11a
223
(K11a
223
)
A knot diagram
1
Linearized knot diagam
6 1 10 8 9 2 11 5 3 4 7
Solving Sequence
2,6
7 1
3,9
10 5 8 4 11
c
6
c
1
c
2
c
9
c
5
c
8
c
4
c
11
c
3
, c
7
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−u
19
+ 2u
18
+ ··· + b + 1, 5u
19
− 11u
18
+ ··· + 2a − 10, u
20
− 3u
19
+ ··· − 6u + 2i
I
u
2
= h−u
11
a + 10u
11
+ ··· + 2a − 13, −2u
10
a + u
11
+ ··· + a
2
+ 1,
u
12
+ u
11
− 3u
10
− 4u
9
+ 3u
8
+ 6u
7
+ 2u
6
− 2u
5
− 4u
4
− 3u
3
+ u
2
+ 2u + 1i
I
u
3
= hb + 1, u
3
− 2u
2
+ 2a + 4, u
4
− 2u
2
+ 2i
I
v
1
= ha, b − 1, v + 1i
* 4 irreducible components of dim
C
= 0, with total 49 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
19
+2u
18
+· · ·+b+1, 5u
19
−11u
18
+· · ·+2a−10, u
20
−3u
19
+· · ·−6u+2i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
9
=
−
5
2
u
19
+
11
2
u
18
+ ··· − 11u + 5
u
19
− 2u
18
+ ··· + 4u − 1
a
10
=
−
3
2
u
19
+
7
2
u
18
+ ··· − 7u + 3
u
19
− 2u
18
+ ··· + 3u − 1
a
5
=
−
1
2
u
19
+
3
2
u
18
+ ··· − 3u + 2
u
18
− u
17
+ ··· − 2u + 1
a
8
=
u
6
− u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
4
=
−
3
2
u
19
+
7
2
u
18
+ ··· − 7u + 3
−u
19
+ 3u
18
+ ··· − 6u + 3
a
11
=
u
3
u
5
− u
3
+ u
a
11
=
u
3
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 2u
19
− 12u
17
+ 6u
16
+ 32u
15
− 30u
14
− 34u
13
+ 66u
12
− 14u
11
−
58u
10
+ 78u
9
− 12u
8
− 62u
7
+ 68u
6
− 8u
5
− 32u
4
+ 38u
3
− 8u
2
− 2u
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
20
− 3u
19
+ ··· − 6u + 2
c
2
u
20
+ 11u
19
+ ··· + 4u + 4
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
20
+ u
19
+ ··· − 2u − 1
c
7
, c
11
u
20
− 9u
19
+ ··· + 110u − 22
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
y
20
− 11y
19
+ ··· − 4y + 4
c
2
y
20
− 3y
19
+ ··· − 208y + 16
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
20
− 27y
19
+ ··· − 10y + 1
c
7
, c
11
y
20
+ 17y
19
+ ··· − 4004y + 484
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.544915 + 0.735723I
a = −0.396672 − 0.140253I
b = 1.50718 − 0.07179I
−8.11263 + 1.41331I −12.03617 − 0.10296I
u = 0.544915 − 0.735723I
a = −0.396672 + 0.140253I
b = 1.50718 + 0.07179I
−8.11263 − 1.41331I −12.03617 + 0.10296I
u = 0.128827 + 0.901492I
a = −0.623064 − 0.737924I
b = −1.60887 + 0.30371I
−14.6659 + 7.5175I −13.03534 − 3.27786I
u = 0.128827 − 0.901492I
a = −0.623064 + 0.737924I
b = −1.60887 − 0.30371I
−14.6659 − 7.5175I −13.03534 + 3.27786I
u = 0.773452 + 0.404695I
a = 0.591002 − 0.705976I
b = 0.108607 + 0.523595I
0.87704 − 1.78379I −2.58390 + 5.68445I
u = 0.773452 − 0.404695I
a = 0.591002 + 0.705976I
b = 0.108607 − 0.523595I
0.87704 + 1.78379I −2.58390 − 5.68445I
u = 0.977557 + 0.624357I
a = −0.48691 + 1.67916I
b = −1.50584 − 0.14245I
−9.37386 − 6.54808I −13.7315 + 5.5285I
u = 0.977557 − 0.624357I
a = −0.48691 − 1.67916I
b = −1.50584 + 0.14245I
−9.37386 + 6.54808I −13.7315 − 5.5285I
u = −1.21457
a = −2.22945
b = −1.62522
−14.1194 −17.9240
u = −1.145210 + 0.438306I
a = −0.641824 − 0.515615I
b = −0.535707 − 0.310794I
−4.07199 + 2.60865I −11.03085 + 0.93775I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −1.145210 − 0.438306I
a = −0.641824 + 0.515615I
b = −0.535707 + 0.310794I
−4.07199 − 2.60865I −11.03085 − 0.93775I
u = 1.160540 + 0.458172I
a = −1.261220 + 0.382851I
b = −0.503696 − 0.478862I
−3.93117 − 5.51600I −10.44810 + 8.22749I
u = 1.160540 − 0.458172I
a = −1.261220 − 0.382851I
b = −0.503696 + 0.478862I
−3.93117 + 5.51600I −10.44810 − 8.22749I
u = −0.695075
a = 0.614797
b = 0.332547
−0.859562 −12.8980
u = −1.280150 + 0.384189I
a = 2.09625 + 0.60113I
b = 1.65612 + 0.28210I
−19.0845 − 3.0881I −16.9887 + 0.4542I
u = −1.280150 − 0.384189I
a = 2.09625 − 0.60113I
b = 1.65612 − 0.28210I
−19.0845 + 3.0881I −16.9887 − 0.4542I
u = 0.058790 + 0.660109I
a = 0.455681 + 0.359674I
b = 0.418244 − 0.389912I
−0.84744 + 1.30386I −6.93259 − 5.24353I
u = 0.058790 − 0.660109I
a = 0.455681 − 0.359674I
b = 0.418244 + 0.389912I
−0.84744 − 1.30386I −6.93259 + 5.24353I
u = 1.236100 + 0.531142I
a = 2.07408 − 1.71089I
b = 1.61029 + 0.34268I
−18.0142 − 12.6981I −15.8020 + 6.4148I
u = 1.236100 − 0.531142I
a = 2.07408 + 1.71089I
b = 1.61029 − 0.34268I
−18.0142 + 12.6981I −15.8020 − 6.4148I
6
II. I
u
2
=
h−u
11
a+10u
11
+· · ·+2a −13, −2u
10
a+u
11
+· · ·+a
2
+1, u
12
+u
11
+· · ·+2u +1i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
9
=
a
1
14
u
11
a −
5
7
u
11
+ ··· −
1
7
a +
13
14
a
10
=
−
1
14
u
11
a −
2
7
u
11
+ ··· +
8
7
a +
1
14
−0.214286au
11
− 1.35714u
11
+ ··· − 0.0714286a + 0.714286
a
5
=
0.285714au
11
− 0.357143u
11
+ ··· + 0.928571a + 1.21429
−0.357143au
11
+ 0.0714286u
11
+ ··· + 0.214286a − 1.14286
a
8
=
u
6
− u
4
+ 1
u
8
− 2u
6
+ 2u
4
a
4
=
−
1
14
u
11
a −
2
7
u
11
+ ··· +
8
7
a +
1
14
1
14
u
11
a +
2
7
u
11
+ ··· −
1
7
a −
15
14
a
11
=
u
3
u
5
− u
3
+ u
a
11
=
u
3
u
5
− u
3
+ u
(ii) Obstruction class = −1
(iii) Cusp Shapes = 4u
10
− 12u
8
− 4u
7
+ 16u
6
+ 8u
5
− 8u
3
− 8u
2
− 4u − 6
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
(u
12
+ u
11
+ ··· + 2u + 1)
2
c
2
(u
12
+ 7u
11
+ ··· + 2u + 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
u
24
+ u
23
+ ··· − 10u + 5
c
7
, c
11
(u
12
+ 3u
11
+ ··· + 2u + 1)
2
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
12
− 7y
11
+ ··· − 2y + 1)
2
c
2
(y
12
− 3y
11
+ ··· + 6y + 1)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y
24
− 21y
23
+ ··· − 220y + 25
c
7
, c
11
(y
12
+ 13y
11
+ ··· + 6y + 1)
2
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 0.961384 + 0.208970I
a = 0.506127 + 0.593369I
b = 0.915862 − 0.401943I
−5.02961 − 0.71593I −15.9565 + 0.6487I
u = 0.961384 + 0.208970I
a = −2.66748 + 1.31736I
b = −1.242690 − 0.150848I
−5.02961 − 0.71593I −15.9565 + 0.6487I
u = 0.961384 − 0.208970I
a = 0.506127 − 0.593369I
b = 0.915862 + 0.401943I
−5.02961 + 0.71593I −15.9565 − 0.6487I
u = 0.961384 − 0.208970I
a = −2.66748 − 1.31736I
b = −1.242690 + 0.150848I
−5.02961 + 0.71593I −15.9565 − 0.6487I
u = −0.958024 + 0.460561I
a = −0.002595 + 0.970301I
b = 0.317703 − 0.537023I
−3.21312 + 4.24921I −9.82351 − 6.98310I
u = −0.958024 + 0.460561I
a = −1.13256 − 1.76796I
b = −1.233460 + 0.149435I
−3.21312 + 4.24921I −9.82351 − 6.98310I
u = −0.958024 − 0.460561I
a = −0.002595 − 0.970301I
b = 0.317703 + 0.537023I
−3.21312 − 4.24921I −9.82351 + 6.98310I
u = −0.958024 − 0.460561I
a = −1.13256 + 1.76796I
b = −1.233460 − 0.149435I
−3.21312 − 4.24921I −9.82351 + 6.98310I
u = −0.049813 + 0.844037I
a = −1.205190 + 0.406247I
b = −1.51479 − 0.10395I
−7.33005 − 3.01307I −11.36825 + 2.63251I
u = −0.049813 + 0.844037I
a = 0.190483 − 0.652317I
b = 0.619350 + 0.907491I
−7.33005 − 3.01307I −11.36825 + 2.63251I
10
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.049813 − 0.844037I
a = −1.205190 − 0.406247I
b = −1.51479 + 0.10395I
−7.33005 + 3.01307I −11.36825 − 2.63251I
u = −0.049813 − 0.844037I
a = 0.190483 + 0.652317I
b = 0.619350 − 0.907491I
−7.33005 + 3.01307I −11.36825 − 2.63251I
u = 1.238640 + 0.435356I
a = −0.178745 + 0.729514I
b = −0.704482 + 0.930610I
−11.20510 − 1.48234I −15.1526 + 0.6754I
u = 1.238640 + 0.435356I
a = 2.56411 − 0.92305I
b = 1.55418 − 0.05622I
−11.20510 − 1.48234I −15.1526 + 0.6754I
u = 1.238640 − 0.435356I
a = −0.178745 − 0.729514I
b = −0.704482 − 0.930610I
−11.20510 + 1.48234I −15.1526 − 0.6754I
u = 1.238640 − 0.435356I
a = 2.56411 + 0.92305I
b = 1.55418 + 0.05622I
−11.20510 + 1.48234I −15.1526 − 0.6754I
u = −1.228550 + 0.484706I
a = −1.41739 − 0.27157I
b = −0.584122 + 0.976162I
−10.84800 + 7.80134I −14.3661 − 5.6398I
u = −1.228550 + 0.484706I
a = 2.49284 + 1.50692I
b = 1.54701 − 0.14731I
−10.84800 + 7.80134I −14.3661 − 5.6398I
u = −1.228550 − 0.484706I
a = −1.41739 + 0.27157I
b = −0.584122 − 0.976162I
−10.84800 − 7.80134I −14.3661 + 5.6398I
u = −1.228550 − 0.484706I
a = 2.49284 − 1.50692I
b = 1.54701 + 0.14731I
−10.84800 − 7.80134I −14.3661 + 5.6398I
11
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = −0.463636 + 0.458719I
a = 1.49987 + 0.51998I
b = −0.312209 − 0.212773I
−1.85256 − 0.35310I −5.33308 + 0.62981I
u = −0.463636 + 0.458719I
a = −0.149462 − 0.021454I
b = 1.137650 + 0.055627I
−1.85256 − 0.35310I −5.33308 + 0.62981I
u = −0.463636 − 0.458719I
a = 1.49987 − 0.51998I
b = −0.312209 + 0.212773I
−1.85256 + 0.35310I −5.33308 − 0.62981I
u = −0.463636 − 0.458719I
a = −0.149462 + 0.021454I
b = 1.137650 − 0.055627I
−1.85256 + 0.35310I −5.33308 − 0.62981I
12
III. I
u
3
= hb + 1, u
3
− 2u
2
+ 2a + 4, u
4
− 2u
2
+ 2i
(i) Arc colorings
a
2
=
0
u
a
6
=
1
0
a
7
=
1
u
2
a
1
=
u
u
a
3
=
−u
3
−u
3
+ u
a
9
=
−
1
2
u
3
+ u
2
− 2
−1
a
10
=
1
2
u
3
+ u
2
− 2
u
3
− u − 1
a
5
=
−
1
2
u
3
+ u
2
− 1
−1
a
8
=
−1
0
a
4
=
−
1
2
u
3
+ u
2
− 2
−1
a
11
=
u
3
u
3
− u
a
11
=
u
3
u
3
− u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u
2
− 20
13
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
4
− 2u
2
+ 2
c
2
(u
2
+ 2u + 2)
2
c
3
, c
8
(u − 1)
4
c
4
, c
5
, c
9
c
10
(u + 1)
4
c
7
, c
11
u
4
+ 2u
2
+ 2
14
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
6
(y
2
− 2y + 2)
2
c
2
(y
2
+ 4)
2
c
3
, c
4
, c
5
c
8
, c
9
, c
10
(y −1)
4
c
7
, c
11
(y
2
+ 2y + 2)
2
15
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
3
√
−1(vol +
√
−1CS) Cusp shape
u = 1.098680 + 0.455090I
a = −1.321800 + 0.223113I
b = −1.00000
−5.75727 − 3.66386I −16.0000 + 4.0000I
u = 1.098680 − 0.455090I
a = −1.321800 − 0.223113I
b = −1.00000
−5.75727 + 3.66386I −16.0000 − 4.0000I
u = −1.098680 + 0.455090I
a = −0.67820 − 1.77689I
b = −1.00000
−5.75727 + 3.66386I −16.0000 − 4.0000I
u = −1.098680 − 0.455090I
a = −0.67820 + 1.77689I
b = −1.00000
−5.75727 − 3.66386I −16.0000 + 4.0000I
16
IV. I
v
1
= ha, b − 1, v + 1i
(i) Arc colorings
a
2
=
−1
0
a
6
=
1
0
a
7
=
1
0
a
1
=
−1
0
a
3
=
−1
0
a
9
=
0
1
a
10
=
−1
1
a
5
=
1
−1
a
8
=
1
0
a
4
=
0
−1
a
11
=
−1
0
a
11
=
−1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes = −12
17
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
u
c
3
, c
8
u + 1
c
4
, c
5
, c
9
c
10
u − 1
18
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
2
, c
6
c
7
, c
11
y
c
3
, c
4
, c
5
c
8
, c
9
, c
10
y −1
19
(vi) Complex Volumes and Cusp Shapes
Solutions to I
v
1
√
−1(vol +
√
−1CS) Cusp shape
v = −1.00000
a = 0
b = 1.00000
−3.28987 −12.0000
20
V. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
, c
6
u(u
4
− 2u
2
+ 2)(u
12
+ u
11
+ ··· + 2u + 1)
2
(u
20
− 3u
19
+ ··· − 6u + 2)
c
2
u(u
2
+ 2u + 2)
2
(u
12
+ 7u
11
+ ··· + 2u + 1)
2
· (u
20
+ 11u
19
+ ··· + 4u + 4)
c
3
, c
8
((u − 1)
4
)(u + 1)(u
20
+ u
19
+ ··· − 2u − 1)(u
24
+ u
23
+ ··· − 10u + 5)
c
4
, c
5
, c
9
c
10
(u − 1)(u + 1)
4
(u
20
+ u
19
+ ··· − 2u − 1)(u
24
+ u
23
+ ··· − 10u + 5)
c
7
, c
11
u(u
4
+ 2u
2
+ 2)(u
12
+ 3u
11
+ ··· + 2u + 1)
2
· (u
20
− 9u
19
+ ··· + 110u − 22)
21
VI. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
6
y(y
2
− 2y + 2)
2
(y
12
− 7y
11
+ ··· − 2y + 1)
2
· (y
20
− 11y
19
+ ··· − 4y + 4)
c
2
y(y
2
+ 4)
2
(y
12
− 3y
11
+ ··· + 6y + 1)
2
(y
20
− 3y
19
+ ··· − 208y + 16)
c
3
, c
4
, c
5
c
8
, c
9
, c
10
((y −1)
5
)(y
20
− 27y
19
+ ··· − 10y + 1)(y
24
− 21y
23
+ ··· − 220y + 25)
c
7
, c
11
y(y
2
+ 2y + 2)
2
(y
12
+ 13y
11
+ ··· + 6y + 1)
2
· (y
20
+ 17y
19
+ ··· − 4004y + 484)
22
