12a
0345
(K12a
0345
)
A knot diagram
1
Linearized knot diagam
3 6 9 7 2 5 11 4 1 12 8 10
Solving Sequence
2,6
3 1 5 7
4,10
9 8 12 11
c
2
c
1
c
5
c
6
c
4
c
9
c
8
c
12
c
10
c
3
, c
7
, c
11
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
9
+ u
7
− u
6
− 3u
5
+ 2u
3
+ b − u, u
12
− u
10
+ 5u
8
− 4u
6
+ 6u
4
− 3u
2
+ a − u + 2,
u
13
− u
12
− u
11
+ 2u
10
+ 4u
9
− 5u
8
− 3u
7
+ 6u
6
+ 4u
5
− 6u
4
− 2u
3
+ 2u
2
+ 2u − 1i
I
u
2
= h−3u
47
+ 17u
46
+ ··· + 2b + 1, −11u
47
+ 29u
46
+ ··· + 2a + 3, u
48
− 3u
47
+ ··· − 2u + 1i
I
u
3
= hb + u, a + u, u
3
+ u
2
− 1i
I
u
4
= hb − a, u
2
a + a
2
+ u
2
+ 2u + 1, u
3
+ u
2
− 1i
* 4 irreducible components of dim
C
= 0, with total 70 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−u
9
+ u
7
− u
6
− 3u
5
+ 2u
3
+ b − u, u
12
− u
10
+ 5u
8
− 4u
6
+ 6u
4
−
3u
2
+ a − u + 2, u
13
− u
12
+ · · · + 2u − 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
7
=
−u
3
−u
3
+ u
a
4
=
u
5
+ u
u
5
− u
3
+ u
a
10
=
−u
12
+ u
10
− 5u
8
+ 4u
6
− 6u
4
+ 3u
2
+ u − 2
u
9
− u
7
+ u
6
+ 3u
5
− 2u
3
+ u
a
9
=
−u
12
+ u
10
− 5u
8
+ 4u
6
− 7u
4
+ 4u
2
+ u − 2
u
9
− u
7
+ 3u
5
− 2u
3
+ u
a
8
=
−u
8
+ u
6
− u
5
− 3u
4
+ 2u
2
− 1
u
12
− 2u
10
+ u
9
+ 5u
8
− u
7
− 6u
6
+ 2u
5
+ 6u
4
− u
3
− 3u
2
+ 1
a
12
=
u
12
− u
11
− u
10
+ u
9
+ 4u
8
− 3u
7
− 4u
6
+ 2u
5
+ 5u
4
− u
3
− 3u
2
− u + 2
−u
11
+ u
9
− u
8
− 3u
7
+ 2u
5
− u
4
− u
3
a
11
=
−u
12
+ u
11
+ u
10
− 2u
9
− 4u
8
+ 4u
7
+ 3u
6
− 5u
5
− 5u
4
+ 3u
3
+ 3u
2
− 2
u
12
− u
10
+ 5u
8
− 4u
6
+ 6u
4
− 2u
2
− u + 1
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
12
−2u
10
+ 18u
8
+ 2u
7
−6u
6
−4u
5
+ 20u
4
+ 6u
3
−2u
2
−12u −6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
, c
12
u
13
+ 3u
12
+ ··· + 8u + 1
c
2
, c
5
, c
7
c
11
u
13
+ u
12
+ ··· + 2u + 1
c
3
, c
8
u
13
− 7u
12
+ ··· − 24u + 8
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
, c
12
y
13
+ 17y
12
+ ··· + 16y −1
c
2
, c
5
, c
7
c
11
y
13
− 3y
12
+ ··· + 8y −1
c
3
, c
8
y
13
+ 7y
12
+ ··· + 128y −64
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.923828 + 0.421970I
a = 0.763252 + 0.399422I
b = 0.261288 − 0.366463I
−0.05055 + 6.40816I −12.2349 − 10.2893I
u = −0.923828 − 0.421970I
a = 0.763252 − 0.399422I
b = 0.261288 + 0.366463I
−0.05055 − 6.40816I −12.2349 + 10.2893I
u = 0.801388 + 0.281223I
a = 0.268761 − 0.576127I
b = −0.168515 + 0.695868I
−2.04094 − 2.18131I −16.1482 + 7.3921I
u = 0.801388 − 0.281223I
a = 0.268761 + 0.576127I
b = −0.168515 − 0.695868I
−2.04094 + 2.18131I −16.1482 − 7.3921I
u = −0.537404 + 0.591513I
a = −0.786204 − 0.602557I
b = −0.632206 − 0.390230I
2.68260 + 1.42666I −3.83184 − 3.78939I
u = −0.537404 − 0.591513I
a = −0.786204 + 0.602557I
b = −0.632206 + 0.390230I
2.68260 − 1.42666I −3.83184 + 3.78939I
u = 0.855993 + 0.936945I
a = −3.04281 − 0.69376I
b = −1.58297 − 3.59981I
18.0409 + 0.9847I −4.10351 + 1.46024I
u = 0.855993 − 0.936945I
a = −3.04281 + 0.69376I
b = −1.58297 + 3.59981I
18.0409 − 0.9847I −4.10351 − 1.46024I
u = −0.928636 + 0.877531I
a = −3.22583 + 2.14654I
b = −0.92820 + 5.06333I
12.42090 + 6.50749I −5.85522 − 4.78409I
u = −0.928636 − 0.877531I
a = −3.22583 − 2.14654I
b = −0.92820 − 5.06333I
12.42090 − 6.50749I −5.85522 + 4.78409I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002580 + 0.857613I
a = −1.90418 − 2.60319I
b = 0.38407 − 4.60433I
17.0728 − 14.2174I −5.54739 + 7.80237I
u = 1.002580 − 0.857613I
a = −1.90418 + 2.60319I
b = 0.38407 + 4.60433I
17.0728 + 14.2174I −5.54739 − 7.80237I
u = 0.459806
a = −1.14598
b = 0.333063
−0.845259 −10.5580
6
II. I
u
2
= h−3u
47
+ 17u
46
+ · · · + 2b + 1, −11u
47
+ 29u
46
+ · · · + 2a + 3, u
48
−
3u
47
+ · · · − 2u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
4
a
5
=
u
u
a
7
=
−u
3
−u
3
+ u
a
4
=
u
5
+ u
u
5
− u
3
+ u
a
10
=
11
2
u
47
−
29
2
u
46
+ ··· −
17
2
u −
3
2
3
2
u
47
−
17
2
u
46
+ ··· −
7
2
u −
1
2
a
9
=
17
2
u
47
− 23u
46
+ ··· − 9u −
5
2
2u
47
−
27
2
u
46
+ ··· −
1
2
u −
7
2
a
8
=
−
1
2
u
47
+ 9u
45
+ ··· − 19u +
9
2
−2u
47
+
13
2
u
46
+ ··· −
19
2
u +
11
2
a
12
=
−
1
2
u
45
+ u
44
+ ··· +
7
2
u
2
+
3
2
−
1
2
u
45
+ u
44
+ ··· +
15
2
u
2
−
1
2
a
11
=
−
1
2
u
47
+ 3u
46
+ ··· + 7u − 5
−
1
2
u
46
− u
45
+ ··· +
13
2
u − 2
(ii) Obstruction class = −1
(iii) Cusp Shapes = −u
47
+
23
2
u
46
+ ··· −
77
2
u +
11
2
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
, c
12
u
48
+ 11u
47
+ ··· + 28u + 1
c
2
, c
5
, c
7
c
11
u
48
+ 3u
47
+ ··· + 2u + 1
c
3
, c
8
(u
24
+ 3u
23
+ ··· + 20u + 8)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
, c
12
y
48
+ 53y
47
+ ··· − 188y + 1
c
2
, c
5
, c
7
c
11
y
48
− 11y
47
+ ··· − 28y + 1
c
3
, c
8
(y
24
+ 21y
23
+ ··· + 496y + 64)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.841794 + 0.516230I
a = −0.724878 − 0.760407I
b = −0.543660 − 0.322602I
1.74731 + 2.75249I −5.85663 − 4.25990I
u = −0.841794 − 0.516230I
a = −0.724878 + 0.760407I
b = −0.543660 + 0.322602I
1.74731 − 2.75249I −5.85663 + 4.25990I
u = 0.930322 + 0.093737I
a = 0.274062 − 0.684145I
b = 0.528392 − 0.143631I
−1.87399 + 1.33150I −14.6087 − 4.8388I
u = 0.930322 − 0.093737I
a = 0.274062 + 0.684145I
b = 0.528392 + 0.143631I
−1.87399 − 1.33150I −14.6087 + 4.8388I
u = 1.074940 + 0.011887I
a = 0.08400 − 1.42217I
b = 0.24511 − 2.16612I
5.18079 + 3.17559I −10.00740 − 2.50769I
u = 1.074940 − 0.011887I
a = 0.08400 + 1.42217I
b = 0.24511 + 2.16612I
5.18079 − 3.17559I −10.00740 + 2.50769I
u = 0.746614 + 0.498009I
a = −0.183156 − 1.279540I
b = −1.014170 + 0.309519I
4.20175 − 4.96532I −9.50591 + 5.64619I
u = 0.746614 − 0.498009I
a = −0.183156 + 1.279540I
b = −1.014170 − 0.309519I
4.20175 + 4.96532I −9.50591 − 5.64619I
u = −0.366415 + 0.805565I
a = −1.17167 + 1.18416I
b = 0.184920 − 0.596372I
10.50330 + 1.42992I −3.65523 − 2.00550I
u = −0.366415 − 0.805565I
a = −1.17167 − 1.18416I
b = 0.184920 + 0.596372I
10.50330 − 1.42992I −3.65523 + 2.00550I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.336222 + 0.803215I
a = 0.87556 − 1.36537I
b = −0.137852 + 0.789514I
10.34730 − 5.08751I −3.92984 + 2.90656I
u = −0.336222 − 0.803215I
a = 0.87556 + 1.36537I
b = −0.137852 − 0.789514I
10.34730 + 5.08751I −3.92984 − 2.90656I
u = −0.880758 + 0.714367I
a = −0.556508 − 0.593812I
b = −0.762556 − 0.329885I
2.31973 + 2.73395I − 60.10 − 0.676902I
u = −0.880758 − 0.714367I
a = −0.556508 + 0.593812I
b = −0.762556 + 0.329885I
2.31973 − 2.73395I − 60.10 + 0.676902I
u = 0.696228 + 0.511765I
a = −0.02775 + 1.48585I
b = 0.840769 − 0.158612I
4.36673 + 1.08082I −8.65197 + 0.09587I
u = 0.696228 − 0.511765I
a = −0.02775 − 1.48585I
b = 0.840769 + 0.158612I
4.36673 − 1.08082I −8.65197 − 0.09587I
u = −1.040760 + 0.477829I
a = −0.022096 + 0.260934I
b = −1.60902 + 0.01527I
8.01455 + 9.71739I −7.98988 − 8.06760I
u = −1.040760 − 0.477829I
a = −0.022096 − 0.260934I
b = −1.60902 − 0.01527I
8.01455 − 9.71739I −7.98988 + 8.06760I
u = −1.035350 + 0.498737I
a = 0.036384 − 0.448348I
b = 1.46608 − 0.47696I
8.29415 + 3.29720I −7.24610 − 3.17160I
u = −1.035350 − 0.498737I
a = 0.036384 + 0.448348I
b = 1.46608 + 0.47696I
8.29415 − 3.29720I −7.24610 + 3.17160I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.751107 + 0.340936I
a = 1.07827 + 0.93654I
b = 1.136920 + 0.261943I
−1.87399 + 1.33150I −14.6087 − 4.8388I
u = −0.751107 − 0.340936I
a = 1.07827 − 0.93654I
b = 1.136920 − 0.261943I
−1.87399 − 1.33150I −14.6087 + 4.8388I
u = −0.864737 + 0.816422I
a = 1.00270 − 1.60989I
b = −0.28591 − 2.01649I
4.36673 + 1.08082I −8.65197 + 0.I
u = −0.864737 − 0.816422I
a = 1.00270 + 1.60989I
b = −0.28591 + 2.01649I
4.36673 − 1.08082I −8.65197 + 0.I
u = −0.917877 + 0.800067I
a = −1.77023 + 0.64969I
b = −1.31308 + 1.88984I
4.20175 + 4.96532I −12.00000 − 5.64619I
u = −0.917877 − 0.800067I
a = −1.77023 − 0.64969I
b = −1.31308 − 1.88984I
4.20175 − 4.96532I −12.00000 + 5.64619I
u = 0.871124 + 0.885192I
a = 1.28643 + 0.63717I
b = 0.88470 + 1.89792I
8.29415 + 3.29720I −12.00000 + 0.I
u = 0.871124 − 0.885192I
a = 1.28643 − 0.63717I
b = 0.88470 − 1.89792I
8.29415 − 3.29720I −12.00000 + 0.I
u = 0.913300 + 0.855274I
a = −0.195582 − 0.163086I
b = 0.346423 + 0.202860I
5.18079 − 3.17559I −12.00000 + 0.I
u = 0.913300 − 0.855274I
a = −0.195582 + 0.163086I
b = 0.346423 − 0.202860I
5.18079 + 3.17559I −12.00000 + 0.I
12
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.845587 + 0.935523I
a = 3.03496 + 1.05311I
b = 1.18613 + 3.70268I
17.5794 + 7.6165I −12.00000 + 0.I
u = 0.845587 − 0.935523I
a = 3.03496 − 1.05311I
b = 1.18613 − 3.70268I
17.5794 − 7.6165I −12.00000 + 0.I
u = 0.898811 + 0.892925I
a = −1.169680 + 0.302901I
b = −1.61570 − 1.19571I
10.50330 − 1.42992I −12.00000 + 0.I
u = 0.898811 − 0.892925I
a = −1.169680 − 0.302901I
b = −1.61570 + 1.19571I
10.50330 + 1.42992I −12.00000 + 0.I
u = −0.917895 + 0.882530I
a = 3.01654 − 2.45632I
b = 0.32556 − 5.04361I
12.4555 0
u = −0.917895 − 0.882530I
a = 3.01654 + 2.45632I
b = 0.32556 + 5.04361I
12.4555 0
u = 0.959444 + 0.848413I
a = −1.18514 − 1.34955I
b = 0.10952 − 1.97053I
8.01455 − 9.71739I 0
u = 0.959444 − 0.848413I
a = −1.18514 + 1.34955I
b = 0.10952 + 1.97053I
8.01455 + 9.71739I 0
u = 0.947556 + 0.870745I
a = 0.33672 + 1.44236I
b = −1.26760 + 1.29298I
10.34730 − 5.08751I 0
u = 0.947556 − 0.870745I
a = 0.33672 − 1.44236I
b = −1.26760 − 1.29298I
10.34730 + 5.08751I 0
13
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.998762 + 0.865177I
a = 1.62848 + 2.70839I
b = −0.93674 + 4.41906I
17.5794 − 7.6165I 0
u = 0.998762 − 0.865177I
a = 1.62848 − 2.70839I
b = −0.93674 − 4.41906I
17.5794 + 7.6165I 0
u = −0.304534 + 0.542389I
a = −0.125842 + 0.286126I
b = 0.557583 + 0.600347I
1.74731 − 2.75249I −5.85663 + 4.25990I
u = −0.304534 − 0.542389I
a = −0.125842 − 0.286126I
b = 0.557583 − 0.600347I
1.74731 + 2.75249I −5.85663 − 4.25990I
u = 0.579212
a = −0.784912
b = 0.245364
−0.838576 −11.3810
u = −0.551506 + 0.061917I
a = 0.27580 + 2.18978I
b = 0.36476 + 1.51138I
2.31973 − 2.73395I 0.279102 + 0.676902I
u = −0.551506 − 0.061917I
a = 0.27580 − 2.18978I
b = 0.36476 − 1.51138I
2.31973 + 2.73395I 0.279102 − 0.676902I
u = 0.273323
a = −1.80982
b = 0.373488
−0.838576 −11.3810
14
III. I
u
3
= hb + u, a + u, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
4
=
1
u
2
a
10
=
−u
−u
a
9
=
−1
−u
2
a
8
=
−1
−u
2
a
12
=
0
−u
a
11
=
−u
u
2
− u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −8u − 12
15
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
9
c
10
u
3
− u
2
+ 2u − 1
c
2
, c
7
u
3
+ u
2
− 1
c
3
, c
8
u
3
c
5
, c
11
u
3
− u
2
+ 1
c
6
, c
12
u
3
+ u
2
+ 2u + 1
16
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
, c
12
y
3
+ 3y
2
+ 2y − 1
c
2
, c
5
, c
7
c
11
y
3
− y
2
+ 2y − 1
c
3
, c
8
y
3
17
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = 0.877439 − 0.744862I
b = 0.877439 − 0.744862I
6.04826 + 5.65624I −4.98049 − 5.95889I
u = −0.877439 − 0.744862I
a = 0.877439 + 0.744862I
b = 0.877439 + 0.744862I
6.04826 − 5.65624I −4.98049 + 5.95889I
u = 0.754878
a = −0.754878
b = −0.754878
−2.22691 −18.0390
18
IV. I
u
4
= hb − a, u
2
a + a
2
+ u
2
+ 2u + 1, u
3
+ u
2
− 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
1
=
−u
2
+ 1
−u
2
− u + 1
a
5
=
u
u
a
7
=
u
2
− 1
u
2
+ u − 1
a
4
=
1
u
2
a
10
=
a
a
a
9
=
u
2
a + au
au
a
8
=
u
2
a + au
au
a
12
=
−u
2
a + a + u + 2
−u
2
a + a + 2
a
11
=
−2u
2
a − au + a + 1
−u
2
a − au + a + 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 8u
2
a + au − u
2
− 8a − 19
19
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
9
c
10
(u
3
− u
2
+ 2u − 1)
2
c
2
, c
7
(u
3
+ u
2
− 1)
2
c
3
, c
8
u
6
c
5
, c
11
(u
3
− u
2
+ 1)
2
c
6
, c
12
(u
3
+ u
2
+ 2u + 1)
2
20
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
, c
12
(y
3
+ 3y
2
+ 2y − 1)
2
c
2
, c
5
, c
7
c
11
(y
3
− y
2
+ 2y − 1)
2
c
3
, c
8
y
6
21
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
4
√
−1(vol +
√
−1CS) Cusp shape
u = −0.877439 + 0.744862I
a = −0.592519 + 0.986732I
b = −0.592519 + 0.986732I
6.04826 −5.39114 + 0.I
u = −0.877439 + 0.744862I
a = 0.377439 + 0.320410I
b = 0.377439 + 0.320410I
1.91067 + 2.82812I −18.8044 − 4.6518I
u = −0.877439 − 0.744862I
a = −0.592519 − 0.986732I
b = −0.592519 − 0.986732I
6.04826 −5.39114 + 0.I
u = −0.877439 − 0.744862I
a = 0.377439 − 0.320410I
b = 0.377439 − 0.320410I
1.91067 − 2.82812I −18.8044 + 4.6518I
u = 0.754878
a = −0.28492 + 1.73159I
b = −0.28492 + 1.73159I
1.91067 + 2.82812I −18.8044 − 4.6518I
u = 0.754878
a = −0.28492 − 1.73159I
b = −0.28492 − 1.73159I
1.91067 − 2.82812I −18.8044 + 4.6518I
22
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
4
, c
9
c
10
((u
3
− u
2
+ 2u − 1)
3
)(u
13
+ 3u
12
+ ··· + 8u + 1)
· (u
48
+ 11u
47
+ ··· + 28u + 1)
c
2
, c
7
((u
3
+ u
2
− 1)
3
)(u
13
+ u
12
+ ··· + 2u + 1)(u
48
+ 3u
47
+ ··· + 2u + 1)
c
3
, c
8
u
9
(u
13
− 7u
12
+ ··· − 24u + 8)(u
24
+ 3u
23
+ ··· + 20u + 8)
2
c
5
, c
11
((u
3
− u
2
+ 1)
3
)(u
13
+ u
12
+ ··· + 2u + 1)(u
48
+ 3u
47
+ ··· + 2u + 1)
c
6
, c
12
((u
3
+ u
2
+ 2u + 1)
3
)(u
13
+ 3u
12
+ ··· + 8u + 1)
· (u
48
+ 11u
47
+ ··· + 28u + 1)
23
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
, c
6
c
9
, c
10
, c
12
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
13
+ 17y
12
+ ··· + 16y − 1)
· (y
48
+ 53y
47
+ ··· − 188y + 1)
c
2
, c
5
, c
7
c
11
((y
3
− y
2
+ 2y − 1)
3
)(y
13
− 3y
12
+ ··· + 8y − 1)
· (y
48
− 11y
47
+ ··· − 28y + 1)
c
3
, c
8
y
9
(y
13
+ 7y
12
+ ··· + 128y − 64)(y
24
+ 21y
23
+ ··· + 496y + 64)
2
24
