11a
347
(K11a
347
)
1
Arc Sequences
9 6 1 10 8 3 11 2 5 4 7
Solving Sequence
2,6
3
7,8
9 1 5 10 4 11
c
2
c
6
c
8
c
1
c
5
c
9
c
4
c
11
c
3
, c
7
, c
10
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu 1, b 1, 2a + 1i
I
u
2
= h4a
2
4a + 3, b + 1, u + 1i
I
u
3
= hb
36
+ 3b
35
+ ··· + 8b + 1, 4.57465 × 10
27
u + 1.53133 × 10
26
b
35
+ ··· + 8.22462 × 10
27
b + 7.22535 × 10
27
,
1.15703 × 10
28
b
35
3.61700 × 10
28
b
34
+ ··· + 1.37239 × 10
28
a 3.84841 × 10
28
i
I
u
4
= hu
22
9u
20
+ ··· 7u 24,
3.95573 × 10
21
u
21
+ 2.98464 × 10
21
u
20
+ ··· + 3.24502 × 10
23
b 6.03328 × 10
21
,
2.52884 × 10
22
u
21
1.36269 × 10
22
u
20
+ ··· + 3.89402 × 10
24
a + 3.16167 × 10
24
i
There are 4 irreducible components with 61 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 1, b 1, 2a + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
7
=
1
0
a
8
=
1
2
1
a
9
=
1
2
1
a
1
=
3
2
1
a
5
=
1
4
1
2
a
10
=
1
2
1
a
4
=
1
4
1
2
a
11
=
1
2
1
a
11
=
1
2
1
(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 = 1.00000
a = 0.500000
b = 1.00000
0 4.50000
3
II. I
u
2
= h4a
2
4a + 3, b + 1, u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
7
=
1
0
a
8
=
a
1
a
9
=
a 1
1
a
1
=
a + 2
1
a
5
=
a +
3
4
a 1
a
10
=
1
2
a
7
4
a
1
2
a
4
=
2a
1
4
a
a
11
=
a + 1
1
a
11
=
a + 1
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 = 1.00000
a = 0.500000 0.707107I
b = 1.00000
4.93480 0
u = 1.00000
a = 0.500000 + 0.707107I
b = 1.00000
4.93480 0
5
III.
I
u
3
= hb
36
+3b
35
+· · ·+8b+1, 4.57×10
27
u+1.53×10
26
b
35
+· · ·+8.22×10
27
b+
7.23 × 10
27
, 1.16× 10
28
b
35
3.62 × 10
28
b
34
+ · · · + 1.37 × 10
28
a 3.85 × 10
28
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
0.0334743b
35
0.323776b
34
+ ··· 1.79787b 1.57943
a
3
=
1
0.607082b
35
+ 1.79253b
34
+ ··· + 10.1695b + 3.04798
a
7
=
0.0334743b
35
+ 0.323776b
34
+ ··· + 1.79787b + 1.57943
0.539109b
35
+ 1.50703b
34
+ ··· + 12.1676b + 2.19831
a
8
=
0.843076b
35
+ 2.63554b
34
+ ··· + 13.8531b + 2.80416
b
a
9
=
0.843076b
35
+ 2.63554b
34
+ ··· + 14.8531b + 2.80416
b
a
1
=
0.106314b
35
+ 0.141825b
34
+ ··· 3.94045b + 0.156924
b
2
a
5
=
0.655892b
35
2.02876b
34
+ ··· 6.73505b + 0.233841
0.0307560b
35
0.231033b
34
+ ··· + 0.296911b 0.975059
a
10
=
0.348678b
35
1.06933b
34
+ ··· 11.2358b 2.35898
0.793273b
35
+ 1.97313b
34
+ ··· + 17.1361b + 2.67981
a
4
=
0.270553b
35
0.603829b
34
+ ··· + 1.16075b + 0.0780805
0.684692b
35
+ 1.91920b
34
+ ··· + 9.04566b + 2.86969
a
11
=
0.605849b
35
+ 1.51358b
34
+ ··· + 8.20307b + 2.16125
1
a
11
=
0.605849b
35
+ 1.51358b
34
+ ··· + 8.20307b + 2.16125
1
(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 = 1.333562 + 0.360812I
a = 0.728667 + 0.348332I
b = 1.211219 0.140810I
0.60037 9.13509I 2.98695 + 5.86478I
u = 1.333562 0.360812I
a = 0.728667 0.348332I
b = 1.211219 + 0.140810I
0.60037 + 9.13509I 2.98695 5.86478I
u = 0.099390 0.822674I
a = 1.50465 0.26347I
b = 0.855022 0.244718I
5.09742 4.87394I 1.52680 + 3.60136I
u = 0.099390 + 0.822674I
a = 1.50465 + 0.26347I
b = 0.855022 + 0.244718I
5.09742 + 4.87394I 1.52680 3.60136I
u = 1.354281 0.099636I
a = 1.110108 0.077757I
b = 0.816163 1.122797I
5.44315 + 3.22673I 7.05526 3.62956I
u = 1.354281 + 0.099636I
a = 1.110108 + 0.077757I
b = 0.816163 + 1.122797I
5.44315 3.22673I 7.05526 + 3.62956I
u = 1.232890 0.279362I
a = 0.747459 + 0.436194I
b = 0.535422 0.229537I
4.41864 + 1.78695I 5.23943 + 0.02251I
u = 1.232890 + 0.279362I
a = 0.747459 0.436194I
b = 0.535422 + 0.229537I
4.41864 1.78695I 5.23943 0.02251I
u = 1.311619 + 0.317206I
a = 0.971024 0.433098I
b = 0.46000 1.36593I
5.17867 + 5.71427I 7.06596 6.05983I
u = 1.311619 0.317206I
a = 0.971024 + 0.433098I
b = 0.46000 + 1.36593I
5.17867 5.71427I 7.06596 + 6.05983I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.424636 + 0.422216I
a = 2.47596 + 2.35989I
b = 0.393324 0.963175I
0.06375 + 1.57187I 1.80878 4.22070I
u = 0.424636 0.422216I
a = 2.47596 2.35989I
b = 0.393324 + 0.963175I
0.06375 1.57187I 1.80878 + 4.22070I
u = 1.354281 + 0.099636I
a = 0.195735 0.375309I
b = 0.27308 1.57039I
5.44315 3.22673I 7.05526 + 3.62956I
u = 1.354281 0.099636I
a = 0.195735 + 0.375309I
b = 0.27308 + 1.57039I
5.44315 + 3.22673I 7.05526 3.62956I
u = 0.064741 0.739221I
a = 0.337485 + 0.752170I
b = 0.259835 0.987292I
0.86368 + 1.88569I 1.68331 3.99357I
u = 0.064741 + 0.739221I
a = 0.337485 0.752170I
b = 0.259835 + 0.987292I
0.86368 1.88569I 1.68331 + 3.99357I
u = 1.160032 + 0.371279I
a = 0.363618 1.040256I
b = 0.197872 0.137215I
1.85527 0.55896I 1.51114 0.25710I
u = 1.160032 0.371279I
a = 0.363618 + 1.040256I
b = 0.197872 + 0.137215I
1.85527 + 0.55896I 1.51114 + 0.25710I
u = 0.424636 0.422216I
a = 1.37670 + 1.71650I
b = 0.181838 1.232261I
0.06375 1.57187I 1.80878 + 4.22070I
u = 0.424636 + 0.422216I
a = 1.37670 1.71650I
b = 0.181838 + 1.232261I
0.06375 + 1.57187I 1.80878 4.22070I
8
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.160032 + 0.371279I
a = 1.010321 0.102773I
b = 0.119141 0.939188I
1.85527 0.55896I 1.51114 0.25710I
u = 1.160032 0.371279I
a = 1.010321 + 0.102773I
b = 0.119141 + 0.939188I
1.85527 + 0.55896I 1.51114 + 0.25710I
u = 0.361873
a = 3.51798 + 2.22902I
b = 0.157243 1.036424I
3.91179 11.9800
u = 0.361873
a = 3.51798 2.22902I
b = 0.157243 + 1.036424I
3.91179 11.9800
u = 1.232890 + 0.279362I
a = 0.969343 0.687856I
b = 0.182954 1.202277I
4.41864 1.78695I 5.23943 0.02251I
u = 1.232890 0.279362I
a = 0.969343 + 0.687856I
b = 0.182954 + 1.202277I
4.41864 + 1.78695I 5.23943 + 0.02251I
u = 0.064741 0.739221I
a = 0.832343 0.603900I
b = 0.430436 0.146579I
0.86368 + 1.88569I 1.68331 3.99357I
u = 0.064741 + 0.739221I
a = 0.832343 + 0.603900I
b = 0.430436 + 0.146579I
0.86368 1.88569I 1.68331 + 3.99357I
u = 1.34147
a = 0.797901 0.342205I
b = 0.53627 1.36483I
9.12242 12.3720
u = 1.34147
a = 0.797901 + 0.342205I
b = 0.53627 + 1.36483I
9.12242 12.3720
9
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.099390 0.822674I
a = 0.793364 + 1.034339I
b = 0.554814 1.110357I
5.09742 4.87394I 1.52680 + 3.60136I
u = 0.099390 + 0.822674I
a = 0.793364 1.034339I
b = 0.554814 + 1.110357I
5.09742 + 4.87394I 1.52680 3.60136I
u = 1.333562 + 0.360812I
a = 1.142449 0.372915I
b = 0.69402 1.37640I
0.60037 9.13509I 2.98695 + 5.86478I
u = 1.333562 0.360812I
a = 1.142449 + 0.372915I
b = 0.69402 + 1.37640I
0.60037 + 9.13509I 2.98695 5.86478I
u = 1.311619 0.317206I
a = 0.768338 0.089735I
b = 1.008893 0.077944I
5.17867 5.71427I 7.06596 + 6.05983I
u = 1.311619 + 0.317206I
a = 0.768338 + 0.089735I
b = 1.008893 + 0.077944I
5.17867 + 5.71427I 7.06596 6.05983I
10
IV. I
u
4
=
hu
22
9u
20
+· · ·7u24, 3.96×10
21
u
21
+2.98×10
21
u
20
+· · ·+3.25×10
23
b
6.03× 10
21
, 2.53 × 10
22
u
21
1.36 ×10
22
u
20
+· · · +3.89 ×10
24
a + 3.16× 10
24
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
3
+ u
a
8
=
0.00649416u
21
+ 0.00349944u
20
+ ··· 2.74830u 0.811929
0.0121902u
21
0.00919760u
20
+ ··· + 0.339941u + 0.0185924
a
9
=
0.0186843u
21
0.00569816u
20
+ ··· 2.40836u 0.793337
0.0121902u
21
0.00919760u
20
+ ··· + 0.339941u + 0.0185924
a
1
=
0.00272475u
21
+ 0.0103464u
20
+ ··· 1.92698u + 0.178658
0.00205819u
21
+ 0.00459894u
20
+ ··· 0.759439u + 0.404174
a
5
=
0.0169020u
21
+ 0.00741262u
20
+ ··· + 0.150666u 0.639956
0.00574750u
21
0.000805504u
20
+ ··· 0.230182u 0.0159975
a
10
=
0.00552350u
21
0.0212193u
20
+ ··· + 1.13251u 0.194895
0.0148709u
21
0.00523811u
20
+ ··· 0.0475394u 0.376699
a
4
=
0.00807918u
21
3.03493 × 10
6
u
20
+ ··· 0.409271u + 0.180131
0.00493091u
21
0.00125393u
20
+ ··· 0.952192u 0.405721
a
11
=
0.00647285u
21
+ 0.0126601u
20
+ ··· 2.03090u 0.113906
0.00537335u
21
0.0000693864u
20
+ ··· 0.418578u + 0.752265
a
11
=
0.00647285u
21
+ 0.0126601u
20
+ ··· 2.03090u 0.113906
0.00537335u
21
0.0000693864u
20
+ ··· 0.418578u + 0.752265
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
11
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.63415 0.33967I
a = 0.205579 + 0.491340I
b = 0.120052 + 0.802501I
1.76897 + 0.99297I 2.13017 4.98419I
u = 1.63415 + 0.33967I
a = 0.205579 0.491340I
b = 0.120052 0.802501I
1.76897 0.99297I 2.13017 + 4.98419I
u = 1.53072 0.47138I
a = 1.004240 0.308031I
b = 0.471805 1.329795I
9.5532 10.8949I 8.89518 + 7.36414I
u = 1.53072 + 0.47138I
a = 1.004240 + 0.308031I
b = 0.471805 + 1.329795I
9.5532 + 10.8949I 8.89518 7.36414I
u = 1.11928
a = 0.613269
b = 1.25729
0.360724 18.5117
u = 0.94726 1.06647I
a = 0.017391 + 0.198314I
b = 0.267923 0.935360I
0.17781 + 3.00927I 2.42568 8.45199I
u = 0.94726 + 1.06647I
a = 0.017391 0.198314I
b = 0.267923 + 0.935360I
0.17781 3.00927I 2.42568 + 8.45199I
u = 0.516469 0.282270I
a = 0.570854 + 0.616463I
b = 0.531392 + 0.374501I
0.81916 1.15831I 3.24306 + 4.91464I
u = 0.516469 + 0.282270I
a = 0.570854 0.616463I
b = 0.531392 0.374501I
0.81916 + 1.15831I 3.24306 4.91464I
u = 0.286033 1.106387I
a = 1.005820 0.564651I
b = 0.519928 + 1.174349I
2.25077 9.90431I 2.68871 + 7.61704I
12
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.286033 + 1.106387I
a = 1.005820 + 0.564651I
b = 0.519928 1.174349I
2.25077 + 9.90431I 2.68871 7.61704I
u = 0.153651 0.420526I
a = 1.72913 + 1.40797I
b = 0.893893 + 0.461606I
7.21746 + 0.43098I 4.81736 2.08890I
u = 0.153651 + 0.420526I
a = 1.72913 1.40797I
b = 0.893893 0.461606I
7.21746 0.43098I 4.81736 + 2.08890I
u = 0.567252 1.239207I
a = 0.639135 0.274799I
b = 0.304231 + 1.040143I
3.20816 + 4.89414I 6.47528 9.10540I
u = 0.567252 + 1.239207I
a = 0.639135 + 0.274799I
b = 0.304231 1.040143I
3.20816 4.89414I 6.47528 + 9.10540I
u = 0.839093
a = 0.647401
b = 0.295995
1.11076 11.5717
u = 1.212209 0.156409I
a = 0.681685 + 0.235402I
b = 1.43217 0.42237I
4.03174 + 1.74144I 4.49330 4.13639I
u = 1.212209 + 0.156409I
a = 0.681685 0.235402I
b = 1.43217 + 0.42237I
4.03174 1.74144I 4.49330 + 4.13639I
u = 1.46819 0.46845I
a = 1.240020 0.398728I
b = 0.60863 1.34961I
3.2485 + 15.4927I 5.17708 7.97844I
u = 1.46819 + 0.46845I
a = 1.240020 + 0.398728I
b = 0.60863 + 1.34961I
3.2485 15.4927I 5.17708 + 7.97844I
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.65341 0.45493I
a = 0.670537 0.283862I
b = 0.313637 1.227102I
8.56742 + 4.93041I 9.24363 4.50732I
u = 1.65341 + 0.45493I
a = 0.670537 + 0.283862I
b = 0.313637 + 1.227102I
8.56742 4.93041I 9.24363 + 4.50732I
13
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
11
(u 1)(u + 1)
2
(u
22
u
21
+ ··· + 7u 3)(u
36
+ 3u
35
+ ··· + 8u + 1)
c
2
(u 1)(u + 1)
2
(1 u + 3u
2
6u
3
14u
6
+ 16u
7
+ 13u
8
11u
9
+ 11u
10
8u
11
27u
12
+ 13u
13
+ 20u
14
6u
15
7u
16
+ u
17
+ u
18
)
2
(u
22
9u
20
+ ··· + 7u 24)
c
3
(2u + 1)(4u
2
+ 4u + 3)(8u
22
20u
21
+ ··· 2u
2
+ 1)
(9u
36
+ 27u
35
+ ··· 20172u + 3559)
c
4
, c
9
, c
10
u(u
2
+ 2)
(1 + 3u u
2
2u
3
+ 4u
4
+ 2u
5
+ 4u
7
+ 9u
8
19u
9
+ 43u
10
36u
11
+ 55u
12
25u
13
+ 32u
14
8u
15
+ 9u
16
u
17
+ u
18
)
2
(u
22
+ 3u
21
+ ··· + 56u + 8)
c
5
(2u 1)(4u
2
4u + 3)(8u
22
20u
21
+ ··· 2u
2
+ 1)
(9u
36
+ 27u
35
+ ··· 20172u + 3559)
c
6
(u 1)
2
(u + 1)
(1 u + 3u
2
6u
3
14u
6
+ 16u
7
+ 13u
8
11u
9
+ 11u
10
8u
11
27u
12
+ 13u
13
+ 20u
14
6u
15
7u
16
+ u
17
+ u
18
)
2
(u
22
9u
20
+ ··· + 7u 24)
c
7
, c
8
(u 1)
2
(u + 1)(u
22
u
21
+ ··· + 7u 3)(u
36
+ 3u
35
+ ··· + 8u + 1)
14
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
, c
8
c
11
(y 1)
3
(y
22
+ 9y
21
+ ··· 55y + 9)(y
36
+ 23y
35
+ ··· 16y + 1)
c
2
, c
6
(y 1)
3
(1 7y 3y
2
8y
3
78y
4
+ 226y
5
+ 168y
6
892y
7
+ 491y
8
+ 1063y
9
1715y
10
+ 536y
11
+ 1003y
12
1451y
13
+ 972y
14
396y
15
+ 101y
16
15y
17
+ y
18
)
2
(y
22
18y
21
+ ··· + 335y + 576)
c
3
, c
5
(4y 1)(16y
2
+ 8y + 9)(64y
22
240y
21
+ ··· 4y + 1)
(81y
36
1377y
35
+ ··· 71011164y + 12666481)
c
4
, c
9
, c
10
y(y + 2)
2
(1 7y + 5y
2
24y
3
18y
4
+ 22y
5
+ 236y
7
+ 683y
8
+ 1011y
9
+ 1769y
10
+ 3136y
11
+ 3843y
12
+ 3073y
13
+ 1628y
14
+ 572y
15
+ 129y
16
+ 17y
17
+ y
18
)
2
(y
22
+ 21y
21
+ ··· 480y + 64)
15