12n
0035
(K12n
0035
)
A knot diagram
1
Linearized knot diagam
3 5 6 8 2 11 12 5 6 7 10 9
Solving Sequence
6,11 2,7
5 3 1 10 12 9 8 4
c
6
c
5
c
2
c
1
c
10
c
11
c
9
c
8
c
4
c
3
, c
7
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
42
− 2u
41
+ ··· + 2b − u, −2u
42
− 4u
41
+ ··· + 2a − 2, u
44
+ 3u
43
+ ··· + 3u + 1i
I
u
2
= h−2u
4
a − 4u
3
a − 2u
4
+ 3u
2
a − 4u
3
− 8au + 3u
2
+ 19b − 7a − 8u − 7,
u
3
a − u
2
a − 2u
3
+ a
2
+ au + 2u
2
− u + 2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
* 2 irreducible components of dim
C
= 0, with total 54 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
42
−2u
41
+· · ·+2b−u, −2u
42
−4u
41
+· · ·+2a−2, u
44
+3u
43
+· · ·+3u+1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
u
42
+ 2u
41
+ ··· +
1
2
u + 1
1
2
u
42
+ u
41
+ ··· + u
2
+
1
2
u
a
7
=
1
−u
2
a
5
=
5
2
u
42
+ 5u
41
+ ··· + 3u + 1
1
2
u
42
+ u
41
+ ··· + 2u
2
+
3
2
u
a
3
=
−3u
42
− 6u
41
+ ··· −
7
2
u − 1
−
3
2
u
42
− 2u
41
+ ··· − 3u
2
−
3
2
u
a
1
=
−u
11
− 2u
9
− 2u
7
− u
3
−u
11
− 3u
9
− 4u
7
− u
5
+ u
3
+ u
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
5
+ u
3
+ u
a
9
=
u
3
u
3
+ u
a
8
=
−u
6
− u
4
+ 1
u
8
+ 2u
6
+ 2u
4
a
4
=
3
2
u
42
+ 4u
41
+ ··· + 2u + 1
3
2
u
42
+ 2u
41
+ ··· + 3u
2
+
3
2
u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
43
+
39
2
u
42
+ ··· + 25u +
23
2
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
44
+ 10u
43
+ ··· + 2u + 1
c
2
, c
5
u
44
+ 6u
43
+ ··· + 6u + 1
c
3
u
44
− 6u
43
+ ··· + 717363u + 73746
c
4
, c
8
u
44
− u
43
+ ··· + 1024u + 1024
c
6
, c
10
u
44
− 3u
43
+ ··· − 3u + 1
c
7
, c
9
u
44
+ 3u
43
+ ··· + 211u + 34
c
11
u
44
+ 23u
43
+ ··· + 3u + 1
c
12
u
44
− u
43
+ ··· + 3u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
44
+ 54y
43
+ ··· + 102y + 1
c
2
, c
5
y
44
+ 10y
43
+ ··· + 2y + 1
c
3
y
44
+ 98y
43
+ ··· + 113290466283y + 5438472516
c
4
, c
8
y
44
− 55y
43
+ ··· − 1048576y + 1048576
c
6
, c
10
y
44
+ 23y
43
+ ··· + 3y + 1
c
7
, c
9
y
44
− 25y
43
+ ··· + 17903y + 1156
c
11
y
44
− y
43
+ ··· + 11y + 1
c
12
y
44
+ 75y
43
+ ··· + 3y + 1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.691393 + 0.770310I
a = −0.066986 + 0.372800I
b = 0.965189 − 0.946149I
11.85400 − 0.89687I 3.89150 − 0.58111I
u = 0.691393 − 0.770310I
a = −0.066986 − 0.372800I
b = 0.965189 + 0.946149I
11.85400 + 0.89687I 3.89150 + 0.58111I
u = 0.681315 + 0.809304I
a = −1.36604 + 1.02725I
b = 0.945813 + 0.981170I
11.73940 + 6.10553I 3.57078 − 5.20880I
u = 0.681315 − 0.809304I
a = −1.36604 − 1.02725I
b = 0.945813 − 0.981170I
11.73940 − 6.10553I 3.57078 + 5.20880I
u = −0.417026 + 0.814240I
a = −0.966326 − 0.411751I
b = 0.276291 − 0.156022I
−0.06080 − 1.78150I 0.19283 + 3.69450I
u = −0.417026 − 0.814240I
a = −0.966326 + 0.411751I
b = 0.276291 + 0.156022I
−0.06080 + 1.78150I 0.19283 − 3.69450I
u = −0.849289 + 0.246416I
a = −0.864419 + 0.930692I
b = 0.878595 + 1.030530I
8.55506 + 8.32906I 2.52276 − 4.33779I
u = −0.849289 − 0.246416I
a = −0.864419 − 0.930692I
b = 0.878595 − 1.030530I
8.55506 − 8.32906I 2.52276 + 4.33779I
u = −0.836265 + 0.281412I
a = −0.206151 − 0.049476I
b = 0.974475 − 0.853175I
9.13585 + 1.52647I 3.42363 + 0.21688I
u = −0.836265 − 0.281412I
a = −0.206151 + 0.049476I
b = 0.974475 + 0.853175I
9.13585 − 1.52647I 3.42363 − 0.21688I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.433525 + 1.047130I
a = −0.326846 + 0.375439I
b = −0.695543 + 0.930432I
−1.29668 + 0.57558I −2.65523 − 2.16561I
u = 0.433525 − 1.047130I
a = −0.326846 − 0.375439I
b = −0.695543 − 0.930432I
−1.29668 − 0.57558I −2.65523 + 2.16561I
u = −0.094054 + 0.853687I
a = −1.11225 + 1.86145I
b = −0.198619 + 0.750506I
−1.66817 − 1.59205I −6.58514 + 4.39514I
u = −0.094054 − 0.853687I
a = −1.11225 − 1.86145I
b = −0.198619 − 0.750506I
−1.66817 + 1.59205I −6.58514 − 4.39514I
u = −0.487271 + 1.047260I
a = −0.779531 + 0.841946I
b = −0.766362 − 0.217429I
−0.23513 − 3.16229I 1.52764 + 3.47706I
u = −0.487271 − 1.047260I
a = −0.779531 − 0.841946I
b = −0.766362 + 0.217429I
−0.23513 + 3.16229I 1.52764 − 3.47706I
u = −0.388702 + 1.120050I
a = −1.03941 − 2.62383I
b = −0.288878 − 1.073200I
−4.05811 − 0.18233I −4.90447 + 0.I
u = −0.388702 − 1.120050I
a = −1.03941 + 2.62383I
b = −0.288878 + 1.073200I
−4.05811 + 0.18233I −4.90447 + 0.I
u = 0.497679 + 1.078530I
a = 1.02995 − 1.29852I
b = −0.800804 − 0.775068I
−0.74030 + 6.20071I 0. − 7.00102I
u = 0.497679 − 1.078530I
a = 1.02995 + 1.29852I
b = −0.800804 + 0.775068I
−0.74030 − 6.20071I 0. + 7.00102I
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.769712 + 0.052444I
a = −0.818253 − 0.833331I
b = 0.257390 − 0.588680I
−2.66109 − 0.97035I −2.65239 − 1.69439I
u = 0.769712 − 0.052444I
a = −0.818253 + 0.833331I
b = 0.257390 + 0.588680I
−2.66109 + 0.97035I −2.65239 + 1.69439I
u = −0.496317 + 1.126200I
a = 1.35629 + 2.84442I
b = −0.396506 + 1.136040I
−3.29384 − 7.52516I 0. + 7.24279I
u = −0.496317 − 1.126200I
a = 1.35629 − 2.84442I
b = −0.396506 − 1.136040I
−3.29384 + 7.52516I 0. − 7.24279I
u = −0.254712 + 1.221060I
a = 0.54185 − 1.65734I
b = 0.918787 − 0.852340I
4.30712 − 1.86418I 0
u = −0.254712 − 1.221060I
a = 0.54185 + 1.65734I
b = 0.918787 + 0.852340I
4.30712 + 1.86418I 0
u = −0.287527 + 1.233810I
a = 0.72859 + 1.68616I
b = 0.853738 + 1.000090I
3.83508 + 4.68909I 0
u = −0.287527 − 1.233810I
a = 0.72859 − 1.68616I
b = 0.853738 − 1.000090I
3.83508 − 4.68909I 0
u = 0.433746 + 1.198270I
a = −0.009131 − 1.365560I
b = 0.240844 − 0.671835I
−6.28026 + 3.26453I 0
u = 0.433746 − 1.198270I
a = −0.009131 + 1.365560I
b = 0.240844 + 0.671835I
−6.28026 − 3.26453I 0
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.474335 + 1.198320I
a = −0.95599 + 1.60970I
b = 0.327962 + 0.610342I
−5.99302 + 5.51262I 0
u = 0.474335 − 1.198320I
a = −0.95599 − 1.60970I
b = 0.327962 − 0.610342I
−5.99302 − 5.51262I 0
u = −0.572998 + 1.165320I
a = 1.218420 − 0.246333I
b = 0.988002 + 0.822945I
6.50379 − 6.73509I 0
u = −0.572998 − 1.165320I
a = 1.218420 + 0.246333I
b = 0.988002 − 0.822945I
6.50379 + 6.73509I 0
u = −0.563792 + 1.182090I
a = −1.12741 − 2.43594I
b = 0.865756 − 1.050490I
5.7613 − 13.5297I 0
u = −0.563792 − 1.182090I
a = −1.12741 + 2.43594I
b = 0.865756 + 1.050490I
5.7613 + 13.5297I 0
u = −0.519184 + 0.444465I
a = 0.243564 − 0.426397I
b = −0.671501 + 0.420740I
1.51555 − 0.97971I 5.31353 + 2.39080I
u = −0.519184 − 0.444465I
a = 0.243564 + 0.426397I
b = −0.671501 − 0.420740I
1.51555 + 0.97971I 5.31353 − 2.39080I
u = 0.363275 + 0.565886I
a = 2.00200 − 1.16729I
b = −0.553077 − 0.974993I
0.28460 + 2.90693I 3.00949 − 1.23686I
u = 0.363275 − 0.565886I
a = 2.00200 + 1.16729I
b = −0.553077 + 0.974993I
0.28460 − 2.90693I 3.00949 + 1.23686I
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.539085 + 0.365328I
a = 0.768774 − 0.591177I
b = −0.715038 + 0.680296I
1.31536 − 1.96169I 3.58238 + 3.39063I
u = 0.539085 − 0.365328I
a = 0.768774 + 0.591177I
b = −0.715038 − 0.680296I
1.31536 + 1.96169I 3.58238 − 3.39063I
u = −0.616927 + 0.185561I
a = 0.74931 − 1.75287I
b = −0.406514 − 1.058610I
−0.68635 + 3.17011I 1.49231 − 4.26381I
u = −0.616927 − 0.185561I
a = 0.74931 + 1.75287I
b = −0.406514 + 1.058610I
−0.68635 − 3.17011I 1.49231 + 4.26381I
9
II. I
u
2
= h−2u
4
a − 2u
4
+ · · · − 7a − 7, u
3
a − u
2
a − 2u
3
+ a
2
+ au + 2u
2
− u +
2, u
5
− u
4
+ 2u
3
− u
2
+ u − 1i
(i) Arc colorings
a
6
=
1
0
a
11
=
0
u
a
2
=
a
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a + 0.368421
a
7
=
1
−u
2
a
5
=
−0.105263au
4
− 0.105263u
4
+ ··· + 0.631579a − 0.368421
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a − 0.631579
a
3
=
u
3
− u
2
+ a + u − 1
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a − 0.631579
a
1
=
−1
0
a
10
=
−u
u
3
+ u
a
12
=
−u
3
u
4
− u
3
+ u
2
+ 1
a
9
=
u
3
u
3
+ u
a
8
=
u
3
u
3
+ u
a
4
=
−0.105263au
4
− 0.105263u
4
+ ··· + 0.631579a − 0.368421
0.105263au
4
+ 0.105263u
4
+ ··· + 0.368421a − 0.631579
(ii) Obstruction class = 1
(iii) Cusp Shapes = −u
4
a − u
3
a + u
2
a + 2u
3
− 3au − 2u
2
− a + u − 2
10
(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
(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
9
, c
12
(u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
c
10
(u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
c
11
(u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
11
(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
10
(y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
c
7
, c
9
, c
12
(y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
c
11
(y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.339110 + 0.822375I
a = 0.523653 + 0.423720I
b = 0.500000 + 0.866025I
−0.329100 + 0.499304I 0.886311 − 0.883423I
u = −0.339110 + 0.822375I
a = −1.39487 − 1.53138I
b = 0.500000 − 0.866025I
−0.32910 − 3.56046I −3.42267 + 7.93863I
u = −0.339110 − 0.822375I
a = 0.523653 − 0.423720I
b = 0.500000 − 0.866025I
−0.329100 − 0.499304I 0.886311 + 0.883423I
u = −0.339110 − 0.822375I
a = −1.39487 + 1.53138I
b = 0.500000 + 0.866025I
−0.32910 + 3.56046I −3.42267 − 7.93863I
u = 0.766826
a = −0.314857 + 1.186700I
b = 0.500000 + 0.866025I
−2.40108 + 2.02988I −0.40252 − 4.16430I
u = 0.766826
a = −0.314857 − 1.186700I
b = 0.500000 − 0.866025I
−2.40108 − 2.02988I −0.40252 + 4.16430I
u = 0.455697 + 1.200150I
a = 0.85051 − 1.45588I
b = 0.500000 − 0.866025I
−5.87256 + 2.37095I −2.86519 + 1.02882I
u = 0.455697 + 1.200150I
a = −0.66443 + 2.33052I
b = 0.500000 + 0.866025I
−5.87256 + 6.43072I −4.19593 − 8.50148I
u = 0.455697 − 1.200150I
a = 0.85051 + 1.45588I
b = 0.500000 + 0.866025I
−5.87256 − 2.37095I −2.86519 − 1.02882I
u = 0.455697 − 1.200150I
a = −0.66443 − 2.33052I
b = 0.500000 − 0.866025I
−5.87256 − 6.43072I −4.19593 + 8.50148I
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u
2
− u + 1)
5
)(u
44
+ 10u
43
+ ··· + 2u + 1)
c
2
((u
2
+ u + 1)
5
)(u
44
+ 6u
43
+ ··· + 6u + 1)
c
3
((u
2
− u + 1)
5
)(u
44
− 6u
43
+ ··· + 717363u + 73746)
c
4
, c
8
u
10
(u
44
− u
43
+ ··· + 1024u + 1024)
c
5
((u
2
− u + 1)
5
)(u
44
+ 6u
43
+ ··· + 6u + 1)
c
6
((u
5
− u
4
+ 2u
3
− u
2
+ u − 1)
2
)(u
44
− 3u
43
+ ··· − 3u + 1)
c
7
((u
5
+ u
4
− 2u
3
− u
2
+ u − 1)
2
)(u
44
+ 3u
43
+ ··· + 211u + 34)
c
9
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
44
+ 3u
43
+ ··· + 211u + 34)
c
10
((u
5
+ u
4
+ 2u
3
+ u
2
+ u + 1)
2
)(u
44
− 3u
43
+ ··· − 3u + 1)
c
11
((u
5
+ 3u
4
+ 4u
3
+ u
2
− u − 1)
2
)(u
44
+ 23u
43
+ ··· + 3u + 1)
c
12
((u
5
− u
4
− 2u
3
+ u
2
+ u + 1)
2
)(u
44
− u
43
+ ··· + 3u + 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y
2
+ y + 1)
5
)(y
44
+ 54y
43
+ ··· + 102y + 1)
c
2
, c
5
((y
2
+ y + 1)
5
)(y
44
+ 10y
43
+ ··· + 2y + 1)
c
3
((y
2
+ y + 1)
5
)(y
44
+ 98y
43
+ ··· + 1.13290 × 10
11
y + 5.43847 × 10
9
)
c
4
, c
8
y
10
(y
44
− 55y
43
+ ··· − 1048576y + 1048576)
c
6
, c
10
((y
5
+ 3y
4
+ 4y
3
+ y
2
− y −1)
2
)(y
44
+ 23y
43
+ ··· + 3y + 1)
c
7
, c
9
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
44
− 25y
43
+ ··· + 17903y + 1156)
c
11
((y
5
− y
4
+ 8y
3
− 3y
2
+ 3y −1)
2
)(y
44
− y
43
+ ··· + 11y + 1)
c
12
((y
5
− 5y
4
+ 8y
3
− 3y
2
− y −1)
2
)(y
44
+ 75y
43
+ ··· + 3y + 1)
15
