12n
0036
(K12n
0036
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 10 11 5 1 12 7 8
Solving Sequence
7,11 4,8
5 12 1 10 6 3 2 9
c
7
c
4
c
11
c
12
c
10
c
6
c
3
c
2
c
9
c
1
, c
5
, c
8
Ideals for irreducible components
2
of X
par
I
u
1
= h4u
54
− 8u
53
+ ··· + u
2
+ 2b, −4u
54
+ 8u
53
+ ··· + 2a + 7, u
55
− 3u
54
+ ··· − 3u + 1i
I
u
2
= h−u
2
a + b, u
4
− u
2
a + 2u
3
+ a
2
− au + 3u
2
− a + 2u + 1, u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 65 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
=
h4u
54
−8u
53
+· · ·+u
2
+2b, −4u
54
+8u
53
+· · ·+2a+7, u
55
−3u
54
+· · ·−3u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
2u
54
− 4u
53
+ ··· + 4u −
7
2
−2u
54
+ 4u
53
+ ··· − u
3
−
1
2
u
2
a
8
=
1
u
2
a
5
=
2u
53
− 4u
52
+ ··· +
1
2
u
2
−
3
2
−u
53
+
5
2
u
52
+ ··· −
9
2
u
2
+ 2u
a
12
=
−u
u
a
1
=
u
3
u
5
+ u
3
+ u
a
10
=
−u
3
u
3
+ u
a
6
=
−u
6
− u
4
+ 1
u
6
+ 2u
4
+ u
2
a
3
=
u
54
−
3
2
u
53
+ ··· + 3u − 2
−u
54
+
3
2
u
53
+ ··· − 2u
2
+
3
2
u
a
2
=
1
2
u
53
− u
52
+ ··· +
7
2
u
3
+ 1
−
1
2
u
53
+ u
52
+ ··· + 2u
3
+
1
2
u
a
9
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
13
− 3u
11
− 5u
9
− 4u
7
− 2u
5
+ u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
54
−
33
2
u
53
+ ··· + 20u +
1
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
55
+ 32u
54
+ ··· − 18u − 1
c
2
, c
5
u
55
+ 6u
54
+ ··· − 6u − 1
c
3
u
55
− 6u
54
+ ··· − 18u − 1
c
4
, c
8
u
55
− u
54
+ ··· + 1024u − 1024
c
6
, c
12
u
55
− 3u
54
+ ··· + 379u − 73
c
7
, c
11
u
55
+ 3u
54
+ ··· − 3u − 1
c
9
u
55
+ 3u
54
+ ··· + 3u − 1
c
10
u
55
− 29u
54
+ ··· − 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
55
− 12y
54
+ ··· − 414y −1
c
2
, c
5
y
55
+ 32y
54
+ ··· − 18y −1
c
3
y
55
− 56y
54
+ ··· − 2y −1
c
4
, c
8
y
55
+ 55y
54
+ ··· − 15728640y −1048576
c
6
, c
12
y
55
− 35y
54
+ ··· − 79739y −5329
c
7
, c
11
y
55
+ 29y
54
+ ··· − 3y −1
c
9
y
55
+ 65y
54
+ ··· − 3y −1
c
10
y
55
− 3y
54
+ ··· + 29y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.629268 + 0.778172I
a = −1.18547 + 2.27384I
b = 2.40744 − 0.92801I
−6.39910 + 2.43818I 1.82381 − 3.17764I
u = −0.629268 − 0.778172I
a = −1.18547 − 2.27384I
b = 2.40744 + 0.92801I
−6.39910 − 2.43818I 1.82381 + 3.17764I
u = −0.665364 + 0.741506I
a = 1.20117 − 2.12038I
b = −2.43399 + 0.82779I
−10.52100 − 2.56871I −1.299980 + 0.183319I
u = −0.665364 − 0.741506I
a = 1.20117 + 2.12038I
b = −2.43399 − 0.82779I
−10.52100 + 2.56871I −1.299980 − 0.183319I
u = 0.495253 + 0.911871I
a = 0.959744 − 0.326051I
b = −1.043700 + 0.351476I
−1.67813 − 2.05989I 0. + 3.35425I
u = 0.495253 − 0.911871I
a = 0.959744 + 0.326051I
b = −1.043700 − 0.351476I
−1.67813 + 2.05989I 0. − 3.35425I
u = −0.648513 + 0.820784I
a = 1.28463 − 2.31577I
b = −2.45034 + 0.98315I
−10.29050 + 7.60349I 0. − 6.23847I
u = −0.648513 − 0.820784I
a = 1.28463 + 2.31577I
b = −2.45034 − 0.98315I
−10.29050 − 7.60349I 0. + 6.23847I
u = 0.833853 + 0.223712I
a = 0.45368 − 1.70191I
b = 1.47926 + 0.68394I
−7.04904 + 9.41227I 0.67128 − 5.21491I
u = 0.833853 − 0.223712I
a = 0.45368 + 1.70191I
b = 1.47926 − 0.68394I
−7.04904 − 9.41227I 0.67128 + 5.21491I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.256410 + 0.818450I
a = −0.479621 − 0.389548I
b = 0.216555 + 0.452958I
0.489184 − 1.277770I 5.06679 + 5.28091I
u = 0.256410 − 0.818450I
a = −0.479621 + 0.389548I
b = 0.216555 − 0.452958I
0.489184 + 1.277770I 5.06679 − 5.28091I
u = 0.795256 + 0.293554I
a = 0.39342 − 1.83875I
b = 1.14407 + 0.90240I
−8.21623 − 0.41016I −0.923933 + 0.833875I
u = 0.795256 − 0.293554I
a = 0.39342 + 1.83875I
b = 1.14407 − 0.90240I
−8.21623 + 0.41016I −0.923933 − 0.833875I
u = −0.394472 + 1.087690I
a = −1.50889 + 0.99463I
b = 1.397460 + 0.027206I
1.89318 − 0.05192I 0
u = −0.394472 − 1.087690I
a = −1.50889 − 0.99463I
b = 1.397460 − 0.027206I
1.89318 + 0.05192I 0
u = 0.795981 + 0.235146I
a = −0.47257 + 1.79828I
b = −1.27580 − 0.65059I
−3.68870 + 4.09212I 3.08841 − 2.21678I
u = 0.795981 − 0.235146I
a = −0.47257 − 1.79828I
b = −1.27580 + 0.65059I
−3.68870 − 4.09212I 3.08841 + 2.21678I
u = 0.522439 + 0.613476I
a = −0.397063 + 1.308290I
b = 0.133932 − 0.914279I
−2.54336 − 2.12347I −1.98097 + 3.91876I
u = 0.522439 − 0.613476I
a = −0.397063 − 1.308290I
b = 0.133932 + 0.914279I
−2.54336 + 2.12347I −1.98097 − 3.91876I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.798934 + 0.043384I
a = 0.0979918 + 0.0882666I
b = −0.341667 − 0.572577I
1.82991 − 1.15915I 1.11016 − 1.39618I
u = −0.798934 − 0.043384I
a = 0.0979918 − 0.0882666I
b = −0.341667 + 0.572577I
1.82991 + 1.15915I 1.11016 + 1.39618I
u = 0.238525 + 1.182470I
a = 0.409365 − 0.546134I
b = −0.256958 − 0.560255I
−3.53580 − 3.51533I 0
u = 0.238525 − 1.182470I
a = 0.409365 + 0.546134I
b = −0.256958 + 0.560255I
−3.53580 + 3.51533I 0
u = 0.428974 + 1.129030I
a = −0.472967 − 0.859039I
b = 1.56190 + 1.45145I
4.01128 − 1.20704I 0
u = 0.428974 − 1.129030I
a = −0.472967 + 0.859039I
b = 1.56190 − 1.45145I
4.01128 + 1.20704I 0
u = 0.306193 + 1.184120I
a = −0.040681 + 0.327783I
b = 0.140846 + 0.893676I
0.680788 + 0.637411I 0
u = 0.306193 − 1.184120I
a = −0.040681 − 0.327783I
b = 0.140846 − 0.893676I
0.680788 − 0.637411I 0
u = 0.466924 + 1.132680I
a = 0.97235 + 1.11590I
b = −2.29775 − 1.19065I
3.73819 − 6.60281I 0
u = 0.466924 − 1.132680I
a = 0.97235 − 1.11590I
b = −2.29775 + 1.19065I
3.73819 + 6.60281I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.445970 + 1.141590I
a = 0.898688 − 0.908912I
b = −0.977538 + 0.234458I
4.49847 + 3.98279I 0
u = −0.445970 − 1.141590I
a = 0.898688 + 0.908912I
b = −0.977538 − 0.234458I
4.49847 − 3.98279I 0
u = −0.504119 + 1.119490I
a = −0.72422 + 1.48566I
b = 1.143880 − 0.723214I
1.05362 + 7.50380I 0
u = −0.504119 − 1.119490I
a = −0.72422 − 1.48566I
b = 1.143880 + 0.723214I
1.05362 − 7.50380I 0
u = −0.006646 + 0.753586I
a = −1.23924 − 0.91553I
b = 0.221110 + 1.001110I
0.94796 − 1.37354I 8.29726 + 4.59305I
u = −0.006646 − 0.753586I
a = −1.23924 + 0.91553I
b = 0.221110 − 1.001110I
0.94796 + 1.37354I 8.29726 − 4.59305I
u = 0.308957 + 1.222630I
a = −0.150304 − 0.576649I
b = 0.145017 − 0.812504I
−2.51298 + 5.72465I 0
u = 0.308957 − 1.222630I
a = −0.150304 + 0.576649I
b = 0.145017 + 0.812504I
−2.51298 − 5.72465I 0
u = 0.562093 + 1.144990I
a = −2.01573 − 0.76521I
b = 2.80550 − 0.09124I
−5.70109 − 4.64931I 0
u = 0.562093 − 1.144990I
a = −2.01573 + 0.76521I
b = 2.80550 + 0.09124I
−5.70109 + 4.64931I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.543221 + 1.166590I
a = 2.03351 + 1.01526I
b = −3.03630 − 0.03630I
−0.93977 − 9.06965I 0
u = 0.543221 − 1.166590I
a = 2.03351 − 1.01526I
b = −3.03630 + 0.03630I
−0.93977 + 9.06965I 0
u = −0.437917 + 1.210660I
a = 0.760392 + 0.093272I
b = −0.473659 − 0.350913I
5.51374 + 3.19066I 0
u = −0.437917 − 1.210660I
a = 0.760392 − 0.093272I
b = −0.473659 + 0.350913I
5.51374 − 3.19066I 0
u = −0.472267 + 1.211330I
a = −0.320209 − 0.705624I
b = −0.049480 + 0.606670I
5.27082 + 5.76680I 0
u = −0.472267 − 1.211330I
a = −0.320209 + 0.705624I
b = −0.049480 − 0.606670I
5.27082 − 5.76680I 0
u = 0.550577 + 1.182870I
a = −2.18265 − 1.04941I
b = 3.13565 − 0.09179I
−4.1957 − 14.5134I 0
u = 0.550577 − 1.182870I
a = −2.18265 + 1.04941I
b = 3.13565 + 0.09179I
−4.1957 + 14.5134I 0
u = −0.627951 + 0.254073I
a = −0.152563 + 0.574638I
b = 1.207510 + 0.084242I
−1.41318 − 3.06748I 0.44741 + 4.00287I
u = −0.627951 − 0.254073I
a = −0.152563 − 0.574638I
b = 1.207510 − 0.084242I
−1.41318 + 3.06748I 0.44741 − 4.00287I
9
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.245896 + 0.622986I
a = 1.09436 + 1.93010I
b = 0.66336 − 1.42809I
0.12059 + 2.86553I 2.35860 + 0.38000I
u = −0.245896 − 0.622986I
a = 1.09436 − 1.93010I
b = 0.66336 + 1.42809I
0.12059 − 2.86553I 2.35860 − 0.38000I
u = −0.626935
a = −0.0900412
b = −0.792661
1.42624 7.14390
u = 0.586127 + 0.078855I
a = −0.17208 + 2.32899I
b = −0.269979 − 0.036468I
0.91267 + 2.50494I 1.11924 − 3.76856I
u = 0.586127 − 0.078855I
a = −0.17208 − 2.32899I
b = −0.269979 + 0.036468I
0.91267 − 2.50494I 1.11924 + 3.76856I
10
II. I
u
2
=
h−u
2
a+b, u
4
−u
2
a+2u
3
+a
2
−au+3u
2
−a+2u+1, u
5
+u
4
+2u
3
+u
2
+u+1i
(i) Arc colorings
a
7
=
1
0
a
11
=
0
u
a
4
=
a
u
2
a
a
8
=
1
u
2
a
5
=
a
u
2
a
a
12
=
−u
u
a
1
=
u
3
−u
4
− u
3
− u
2
− 1
a
10
=
−u
3
u
3
+ u
a
6
=
−u
3
u
4
+ u
3
+ u
2
+ 1
a
3
=
−u
3
a
u
3
a + au
a
2
=
−u
3
a − u
2
− u − 1
u
3
a − u
4
− u
3
+ au − u
2
a
9
=
1
u
2
(ii) Obstruction class = 1
(iii) Cusp Shapes = −5u
3
a + u
4
+ 6u
3
− 6au + 7u
2
− a + 6u + 7
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
3
, c
5
(u
2
− u + 1)
5
c
2
(u
2
+ u + 1)
5
c
4
, c
8
u
10
c
6
, c
9
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
7
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
10
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
c
11
(u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
c
12
(u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
3
c
5
(y
2
+ y + 1)
5
c
4
, c
8
y
10
c
6
, c
9
, c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
7
, c
11
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
10
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.339110 + 0.822375I
a = −0.80632 + 1.36366I
b = −0.307991 − 1.215160I
0.32910 − 3.56046I 5.91654 + 9.74472I
u = 0.339110 + 0.822375I
a = 1.58413 + 0.01647I
b = −0.898363 + 0.874307I
0.329100 + 0.499304I 1.60756 + 0.92266I
u = 0.339110 − 0.822375I
a = −0.80632 − 1.36366I
b = −0.307991 + 1.215160I
0.32910 + 3.56046I 5.91654 − 9.74472I
u = 0.339110 − 0.822375I
a = 1.58413 − 0.01647I
b = −0.898363 − 0.874307I
0.329100 − 0.499304I 1.60756 − 0.92266I
u = −0.766826
a = 0.410598 + 0.711177I
b = 0.241441 + 0.418187I
2.40108 − 2.02988I 6.55976 + 4.16430I
u = −0.766826
a = 0.410598 − 0.711177I
b = 0.241441 − 0.418187I
2.40108 + 2.02988I 6.55976 − 4.16430I
u = −0.455697 + 1.200150I
a = −0.252108 + 0.649344I
b = 1.021040 − 0.524691I
5.87256 + 6.43072I 10.62344 − 8.02599I
u = −0.455697 + 1.200150I
a = −0.436295 − 0.543004I
b = −0.056121 + 1.146590I
5.87256 + 2.37095I 9.29269 + 1.50431I
u = −0.455697 − 1.200150I
a = −0.252108 − 0.649344I
b = 1.021040 + 0.524691I
5.87256 − 6.43072I 10.62344 + 8.02599I
u = −0.455697 − 1.200150I
a = −0.436295 + 0.543004I
b = −0.056121 − 1.146590I
5.87256 − 2.37095I 9.29269 − 1.50431I
14
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
55
+ 32u
54
+ ··· − 18u − 1)
c
2
((u
2
+ u + 1)
5
)(u
55
+ 6u
54
+ ··· − 6u − 1)
c
3
((u
2
− u + 1)
5
)(u
55
− 6u
54
+ ··· − 18u − 1)
c
4
, c
8
u
10
(u
55
− u
54
+ ··· + 1024u − 1024)
c
5
((u
2
− u + 1)
5
)(u
55
+ 6u
54
+ ··· − 6u − 1)
c
6
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
55
− 3u
54
+ ··· + 379u − 73)
c
7
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
55
+ 3u
54
+ ··· − 3u − 1)
c
9
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
55
+ 3u
54
+ ··· + 3u − 1)
c
10
((u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
)(u
55
− 29u
54
+ ··· − 3u + 1)
c
11
((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
)(u
55
+ 3u
54
+ ··· − 3u − 1)
c
12
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
55
− 3u
54
+ ··· + 379u − 73)
15
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
55
− 12y
54
+ ··· − 414y −1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
55
+ 32y
54
+ ··· − 18y −1)
c
3
((y
2
+ y + 1)
5
)(y
55
− 56y
54
+ ··· − 2y −1)
c
4
, c
8
y
10
(y
55
+ 55y
54
+ ··· − 1.57286 × 10
7
y −1048576)
c
6
, c
12
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
55
− 35y
54
+ ··· − 79739y −5329)
c
7
, c
11
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
55
+ 29y
54
+ ··· − 3y −1)
c
9
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
55
+ 65y
54
+ ··· − 3y −1)
c
10
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
55
− 3y
54
+ ··· + 29y −1)
16
