11n
24
(K11n
24
)
1
Arc Sequences
4 1 7 2 10 3 5 11 1 5 9
Solving Sequence
1,4
2 3
5,9
10 6 7 11 8
c
1
c
2
c
4
c
9
c
5
c
6
c
11
c
8
c
3
, c
7
, c
10
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
6
+ u
5
u
4
2u
3
+ u + 1, b + 1, u
4
u
3
+ a + u + 1i
I
u
2
= hu
17
2u
16
+ ··· + u + 1, 44u
16
153u
15
+ ··· + 4229b + 2570,
15u
16
2455u
15
+ ··· + 4229a 4314i
There are 2 irreducible components with 23 representations.
1
The knot diagram image is adapter from “C. Livingston and A. H. Moore, KnotInfo: Table of Knot
Invariants, http://www.indiana.edu/ knotinfo”
1
I. I
u
1
= hu
6
+ u
5
u
4
2u
3
+ u + 1, b + 1, u
4
u
3
+ a + u + 1i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
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
6
=
u
u
3
+ u
a
7
=
u
4
+ u
2
1
u
5
+ u
4
2u
3
u
2
+ u + 1
a
11
=
u
4
+ u
3
u
1
a
8
=
1
0
a
8
=
1
0
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
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
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 = 1.002193 0.295542I
a = 0.76815 1.65564I
b = 1.00000
3.53554 + 0.92430I 5.77331 + 0.83820I
u = 1.002193 + 0.295542I
a = 0.76815 + 1.65564I
b = 1.00000
3.53554 0.92430I 5.77331 0.83820I
3
II. I
u
2
= hu
17
2u
16
+ · · · + u + 1, 44u
16
153u
15
+ · · · + 4229b +
2570, 15u
16
2455u
15
+ · · · + 4229a 4314i
(i) Arc colorings
a
1
=
1
0
a
4
=
0
u
a
2
=
1
u
2
a
3
=
u
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
6
=
0.0300307u
16
0.581698u
15
+ ··· + 3.23859u + 0.163159
0.301490u
16
+ 0.343817u
15
+ ··· 0.639395u 0.291558
a
7
=
0.0134784u
16
0.205959u
15
+ ··· + 2.68976u 0.0763774
0.0520218u
16
+ 0.180894u
15
+ ··· 0.153464u 0.0385434
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
8
=
0.933318u
16
0.913691u
15
+ ··· + 2.09671u 0.377867
0.895956u
16
0.361788u
15
+ ··· 1.69307u 0.922913
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
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 = 1.11899
a = 3.55652
b = 1.14890
3.74765 10.6759
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 = 0.225548 0.312459I
a = 1.92533 + 0.42540I
b = 1.028559 0.314868I
1.91105 0.93427I 3.47646 + 1.18545I
u = 0.225548 + 0.312459I
a = 1.92533 0.42540I
b = 1.028559 + 0.314868I
1.91105 + 0.93427I 3.47646 1.18545I
u = 0.007198 1.101267I
a = 0.0948459 0.0860640I
b = 2.00423 + 0.17609I
14.0076 4.0781I 2.81048 + 2.03189I
u = 0.007198 + 1.101267I
a = 0.0948459 + 0.0860640I
b = 2.00423 0.17609I
14.0076 + 4.0781I 2.81048 2.03189I
u = 0.397422 0.534657I
a = 0.841166 + 0.259861I
b = 0.289585 0.225601I
1.46199 0.53067I 6.34861 + 0.44801I
5
Solution to I
u
2
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 = 1.094409 0.448256I
a = 0.297387 + 0.400102I
b = 0.394810 + 0.688088I
0.57451 + 4.58866I 0.38155 5.05474I
u = 1.094409 + 0.448256I
a = 0.297387 0.400102I
b = 0.394810 0.688088I
0.57451 4.58866I 0.38155 + 5.05474I
u = 1.279611 0.127778I
a = 1.75394 + 0.15484I
b = 1.65978 + 0.99674I
6.36531 + 2.55518I 9.15621 3.45666I
u = 1.279611 + 0.127778I
a = 1.75394 0.15484I
b = 1.65978 0.99674I
6.36531 2.55518I 9.15621 + 3.45666I
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
6
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
4
(u
6
u
5
+ ··· u + 1)(u
17
+ 2u
16
+ ··· + u 1)
c
2
(u
6
+ 3u
5
+ ··· + u + 1)(u
17
+ 12u
16
+ ··· u + 1)
c
3
, c
6
(u
6
+ u
5
+ ··· + u + 1)(u
17
+ 2u
16
+ ··· + u + 1)
c
5
, c
10
u
6
(u
17
+ 3u
16
+ ··· + 128u 64)
c
7
(u
6
3u
5
+ ··· u + 1)(u
17
+ 6u
16
+ ··· + 3897u + 1609)
c
8
, c
9
, c
11
(u 1)
6
(u
17
+ 7u
16
+ ··· 18u
2
1)
7
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
6
3y
5
+ ··· y + 1)(y
17
12y
16
+ ··· y 1)
c
2
(y
6
+ y
5
+ ··· + 3y + 1)(y
17
12y
16
+ ··· 77y 1)
c
3
, c
6
(y
6
3y
5
+ ··· 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)
8