12a
0822
(K12a
0822
)
A knot diagram
1
Linearized knot diagam
4 5 8 2 9 11 12 3 1 6 7 10
Solving Sequence
7,12 3,8
4 9 11 6 5 2 10 1
c
7
c
3
c
8
c
11
c
6
c
5
c
2
c
10
c
12
c
1
, c
4
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= hu
64
− 34u
62
+ ··· + b + u, −u
63
+ 34u
61
+ ··· + a − 2, u
65
+ 2u
64
+ ··· − u + 1i
I
u
2
= hu
5
− 2u
3
− u
2
+ b + 1, u
5
− 3u
3
− u
2
+ a + 2u + 2, u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 71 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
64
−34u
62
+· · ·+b+u, −u
63
+34u
61
+· · ·+a−2, u
65
+2u
64
+· · ·−u+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
3
=
u
63
− 34u
61
+ ··· − 10u + 2
−u
64
+ 34u
62
+ ··· + 7u
2
− u
a
8
=
1
u
2
a
4
=
u
64
+ u
63
+ ··· − 12u + 3
2u
64
+ 2u
63
+ ··· − 4u + 2
a
9
=
−u
11
+ 6u
9
− 12u
7
+ 8u
5
− u
3
+ 2u
−u
11
+ 5u
9
− 8u
7
+ 5u
5
− 3u
3
+ u
a
11
=
u
u
a
6
=
−u
2
+ 1
−u
2
a
5
=
u
20
− 11u
18
+ ··· − 3u
2
+ 1
u
20
− 10u
18
+ 40u
16
− 82u
14
+ 95u
12
− 72u
10
+ 44u
8
− 18u
6
+ 5u
4
− 2u
2
a
2
=
u
64
+ u
63
+ ··· − 11u + 3
u
64
+ u
63
+ ··· − 2u + 1
a
10
=
−u
3
+ 2u
−u
3
+ u
a
1
=
−u
7
+ 4u
5
− 4u
3
−u
7
+ 3u
5
− 2u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 8u
64
+ 11u
63
+ ··· − 22u − 6
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
4
u
65
− 7u
64
+ ··· − 8u + 1
c
3
, c
8
u
65
+ u
64
+ ··· + 128u + 64
c
5
u
65
+ 2u
64
+ ··· − 5425u − 1549
c
6
, c
7
, c
10
c
11
u
65
− 2u
64
+ ··· − u − 1
c
9
, c
12
u
65
− 12u
64
+ ··· + 69u + 73
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
y
65
− 63y
64
+ ··· + 52y −1
c
3
, c
8
y
65
+ 39y
64
+ ··· + 12288y −4096
c
5
y
65
− 24y
64
+ ··· + 86536059y −2399401
c
6
, c
7
, c
10
c
11
y
65
− 72y
64
+ ··· + 19y −1
c
9
, c
12
y
65
+ 36y
64
+ ··· + 52795y −5329
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.849078 + 0.151759I
a = −0.52962 + 2.16096I
b = −0.290266 + 0.275339I
−10.18900 + 5.61738I −17.6543 − 4.7770I
u = −0.849078 − 0.151759I
a = −0.52962 − 2.16096I
b = −0.290266 − 0.275339I
−10.18900 − 5.61738I −17.6543 + 4.7770I
u = 0.612809 + 0.581694I
a = −2.23569 − 1.18411I
b = 0.116075 − 0.314062I
−5.66682 − 11.33370I −12.3674 + 8.6652I
u = 0.612809 − 0.581694I
a = −2.23569 + 1.18411I
b = 0.116075 + 0.314062I
−5.66682 + 11.33370I −12.3674 − 8.6652I
u = 0.591829 + 0.558370I
a = 2.06155 + 1.34040I
b = −0.355492 + 0.274934I
0.33245 − 7.21157I −9.35363 + 8.77748I
u = 0.591829 − 0.558370I
a = 2.06155 − 1.34040I
b = −0.355492 − 0.274934I
0.33245 + 7.21157I −9.35363 − 8.77748I
u = 0.664359 + 0.450292I
a = 1.56056 + 0.64097I
b = −0.296279 − 0.301108I
−8.31384 − 0.16938I −15.5304 + 3.0150I
u = 0.664359 − 0.450292I
a = 1.56056 − 0.64097I
b = −0.296279 + 0.301108I
−8.31384 + 0.16938I −15.5304 − 3.0150I
u = −0.591992 + 0.536176I
a = −0.33537 + 1.45599I
b = −0.242545 + 0.593048I
−2.33808 + 4.89491I −11.39737 − 6.04524I
u = −0.591992 − 0.536176I
a = −0.33537 − 1.45599I
b = −0.242545 − 0.593048I
−2.33808 − 4.89491I −11.39737 + 6.04524I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.494585 + 0.625336I
a = 0.779520 + 0.719093I
b = 0.361458 + 0.362562I
−0.05575 + 2.11893I −12.62240 − 3.49033I
u = −0.494585 − 0.625336I
a = 0.779520 − 0.719093I
b = 0.361458 − 0.362562I
−0.05575 − 2.11893I −12.62240 + 3.49033I
u = 0.774602
a = 0.765459
b = −0.650676
−5.70560 −17.2280
u = −0.529854 + 0.558575I
a = 0.192170 − 0.901830I
b = 0.116273 − 0.391001I
2.87311 + 2.88923I −3.42187 − 4.67269I
u = −0.529854 − 0.558575I
a = 0.192170 + 0.901830I
b = 0.116273 + 0.391001I
2.87311 − 2.88923I −3.42187 + 4.67269I
u = 0.573094 + 0.509493I
a = −1.56952 − 1.35973I
b = 0.564918 + 0.034458I
−0.92967 − 2.31652I −12.22093 + 3.86577I
u = 0.573094 − 0.509493I
a = −1.56952 + 1.35973I
b = 0.564918 − 0.034458I
−0.92967 + 2.31652I −12.22093 − 3.86577I
u = −0.761928 + 0.069316I
a = 0.34486 − 2.57712I
b = 0.144740 − 0.449027I
−3.63927 + 2.31291I −16.5119 − 4.5602I
u = −0.761928 − 0.069316I
a = 0.34486 + 2.57712I
b = 0.144740 + 0.449027I
−3.63927 − 2.31291I −16.5119 + 4.5602I
u = −0.444280 + 0.563851I
a = −0.979188 + 0.166649I
b = −0.439103 + 0.089481I
3.12509 + 0.97162I −2.56463 − 3.01871I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.444280 − 0.563851I
a = −0.979188 − 0.166649I
b = −0.439103 − 0.089481I
3.12509 − 0.97162I −2.56463 + 3.01871I
u = 0.345558 + 0.625959I
a = 0.597480 + 0.424044I
b = 0.26840 + 1.44188I
−4.88166 + 7.23122I −10.46990 − 2.83323I
u = 0.345558 − 0.625959I
a = 0.597480 − 0.424044I
b = 0.26840 − 1.44188I
−4.88166 − 7.23122I −10.46990 + 2.83323I
u = 0.360635 + 0.581410I
a = −0.519658 + 0.056262I
b = −0.363851 − 1.310020I
1.00659 + 3.30244I −7.13975 − 2.59787I
u = 0.360635 − 0.581410I
a = −0.519658 − 0.056262I
b = −0.363851 + 1.310020I
1.00659 − 3.30244I −7.13975 + 2.59787I
u = −0.347239 + 0.544294I
a = 1.54475 − 0.29359I
b = 0.667710 − 0.250220I
−1.63173 − 1.15310I −8.96204 − 0.62633I
u = −0.347239 − 0.544294I
a = 1.54475 + 0.29359I
b = 0.667710 + 0.250220I
−1.63173 + 1.15310I −8.96204 + 0.62633I
u = 0.397848 + 0.479108I
a = −0.152635 − 0.791660I
b = 0.493556 + 0.869041I
−0.383917 − 1.172590I −10.77474 + 3.53972I
u = 0.397848 − 0.479108I
a = −0.152635 + 0.791660I
b = 0.493556 − 0.869041I
−0.383917 + 1.172590I −10.77474 − 3.53972I
u = −1.40989 + 0.11554I
a = −0.085739 − 0.727029I
b = −0.963524 − 0.159703I
−10.37080 − 4.63328I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.40989 − 0.11554I
a = −0.085739 + 0.727029I
b = −0.963524 + 0.159703I
−10.37080 + 4.63328I 0
u = 0.163296 + 0.550521I
a = 0.765667 − 0.368871I
b = 0.216553 − 1.080350I
−6.83780 − 3.22697I −11.27004 + 3.16299I
u = 0.163296 − 0.550521I
a = 0.765667 + 0.368871I
b = 0.216553 + 1.080350I
−6.83780 + 3.22697I −11.27004 − 3.16299I
u = 0.544767
a = −0.440631
b = 0.243537
−0.894856 −10.6250
u = −1.46159 + 0.10569I
a = 0.181952 + 0.962693I
b = 1.16904 + 1.01325I
−4.78459 − 1.02708I 0
u = −1.46159 − 0.10569I
a = 0.181952 − 0.962693I
b = 1.16904 − 1.01325I
−4.78459 + 1.02708I 0
u = 1.49086 + 0.09279I
a = −0.707400 − 0.841672I
b = −2.22656 − 1.64975I
−7.55431 − 0.79217I 0
u = 1.49086 − 0.09279I
a = −0.707400 + 0.841672I
b = −2.22656 + 1.64975I
−7.55431 + 0.79217I 0
u = 1.50040 + 0.14327I
a = 0.388062 + 0.569649I
b = 1.27751 + 0.96561I
−3.25528 − 3.41758I 0
u = 1.50040 − 0.14327I
a = 0.388062 − 0.569649I
b = 1.27751 − 0.96561I
−3.25528 + 3.41758I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.51289 + 0.11377I
a = 0.01624 − 1.55519I
b = −0.88043 − 2.69896I
−6.78772 + 3.13550I 0
u = −1.51289 − 0.11377I
a = 0.01624 + 1.55519I
b = −0.88043 + 2.69896I
−6.78772 − 3.13550I 0
u = 1.50743 + 0.18519I
a = −0.660901 − 0.183350I
b = −1.47488 + 0.16745I
−6.61480 − 5.02407I 0
u = 1.50743 − 0.18519I
a = −0.660901 + 0.183350I
b = −1.47488 − 0.16745I
−6.61480 + 5.02407I 0
u = 1.53689 + 0.16083I
a = 0.347709 − 0.503738I
b = 0.381409 − 1.252670I
−3.99527 − 5.46572I 0
u = 1.53689 − 0.16083I
a = 0.347709 + 0.503738I
b = 0.381409 + 1.252670I
−3.99527 + 5.46572I 0
u = −1.55420
a = 0.565237
b = 0.840718
−8.08338 0
u = −1.55791 + 0.14949I
a = 1.11591 − 2.21291I
b = 1.77120 − 4.71029I
−8.07881 + 4.70967I 0
u = −1.55791 − 0.14949I
a = 1.11591 + 2.21291I
b = 1.77120 + 4.71029I
−8.07881 − 4.70967I 0
u = −1.56117 + 0.16671I
a = −1.34382 + 2.19591I
b = −2.54714 + 4.80810I
−6.86256 + 9.85745I 0
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.56117 − 0.16671I
a = −1.34382 − 2.19591I
b = −2.54714 − 4.80810I
−6.86256 − 9.85745I 0
u = 1.56229 + 0.15874I
a = −0.629847 + 0.881225I
b = −0.75213 + 2.10739I
−9.55235 − 7.42892I 0
u = 1.56229 − 0.15874I
a = −0.629847 − 0.881225I
b = −0.75213 − 2.10739I
−9.55235 + 7.42892I 0
u = −1.56850 + 0.17675I
a = 1.36052 − 2.11880I
b = 2.79633 − 4.55461I
−12.9536 + 14.1196I 0
u = −1.56850 − 0.17675I
a = 1.36052 + 2.11880I
b = 2.79633 + 4.55461I
−12.9536 − 14.1196I 0
u = −1.58211 + 0.12862I
a = −1.21036 + 1.81529I
b = −2.00477 + 3.54223I
−15.8923 + 2.2931I 0
u = −1.58211 − 0.12862I
a = −1.21036 − 1.81529I
b = −2.00477 − 3.54223I
−15.8923 − 2.2931I 0
u = 1.59363 + 0.01214I
a = −0.09196 − 2.88914I
b = −0.32386 − 6.15827I
−11.62670 − 2.56811I 0
u = 1.59363 − 0.01214I
a = −0.09196 + 2.88914I
b = −0.32386 + 6.15827I
−11.62670 + 2.56811I 0
u = −1.59618
a = −1.28300
b = −2.03922
−13.7436 0
10
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.220404 + 0.332017I
a = −0.864656 − 0.515239I
b = −0.008971 + 0.638980I
−0.505584 − 0.990498I −8.02418 + 6.33094I
u = 0.220404 − 0.332017I
a = −0.864656 + 0.515239I
b = −0.008971 − 0.638980I
−0.505584 + 0.990498I −8.02418 − 6.33094I
u = 1.61071 + 0.02844I
a = 0.00288 + 2.74367I
b = 0.26375 + 5.67324I
−18.5211 − 6.1970I 0
u = 1.61071 − 0.02844I
a = 0.00288 − 2.74367I
b = 0.26375 − 5.67324I
−18.5211 + 6.1970I 0
u = −0.287026
a = 3.70600
b = 0.727372
−2.04090 0.885750
11
II. I
u
2
=
hu
5
−2u
3
−u
2
+b+1, u
5
−3u
3
−u
2
+a+2u+2, u
6
−u
5
−3u
4
+2u
3
+2u
2
+u−1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
3
=
−u
5
+ 3u
3
+ u
2
− 2u − 2
−u
5
+ 2u
3
+ u
2
− 1
a
8
=
1
u
2
a
4
=
−u
5
+ 3u
3
+ u
2
− 2u − 2
−u
5
+ 2u
3
+ u
2
− 1
a
9
=
1
u
2
a
11
=
u
u
a
6
=
−u
2
+ 1
−u
2
a
5
=
u
4
− 3u
2
+ 1
u
5
+ u
4
− 2u
3
− 3u
2
− u + 1
a
2
=
−u
5
− u
4
+ 3u
3
+ 4u
2
− 2u − 3
−2u
5
− u
4
+ 4u
3
+ 4u
2
+ u − 2
a
10
=
−u
3
+ 2u
−u
3
+ u
a
1
=
−u
4
+ 3u
2
− 1
−u
5
− u
4
+ 2u
3
+ 3u
2
+ u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −3u
5
+ u
4
+ 14u
3
− u
2
− 14u − 18
12
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
6
c
3
, c
8
u
6
c
4
(u + 1)
6
c
5
, c
9
u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
6
, c
7
u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1
c
10
, c
11
u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1
c
12
u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1
13
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
(y −1)
6
c
3
, c
8
y
6
c
5
, c
9
, c
12
y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1
c
6
, c
7
, c
10
c
11
y
6
− 7y
5
+ 17y
4
− 16y
3
+ 6y
2
− 5y + 1
14
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.493180 + 0.575288I
a = −0.089969 − 0.799962I
b = −0.446039 + 0.121233I
1.31531 + 1.97241I −6.43930 − 3.48596I
u = −0.493180 − 0.575288I
a = −0.089969 + 0.799962I
b = −0.446039 − 0.121233I
1.31531 − 1.97241I −6.43930 + 3.48596I
u = 0.483672
a = −2.42043
b = −0.566232
−2.38379 −23.4460
u = 1.52087 + 0.16310I
a = 0.227586 − 0.710576I
b = 0.87287 − 1.51178I
−5.34051 − 4.59213I −10.66600 + 2.48468I
u = 1.52087 − 0.16310I
a = 0.227586 + 0.710576I
b = 0.87287 + 1.51178I
−5.34051 + 4.59213I −10.66600 − 2.48468I
u = −1.53904
a = 1.14519
b = 2.71257
−9.30502 −18.3430
15
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
2
((u − 1)
6
)(u
65
− 7u
64
+ ··· − 8u + 1)
c
3
, c
8
u
6
(u
65
+ u
64
+ ··· + 128u + 64)
c
4
((u + 1)
6
)(u
65
− 7u
64
+ ··· − 8u + 1)
c
5
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
65
+ 2u
64
+ ··· − 5425u − 1549)
c
6
, c
7
(u
6
− u
5
− 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
65
− 2u
64
+ ··· − u − 1)
c
9
(u
6
+ u
5
+ 3u
4
+ 2u
3
+ 2u
2
+ u − 1)(u
65
− 12u
64
+ ··· + 69u + 73)
c
10
, c
11
(u
6
+ u
5
− 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
65
− 2u
64
+ ··· − u − 1)
c
12
(u
6
− u
5
+ 3u
4
− 2u
3
+ 2u
2
− u − 1)(u
65
− 12u
64
+ ··· + 69u + 73)
16
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
4
((y −1)
6
)(y
65
− 63y
64
+ ··· + 52y −1)
c
3
, c
8
y
6
(y
65
+ 39y
64
+ ··· + 12288y −4096)
c
5
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
65
− 24y
64
+ ··· + 86536059y −2399401)
c
6
, c
7
, c
10
c
11
(y
6
− 7y
5
+ ··· − 5y + 1)(y
65
− 72y
64
+ ··· + 19y −1)
c
9
, c
12
(y
6
+ 5y
5
+ 9y
4
+ 4y
3
− 6y
2
− 5y + 1)
· (y
65
+ 36y
64
+ ··· + 52795y −5329)
17
