11n
98
(K11n
98
)
1
Arc Sequences
7 1 8 7 10 2 4 11 6 8 9
Solving Sequence
8,10
11 9
1,4
3 2 7 5 6
c
10
c
8
c
11
c
3
c
2
c
7
c
4
c
5
c
1
, c
6
, c
9
Representation Ideals
I =
4
\
i=1
I
u
i
I
u
1
= hu + 1, a 1, 2b + 1i
I
u
2
= ha
4
+ 3a
2
+ 1, a
3
2a + u, a
3
+ a
2
+ b 2a + 2i
I
u
3
= hu
17
u
16
+ ··· 4u
2
1, 11u
16
+ 14u
15
+ ··· + 32b + 3, 17u
16
26u
15
+ ··· + 16a + 7i
I
u
4
= hu
22
+ 3u
21
+ ··· + 24u + 9,
749178430u
21
3147363456u
20
+ ··· + 7660380639b + 10655174784,
2694752718u
21
5680799296u
20
+ ··· + 7660380639a 11072995247i
There are 4 irreducible components with 44 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 + 1, a 1, 2b + 1i
(i) Arc colorings
a
8
=
0
1
a
10
=
1
1
2
a
11
=
1
1
2
a
9
=
1
1
2
a
1
=
0
1
a
4
=
1
0
a
3
=
1
1
a
2
=
1
0
a
7
=
1
1
a
5
=
2
1
a
6
=
2
1
a
6
=
2
1
(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 = 1.00000
a = 1.00000
b = 0.500000
0 2.25000
3
II. I
u
2
= ha
4
+ 3a
2
+ 1, a
3
2a + u, a
3
+ a
2
+ b 2a + 2i
(i) Arc colorings
a
8
=
0
a
3
+ 2a
a
10
=
a
a
3
a
2
+ 2a 2
a
11
=
a
a
3
a
2
+ a 2
a
9
=
a
3
+ a
2a 1
a
1
=
a
3
+ a
a
3
a 1
a
4
=
1
0
a
3
=
1
1
a
2
=
a
3
+ a + 1
a
3
a 2
a
7
=
a
3
2a
a
3
+ 2a
a
5
=
0
1
a
6
=
a
2
a
3
a
2
2a
a
6
=
a
2
a
3
a
2
2a
(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 = 1.00000I
a = 0.618034I
b = 1.61803 1.00000I
4.27683 8.00000
u = 1.00000I
a = 0.618034I
b = 1.61803 + 1.00000I
4.27683 8.00000
u = 1.00000I
a = 1.61803I
b = 0.618034 + 1.000000I
12.1725 8.00000
u = 1.00000I
a = 1.61803I
b = 0.618034 1.000000I
12.1725 8.00000
5
III. I
u
3
= hu
17
u
16
+ · · · 4u
2
1, 11u
16
+ 14u
15
+ · · · + 32b +
3, 17u
16
26u
15
+ · · · + 16a + 7i
(i) Arc colorings
a
8
=
0
u
a
10
=
1.06250u
16
+ 1.62500u
15
+ ··· + 5.43750u 0.437500
0.343750u
16
0.437500u
15
+ ··· 1.15625u 0.0937500
a
11
=
1.06250u
16
+ 1.62500u
15
+ ··· + 5.43750u 0.437500
0.781250u
16
1.31250u
15
+ ··· 2.21875u + 0.468750
a
9
=
0.0625000u
16
0.125000u
15
+ ··· + 3.93750u 0.687500
0.0312500u
16
+ 0.562500u
15
+ ··· 0.0312500u 0.468750
a
1
=
1
4
u
15
+
1
4
u
14
+ ···
5
4
u
2
+
5
4
1
4
u
15
1
4
u
14
+ ··· +
5
4
u
2
1
4
a
4
=
1
0
a
3
=
1
u
2
a
2
=
1
4
u
15
+
1
4
u
14
+ ···
1
4
u
2
+
5
4
1
4
u
15
1
4
u
14
+ ··· +
1
4
u
2
1
4
a
7
=
u
u
a
5
=
u
2
+ 1
u
2
a
6
=
1
4
u
16
+
1
4
u
15
+ ···
1
4
u
3
+
9
4
u
1
4
u
16
1
4
u
15
+ ··· +
1
4
u
3
5
4
u
a
6
=
1
4
u
16
+
1
4
u
15
+ ···
1
4
u
3
+
9
4
u
1
4
u
16
1
4
u
15
+ ··· +
1
4
u
3
5
4
u
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
6
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.678472 1.240021I
a = 0.225489 + 1.034292I
b = 1.54769 1.47366I
4.6382 + 13.7590I 2.60534 8.09148I
u = 0.678472 + 1.240021I
a = 0.225489 1.034292I
b = 1.54769 + 1.47366I
4.6382 13.7590I 2.60534 + 8.09148I
u = 0.678341 1.093893I
a = 0.544216 0.556925I
b = 0.747437 + 1.180736I
0.65577 + 8.75138I 0.84209 7.19652I
u = 0.678341 + 1.093893I
a = 0.544216 + 0.556925I
b = 0.747437 1.180736I
0.65577 8.75138I 0.84209 + 7.19652I
u = 0.583285 0.914805I
a = 1.015806 0.479229I
b = 0.083408 0.398282I
1.75655 + 3.15519I 1.08772 4.41422I
u = 0.583285 + 0.914805I
a = 1.015806 + 0.479229I
b = 0.083408 + 0.398282I
1.75655 3.15519I 1.08772 + 4.41422I
u = 0.239314 0.869369I
a = 0.54621 + 1.78172I
b = 0.092556 1.046207I
11.59492 + 1.04318I 1.35194 7.04363I
u = 0.239314 + 0.869369I
a = 0.54621 1.78172I
b = 0.092556 + 1.046207I
11.59492 1.04318I 1.35194 + 7.04363I
u = 0.187558 0.379982I
a = 0.37683 2.21344I
b = 0.554103 + 0.502263I
1.69658 + 0.80451I 2.69480 2.52231I
u = 0.187558 + 0.379982I
a = 0.37683 + 2.21344I
b = 0.554103 0.502263I
1.69658 0.80451I 2.69480 + 2.52231I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 0.606656 1.024999I
a = 0.400728 1.170949I
b = 0.98051 + 1.70447I
2.68580 6.31167I 1.34813 + 5.64607I
u = 0.606656 + 1.024999I
a = 0.400728 + 1.170949I
b = 0.98051 1.70447I
2.68580 + 6.31167I 1.34813 5.64607I
u = 0.635251
a = 0.465730
b = 0.236283
0.963952 10.6381
u = 0.784905 0.787523I
a = 0.294224 + 0.608642I
b = 0.240378 1.166900I
2.78581 2.60100I 4.98680 + 3.49505I
u = 0.784905 + 0.787523I
a = 0.294224 0.608642I
b = 0.240378 + 1.166900I
2.78581 + 2.60100I 4.98680 3.49505I
u = 1.157785 0.487195I
a = 0.885367 + 0.183583I
b = 0.530392 + 0.376632I
0.354258 + 0.834124I 0.06498 5.84789I
u = 1.157785 + 0.487195I
a = 0.885367 0.183583I
b = 0.530392 0.376632I
0.354258 0.834124I 0.06498 + 5.84789I
8
IV.
I
u
4
= hu
22
+ 3u
21
+ · · · + 24u + 9, 7.49 × 10
8
u
21
3.15 × 10
9
u
20
+ · · · + 7.66 ×
10
9
b+1.07×10
10
, 2.69×10
9
u
21
5.68×10
9
u
20
+· · ·+7.66×10
9
a1.11×10
10
i
(i) Arc colorings
a
8
=
0
u
a
10
=
0.351778u
21
+ 0.741582u
20
+ ··· + 5.88891u + 1.44549
0.0977991u
21
+ 0.410863u
20
+ ··· 0.0725632u 1.39095
a
11
=
0.351778u
21
+ 0.741582u
20
+ ··· + 5.88891u + 1.44549
0.109720u
21
+ 0.727196u
20
+ ··· + 4.29148u + 1.43282
a
9
=
0.205132u
21
+ 0.242650u
20
+ ··· + 0.665474u 1.10003
0.0568315u
21
+ 0.584282u
20
+ ··· + 4.33085u + 2.13109
a
1
=
0.242379u
21
+ 0.907777u
20
+ ··· + 9.08858u + 4.51788
1
a
4
=
1
0
a
3
=
1
u
2
a
2
=
0.106015u
21
0.00995851u
20
+ ··· + 2.57184u + 2.89213
0.348395u
21
+ 0.917735u
20
+ ··· + 6.51674u + 0.625748
a
7
=
u
u
a
5
=
u
2
+ 1
u
2
a
6
=
0.0163379u
21
0.197272u
20
+ ··· 1.51350u 0.468886
0.0531897u
21
+ 0.337084u
20
+ ··· + 9.03495u + 5.31697
a
6
=
0.0163379u
21
0.197272u
20
+ ··· 1.51350u 0.468886
0.0531897u
21
+ 0.337084u
20
+ ··· + 9.03495u + 5.31697
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
9
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.081771 0.344108I
a = 1.198122 + 0.024840I
b = 0.500831 + 0.714833I
1.85809 7.47524I 0.22908 + 5.55460I
u = 1.081771 + 0.344108I
a = 1.198122 0.024840I
b = 0.500831 0.714833I
1.85809 + 7.47524I 0.22908 5.55460I
u = 0.853835 0.533591I
a = 0.484207 + 0.691979I
b = 0.094966 1.198131I
2.35273 3.04152I 4.06121 + 2.82242I
u = 0.853835 + 0.533591I
a = 0.484207 0.691979I
b = 0.094966 + 1.198131I
2.35273 + 3.04152I 4.06121 2.82242I
u = 0.752651 0.945347I
a = 0.595091 + 0.919581I
b = 1.60394 0.21728I
8.47148 + 2.94672I 5.79937 4.11787I
u = 0.752651 + 0.945347I
a = 0.595091 0.919581I
b = 1.60394 + 0.21728I
8.47148 2.94672I 5.79937 + 4.11787I
u = 0.555909 0.782909I
a = 0.409273 1.283123I
b = 0.99043 + 1.31114I
1.31282 + 1.41699I 0.791306 0.633731I
u = 0.555909 + 0.782909I
a = 0.409273 + 1.283123I
b = 0.99043 1.31114I
1.31282 1.41699I 0.791306 + 0.633731I
u = 0.285332 0.830788I
a = 0.314160 0.435233I
b = 1.67105 + 0.02430I
3.59460 + 1.13130I 0.01220 6.05785I
u = 0.285332 + 0.830788I
a = 0.314160 + 0.435233I
b = 1.67105 0.02430I
3.59460 1.13130I 0.01220 + 6.05785I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.14927 1.48798I
a = 0.174119 + 0.867773I
b = 0.067031 0.806341I
8.47148 2.94672I 5.79937 + 4.11787I
u = 0.14927 + 1.48798I
a = 0.174119 0.867773I
b = 0.067031 + 0.806341I
8.47148 + 2.94672I 5.79937 4.11787I
u = 0.087548 1.187665I
a = 0.339357 0.203960I
b = 0.011775 0.775560I
3.59460 1.13130I 0.01220 + 6.05785I
u = 0.087548 + 1.187665I
a = 0.339357 + 0.203960I
b = 0.011775 + 0.775560I
3.59460 + 1.13130I 0.01220 6.05785I
u = 0.181376 1.048185I
a = 0.183886 1.062695I
b = 2.71263 + 2.60065I
5.48524 0.376257
u = 0.181376 + 1.048185I
a = 0.183886 + 1.062695I
b = 2.71263 2.60065I
5.48524 0.376257
u = 0.623653 0.552777I
a = 1.41909 0.62786I
b = 0.134458 0.204944I
1.31282 + 1.41699I 0.791306 0.633731I
u = 0.623653 + 0.552777I
a = 1.41909 + 0.62786I
b = 0.134458 + 0.204944I
1.31282 1.41699I 0.791306 + 0.633731I
u = 0.714099 0.923041I
a = 0.552364 0.475215I
b = 0.693911 + 0.889315I
2.35273 3.04152I 4.06121 + 2.82242I
u = 0.714099 + 0.923041I
a = 0.552364 + 0.475215I
b = 0.693911 0.889315I
2.35273 + 3.04152I 4.06121 2.82242I
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.747184 1.181246I
a = 0.265936 + 0.936246I
b = 1.36561 1.12781I
1.85809 7.47524I 0.22908 + 5.55460I
u = 0.747184 + 1.181246I
a = 0.265936 0.936246I
b = 1.36561 + 1.12781I
1.85809 + 7.47524I 0.22908 5.55460I
11
V. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
3
, c
4
(u + 1)(u
2
+ 1)
2
(u
17
u
16
+ ··· 4u
2
1)(u
22
+ 3u
21
+ ··· + 24u + 9)
c
2
(u 1)(u + 1)
4
(u
17
+ 5u
16
+ ··· 8u 1)
(u
22
+ 11u
21
+ ··· + 432u + 81)
c
5
, c
9
u(u
4
+ 3u
2
+ 1)
(u
11
u
10
+ 2u
9
u
8
+ 4u
7
2u
6
+ 4u
5
u
4
+ 3u
3
+ u
2
+ 1)
2
(u
17
+ 3u
16
+ ··· + 22u + 8)
c
6
, c
7
(u 1)(u
2
+ 1)
2
(u
17
u
16
+ ··· 4u
2
1)(u
22
+ 3u
21
+ ··· + 24u + 9)
c
8
(u 1)(u
2
+ u 1)
2
(u
11
+ u
10
4u
9
3u
8
+ 6u
7
+ 2u
6
2u
5
+ 3u
4
3u
3
3u
2
+ 2u 1)
2
(u
17
+ 2u
16
+ ··· + 13u + 4)
c
10
, c
11
(u + 1)(u
2
u 1)
2
(u
11
+ u
10
4u
9
3u
8
+ 6u
7
+ 2u
6
2u
5
+ 3u
4
3u
3
3u
2
+ 2u 1)
2
(u
17
+ 2u
16
+ ··· + 13u + 4)
12
VI. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
3
, c
4
c
6
, c
7
(y 1)(y + 1)
4
(y
17
+ 5y
16
+ ··· 8y 1)
(y
22
+ 11y
21
+ ··· + 432y + 81)
c
2
(y 1)
5
(y
17
+ 13y
16
+ ··· 12y 1)
(y
22
y
21
+ ··· + 35640y + 6561)
c
5
, c
9
y(y
2
+ 3y + 1)
2
(1 2y + y
2
+ 15y
3
+ 29y
4
+ 40y
5
+ 40y
6
+ 32y
7
+ 19y
8
+ 10y
9
+ 3y
10
+ y
11
)
2
(y
17
+ 9y
16
+ ··· 12y 64)
c
8
, c
10
, c
11
(y 1)(y
2
3y + 1)
2
(1 2y 15y
2
+ 23y
3
+ 33y
4
76y
5
+ 24y
6
+ 52y
7
65y
8
+ 34y
9
9y
10
+ y
11
)
2
(y
17
16y
16
+ ··· + 209y 16)
13