12n
0340
(K12n
0340
)
A knot diagram
1
Linearized knot diagam
3 6 7 8 2 11 4 5 6 12 7 10
Solving Sequence
4,7
8 5
9,11
12 3 6 2 1 10
c
7
c
4
c
8
c
11
c
3
c
6
c
2
c
1
c
10
c
5
, c
9
, c
12
Ideals for irreducible components
2
of X
par
I
u
1
= h−62822499u
16
+ 98950048u
15
+ ··· + 179809420b + 1255421292,
75505587u
16
− 95682099u
15
+ ··· + 179809420a − 2254628556, u
17
− u
16
+ ··· − 40u − 8i
I
u
2
= h2a
2
+ 2au + 5b + 4a + 1, 4a
3
+ 4a
2
− 2au + 6a − 7u + 8, u
2
− 2i
I
v
1
= ha, b + v + 1, v
3
+ 2v
2
+ v + 1i
* 3 irreducible components of dim
C
= 0, with total 26 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−6.28 × 10
7
u
16
+ 9.90 × 10
7
u
15
+ · · · + 1.80 × 10
8
b + 1.26 × 10
9
, 7.55 ×
10
7
u
16
−9.57×10
7
u
15
+· · ·+1.80×10
8
a−2.25×10
9
, u
17
−u
16
+· · ·−40u −8i
(i) Arc colorings
a
4
=
0
u
a
7
=
1
0
a
8
=
1
−u
2
a
5
=
u
−u
3
+ u
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
11
=
−0.419920u
16
+ 0.532131u
15
+ ··· + 20.6991u + 12.5390
0.349384u
16
− 0.550305u
15
+ ··· − 19.0383u − 6.98196
a
12
=
−0.769304u
16
+ 1.08244u
15
+ ··· + 39.7373u + 19.5209
0.349384u
16
− 0.550305u
15
+ ··· − 19.0383u − 6.98196
a
3
=
−u
u
a
6
=
−0.706889u
16
+ 0.968606u
15
+ ··· + 35.6654u + 16.9023
−0.0874591u
16
+ 0.0795210u
15
+ ··· + 3.47438u + 1.11299
a
2
=
0.706889u
16
− 0.968606u
15
+ ··· − 35.6654u − 16.9023
−0.159634u
16
+ 0.215306u
15
+ ··· + 8.28792u + 3.20672
a
1
=
0.796542u
16
− 1.07340u
15
+ ··· − 39.5291u − 18.5507
−0.249287u
16
+ 0.320105u
15
+ ··· + 12.1516u + 4.85508
a
10
=
−0.267792u
16
+ 0.361424u
15
+ ··· + 13.1869u + 7.51366
0.153138u
16
− 0.238805u
15
+ ··· − 8.95604u − 3.24946
(ii) Obstruction class = −1
(iii) Cusp Shapes = −
54399547
25687060
u
16
+
11504827
3669580
u
15
+ ··· +
93333338
917395
u +
275633814
6421765
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
17
− 4u
16
+ ··· + 3757u + 529
c
2
, c
5
u
17
+ 4u
16
+ ··· − 59u + 23
c
3
, c
4
, c
7
c
8
u
17
− u
16
+ ··· − 40u − 8
c
6
, c
11
u
17
− 2u
16
+ ··· − 4u + 1
c
9
u
17
+ 2u
16
+ ··· − 284u + 1429
c
10
, c
12
u
17
+ 2u
16
+ ··· + 10u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
17
+ 60y
16
+ ··· + 3317101y −279841
c
2
, c
5
y
17
+ 4y
16
+ ··· + 3757y −529
c
3
, c
4
, c
7
c
8
y
17
− 31y
16
+ ··· + 960y −64
c
6
, c
11
y
17
− 2y
16
+ ··· + 10y −1
c
9
y
17
+ 102y
16
+ ··· + 30338302y −2042041
c
10
, c
12
y
17
+ 30y
16
+ ··· + 34y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.030470 + 0.189629I
a = −0.224345 − 0.523213I
b = −0.453303 + 0.618424I
2.71496 + 0.03038I 1.85613 − 0.36758I
u = 1.030470 − 0.189629I
a = −0.224345 + 0.523213I
b = −0.453303 − 0.618424I
2.71496 − 0.03038I 1.85613 + 0.36758I
u = −0.761878 + 0.176219I
a = −0.973738 + 0.882302I
b = −0.880524 − 0.587541I
1.72734 − 4.32421I −0.54590 + 7.81400I
u = −0.761878 − 0.176219I
a = −0.973738 − 0.882302I
b = −0.880524 + 0.587541I
1.72734 + 4.32421I −0.54590 − 7.81400I
u = 1.35959
a = −0.625953
b = −0.396497
3.15281 3.23020
u = −0.013551 + 0.593749I
a = 0.794033 − 0.406646I
b = 0.490534 − 0.506411I
−0.43270 + 1.37617I −3.71871 − 4.10562I
u = −0.013551 − 0.593749I
a = 0.794033 + 0.406646I
b = 0.490534 + 0.506411I
−0.43270 − 1.37617I −3.71871 + 4.10562I
u = −0.456807
a = 1.47926
b = 0.882527
−1.65919 −3.40410
u = −0.340015
a = 4.30114
b = −0.599895
−2.41310 6.57470
u = 1.70987 + 0.34907I
a = −0.035901 − 0.854639I
b = 1.105810 + 0.797473I
10.32630 − 2.94049I 0.85532 + 2.11383I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.70987 − 0.34907I
a = −0.035901 + 0.854639I
b = 1.105810 − 0.797473I
10.32630 + 2.94049I 0.85532 − 2.11383I
u = −1.64126 + 0.59755I
a = 0.137352 − 1.120570I
b = 0.823579 + 1.043060I
11.40790 − 3.91728I 1.42724 + 2.75158I
u = −1.64126 − 0.59755I
a = 0.137352 + 1.120570I
b = 0.823579 − 1.043060I
11.40790 + 3.91728I 1.42724 − 2.75158I
u = 1.96918 + 0.29196I
a = −0.119407 − 1.324750I
b = −1.10155 + 0.95457I
−15.4344 + 9.3275I 0.15944 − 3.93386I
u = 1.96918 − 0.29196I
a = −0.119407 + 1.324750I
b = −1.10155 − 0.95457I
−15.4344 − 9.3275I 0.15944 + 3.93386I
u = −2.07422 + 0.20650I
a = 0.344781 − 0.929300I
b = −0.92761 + 1.10685I
−14.7845 − 1.8072I 0.766090 − 0.113701I
u = −2.07422 − 0.20650I
a = 0.344781 + 0.929300I
b = −0.92761 − 1.10685I
−14.7845 + 1.8072I 0.766090 + 0.113701I
6
II. I
u
2
= h2a
2
+ 2au + 5b + 4a + 1, 4a
3
+ 4a
2
− 2au + 6a − 7u + 8, u
2
− 2i
(i) Arc colorings
a
4
=
0
u
a
7
=
1
0
a
8
=
1
−2
a
5
=
u
−u
a
9
=
−1
0
a
11
=
a
−
2
5
a
2
−
2
5
au −
4
5
a −
1
5
a
12
=
2
5
a
2
+
2
5
au +
9
5
a +
1
5
−
2
5
a
2
−
2
5
au −
4
5
a −
1
5
a
3
=
−u
u
a
6
=
2
5
a
2
u +
1
5
au + ··· −
2
5
a +
1
5
−
2
5
a
2
u −
1
5
au + ··· +
2
5
a −
1
5
a
2
=
2
5
a
2
u +
1
5
au + ··· −
2
5
a +
1
5
−
2
5
a
2
u −
1
5
au + ··· +
2
5
a −
1
5
a
1
=
2
5
a
2
u +
1
5
au + ··· −
2
5
a +
1
5
−
2
5
a
2
u −
1
5
au + ··· +
2
5
a −
1
5
a
10
=
3
5
a
2
u +
2
5
au + ··· +
3
5
a −
7
5
−
2
5
a
2
u −
3
5
au + ··· −
2
5
a +
3
5
(ii) Obstruction class = 1
(iii) Cusp Shapes = −
8
5
a
2
−
8
5
au −
16
5
a −
24
5
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
5
(u − 1)
6
c
2
(u + 1)
6
c
3
, c
4
, c
7
c
8
(u
2
− 2)
3
c
6
(u
3
− u
2
+ 1)
2
c
9
, c
10
(u
3
− u
2
+ 2u − 1)
2
c
11
(u
3
+ u
2
− 1)
2
c
12
(u
3
+ u
2
+ 2u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y −1)
6
c
3
, c
4
, c
7
c
8
(y −2)
6
c
6
, c
11
(y
3
− y
2
+ 2y −1)
2
c
9
, c
10
, c
12
(y
3
+ 3y
2
+ 2y −1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.41421
a = −0.683438 + 0.909550I
b = 0.877439 − 0.744862I
6.31400 + 2.82812I −0.49024 − 2.97945I
u = 1.41421
a = −0.683438 − 0.909550I
b = 0.877439 + 0.744862I
6.31400 − 2.82812I −0.49024 + 2.97945I
u = 1.41421
a = 0.366877
b = −0.754878
2.17641 −7.01950
u = −1.41421
a = −1.50656
b = −0.754878
2.17641 −7.01950
u = −1.41421
a = 0.25328 + 1.70473I
b = 0.877439 − 0.744862I
6.31400 + 2.82812I −0.49024 − 2.97945I
u = −1.41421
a = 0.25328 − 1.70473I
b = 0.877439 + 0.744862I
6.31400 − 2.82812I −0.49024 + 2.97945I
10
III. I
v
1
= ha, b + v + 1, v
3
+ 2v
2
+ v + 1i
(i) Arc colorings
a
4
=
v
0
a
7
=
1
0
a
8
=
1
0
a
5
=
v
0
a
9
=
1
0
a
11
=
0
−v − 1
a
12
=
v + 1
−v − 1
a
3
=
v
0
a
6
=
1
−v
2
− 2v − 1
a
2
=
v − 1
v
2
+ 2v + 1
a
1
=
−1
v
2
+ 2v + 1
a
10
=
−v
2
− 2v
v
2
+ v − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = −4v
2
+ 2v − 2
11
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
(u − 1)
3
c
3
, c
4
, c
7
c
8
u
3
c
5
(u + 1)
3
c
6
u
3
+ u
2
− 1
c
9
, c
12
u
3
+ u
2
+ 2u + 1
c
10
u
3
− u
2
+ 2u − 1
c
11
u
3
− u
2
+ 1
12
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
5
(y − 1)
3
c
3
, c
4
, c
7
c
8
y
3
c
6
, c
11
y
3
− y
2
+ 2y − 1
c
9
, c
10
, c
12
y
3
+ 3y
2
+ 2y − 1
13
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −0.122561 + 0.744862I
a = 0
b = −0.877439 − 0.744862I
1.37919 − 2.82812I −0.08593 + 2.22005I
v = −0.122561 − 0.744862I
a = 0
b = −0.877439 + 0.744862I
1.37919 + 2.82812I −0.08593 − 2.22005I
v = −1.75488
a = 0
b = 0.754878
−2.75839 −17.8280
14
IV. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
((u − 1)
9
)(u
17
− 4u
16
+ ··· + 3757u + 529)
c
2
((u − 1)
3
)(u + 1)
6
(u
17
+ 4u
16
+ ··· − 59u + 23)
c
3
, c
4
, c
7
c
8
u
3
(u
2
− 2)
3
(u
17
− u
16
+ ··· − 40u − 8)
c
5
((u − 1)
6
)(u + 1)
3
(u
17
+ 4u
16
+ ··· − 59u + 23)
c
6
((u
3
− u
2
+ 1)
2
)(u
3
+ u
2
− 1)(u
17
− 2u
16
+ ··· − 4u + 1)
c
9
((u
3
− u
2
+ 2u − 1)
2
)(u
3
+ u
2
+ 2u + 1)(u
17
+ 2u
16
+ ··· − 284u + 1429)
c
10
((u
3
− u
2
+ 2u − 1)
3
)(u
17
+ 2u
16
+ ··· + 10u + 1)
c
11
(u
3
− u
2
+ 1)(u
3
+ u
2
− 1)
2
(u
17
− 2u
16
+ ··· − 4u + 1)
c
12
((u
3
+ u
2
+ 2u + 1)
3
)(u
17
+ 2u
16
+ ··· + 10u + 1)
15
V. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
((y − 1)
9
)(y
17
+ 60y
16
+ ··· + 3317101y − 279841)
c
2
, c
5
((y − 1)
9
)(y
17
+ 4y
16
+ ··· + 3757y − 529)
c
3
, c
4
, c
7
c
8
y
3
(y − 2)
6
(y
17
− 31y
16
+ ··· + 960y − 64)
c
6
, c
11
((y
3
− y
2
+ 2y − 1)
3
)(y
17
− 2y
16
+ ··· + 10y − 1)
c
9
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
17
+ 102y
16
+ ··· + 3.03383 × 10
7
y − 2042041)
c
10
, c
12
((y
3
+ 3y
2
+ 2y − 1)
3
)(y
17
+ 30y
16
+ ··· + 34y − 1)
16
