11n
24
(K11n
24
)
A knot diagram
1
Linearized knot diagam
4 1 7 2 10 3 5 11 1 5 9
Solving Sequence
1,4
2
5,9
10 11 8 7 3 6
c
1
c
4
c
9
c
11
c
8
c
7
c
3
c
6
c
2
, c
5
, c
10
Ideals for irreducible components
2
of X
par
I
u
1
= h−44u
16
+ 153u
15
+ ··· + 4229b + 2570, −15u
16
+ 2455u
15
+ ··· + 4229a − 4314,
u
17
+ 2u
16
+ ··· + u − 1i
I
u
2
= hb + 1, −u
4
− u
3
+ a + u + 1, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
* 2 irreducible components of dim
C
= 0, with total 23 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−44u
16
+ 153u
15
+ · · · + 4229b + 2570, −15u
16
+ 2455u
15
+ · · · +
4229a − 4314, u
17
+ 2u
16
+ · · · + u − 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
9
=
0.00354694u
16
− 0.580515u
15
+ ··· + 1.76046u + 1.02010
0.0104044u
16
− 0.0361788u
15
+ ··· − 1.16931u − 0.607709
a
10
=
−0.00685741u
16
− 0.544337u
15
+ ··· + 2.92977u + 1.62781
0.0104044u
16
− 0.0361788u
15
+ ··· − 1.16931u − 0.607709
a
11
=
−0.0245921u
16
− 0.641759u
15
+ ··· + 3.12745u + 1.52731
−0.0416174u
16
+ 0.144715u
15
+ ··· − 1.32277u − 0.569165
a
8
=
0.933318u
16
+ 0.913691u
15
+ ··· − 2.09671u − 0.377867
0.895956u
16
+ 0.361788u
15
+ ··· + 1.69307u − 0.922913
a
7
=
−0.0134784u
16
+ 0.205959u
15
+ ··· − 2.68976u − 0.0763774
0.0520218u
16
− 0.180894u
15
+ ··· + 0.153464u − 0.0385434
a
3
=
−u
2
+ 1
u
2
a
6
=
−0.0300307u
16
+ 0.581698u
15
+ ··· − 3.23859u + 0.163159
0.301490u
16
− 0.343817u
15
+ ··· + 0.639395u − 0.291558
a
6
=
−0.0300307u
16
+ 0.581698u
15
+ ··· − 3.23859u + 0.163159
0.301490u
16
− 0.343817u
15
+ ··· + 0.639395u − 0.291558
(ii) Obstruction class = −1
(iii) Cusp Shapes =
7665
4229
u
16
+
5737
4229
u
15
+ ··· −
1705
4229
u −
11542
4229
2
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
4
u
17
− 2u
16
+ ··· + u + 1
c
2
u
17
+ 12u
16
+ ··· − u + 1
c
3
, c
6
u
17
− 2u
16
+ ··· + u − 1
c
5
, c
10
u
17
+ 3u
16
+ ··· + 128u − 64
c
7
u
17
+ 6u
16
+ ··· + 3897u + 1609
c
8
, c
9
, c
11
u
17
− 7u
16
+ ··· + 18u
2
+ 1
3
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
4
y
17
− 12y
16
+ ··· − y −1
c
2
y
17
− 12y
16
+ ··· − 77y −1
c
3
, c
6
y
17
+ 18y
15
+ ··· − y −1
c
5
, c
10
y
17
+ 39y
16
+ ··· + 8192y −4096
c
7
y
17
+ 68y
16
+ ··· − 35776857y −2588881
c
8
, c
9
, c
11
y
17
− 31y
16
+ ··· − 36y −1
4
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = 0.977894 + 0.194640I
a = 0.644273 + 0.271764I
b = −0.074423 − 0.157594I
−1.75571 − 0.69092I −4.20927 − 0.03881I
u = 0.977894 − 0.194640I
a = 0.644273 − 0.271764I
b = −0.074423 + 0.157594I
−1.75571 + 0.69092I −4.20927 + 0.03881I
u = 0.007198 + 1.101270I
a = 0.0948459 − 0.0860640I
b = 2.00423 + 0.17609I
−14.0076 − 4.0781I −2.81048 + 2.03189I
u = 0.007198 − 1.101270I
a = 0.0948459 + 0.0860640I
b = 2.00423 − 0.17609I
−14.0076 + 4.0781I −2.81048 − 2.03189I
u = 1.11899
a = −3.55652
b = −1.14890
−3.74765 10.6760
u = −1.094410 + 0.448256I
a = 0.297387 + 0.400102I
b = 0.394810 + 0.688088I
−0.57451 + 4.58866I −0.38155 − 5.05474I
u = −1.094410 − 0.448256I
a = 0.297387 − 0.400102I
b = 0.394810 − 0.688088I
−0.57451 − 4.58866I −0.38155 + 5.05474I
u = −1.279610 + 0.127778I
a = −1.75394 + 0.15484I
b = −1.65978 + 0.99674I
−6.36531 + 2.55518I −9.15621 − 3.45666I
u = −1.279610 − 0.127778I
a = −1.75394 − 0.15484I
b = −1.65978 − 0.99674I
−6.36531 − 2.55518I −9.15621 + 3.45666I
u = −0.397422 + 0.534657I
a = 0.841166 + 0.259861I
b = 0.289585 − 0.225601I
1.46199 − 0.53067I 6.34861 + 0.44801I
5
Solutions to I
u
1
√
−1(vol +
√
−1CS) Cusp shape
u = −0.397422 − 0.534657I
a = 0.841166 − 0.259861I
b = 0.289585 + 0.225601I
1.46199 + 0.53067I 6.34861 − 0.44801I
u = −1.38093 + 0.53931I
a = 1.72060 + 1.08326I
b = 2.05747 − 0.40594I
−18.3559 + 9.9055I −5.27294 − 4.68483I
u = −1.38093 − 0.53931I
a = 1.72060 − 1.08326I
b = 2.05747 + 0.40594I
−18.3559 − 9.9055I −5.27294 + 4.68483I
u = 1.38223 + 0.54918I
a = 1.50860 − 1.17337I
b = 2.09111 + 0.05760I
−18.3031 − 1.7986I −5.37966 + 0.73401I
u = 1.38223 − 0.54918I
a = 1.50860 + 1.17337I
b = 2.09111 − 0.05760I
−18.3031 + 1.7986I −5.37966 − 0.73401I
u = 0.225548 + 0.312459I
a = 1.92533 + 0.42540I
b = −1.028560 − 0.314868I
−1.91105 − 0.93427I −3.47646 + 1.18545I
u = 0.225548 − 0.312459I
a = 1.92533 − 0.42540I
b = −1.028560 + 0.314868I
−1.91105 + 0.93427I −3.47646 − 1.18545I
6
II. I
u
2
= hb + 1, −u
4
− u
3
+ a + u + 1, u
6
+ u
5
− u
4
− 2u
3
+ u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
5
=
−u
−u
3
+ u
a
9
=
u
4
+ u
3
− u − 1
−1
a
10
=
u
4
+ u
3
− u
−1
a
11
=
u
4
+ u
3
− u
−1
a
8
=
−1
0
a
7
=
−u
4
+ u
2
− 1
u
5
+ u
4
− 2u
3
− u
2
+ u + 1
a
3
=
−u
2
+ 1
u
2
a
6
=
−u
−u
3
+ u
a
6
=
−u
−u
3
+ u
(ii) Obstruction class = 1
(iii) Cusp Shapes = 3u
4
− 2u
3
− 3u
2
− 2u − 1
7
(iv) u-Polynomials at the component
Crossings u-Polynomials at each crossing
c
1
, c
6
u
6
+ u
5
− u
4
− 2u
3
+ u + 1
c
2
u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1
c
3
, c
4
u
6
− u
5
− u
4
+ 2u
3
− u + 1
c
5
, c
10
u
6
c
7
u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1
c
8
, c
9
(u − 1)
6
c
11
(u + 1)
6
8
(v) Riley Polynomials at the component
Crossings Riley Polynomials at each crossing
c
1
, c
3
, c
4
c
6
y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1
c
2
, c
7
y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1
c
5
, c
10
y
6
c
8
, c
9
, c
11
(y −1)
6
9
(vi) Complex Volumes and Cusp Shapes
Solutions to I
u
2
√
−1(vol +
√
−1CS) Cusp shape
u = 1.002190 + 0.295542I
a = −0.76815 + 1.65564I
b = −1.00000
−3.53554 − 0.92430I −5.77331 − 0.83820I
u = 1.002190 − 0.295542I
a = −0.76815 − 1.65564I
b = −1.00000
−3.53554 + 0.92430I −5.77331 + 0.83820I
u = −0.428243 + 0.664531I
a = −0.340228 − 0.298454I
b = −1.00000
0.245672 − 0.924305I −1.11831 + 1.11590I
u = −0.428243 − 0.664531I
a = −0.340228 + 0.298454I
b = −1.00000
0.245672 + 0.924305I −1.11831 − 1.11590I
u = −1.073950 + 0.558752I
a = −0.891622 − 0.818891I
b = −1.00000
−1.64493 + 5.69302I −3.10838 − 7.09196I
u = −1.073950 − 0.558752I
a = −0.891622 + 0.818891I
b = −1.00000
−1.64493 − 5.69302I −3.10838 + 7.09196I
10
III. u-Polynomials
Crossings u-Polynomials at each crossing
c
1
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
17
− 2u
16
+ ··· + u + 1)
c
2
(u
6
+ 3u
5
+ 5u
4
+ 4u
3
+ 2u
2
+ u + 1)(u
17
+ 12u
16
+ ··· − u + 1)
c
3
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
17
− 2u
16
+ ··· + u − 1)
c
4
(u
6
− u
5
− u
4
+ 2u
3
− u + 1)(u
17
− 2u
16
+ ··· + u + 1)
c
5
, c
10
u
6
(u
17
+ 3u
16
+ ··· + 128u − 64)
c
6
(u
6
+ u
5
− u
4
− 2u
3
+ u + 1)(u
17
− 2u
16
+ ··· + u − 1)
c
7
(u
6
− 3u
5
+ 5u
4
− 4u
3
+ 2u
2
− u + 1)(u
17
+ 6u
16
+ ··· + 3897u + 1609)
c
8
, c
9
((u − 1)
6
)(u
17
− 7u
16
+ ··· + 18u
2
+ 1)
c
11
((u + 1)
6
)(u
17
− 7u
16
+ ··· + 18u
2
+ 1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossing
c
1
, c
4
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
17
− 12y
16
+ ··· − y −1)
c
2
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)(y
17
− 12y
16
+ ··· − 77y −1)
c
3
, c
6
(y
6
− 3y
5
+ 5y
4
− 4y
3
+ 2y
2
− y + 1)(y
17
+ 18y
15
+ ··· − y −1)
c
5
, c
10
y
6
(y
17
+ 39y
16
+ ··· + 8192y −4096)
c
7
(y
6
+ y
5
+ 5y
4
+ 6y
2
+ 3y + 1)
· (y
17
+ 68y
16
+ ··· − 35776857y −2588881)
c
8
, c
9
, c
11
((y −1)
6
)(y
17
− 31y
16
+ ··· − 36y −1)
12
