12a
0859
(K12a
0859
)
A knot diagram
1
Linearized knot diagam
4 5 10 9 3 11 12 1 2 6 7 8
Solving Sequence
7,12
8
1,4
2 9 5 11 6 10 3
c
7
c
12
c
1
c
8
c
4
c
11
c
6
c
10
c
3
c
2
, c
5
, c
9
Ideals for irreducible components
2
of X
par
I
u
1
= h−6396631669231u
40
+ 14505431853193u
39
+ ··· + 4256439997929b − 11189757056882,
− 1233760383109u
40
+ 3011254162855u
39
+ ··· + 608062856847a − 1195143358109,
u
41
− 2u
40
+ ··· + 5u + 1i
I
u
2
= hb + 1, a − u + 1, u
2
− u − 1i
* 2 irreducible components of dim
C
= 0, with total 43 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−6.40×10
12
u
40
+1.45×10
13
u
39
+· · ·+4.26×10
12
b−1.12×10
13
, −1.23×
10
12
u
40
+3.01×10
12
u
39
+· · ·+6.08×10
11
a−1.20×10
12
, u
41
−2u
40
+· · ·+5u+1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
−u
2
a
1
=
u
−u
3
+ u
a
4
=
2.02900u
40
− 4.95221u
39
+ ··· + 15.4220u + 1.96549
1.50281u
40
− 3.40788u
39
+ ··· + 11.3789u + 2.62890
a
2
=
0.644891u
40
− 0.202904u
39
+ ··· − 3.45077u + 0.277543
−1.08688u
40
− 0.397073u
39
+ ··· + 2.94691u + 0.644891
a
9
=
−u
2
+ 1
u
4
− 2u
2
a
5
=
3.70468u
40
− 8.56110u
39
+ ··· + 27.1650u + 4.70761
−0.0447870u
40
− 1.39889u
39
+ ··· + 9.37521u + 1.50458
a
11
=
−u
u
a
6
=
−u
2
+ 1
u
2
a
10
=
u
3
− 2u
−u
3
+ u
a
3
=
3.76543u
40
− 8.75980u
39
+ ··· + 26.5272u + 5.57736
0.0237411u
40
− 1.60020u
39
+ ··· + 9.41064u + 1.56557
(ii) Obstruction class = −1
(iii) Cusp Shapes
= −
47359814646430
1418813332643
u
40
+
116172431816127
1418813332643
u
39
+ ··· −
243964640346834
1418813332643
u −
78234342263103
1418813332643
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
41
− 7u
40
+ ··· + 20u + 4
c
2
, c
5
u
41
+ 3u
40
+ ··· + 14u − 1
c
3
u
41
+ 2u
40
+ ··· − 7931u + 3953
c
4
u
41
+ 4u
40
+ ··· + 197u − 19
c
6
, c
7
, c
8
c
10
, c
11
, c
12
u
41
+ 2u
40
+ ··· + 5u − 1
c
9
u
41
− 2u
40
+ ··· − u + 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
41
+ 15y
40
+ ··· + 104y −16
c
2
, c
5
y
41
− 35y
40
+ ··· + 102y −1
c
3
y
41
− 12y
40
+ ··· − 126297725y −15626209
c
4
y
41
− 56y
40
+ ··· + 33831y −361
c
6
, c
7
, c
8
c
10
, c
11
, c
12
y
41
− 60y
40
+ ··· + 3y −1
c
9
y
41
+ 4y
40
+ ··· + 3y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.650909 + 0.546359I
a = −0.342020 + 0.264851I
b = 0.419018 − 0.668068I
5.06652 + 0.73720I 15.9081 − 0.1824I
u = −0.650909 − 0.546359I
a = −0.342020 − 0.264851I
b = 0.419018 + 0.668068I
5.06652 − 0.73720I 15.9081 + 0.1824I
u = 0.698668 + 0.479441I
a = 0.153099 − 0.435306I
b = −0.188432 + 1.146320I
5.52631 + 8.23953I 13.0991 − 8.0874I
u = 0.698668 − 0.479441I
a = 0.153099 + 0.435306I
b = −0.188432 − 1.146320I
5.52631 − 8.23953I 13.0991 + 8.0874I
u = −0.814283
a = 0.0171192
b = −0.681057
1.28070 6.70620
u = 1.247790 + 0.069918I
a = 0.862844 + 1.105030I
b = −0.437368 − 0.248810I
6.66501 + 1.00107I 0
u = 1.247790 − 0.069918I
a = 0.862844 − 1.105030I
b = −0.437368 + 0.248810I
6.66501 − 1.00107I 0
u = 1.28100
a = −3.02459
b = 2.36170
8.47175 0
u = −1.290140 + 0.118342I
a = 0.175331 − 1.312220I
b = −0.325850 − 0.263604I
6.85332 − 5.09177I 0
u = −1.290140 − 0.118342I
a = 0.175331 + 1.312220I
b = −0.325850 + 0.263604I
6.85332 + 5.09177I 0
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.328540 + 0.037966I
a = 0.004954 − 0.826516I
b = 1.188080 − 0.254233I
10.93780 − 2.08245I 0
u = −1.328540 − 0.037966I
a = 0.004954 + 0.826516I
b = 1.188080 + 0.254233I
10.93780 + 2.08245I 0
u = 0.659491 + 0.101039I
a = 1.78864 + 0.62056I
b = −0.599129 − 0.924522I
4.32500 + 1.61112I 18.8008 − 4.7797I
u = 0.659491 − 0.101039I
a = 1.78864 − 0.62056I
b = −0.599129 + 0.924522I
4.32500 − 1.61112I 18.8008 + 4.7797I
u = 0.576581 + 0.288116I
a = −0.354068 + 1.004510I
b = 0.372702 − 1.123810I
0.71083 + 3.70513I 10.09368 − 9.17012I
u = 0.576581 − 0.288116I
a = −0.354068 − 1.004510I
b = 0.372702 + 1.123810I
0.71083 − 3.70513I 10.09368 + 9.17012I
u = −0.043590 + 0.639393I
a = 1.046310 − 0.645722I
b = 0.0798501 + 0.0811079I
3.26424 − 4.59923I 10.62501 + 5.19359I
u = −0.043590 − 0.639393I
a = 1.046310 + 0.645722I
b = 0.0798501 − 0.0811079I
3.26424 + 4.59923I 10.62501 − 5.19359I
u = −1.344360 + 0.232182I
a = −0.353369 + 1.278910I
b = −0.006247 − 0.161367I
12.2007 − 10.7908I 0
u = −1.344360 − 0.232182I
a = −0.353369 − 1.278910I
b = −0.006247 + 0.161367I
12.2007 + 10.7908I 0
6
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 1.348660 + 0.276814I
a = −0.414645 − 0.555057I
b = −0.165637 − 0.010708I
11.56960 + 2.23576I 0
u = 1.348660 − 0.276814I
a = −0.414645 + 0.555057I
b = −0.165637 + 0.010708I
11.56960 − 2.23576I 0
u = −0.497845 + 0.153887I
a = −0.345928 − 0.229979I
b = −0.477639 + 0.446316I
0.963123 − 0.222847I 11.33687 + 1.79100I
u = −0.497845 − 0.153887I
a = −0.345928 + 0.229979I
b = −0.477639 − 0.446316I
0.963123 + 0.222847I 11.33687 − 1.79100I
u = −0.503963
a = 4.08982
b = 2.61045
2.49303 −73.8380
u = 0.041427 + 0.380391I
a = −1.47501 + 1.25938I
b = 0.223887 + 0.085512I
−0.87754 − 1.43971I 1.57134 + 2.80893I
u = 0.041427 − 0.380391I
a = −1.47501 − 1.25938I
b = 0.223887 − 0.085512I
−0.87754 + 1.43971I 1.57134 − 2.80893I
u = 1.66053
a = 0.443601
b = −0.446168
10.1068 0
u = −0.150168 + 0.199500I
a = −3.03352 + 0.19155I
b = −0.797576 + 0.621794I
1.94826 − 0.63992I 4.80549 − 1.01730I
u = −0.150168 − 0.199500I
a = −3.03352 − 0.19155I
b = −0.797576 − 0.621794I
1.94826 + 0.63992I 4.80549 + 1.01730I
7
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.80288 + 0.01775I
a = −0.75332 + 2.11781I
b = 1.72917 − 4.38308I
17.9481 − 1.4034I 0
u = −1.80288 − 0.01775I
a = −0.75332 − 2.11781I
b = 1.72917 + 4.38308I
17.9481 + 1.4034I 0
u = −1.81126
a = 2.44844
b = −5.97707
−19.5212 0
u = 1.81283 + 0.02856I
a = 0.40071 − 2.86621I
b = −0.63518 + 5.58995I
18.3589 + 5.7706I 0
u = 1.81283 − 0.02856I
a = 0.40071 + 2.86621I
b = −0.63518 − 5.58995I
18.3589 − 5.7706I 0
u = 1.82183 + 0.00897I
a = −1.28360 − 2.01039I
b = 1.99257 + 3.92606I
−16.7931 + 2.3034I 0
u = 1.82183 − 0.00897I
a = −1.28360 + 2.01039I
b = 1.99257 − 3.92606I
−16.7931 − 2.3034I 0
u = 1.82503 + 0.05956I
a = 0.04404 + 2.55326I
b = −0.10800 − 5.18854I
−15.5464 + 12.1966I 0
u = 1.82503 − 0.05956I
a = 0.04404 − 2.55326I
b = −0.10800 + 5.18854I
−15.5464 − 12.1966I 0
u = −1.82989 + 0.06825I
a = 0.39236 − 1.61771I
b = −0.69814 + 3.25968I
−16.1413 − 3.8895I 0
8
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.82989 − 0.06825I
a = 0.39236 + 1.61771I
b = −0.69814 − 3.25968I
−16.1413 + 3.8895I 0
9
II. I
u
2
= hb + 1, a − u + 1, u
2
− u − 1i
(i) Arc colorings
a
7
=
1
0
a
12
=
0
u
a
8
=
1
−u − 1
a
1
=
u
−u − 1
a
4
=
u − 1
−1
a
2
=
u
−u − 1
a
9
=
−u
u
a
5
=
u − 2
0
a
11
=
−u
u
a
6
=
−u
u + 1
a
10
=
1
−u − 1
a
3
=
2u − 2
−u − 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 21
10
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
u
2
c
2
(u + 1)
2
c
3
, c
4
, c
6
c
7
, c
8
, c
9
u
2
− u − 1
c
5
(u − 1)
2
c
10
, c
11
, c
12
u
2
+ u − 1
11
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
y
2
c
2
, c
5
(y −1)
2
c
3
, c
4
, c
6
c
7
, c
8
, c
9
c
10
, c
11
, c
12
y
2
− 3y + 1
12
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.618034
a = −1.61803
b = −1.00000
2.63189 21.0000
u = 1.61803
a = 0.618034
b = −1.00000
10.5276 21.0000
13
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
u
2
(u
41
− 7u
40
+ ··· + 20u + 4)
c
2
((u + 1)
2
)(u
41
+ 3u
40
+ ··· + 14u − 1)
c
3
(u
2
− u − 1)(u
41
+ 2u
40
+ ··· − 7931u + 3953)
c
4
(u
2
− u − 1)(u
41
+ 4u
40
+ ··· + 197u − 19)
c
5
((u − 1)
2
)(u
41
+ 3u
40
+ ··· + 14u − 1)
c
6
, c
7
, c
8
(u
2
− u − 1)(u
41
+ 2u
40
+ ··· + 5u − 1)
c
9
(u
2
− u − 1)(u
41
− 2u
40
+ ··· − u + 1)
c
10
, c
11
, c
12
(u
2
+ u − 1)(u
41
+ 2u
40
+ ··· + 5u − 1)
14
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
y
2
(y
41
+ 15y
40
+ ··· + 104y −16)
c
2
, c
5
((y −1)
2
)(y
41
− 35y
40
+ ··· + 102y −1)
c
3
(y
2
− 3y + 1)(y
41
− 12y
40
+ ··· − 1.26298 × 10
8
y −1.56262 × 10
7
)
c
4
(y
2
− 3y + 1)(y
41
− 56y
40
+ ··· + 33831y −361)
c
6
, c
7
, c
8
c
10
, c
11
, c
12
(y
2
− 3y + 1)(y
41
− 60y
40
+ ··· + 3y −1)
c
9
(y
2
− 3y + 1)(y
41
+ 4y
40
+ ··· + 3y −1)
15
