+ 3u
25
+ ··· + 9u + 1)
c
4
(u
3
+ u
2
+ 1)
2
(u
6
2u
5
+ 3u
4
2u
3
+ 5u
2
7u + 3)
(5 + 8u 20u
2
+ 23u
3
26u
4
+ 29u
5
17u
6
+ 25u
7
7u
8
+ 14u
9
2u
10
+ 5u
11
u
12
+ u
13
)
2
(u
13
+ 11u
12
+ ··· 40u 4)
c
6
, c
8
(u
6
3u
5
+ ··· + 2u + 1)(u
6
+ u
5
+ ··· + u + 1)
(u
13
u
12
+ ··· 11u + 3)(u
26
+ 4u
25
+ ··· + 2400u + 1353)
c
7
, c
11
(u
6
2u
5
+ 4u
4
6u
3
+ 6u
2
5u + 3)
(u
6
u
5
+ ··· 2u + 1)(u
13
+ u
12
+ ··· + 4u 1)
(u
26
+ 3u
25
+ ··· + 9u + 1)
c
9
(u
3
+ u
2
+ 1)
2
(u
6
3u
5
+ 2u
4
+ u
2
u + 1)
(3 + 2u + 20u
2
23u
3
32u
4
+ 63u
5
7u
6
+ 7u
7
39u
8
+ 4u
9
+ 20u
10
5u
11
3u
12
+ u
13
)
2
(u
13
+ 8u
12
+ ··· + 6u 12)
13
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
(y
3
5y
2
+ 2y 1)
2
(y
6
y
5
+ 22y
4
60y
3
+ 25y
2
+ 17y + 9)
(1 + 2493y 1.10 × 10
4
y
2
+ 2.59 × 10
4
y
3
4.00 × 10
4
y
4
+ 4.43 × 10
4
y
5
3.63 × 10
4
y
6
+ 2.19 × 10
4
y
7
9857y
8
+ 3288y
9
804y
10
+ 140y
11
16y
12
+ y
13
)
2
(y
13
12y
12
+ ··· + 59904y 1024)
c
2
, c
5
(y
6
+ 4y
5
+ ··· 8y + 1)(y
6
+ 6y
5
+ ··· y + 9)
(y
13
+ 24y
12
+ ··· 14y 1)(y
26
+ 33y
25
+ ··· + 38147y + 3721)
c
3
, c
7
, c
10
c
11
(y
6
+ 4y
5
+ ··· + 11y + 9)(y
6
+ 5y
5
+ ··· + 2y + 1)
(y
13
+ 13y
12
+ ··· + 20y 1)(y
26
+ 19y
25
+ ··· y + 1)
c
4
(y
3
y
2
2y 1)
2
(y
6
+ 2y
5
+ 11y
4
+ 4y
3
+ 15y
2
19y + 9)
(y
13
3y
12
+ ··· + 456y 16)
(25 136y 292y
2
217y
3
+ 308y
4
+ 1031y
5
+ 1431y
6
+ 1301y
7
+ 867y
8
+ 442y
9
+ 172y
10
+ 49y
11
+ 9y
12
+ y
13
)
2
c
6
(y
6
+ y
5
+ 11y
4
+ 19y
2
6y + 1)(y
6
+ 3y
5
+ 2y
4
+ y
2
+ y + 1)
(y
13
+ 11y
12
+ ··· 35y 9)
(y
26
+ 20y
25
+ ··· + 9336774y + 1830609)
c
8
(y
6
+ y
5
+ 11y
4
+ 19y
2
6y + 1)(y
6
+ 3y
5
+ 2y
4
+ y
2
+ y + 1)
(y
13
+ 11y
12
+ ··· 35y 9)
(y
26
+ 20y
25
+ ··· + 9336774y + 1830609)
c
9
(y
3
y
2
2y 1)
2
(y
6
5y
5
+ 6y
4
+ 5y
2
+ y + 1)
(9 + 124y 684y
2
+ 2019y
3
3848y
4
+ 4895y
5
2685y
6
+ 1641y
7
2053y
8
+ 1590y
9
660y
10
+ 153y
11
19y
12
+ y
13
)
2
(y
13
14y
12
+ ··· + 732y 144)
14