12a
0531
(K12a
0531
)
A knot diagram
1
Linearized knot diagam
3 7 8 10 11 2 6 12 1 5 4 9
Solving Sequence
2,6
7 3 8
4,11
12 1 5 10 9
c
6
c
2
c
7
c
3
c
11
c
1
c
5
c
10
c
9
c
4
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h1.50790 × 10
28
u
83
2.04992 × 10
28
u
82
+ ··· + 4.86000 × 10
27
b + 2.90878 × 10
28
,
5.01481 × 10
27
u
83
+ 7.71256 × 10
27
u
82
+ ··· + 4.86000 × 10
27
a + 1.48950 × 10
28
, u
84
2u
83
+ ··· + 8u 1i
I
u
2
= hu
2
a + 2au 2u
2
+ b + 2a 4u 3, a
2
2au u
2
2a + 6u 1, u
3
+ u
2
1i
I
u
3
= hb, a u + 1, u
3
u
2
+ 1i
* 3 irreducible components of dim
C
= 0, with total 93 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
= h1.51×10
28
u
83
2.05×10
28
u
82
+· · ·+4.86×10
27
b+2.91×10
28
, 5.01×
10
27
u
83
+7.71×10
27
u
82
+· · ·+4.86×10
27
a+1.49×10
28
, u
84
2u
83
+· · ·+8u1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
u
u
3
+ u
a
8
=
u
2
+ 1
u
2
a
4
=
u
7
2u
5
+ 2u
3
2u
u
7
+ u
5
2u
3
+ u
a
11
=
1.03185u
83
1.58695u
82
+ ··· 4.40283u 3.06481
3.10267u
83
+ 4.21794u
82
+ ··· + 29.3257u 5.98515
a
12
=
0.620344u
83
0.533770u
82
+ ··· + 1.10673u 4.48153
2.17358u
83
+ 3.18151u
82
+ ··· + 22.4583u 4.80039
a
1
=
u
3
u
5
u
3
+ u
a
5
=
2.85766u
83
6.27375u
82
+ ··· 51.4385u + 14.7969
2.17915u
83
3.63605u
82
+ ··· 23.9333u + 5.83697
a
10
=
6.60984u
83
9.43458u
82
+ ··· 80.2900u + 17.7701
1.88651u
83
2.65258u
82
+ ··· 18.0252u + 4.01556
a
9
=
5.50731u
83
8.24850u
82
+ ··· 71.9841u + 16.5651
0.919387u
83
1.31392u
82
+ ··· 7.17730u + 2.14577
(ii) Obstruction class = 1
(iii) Cusp Shapes = 0.957516u
83
+ 4.31532u
82
+ ··· + 23.1863u 1.43761
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
7
u
84
+ 28u
83
+ ··· + 28u + 1
c
2
, c
6
u
84
2u
83
+ ··· + 8u 1
c
3
u
84
+ 2u
83
+ ··· 4588u 793
c
4
, c
5
, c
10
u
84
+ u
83
+ ··· 40u 8
c
8
, c
9
, c
12
u
84
+ 4u
83
+ ··· + 163u 23
c
11
u
84
3u
83
+ ··· + 1464u 872
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
7
y
84
+ 60y
83
+ ··· 316y + 1
c
2
, c
6
y
84
28y
83
+ ··· 28y + 1
c
3
y
84
12y
83
+ ··· 50872888y + 628849
c
4
, c
5
, c
10
y
84
77y
83
+ ··· 64y + 64
c
8
, c
9
, c
12
y
84
78y
83
+ ··· 763y + 529
c
11
y
84
+ 7y
83
+ ··· 34832832y + 760384
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.005860 + 0.053427I
a = 1.18941 + 1.31282I
b = 1.213460 + 0.177851I
0.638993 + 0.814219I 0
u = 1.005860 0.053427I
a = 1.18941 1.31282I
b = 1.213460 0.177851I
0.638993 0.814219I 0
u = 1.009640 + 0.062094I
a = 0.30606 1.46538I
b = 0.119169 0.661433I
3.80257 2.20400I 0
u = 1.009640 0.062094I
a = 0.30606 + 1.46538I
b = 0.119169 + 0.661433I
3.80257 + 2.20400I 0
u = 0.776499 + 0.653312I
a = 0.222981 0.644766I
b = 0.153963 0.553953I
0.07926 + 2.02066I 0
u = 0.776499 0.653312I
a = 0.222981 + 0.644766I
b = 0.153963 + 0.553953I
0.07926 2.02066I 0
u = 0.695537 + 0.739243I
a = 1.61519 0.94961I
b = 1.345440 0.210053I
4.80676 + 0.75078I 0
u = 0.695537 0.739243I
a = 1.61519 + 0.94961I
b = 1.345440 + 0.210053I
4.80676 0.75078I 0
u = 0.624355 + 0.759958I
a = 1.325510 + 0.022658I
b = 0.772721 0.476917I
2.12749 + 1.40116I 0
u = 0.624355 0.759958I
a = 1.325510 0.022658I
b = 0.772721 + 0.476917I
2.12749 1.40116I 0
5
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.728167 + 0.715851I
a = 3.24024 0.97356I
b = 1.51855 0.09042I
5.33778 + 0.28990I 0
u = 0.728167 0.715851I
a = 3.24024 + 0.97356I
b = 1.51855 + 0.09042I
5.33778 0.28990I 0
u = 0.688413 + 0.754788I
a = 0.551967 + 0.024503I
b = 0.262258 + 0.664682I
1.80955 2.15614I 0
u = 0.688413 0.754788I
a = 0.551967 0.024503I
b = 0.262258 0.664682I
1.80955 + 2.15614I 0
u = 0.976324
a = 0.614902
b = 1.53549
0.457615 10.5000
u = 0.891298 + 0.536778I
a = 0.139782 + 0.390452I
b = 1.225720 0.173090I
2.94410 0.08006I 0
u = 0.891298 0.536778I
a = 0.139782 0.390452I
b = 1.225720 + 0.173090I
2.94410 + 0.08006I 0
u = 0.643339 + 0.823548I
a = 0.530313 + 0.337551I
b = 0.293170 0.790812I
3.73484 5.86407I 0
u = 0.643339 0.823548I
a = 0.530313 0.337551I
b = 0.293170 + 0.790812I
3.73484 + 5.86407I 0
u = 1.042210 + 0.126083I
a = 0.23452 1.89207I
b = 1.335370 0.268074I
0.78059 + 5.59119I 0
6
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.042210 0.126083I
a = 0.23452 + 1.89207I
b = 1.335370 + 0.268074I
0.78059 5.59119I 0
u = 0.771829 + 0.720791I
a = 0.846377 + 0.082883I
b = 0.526872 + 0.345121I
3.03488 1.28276I 0
u = 0.771829 0.720791I
a = 0.846377 0.082883I
b = 0.526872 0.345121I
3.03488 + 1.28276I 0
u = 0.684662 + 0.805563I
a = 2.20441 + 0.70141I
b = 1.39828 + 0.26713I
7.09037 + 5.56514I 0
u = 0.684662 0.805563I
a = 2.20441 0.70141I
b = 1.39828 0.26713I
7.09037 5.56514I 0
u = 0.669349 + 0.848194I
a = 2.39133 0.39984I
b = 1.42903 0.31945I
1.75823 + 9.89270I 0
u = 0.669349 0.848194I
a = 2.39133 + 0.39984I
b = 1.42903 + 0.31945I
1.75823 9.89270I 0
u = 1.085550 + 0.056336I
a = 0.275785 + 0.475047I
b = 0.940188 + 0.467824I
7.95233 + 0.71366I 0
u = 1.085550 0.056336I
a = 0.275785 0.475047I
b = 0.940188 0.467824I
7.95233 0.71366I 0
u = 1.099610 + 0.112323I
a = 0.415514 + 1.063700I
b = 0.222123 + 0.831690I
10.19880 5.33031I 0
7
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.099610 0.112323I
a = 0.415514 1.063700I
b = 0.222123 0.831690I
10.19880 + 5.33031I 0
u = 1.099630 + 0.155067I
a = 0.29211 + 1.63269I
b = 1.39491 + 0.34897I
5.08068 + 9.58817I 0
u = 1.099630 0.155067I
a = 0.29211 1.63269I
b = 1.39491 0.34897I
5.08068 9.58817I 0
u = 0.928621 + 0.619657I
a = 0.100165 1.173450I
b = 1.083530 0.032967I
2.58787 4.44451I 0
u = 0.928621 0.619657I
a = 0.100165 + 1.173450I
b = 1.083530 + 0.032967I
2.58787 + 4.44451I 0
u = 1.019450 + 0.457537I
a = 0.462210 0.527152I
b = 1.317910 + 0.341260I
3.24713 + 2.82631I 0
u = 1.019450 0.457537I
a = 0.462210 + 0.527152I
b = 1.317910 0.341260I
3.24713 2.82631I 0
u = 0.806273 + 0.783156I
a = 2.88994 + 0.82117I
b = 1.43003 + 0.12335I
9.14541 + 2.98358I 0
u = 0.806273 0.783156I
a = 2.88994 0.82117I
b = 1.43003 0.12335I
9.14541 2.98358I 0
u = 0.862705
a = 0.275443
b = 0.403590
1.54513 4.21480
8
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.018780 + 0.525725I
a = 0.625001 + 0.295281I
b = 0.109685 + 0.817642I
7.70761 + 1.33555I 0
u = 1.018780 0.525725I
a = 0.625001 0.295281I
b = 0.109685 0.817642I
7.70761 1.33555I 0
u = 0.946108 + 0.648190I
a = 0.855038 0.839227I
b = 0.074916 0.634094I
0.46603 + 3.04786I 0
u = 0.946108 0.648190I
a = 0.855038 + 0.839227I
b = 0.074916 + 0.634094I
0.46603 3.04786I 0
u = 0.937402 + 0.690689I
a = 0.234099 0.621016I
b = 0.601970 + 0.256880I
2.52543 4.12355I 0
u = 0.937402 0.690689I
a = 0.234099 + 0.621016I
b = 0.601970 0.256880I
2.52543 + 4.12355I 0
u = 1.022020 + 0.590072I
a = 1.13586 + 1.40697I
b = 1.091450 + 0.410000I
4.68766 5.74442I 0
u = 1.022020 0.590072I
a = 1.13586 1.40697I
b = 1.091450 0.410000I
4.68766 + 5.74442I 0
u = 0.840965 + 0.836084I
a = 2.72881 0.61998I
b = 1.328120 0.208438I
4.91735 + 5.68260I 0
u = 0.840965 0.836084I
a = 2.72881 + 0.61998I
b = 1.328120 + 0.208438I
4.91735 5.68260I 0
9
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.969325 + 0.683438I
a = 2.45171 + 1.35359I
b = 1.54495 0.07716I
4.60204 + 5.08829I 0
u = 0.969325 0.683438I
a = 2.45171 1.35359I
b = 1.54495 + 0.07716I
4.60204 5.08829I 0
u = 0.433786 + 0.682887I
a = 1.51254 0.21200I
b = 1.025860 + 0.318582I
3.05008 + 0.92169I 4.35708 0.77356I
u = 0.433786 0.682887I
a = 1.51254 + 0.21200I
b = 1.025860 0.318582I
3.05008 0.92169I 4.35708 + 0.77356I
u = 0.881987 + 0.801324I
a = 0.380306 + 0.465520I
b = 0.031624 0.527614I
0.55770 2.99594I 0
u = 0.881987 0.801324I
a = 0.380306 0.465520I
b = 0.031624 + 0.527614I
0.55770 + 2.99594I 0
u = 0.933284 + 0.748392I
a = 2.46322 1.14066I
b = 1.43177 + 0.10143I
8.75543 + 2.78886I 0
u = 0.933284 0.748392I
a = 2.46322 + 1.14066I
b = 1.43177 0.10143I
8.75543 2.78886I 0
u = 0.988622 + 0.690582I
a = 1.26017 2.60390I
b = 1.321940 0.243185I
3.92465 6.21786I 0
u = 0.988622 0.690582I
a = 1.26017 + 2.60390I
b = 1.321940 + 0.243185I
3.92465 + 6.21786I 0
10
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.995284 + 0.694853I
a = 1.28752 + 0.63214I
b = 0.242296 + 0.706638I
0.88560 + 7.67713I 0
u = 0.995284 0.694853I
a = 1.28752 0.63214I
b = 0.242296 0.706638I
0.88560 7.67713I 0
u = 1.021630 + 0.682556I
a = 0.225119 + 0.797631I
b = 0.807645 0.537339I
3.30063 6.88949I 0
u = 1.021630 0.682556I
a = 0.225119 0.797631I
b = 0.807645 + 0.537339I
3.30063 + 6.88949I 0
u = 0.934862 + 0.804107I
a = 2.30600 + 1.00101I
b = 1.307880 0.193046I
4.62756 + 0.42572I 0
u = 0.934862 0.804107I
a = 2.30600 1.00101I
b = 1.307880 + 0.193046I
4.62756 0.42572I 0
u = 1.012340 + 0.716613I
a = 1.90120 + 2.49253I
b = 1.39597 + 0.28620I
6.09591 11.29490I 0
u = 1.012340 0.716613I
a = 1.90120 2.49253I
b = 1.39597 0.28620I
6.09591 + 11.29490I 0
u = 0.283981 + 0.692331I
a = 0.503996 0.319424I
b = 0.173149 + 0.760116I
5.64015 + 3.06330I 7.15320 3.71038I
u = 0.283981 0.692331I
a = 0.503996 + 0.319424I
b = 0.173149 0.760116I
5.64015 3.06330I 7.15320 + 3.71038I
11
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 1.035440 + 0.709315I
a = 1.36459 0.38464I
b = 0.304641 0.819504I
4.92175 + 11.61130I 0
u = 1.035440 0.709315I
a = 1.36459 + 0.38464I
b = 0.304641 + 0.819504I
4.92175 11.61130I 0
u = 0.196384 + 0.718380I
a = 2.39252 + 0.55518I
b = 1.366140 + 0.311678I
0.77281 6.93544I 2.29673 + 5.13999I
u = 0.196384 0.718380I
a = 2.39252 0.55518I
b = 1.366140 0.311678I
0.77281 + 6.93544I 2.29673 5.13999I
u = 1.035180 + 0.729097I
a = 2.15510 2.20542I
b = 1.43822 0.33088I
0.6389 15.7800I 0
u = 1.035180 0.729097I
a = 2.15510 + 2.20542I
b = 1.43822 + 0.33088I
0.6389 + 15.7800I 0
u = 0.185341 + 0.584468I
a = 2.26406 1.07778I
b = 1.352270 0.198800I
4.67557 3.47099I 2.75649 + 4.83228I
u = 0.185341 0.584468I
a = 2.26406 + 1.07778I
b = 1.352270 + 0.198800I
4.67557 + 3.47099I 2.75649 4.83228I
u = 0.492916 + 0.333326I
a = 0.555211 + 0.726800I
b = 1.300790 0.012081I
3.26707 + 0.11469I 0.147841 + 0.385810I
u = 0.492916 0.333326I
a = 0.555211 0.726800I
b = 1.300790 + 0.012081I
3.26707 0.11469I 0.147841 0.385810I
12
Solutions to I
u
1
1(vol +
1CS) Cusp shape
u = 0.537371
a = 2.45205
b = 0.372992
2.35654 3.13870
u = 0.208703 + 0.374292I
a = 0.716811 0.283596I
b = 0.169135 0.458913I
0.132657 + 0.963114I 2.58462 6.99680I
u = 0.208703 0.374292I
a = 0.716811 + 0.283596I
b = 0.169135 + 0.458913I
0.132657 0.963114I 2.58462 + 6.99680I
u = 0.230433
a = 3.01588
b = 1.37504
3.21591 2.14730
13
II.
I
u
2
= hu
2
a+2au2u
2
+b+2a4u 3, a
2
2auu
2
2a+6u1, u
3
+u
2
1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
u
u
2
+ u 1
a
8
=
u
2
+ 1
u
2
a
4
=
1
0
a
11
=
a
u
2
a 2au + 2u
2
2a + 4u + 3
a
12
=
u
2
a 2au + 2u
2
a + 4u + 3
u
2
a 2au + 2u
2
2a + 4u + 3
a
1
=
u
2
+ 1
u
2
a
5
=
2u
2
a 4au + 4u
2
3a + 9u + 4
2
a
10
=
u
2
a + 2au 2u
2
+ a 4u 3
u
2
a + 2au 2u
2
+ 2a 4u 3
a
9
=
u
2
a + 2au 3u
2
+ a 4u 2
u
2
a + 2au u
2
+ 2a 4u 3
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u 4
14
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
(u
3
u
2
+ 2u 1)
2
c
2
(u
3
u
2
+ 1)
2
c
3
, c
7
(u
3
+ u
2
+ 2u + 1)
2
c
4
, c
5
, c
10
c
11
(u
2
2)
3
c
6
(u
3
+ u
2
1)
2
c
8
, c
9
(u + 1)
6
c
12
(u 1)
6
15
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
(y
3
+ 3y
2
+ 2y 1)
2
c
2
, c
6
(y
3
y
2
+ 2y 1)
2
c
4
, c
5
, c
10
c
11
(y 2)
6
c
8
, c
9
, c
12
(y 1)
6
16
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
1(vol +
1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 2.77766 0.30853I
b = 1.41421
6.31400 + 2.82812I 0.49024 2.97945I
u = 0.877439 + 0.744862I
a = 2.53254 + 1.79826I
b = 1.41421
6.31400 + 2.82812I 0.49024 2.97945I
u = 0.877439 0.744862I
a = 2.77766 + 0.30853I
b = 1.41421
6.31400 2.82812I 0.49024 + 2.97945I
u = 0.877439 0.744862I
a = 2.53254 1.79826I
b = 1.41421
6.31400 2.82812I 0.49024 + 2.97945I
u = 0.754878
a = 1.40822
b = 1.41421
2.17641 7.01950
u = 0.754878
a = 2.10153
b = 1.41421
2.17641 7.01950
17
III. I
u
3
= hb, a u + 1, u
3
u
2
+ 1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
3
=
u
u
2
+ u + 1
a
8
=
u
2
+ 1
u
2
a
4
=
1
0
a
11
=
u 1
0
a
12
=
u 1
0
a
1
=
u
2
1
u
2
a
5
=
1
0
a
10
=
u 1
0
a
9
=
u
2
+ u
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
+ 10u 8
18
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
u
3
u
2
+ 2u 1
c
2
u
3
+ u
2
1
c
4
, c
5
, c
10
c
11
u
3
c
6
u
3
u
2
+ 1
c
7
u
3
+ u
2
+ 2u + 1
c
8
, c
9
(u 1)
3
c
12
(u + 1)
3
19
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
7
y
3
+ 3y
2
+ 2y 1
c
2
, c
6
y
3
y
2
+ 2y 1
c
4
, c
5
, c
10
c
11
y
3
c
8
, c
9
, c
12
(y 1)
3
20
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
1(vol +
1CS) Cusp shape
u = 0.877439 + 0.744862I
a = 0.122561 + 0.744862I
b = 0
1.37919 2.82812I 0.08593 + 2.22005I
u = 0.877439 0.744862I
a = 0.122561 0.744862I
b = 0
1.37919 + 2.82812I 0.08593 2.22005I
u = 0.754878
a = 1.75488
b = 0
2.75839 17.8280
21
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
3
u
2
+ 2u 1)
3
)(u
84
+ 28u
83
+ ··· + 28u + 1)
c
2
((u
3
u
2
+ 1)
2
)(u
3
+ u
2
1)(u
84
2u
83
+ ··· + 8u 1)
c
3
(u
3
u
2
+ 2u 1)(u
3
+ u
2
+ 2u + 1)
2
(u
84
+ 2u
83
+ ··· 4588u 793)
c
4
, c
5
, c
10
u
3
(u
2
2)
3
(u
84
+ u
83
+ ··· 40u 8)
c
6
(u
3
u
2
+ 1)(u
3
+ u
2
1)
2
(u
84
2u
83
+ ··· + 8u 1)
c
7
((u
3
+ u
2
+ 2u + 1)
3
)(u
84
+ 28u
83
+ ··· + 28u + 1)
c
8
, c
9
((u 1)
3
)(u + 1)
6
(u
84
+ 4u
83
+ ··· + 163u 23)
c
11
u
3
(u
2
2)
3
(u
84
3u
83
+ ··· + 1464u 872)
c
12
((u 1)
6
)(u + 1)
3
(u
84
+ 4u
83
+ ··· + 163u 23)
22
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
7
((y
3
+ 3y
2
+ 2y 1)
3
)(y
84
+ 60y
83
+ ··· 316y + 1)
c
2
, c
6
((y
3
y
2
+ 2y 1)
3
)(y
84
28y
83
+ ··· 28y + 1)
c
3
((y
3
+ 3y
2
+ 2y 1)
3
)(y
84
12y
83
+ ··· 5.08729 × 10
7
y + 628849)
c
4
, c
5
, c
10
y
3
(y 2)
6
(y
84
77y
83
+ ··· 64y + 64)
c
8
, c
9
, c
12
((y 1)
9
)(y
84
78y
83
+ ··· 763y + 529)
c
11
y
3
(y 2)
6
(y
84
+ 7y
83
+ ··· 3.48328 × 10
7
y + 760384)
23