12a
0681
(K12a
0681
)
A knot diagram
1
Linearized knot diagam
3 7 11 12 8 9 2 6 1 5 4 10
Solving Sequence
3,11
4 12
5,7
2 8 1 10 9 6
c
3
c
11
c
4
c
2
c
7
c
1
c
10
c
9
c
6
c
5
, c
8
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= hu
76
− 35u
74
+ ··· + b − 1, u
75
+ u
74
+ ··· + a + 1, u
77
+ 2u
76
+ ··· − u − 1i
I
u
2
= hb, u
4
− u
3
− u
2
+ a + u, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 82 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
= hu
76
− 35u
74
+ · · · + b − 1, u
75
+ u
74
+ · · · + a + 1, u
77
+ 2u
76
+ · · · − u − 1i
(i) Arc colorings
a
3
=
1
0
a
11
=
0
u
a
4
=
1
u
2
a
12
=
−u
−u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
−u
75
− u
74
+ ··· − 5u − 1
−u
76
+ 35u
74
+ ··· + u + 1
a
2
=
u
11
− 6u
9
+ 12u
7
− 8u
5
+ u
3
− 2u
−u
11
+ 5u
9
− 8u
7
+ 3u
5
+ u
3
+ u
a
8
=
−u
75
− u
74
+ ··· − 10u
2
− 4u
u
76
− 35u
74
+ ··· + 9u
2
− 1
a
1
=
−u
9
+ 4u
7
− 5u
5
+ 2u
3
− u
−u
11
+ 5u
9
− 8u
7
+ 3u
5
+ u
3
+ u
a
10
=
u
5
− 2u
3
+ u
u
7
− 3u
5
+ 2u
3
+ u
a
9
=
u
13
− 6u
11
+ 13u
9
− 12u
7
+ 6u
5
− 4u
3
+ u
u
15
− 7u
13
+ 18u
11
− 19u
9
+ 6u
7
− 2u
5
+ 4u
3
+ u
a
6
=
−u
75
− u
74
+ ··· − 10u
2
− 4u
u
37
− 17u
35
+ ··· + 6u
2
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
76
+ 4u
75
+ ··· − 9u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
77
+ 33u
76
+ ··· − 7680u − 1024
c
2
, c
7
u
77
+ u
76
+ ··· − 96u − 32
c
3
, c
4
, c
11
u
77
+ 2u
76
+ ··· − u − 1
c
5
, c
6
, c
8
u
77
− 6u
76
+ ··· + u − 1
c
9
, c
12
u
77
+ 12u
76
+ ··· + 1689u + 73
c
10
u
77
− 6u
76
+ ··· + 813u − 935
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
77
+ 13y
76
+ ··· + 11141120y −1048576
c
2
, c
7
y
77
+ 33y
76
+ ··· − 7680y −1024
c
3
, c
4
, c
11
y
77
− 72y
76
+ ··· + 7y −1
c
5
, c
6
, c
8
y
77
− 68y
76
+ ··· − 7y −1
c
9
, c
12
y
77
+ 60y
76
+ ··· + 148363y −5329
c
10
y
77
− 24y
76
+ ··· + 24711039y −874225
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.118910 + 0.179695I
a = −0.440072 + 1.213090I
b = 0.513943 − 0.982821I
−4.41825 − 2.77630I 0
u = −1.118910 − 0.179695I
a = −0.440072 − 1.213090I
b = 0.513943 + 0.982821I
−4.41825 + 2.77630I 0
u = 0.388896 + 0.697285I
a = 2.58400 + 0.08701I
b = −0.694399 + 1.197770I
−6.76637 − 11.69120I −8.55592 + 8.80415I
u = 0.388896 − 0.697285I
a = 2.58400 − 0.08701I
b = −0.694399 − 1.197770I
−6.76637 + 11.69120I −8.55592 − 8.80415I
u = 0.467253 + 0.642938I
a = −1.072070 + 0.222088I
b = 0.025679 − 1.363990I
−11.71880 − 2.13120I −12.50690 + 3.22543I
u = 0.467253 − 0.642938I
a = −1.072070 − 0.222088I
b = 0.025679 + 1.363990I
−11.71880 + 2.13120I −12.50690 − 3.22543I
u = −1.199980 + 0.164041I
a = 0.765140 − 0.964895I
b = −0.662389 + 0.724093I
−0.198946 + 0.491194I 0
u = −1.199980 − 0.164041I
a = 0.765140 + 0.964895I
b = −0.662389 − 0.724093I
−0.198946 − 0.491194I 0
u = 0.550317 + 0.554367I
a = −0.557825 − 0.493865I
b = 0.660792 + 1.201360I
−7.39452 + 7.48341I −10.14800 − 2.91847I
u = 0.550317 − 0.554367I
a = −0.557825 + 0.493865I
b = 0.660792 − 1.201360I
−7.39452 − 7.48341I −10.14800 + 2.91847I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.382467 + 0.674572I
a = −2.43549 − 0.42583I
b = 0.583029 − 1.060690I
−1.41933 − 7.47214I −5.05697 + 8.34642I
u = 0.382467 − 0.674572I
a = −2.43549 + 0.42583I
b = 0.583029 + 1.060690I
−1.41933 + 7.47214I −5.05697 − 8.34642I
u = −0.391946 + 0.660763I
a = 1.07868 − 1.47152I
b = −1.058370 + 0.462731I
−4.41478 + 5.37995I −7.77105 − 5.68914I
u = −0.391946 − 0.660763I
a = 1.07868 + 1.47152I
b = −1.058370 − 0.462731I
−4.41478 − 5.37995I −7.77105 + 5.68914I
u = 0.392387 + 0.638561I
a = 1.89360 + 0.81444I
b = −0.337099 + 0.970309I
−3.46473 − 2.75353I −9.00366 + 4.34907I
u = 0.392387 − 0.638561I
a = 1.89360 − 0.81444I
b = −0.337099 − 0.970309I
−3.46473 + 2.75353I −9.00366 − 4.34907I
u = −1.25296
a = −0.489598
b = 0.440012
−2.70327 0
u = 1.247280 + 0.146275I
a = −2.07717 − 0.81765I
b = 0.556055 − 0.712579I
−3.52703 − 1.49966I 0
u = 1.247280 − 0.146275I
a = −2.07717 + 0.81765I
b = 0.556055 + 0.712579I
−3.52703 + 1.49966I 0
u = 0.512426 + 0.536307I
a = 0.551321 + 0.026348I
b = −0.538012 − 1.042190I
−1.97887 + 3.43479I −6.68994 − 2.31052I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.512426 − 0.536307I
a = 0.551321 − 0.026348I
b = −0.538012 + 1.042190I
−1.97887 − 3.43479I −6.68994 + 2.31052I
u = −0.488346 + 0.553137I
a = −1.48960 + 0.38906I
b = 1.044170 + 0.400052I
−4.84664 − 1.37205I −9.12609 − 0.79115I
u = −0.488346 − 0.553137I
a = −1.48960 − 0.38906I
b = 1.044170 − 0.400052I
−4.84664 + 1.37205I −9.12609 + 0.79115I
u = 1.254630 + 0.207259I
a = 2.05786 + 0.27148I
b = −0.637359 + 0.904548I
−0.73681 − 5.52218I 0
u = 1.254630 − 0.207259I
a = 2.05786 − 0.27148I
b = −0.637359 − 0.904548I
−0.73681 + 5.52218I 0
u = 0.453993 + 0.561890I
a = 0.036223 + 0.618083I
b = 0.244719 + 0.964800I
−3.76621 − 1.14578I −10.36985 + 2.95072I
u = 0.453993 − 0.561890I
a = 0.036223 − 0.618083I
b = 0.244719 − 0.964800I
−3.76621 + 1.14578I −10.36985 − 2.95072I
u = −0.345053 + 0.634395I
a = −0.540936 + 1.304630I
b = 0.687430 − 0.491293I
0.27587 + 2.54213I −1.22958 − 3.86119I
u = −0.345053 − 0.634395I
a = −0.540936 − 1.304630I
b = 0.687430 + 0.491293I
0.27587 − 2.54213I −1.22958 + 3.86119I
u = −1.266550 + 0.180413I
a = −1.16776 + 0.80875I
b = 0.835295 − 0.561310I
−3.88660 + 3.89208I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.266550 − 0.180413I
a = −1.16776 − 0.80875I
b = 0.835295 + 0.561310I
−3.88660 − 3.89208I 0
u = −0.263295 + 0.667225I
a = −0.30263 − 1.80362I
b = −0.322531 + 0.936487I
−3.32754 + 0.27451I −8.04091 − 0.77276I
u = −0.263295 − 0.667225I
a = −0.30263 + 1.80362I
b = −0.322531 − 0.936487I
−3.32754 − 0.27451I −8.04091 + 0.77276I
u = 1.268150 + 0.243748I
a = −2.03646 + 0.04790I
b = 0.637081 − 1.071140I
−5.48995 − 9.38623I 0
u = 1.268150 − 0.243748I
a = −2.03646 − 0.04790I
b = 0.637081 + 1.071140I
−5.48995 + 9.38623I 0
u = −0.623273 + 0.290139I
a = −0.841020 − 0.501914I
b = 0.400346 + 1.029660I
−4.77164 + 3.23057I −10.85312 − 4.67428I
u = −0.623273 − 0.290139I
a = −0.841020 + 0.501914I
b = 0.400346 − 1.029660I
−4.77164 − 3.23057I −10.85312 + 4.67428I
u = −0.076390 + 0.675447I
a = 2.15894 + 1.81721I
b = −0.585185 − 1.035670I
−1.33875 + 6.04370I −3.74417 − 6.13811I
u = −0.076390 − 0.675447I
a = 2.15894 − 1.81721I
b = −0.585185 + 1.035670I
−1.33875 − 6.04370I −3.74417 + 6.13811I
u = 1.352570 + 0.042876I
a = −0.466208 − 1.294710I
b = 0.174346 − 0.892035I
−5.20171 − 1.91816I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.352570 − 0.042876I
a = −0.466208 + 1.294710I
b = 0.174346 + 0.892035I
−5.20171 + 1.91816I 0
u = −1.35955
a = 0.837136
b = −0.928982
−7.10233 0
u = −0.040790 + 0.631699I
a = −2.35095 − 1.38726I
b = 0.630731 + 0.823161I
3.22737 + 2.45907I 2.41719 − 4.22669I
u = −0.040790 − 0.631699I
a = −2.35095 + 1.38726I
b = 0.630731 − 0.823161I
3.22737 − 2.45907I 2.41719 + 4.22669I
u = −0.385095 + 0.464622I
a = 0.995000 − 0.148955I
b = −0.526774 − 0.405932I
−0.270419 + 0.973560I −2.30158 − 4.17778I
u = −0.385095 − 0.464622I
a = 0.995000 + 0.148955I
b = −0.526774 + 0.405932I
−0.270419 − 0.973560I −2.30158 + 4.17778I
u = 0.033568 + 0.581274I
a = 2.74191 + 0.87771I
b = −0.687117 − 0.568792I
0.094766 − 1.112510I −0.414963 + 0.658190I
u = 0.033568 − 0.581274I
a = 2.74191 − 0.87771I
b = −0.687117 + 0.568792I
0.094766 + 1.112510I −0.414963 − 0.658190I
u = 1.39972 + 0.25277I
a = 0.700734 − 0.638857I
b = 0.251532 + 0.891469I
−8.62598 − 3.61399I 0
u = 1.39972 − 0.25277I
a = 0.700734 + 0.638857I
b = 0.251532 − 0.891469I
−8.62598 + 3.61399I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.42729 + 0.06631I
a = 0.417873 + 0.696599I
b = −0.374341 + 1.178740I
−11.08620 − 4.31379I 0
u = 1.42729 − 0.06631I
a = 0.417873 − 0.696599I
b = −0.374341 − 1.178740I
−11.08620 + 4.31379I 0
u = 1.43420 + 0.20084I
a = −0.399133 − 0.452591I
b = 0.643759 − 0.302963I
−6.06731 − 3.57322I 0
u = 1.43420 − 0.20084I
a = −0.399133 + 0.452591I
b = 0.643759 + 0.302963I
−6.06731 + 3.57322I 0
u = 1.43740 + 0.24134I
a = −0.080952 + 0.770503I
b = −0.743107 − 0.462573I
−5.45133 − 5.75261I 0
u = 1.43740 − 0.24134I
a = −0.080952 − 0.770503I
b = −0.743107 + 0.462573I
−5.45133 + 5.75261I 0
u = −1.45356 + 0.23958I
a = −1.51331 + 1.62581I
b = 0.362141 + 1.020760I
−9.40303 + 5.97378I 0
u = −1.45356 − 0.23958I
a = −1.51331 − 1.62581I
b = 0.362141 − 1.020760I
−9.40303 − 5.97378I 0
u = −1.46014 + 0.20569I
a = −0.29342 + 1.41422I
b = −0.220303 + 1.036640I
−9.90965 + 3.96691I 0
u = −1.46014 − 0.20569I
a = −0.29342 − 1.41422I
b = −0.220303 − 1.036640I
−9.90965 − 3.96691I 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.45430 + 0.25332I
a = 1.82596 − 1.39983I
b = −0.594804 − 1.088050I
−7.32899 + 10.86290I 0
u = −1.45430 − 0.25332I
a = 1.82596 + 1.39983I
b = −0.594804 + 1.088050I
−7.32899 − 10.86290I 0
u = 1.45612 + 0.24706I
a = −0.082844 − 0.995707I
b = 1.093950 + 0.480855I
−10.36290 − 8.70087I 0
u = 1.45612 − 0.24706I
a = −0.082844 + 0.995707I
b = 1.093950 − 0.480855I
−10.36290 + 8.70087I 0
u = −1.46657 + 0.18641I
a = −0.067219 − 0.960194I
b = 0.502448 − 1.083330I
−8.31617 − 0.82132I 0
u = −1.46657 − 0.18641I
a = −0.067219 + 0.960194I
b = 0.502448 + 1.083330I
−8.31617 + 0.82132I 0
u = 1.46582 + 0.19639I
a = 0.512261 + 0.674143I
b = −1.098560 + 0.367386I
−11.11540 − 1.35977I 0
u = 1.46582 − 0.19639I
a = 0.512261 − 0.674143I
b = −1.098560 − 0.367386I
−11.11540 + 1.35977I 0
u = −1.45965 + 0.26153I
a = −1.90826 + 1.20800I
b = 0.712807 + 1.210690I
−12.7169 + 15.1904I 0
u = −1.45965 − 0.26153I
a = −1.90826 − 1.20800I
b = 0.712807 − 1.210690I
−12.7169 − 15.1904I 0
11
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.48110 + 0.17882I
a = 0.014011 + 0.628968I
b = −0.647796 + 1.238820I
−13.9447 − 4.8720I 0
u = −1.48110 − 0.17882I
a = 0.014011 − 0.628968I
b = −0.647796 − 1.238820I
−13.9447 + 4.8720I 0
u = −1.47901 + 0.22669I
a = 0.999876 − 0.986609I
b = −0.050785 − 1.399520I
−18.0111 + 5.2978I 0
u = −1.47901 − 0.22669I
a = 0.999876 + 0.986609I
b = −0.050785 + 1.399520I
−18.0111 − 5.2978I 0
u = −0.303521 + 0.301278I
a = 1.018990 − 0.220898I
b = −0.312600 − 0.550494I
−0.201500 + 0.933792I −4.38363 − 6.63917I
u = −0.303521 − 0.301278I
a = 1.018990 + 0.220898I
b = −0.312600 + 0.550494I
−0.201500 − 0.933792I −4.38363 + 6.63917I
u = 0.278524
a = −2.80564
b = 0.551515
−2.11486 −4.47090
12
II. I
u
2
= hb, u
4
− u
3
− u
2
+ a + u, u
5
− u
4
− 2u
3
+ u
2
+ u + 1i
(i) Arc colorings
a
3
=
1
0
a
11
=
0
u
a
4
=
1
u
2
a
12
=
−u
−u
3
+ u
a
5
=
−u
2
+ 1
−u
4
+ 2u
2
a
7
=
−u
4
+ u
3
+ u
2
− u
0
a
2
=
1
0
a
8
=
−u
4
+ u
3
+ u
2
− u
0
a
1
=
1
0
a
10
=
u
4
− u
2
− 1
u
4
− 2u
2
a
9
=
u
2
− 1
u
4
− 2u
2
a
6
=
−u
4
+ u
3
− u + 1
−u
4
+ 2u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = 5u
3
− u
2
− 8u − 9
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
7
u
5
c
3
, c
4
u
5
− u
4
− 2u
3
+ u
2
+ u + 1
c
5
, c
6
(u − 1)
5
c
8
(u + 1)
5
c
9
u
5
− u
4
+ 2u
3
− u
2
+ u − 1
c
10
u
5
− 3u
4
+ 4u
3
− u
2
− u + 1
c
11
u
5
+ u
4
− 2u
3
− u
2
+ u − 1
c
12
u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
7
y
5
c
3
, c
4
, c
11
y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1
c
5
, c
6
, c
8
(y −1)
5
c
9
, c
12
y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1
c
10
y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −1.21774
a = −1.30408
b = 0
−4.04602 −9.76980
u = −0.309916 + 0.549911I
a = 0.428550 − 1.039280I
b = 0
−1.97403 + 1.53058I −5.05737 − 4.09764I
u = −0.309916 − 0.549911I
a = 0.428550 + 1.039280I
b = 0
−1.97403 − 1.53058I −5.05737 + 4.09764I
u = 1.41878 + 0.21917I
a = −0.276511 − 0.728237I
b = 0
−7.51750 − 4.40083I −9.05774 + 4.18967I
u = 1.41878 − 0.21917I
a = −0.276511 + 0.728237I
b = 0
−7.51750 + 4.40083I −9.05774 − 4.18967I
16
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
5
(u
77
+ 33u
76
+ ··· − 7680u − 1024)
c
2
, c
7
u
5
(u
77
+ u
76
+ ··· − 96u − 32)
c
3
, c
4
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)(u
77
+ 2u
76
+ ··· − u − 1)
c
5
, c
6
((u − 1)
5
)(u
77
− 6u
76
+ ··· + u − 1)
c
8
((u + 1)
5
)(u
77
− 6u
76
+ ··· + u − 1)
c
9
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)(u
77
+ 12u
76
+ ··· + 1689u + 73)
c
10
(u
5
− 3u
4
+ 4u
3
− u
2
− u + 1)(u
77
− 6u
76
+ ··· + 813u − 935)
c
11
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)(u
77
+ 2u
76
+ ··· − u − 1)
c
12
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)(u
77
+ 12u
76
+ ··· + 1689u + 73)
17
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
5
(y
77
+ 13y
76
+ ··· + 1.11411 × 10
7
y −1048576)
c
2
, c
7
y
5
(y
77
+ 33y
76
+ ··· − 7680y −1024)
c
3
, c
4
, c
11
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)(y
77
− 72y
76
+ ··· + 7y −1)
c
5
, c
6
, c
8
((y −1)
5
)(y
77
− 68y
76
+ ··· − 7y −1)
c
9
, c
12
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)(y
77
+ 60y
76
+ ··· + 148363y −5329)
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
· (y
77
− 24y
76
+ ··· + 24711039y −874225)
18
