12n
0579
(K12n
0579
)
A knot diagram
1
Linearized knot diagam
3 7 9 10 8 11 2 5 1 5 7 10
Solving Sequence
2,8
7 3
1,11
6 5 9 4 10 12
c
7
c
2
c
1
c
6
c
5
c
8
c
3
c
10
c
12
c
4
, c
9
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−11u
19
+ 125u
18
+ ··· + 8b + 264, −11u
19
+ 91u
18
+ ··· + 16a − 128, u
20
− 11u
19
+ ··· − 80u + 16i
I
u
2
= hu
15
+ 2u
14
+ ··· + b + 1, −2u
15
− 2u
14
+ ··· + a + 4,
u
16
+ u
15
+ 4u
14
+ 3u
13
+ 10u
12
+ 5u
11
+ 15u
10
+ 4u
9
+ 17u
8
+ u
7
+ 14u
6
− 2u
5
+ 10u
4
− 2u
3
+ 4u
2
− u + 1i
* 2 irreducible components of dim
C
= 0, with total 36 representations.
1
The image of knot diagram is generated by the software “Draw programme” developed by An-
drew Bartholomew(http://www.layer8.co.uk/maths/draw/index.htm#Running-draw), where we modi-
fied some parts for our purpose(https://github.com/CATsTAILs/LinksPainter).
2
All coefficients of polynomials are rational numbers. But the coefficients are sometimes approximated
in decimal forms when there is not enough margin.
1
I. I
u
1
= h−11u
19
+ 125u
18
+ · · · + 8b + 264, −11u
19
+ 91u
18
+ · · · + 16a −
128, u
20
− 11u
19
+ · · · − 80u + 16i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
11
=
11
16
u
19
−
91
16
u
18
+ ··· − 23u + 8
11
8
u
19
−
125
8
u
18
+ ··· + 140u − 33
a
6
=
−
1
4
u
19
+
5
2
u
18
+ ··· − 10u +
5
2
1
4
u
19
−
9
4
u
18
+ ··· −
15
2
u
2
+
3
2
u
a
5
=
1
4
u
18
−
9
4
u
17
+ ··· −
17
2
u +
5
2
1
4
u
19
−
9
4
u
18
+ ··· −
15
2
u
2
+
3
2
u
a
9
=
2u
19
−
83
4
u
18
+ ··· +
419
4
u − 21
5
4
u
19
−
29
2
u
18
+ ··· + 122u − 28
a
4
=
−
59
16
u
19
+
475
16
u
18
+ ··· + 147u − 48
−
79
8
u
19
+
769
8
u
18
+ ··· − 274u + 45
a
10
=
7u
19
−
291
4
u
18
+ ··· +
1491
4
u − 77
4u
19
−
193
4
u
18
+ ··· + 478u − 112
a
12
=
1
16
u
19
+
3
16
u
18
+ ··· − 22u + 5
7
8
u
19
−
73
8
u
18
+ ··· + 70u − 17
(ii) Obstruction class = −1
(iii) Cusp Shapes =
19
2
u
19
−
231
2
u
18
+ 698u
17
−
5601
2
u
16
+ 8288u
15
−
38259
2
u
14
+
71281
2
u
13
− 54910u
12
+
142795
2
u
11
−
159597
2
u
10
+ 77978u
9
− 67433u
8
+ 51758u
7
−
35082u
6
+ 20913u
5
−
22571
2
u
4
+
11687
2
u
3
− 2983u
2
+ 1252u − 290
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
20
+ 7u
19
+ ··· + 640u + 256
c
2
, c
7
u
20
+ 11u
19
+ ··· + 80u + 16
c
3
u
20
+ 2u
19
+ ··· + 55u + 1477
c
4
, c
10
u
20
+ 21u
18
+ ··· + 59u + 42
c
5
, c
8
u
20
− 3u
19
+ ··· − 2u + 1
c
6
, c
11
u
20
+ u
19
+ ··· + u + 1
c
9
, c
12
u
20
− 3u
19
+ ··· − 4u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
20
+ 11y
19
+ ··· + 1155072y + 65536
c
2
, c
7
y
20
+ 7y
19
+ ··· + 640y + 256
c
3
y
20
− 22y
19
+ ··· − 6941971y + 2181529
c
4
, c
10
y
20
+ 42y
19
+ ··· + 22475y + 1764
c
5
, c
8
y
20
− 43y
19
+ ··· + 54y + 1
c
6
, c
11
y
20
+ 35y
19
+ ··· − 7y + 1
c
9
, c
12
y
20
+ 3y
19
+ ··· + 10y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.374976 + 0.845868I
a = 0.304346 − 0.429032I
b = −0.004516 + 0.301455I
−0.45102 − 2.09204I 1.92279 + 3.74231I
u = −0.374976 − 0.845868I
a = 0.304346 + 0.429032I
b = −0.004516 − 0.301455I
−0.45102 + 2.09204I 1.92279 − 3.74231I
u = 0.282023 + 1.083820I
a = 0.39008 − 1.59752I
b = 0.80004 + 1.16903I
−3.81050 + 0.28255I −5.79309 − 4.21842I
u = 0.282023 − 1.083820I
a = 0.39008 + 1.59752I
b = 0.80004 − 1.16903I
−3.81050 − 0.28255I −5.79309 + 4.21842I
u = 0.684540 + 0.363080I
a = 0.592020 + 0.442074I
b = −0.478548 + 1.089890I
0.21798 − 2.21625I 4.78739 + 3.65383I
u = 0.684540 − 0.363080I
a = 0.592020 − 0.442074I
b = −0.478548 − 1.089890I
0.21798 + 2.21625I 4.78739 − 3.65383I
u = 0.854003 + 0.892169I
a = 0.506066 + 0.028446I
b = −0.446717 + 0.529073I
7.04464 + 1.66022I 3.68035 + 3.33714I
u = 0.854003 − 0.892169I
a = 0.506066 − 0.028446I
b = −0.446717 − 0.529073I
7.04464 − 1.66022I 3.68035 − 3.33714I
u = 0.557284 + 1.103710I
a = −0.91469 + 1.63155I
b = −0.64503 − 1.83466I
−1.93145 + 7.03308I 4.92278 − 7.07574I
u = 0.557284 − 1.103710I
a = −0.91469 − 1.63155I
b = −0.64503 + 1.83466I
−1.93145 − 7.03308I 4.92278 + 7.07574I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.843906 + 0.926863I
a = −0.147807 + 0.491044I
b = −0.393488 − 0.559354I
6.93872 + 4.64909I 6.55707 − 9.56183I
u = 0.843906 − 0.926863I
a = −0.147807 − 0.491044I
b = −0.393488 + 0.559354I
6.93872 − 4.64909I 6.55707 + 9.56183I
u = −0.408387 + 0.451294I
a = 0.739810 + 0.208470I
b = −0.306448 + 0.078414I
0.672383 − 0.998162I 5.05990 + 5.48797I
u = −0.408387 − 0.451294I
a = 0.739810 − 0.208470I
b = −0.306448 − 0.078414I
0.672383 + 0.998162I 5.05990 − 5.48797I
u = 1.50042 + 0.03858I
a = 0.039797 − 0.614362I
b = −0.02477 + 1.84088I
−10.85850 − 3.49800I 1.91176 + 2.12217I
u = 1.50042 − 0.03858I
a = 0.039797 + 0.614362I
b = −0.02477 − 1.84088I
−10.85850 + 3.49800I 1.91176 − 2.12217I
u = 0.75634 + 1.50493I
a = 0.88084 − 1.46544I
b = 0.16039 + 1.88216I
−15.5277 + 4.3237I − 60.10 − 0.824539I
u = 0.75634 − 1.50493I
a = 0.88084 + 1.46544I
b = 0.16039 − 1.88216I
−15.5277 − 4.3237I − 60.10 + 0.824539I
u = 0.80485 + 1.48779I
a = −0.89046 + 1.47434I
b = −0.16092 − 1.93495I
−15.1932 + 11.4769I 0. − 4.87637I
u = 0.80485 − 1.48779I
a = −0.89046 − 1.47434I
b = −0.16092 + 1.93495I
−15.1932 − 11.4769I 0. + 4.87637I
6
II.
I
u
2
= hu
15
+2u
14
+· · ·+b+1, −2u
15
−2u
14
+· · ·+a+4, u
16
+u
15
+· · ·−u+1i
(i) Arc colorings
a
2
=
0
u
a
8
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
1
=
u
3
u
5
+ u
3
+ u
a
11
=
2u
15
+ 2u
14
+ ··· + 5u − 4
−u
15
− 2u
14
+ ··· − 4u − 1
a
6
=
u
14
+ u
13
+ ··· − 2u + 4
u
15
+ u
14
+ ··· + 3u − 1
a
5
=
u
15
+ 2u
14
+ ··· + u + 3
u
15
+ u
14
+ ··· + 3u − 1
a
9
=
2u
15
+ u
14
+ ··· + u − 2
−u
15
− 2u
14
+ ··· − 2u − 1
a
4
=
−u
15
+ u
14
+ ··· − 7u + 5
u
12
+ u
11
+ ··· + 2u + 1
a
10
=
2u
15
+ u
14
+ ··· + 2u − 2
−2u
15
− 3u
14
+ ··· − 2u
2
− 3u
a
12
=
u
15
+ u
14
+ ··· + 3u − 5
−u
15
− 3u
14
+ ··· − 3u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
14
− 7u
13
− 13u
12
− 19u
11
− 31u
10
− 41u
9
− 44u
8
− 45u
7
−
43u
6
− 39u
5
− 25u
4
− 13u
3
− 10u
2
− 9u − 2
7
(iv) u-Polynomials at the component
8
Crossings u-Polynomials at each crossing
c
1
u
16
− 7u
15
+ ··· − 7u + 1
c
2
u
16
− u
15
+ ··· + u + 1
c
3
u
16
− u
15
+ ··· − 15u + 5
c
4
u
16
− u
15
+ ··· + 9u
2
+ 5
c
5
u
16
− 2u
15
+ ··· − 2u
2
+ 1
c
6
u
16
− 3u
14
+ ··· + u + 1
c
7
u
16
+ u
15
+ ··· − u + 1
c
8
u
16
+ 2u
15
+ ··· − 2u
2
+ 1
c
9
u
16
− 6u
15
+ ··· − 4u + 1
c
10
u
16
+ u
15
+ ··· + 9u
2
+ 5
c
11
u
16
− 3u
14
+ ··· − u + 1
c
12
u
16
+ 6u
15
+ ··· + 4u + 1
9
10
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
16
+ 11y
15
+ ··· + 15y + 1
c
2
, c
7
y
16
+ 7y
15
+ ··· + 7y + 1
c
3
y
16
+ 13y
15
+ ··· + 415y + 25
c
4
, c
10
y
16
− 3y
15
+ ··· + 90y + 25
c
5
, c
8
y
16
+ 4y
15
+ ··· − 4y + 1
c
6
, c
11
y
16
− 6y
15
+ ··· + 7y + 1
c
9
, c
12
y
16
+ 2y
15
+ ··· + 8y + 1
11
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.410507 + 1.049820I
a = 0.365556 − 1.223390I
b = 0.216644 + 0.789371I
3.41967 − 3.66276I −0.18386 + 3.97608I
u = −0.410507 − 1.049820I
a = 0.365556 + 1.223390I
b = 0.216644 − 0.789371I
3.41967 + 3.66276I −0.18386 − 3.97608I
u = 0.373058 + 1.082120I
a = 0.93325 − 1.77713I
b = 0.67039 + 1.71046I
−3.80708 − 0.42521I −5.76487 + 5.34400I
u = 0.373058 − 1.082120I
a = 0.93325 + 1.77713I
b = 0.67039 − 1.71046I
−3.80708 + 0.42521I −5.76487 − 5.34400I
u = 0.592477 + 0.599555I
a = 1.53653 + 0.05346I
b = −0.40603 + 1.74092I
−1.07065 − 2.50055I −0.64235 + 5.46517I
u = 0.592477 − 0.599555I
a = 1.53653 − 0.05346I
b = −0.40603 − 1.74092I
−1.07065 + 2.50055I −0.64235 − 5.46517I
u = −0.847455 + 0.790735I
a = 0.510736 + 0.409085I
b = −0.357187 − 0.121933I
7.39011 − 2.39888I 8.56082 + 4.71015I
u = −0.847455 − 0.790735I
a = 0.510736 − 0.409085I
b = −0.357187 + 0.121933I
7.39011 + 2.39888I 8.56082 − 4.71015I
u = 0.558989 + 1.054820I
a = −1.41911 + 1.69221I
b = −0.42524 − 2.26646I
−2.54997 + 7.14014I −8.11047 − 10.17470I
u = 0.558989 − 1.054820I
a = −1.41911 − 1.69221I
b = −0.42524 + 2.26646I
−2.54997 − 7.14014I −8.11047 + 10.17470I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.293434 + 0.723430I
a = −0.07923 + 1.82784I
b = −0.569488 − 0.777509I
4.75010 + 0.63044I 3.31942 − 3.05192I
u = −0.293434 − 0.723430I
a = −0.07923 − 1.82784I
b = −0.569488 + 0.777509I
4.75010 − 0.63044I 3.31942 + 3.05192I
u = −0.858001 + 0.997836I
a = 0.357745 − 0.374169I
b = −0.124363 + 0.298124I
6.77645 − 3.96119I 3.55545 − 1.03144I
u = −0.858001 − 0.997836I
a = 0.357745 + 0.374169I
b = −0.124363 − 0.298124I
6.77645 + 3.96119I 3.55545 + 1.03144I
u = 0.384873 + 0.519863I
a = −1.70547 + 0.11584I
b = −0.004726 − 1.372860I
−1.74916 + 3.71981I −1.23414 − 3.33267I
u = 0.384873 − 0.519863I
a = −1.70547 − 0.11584I
b = −0.004726 + 1.372860I
−1.74916 − 3.71981I −1.23414 + 3.33267I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
16
− 7u
15
+ ··· − 7u + 1)(u
20
+ 7u
19
+ ··· + 640u + 256)
c
2
(u
16
− u
15
+ ··· + u + 1)(u
20
+ 11u
19
+ ··· + 80u + 16)
c
3
(u
16
− u
15
+ ··· − 15u + 5)(u
20
+ 2u
19
+ ··· + 55u + 1477)
c
4
(u
16
− u
15
+ ··· + 9u
2
+ 5)(u
20
+ 21u
18
+ ··· + 59u + 42)
c
5
(u
16
− 2u
15
+ ··· − 2u
2
+ 1)(u
20
− 3u
19
+ ··· − 2u + 1)
c
6
(u
16
− 3u
14
+ ··· + u + 1)(u
20
+ u
19
+ ··· + u + 1)
c
7
(u
16
+ u
15
+ ··· − u + 1)(u
20
+ 11u
19
+ ··· + 80u + 16)
c
8
(u
16
+ 2u
15
+ ··· − 2u
2
+ 1)(u
20
− 3u
19
+ ··· − 2u + 1)
c
9
(u
16
− 6u
15
+ ··· − 4u + 1)(u
20
− 3u
19
+ ··· − 4u + 1)
c
10
(u
16
+ u
15
+ ··· + 9u
2
+ 5)(u
20
+ 21u
18
+ ··· + 59u + 42)
c
11
(u
16
− 3u
14
+ ··· − u + 1)(u
20
+ u
19
+ ··· + u + 1)
c
12
(u
16
+ 6u
15
+ ··· + 4u + 1)(u
20
− 3u
19
+ ··· − 4u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
(y
16
+ 11y
15
+ ··· + 15y + 1)(y
20
+ 11y
19
+ ··· + 1155072y + 65536)
c
2
, c
7
(y
16
+ 7y
15
+ ··· + 7y + 1)(y
20
+ 7y
19
+ ··· + 640y + 256)
c
3
(y
16
+ 13y
15
+ ··· + 415y + 25)
· (y
20
− 22y
19
+ ··· − 6941971y + 2181529)
c
4
, c
10
(y
16
− 3y
15
+ ··· + 90y + 25)(y
20
+ 42y
19
+ ··· + 22475y + 1764)
c
5
, c
8
(y
16
+ 4y
15
+ ··· − 4y + 1)(y
20
− 43y
19
+ ··· + 54y + 1)
c
6
, c
11
(y
16
− 6y
15
+ ··· + 7y + 1)(y
20
+ 35y
19
+ ··· − 7y + 1)
c
9
, c
12
(y
16
+ 2y
15
+ ··· + 8y + 1)(y
20
+ 3y
19
+ ··· + 10y + 1)
15
