11n
130
(K11n
130
)
1
Arc Sequences
7 1 10 8 9 2 11 1 7 4 5
Solving Sequence
5,11 1,8
9 6 4 7 2 10 3
c
11
c
8
c
5
c
4
c
7
c
1
c
10
c
3
c
2
, c
6
, c
9
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
11
2u
9
3u
8
+ 7u
7
+ 6u
6
3u
5
12u
4
+ 3u
3
+ 6u
2
u 1,
10u
10
+ 5u
9
+ 17u
8
+ 21u
7
80u
6
18u
5
+ 38u
4
+ 97u
3
82u
2
+ b 17u + 21,
18u
10
+ 11u
9
+ 32u
8
+ 35u
7
152u
6
24u
5
+ 85u
4
+ 181u
3
165u
2
+ a 35u + 43i
I
u
2
= hu
37
3u
36
+ ··· + 12u + 1,
5.80282 × 10
54
u
36
+ 1.84483 × 10
55
u
35
+ ··· + 9.07907 × 10
53
b 1.06031 × 10
55
,
1.67334 × 10
55
u
36
5.67719 × 10
55
u
35
+ ··· + 9.07907 × 10
53
a + 5.36327 × 10
55
i
There are 2 irreducible components with 48 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
11
2u
9
+· · ·u1, 10u
10
+5u
9
+· · ·+b+21, 18u
10
+11u
9
+· · ·+a+43i
(i) Arc colorings
a
5
=
1
0
a
11
=
18u
10
11u
9
+ ··· + 35u 43
10u
10
5u
9
+ ··· + 17u 21
a
1
=
8u
10
6u
9
+ ··· + 18u 22
10u
10
5u
9
+ ··· + 17u 21
a
8
=
0
u
a
9
=
20u
10
9u
9
+ ··· + 44u 38
6u
10
2u
9
+ ··· + 11u 10
a
6
=
7u
10
2u
9
+ ··· + 17u 11
12u
10
3u
9
+ ··· + 25u 16
a
4
=
1
u
2
a
7
=
21u
10
7u
9
+ ··· + 45u 33
5u
10
+ 4u
9
+ ··· 8u + 15
a
2
=
19u
10
10u
9
+ ··· + 43u 40
11u
10
6u
9
+ ··· + 19u 24
a
10
=
12u
10
9u
9
+ ··· + 25u 33
8u
10
5u
9
+ ··· + 13u 19
a
3
=
6u
10
4u
9
+ ··· + 14u 16
17u
10
8u
9
+ ··· + 35u 32
a
3
=
6u
10
4u
9
+ ··· + 14u 16
17u
10
8u
9
+ ··· + 35u 32
(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 = 1.08234 0.99501I
a = 0.042389 0.586060I
b = 0.470761 + 0.380594I
9.12876 2.79567I 1.55052 0.06651I
u = 1.08234 + 0.99501I
a = 0.042389 + 0.586060I
b = 0.470761 0.380594I
9.12876 + 2.79567I 1.55052 + 0.06651I
u = 0.852955 1.045330I
a = 0.411490 + 0.904328I
b = 0.575448 + 0.685098I
2.53579 + 3.27380I 4.40962 3.96450I
u = 0.852955 + 1.045330I
a = 0.411490 0.904328I
b = 0.575448 0.685098I
2.53579 3.27380I 4.40962 + 3.96450I
u = 0.496775 0.005375I
a = 0.92893 + 1.31151I
b = 1.132076 + 0.689250I
2.10419 + 0.48851I 1.91182 5.66309I
u = 0.496775 + 0.005375I
a = 0.92893 1.31151I
b = 1.132076 0.689250I
2.10419 0.48851I 1.91182 + 5.66309I
u = 0.726903 0.211808I
a = 0.302063 0.767741I
b = 1.18464 1.06749I
0.02345 3.01225I 6.59416 + 5.79825I
u = 0.726903 + 0.211808I
a = 0.302063 + 0.767741I
b = 1.18464 + 1.06749I
0.02345 + 3.01225I 6.59416 5.79825I
u = 0.773734
a = 1.80544
b = 0.547799
2.17580 12.3107
u = 1.31830 0.76405I
a = 0.264612 + 0.924298I
b = 0.931149 + 0.725260I
2.26337 5.93413I 0.97958 + 6.14906I
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 1.31830 + 0.76405I
a = 0.264612 0.924298I
b = 0.931149 0.725260I
2.26337 + 5.93413I 0.97958 6.14906I
3
II. I
u
2
=
hu
37
3u
36
+· · ·+12u+1, 5.80×10
54
u
36
+1.84×10
55
u
35
+· · ·+9.08×10
53
b
1.06 × 10
55
, 1.67 × 10
55
u
36
5.68 × 10
55
u
35
+ · · · + 9.08 × 10
53
a + 5.36 × 10
55
i
(i) Arc colorings
a
5
=
1
0
a
11
=
18.4307u
36
+ 62.5306u
35
+ ··· 497.870u 59.0729
6.39143u
36
20.3196u
35
+ ··· + 146.977u + 11.6786
a
1
=
24.8222u
36
+ 82.8501u
35
+ ··· 644.847u 70.7515
6.39143u
36
20.3196u
35
+ ··· + 146.977u + 11.6786
a
8
=
0
u
a
9
=
24.9654u
36
+ 86.7125u
35
+ ··· 590.690u 61.5080
12.8061u
36
+ 45.4195u
35
+ ··· 244.776u 25.4763
a
6
=
29.6942u
36
106.259u
35
+ ··· + 356.560u + 28.4739
8.38731u
36
+ 32.2204u
35
+ ··· 58.7679u 3.79749
a
4
=
1
u
2
a
7
=
33.7615u
36
+ 116.156u
35
+ ··· 837.697u 91.4706
4.01009u
36
15.9758u
35
+ ··· 0.231137u 4.48625
a
2
=
33.6666u
36
+ 119.378u
35
+ ··· 598.309u 57.2702
15.3168u
36
+ 52.5724u
35
+ ··· 367.582u 39.3006
a
10
=
28.1959u
36
+ 94.5719u
35
+ ··· 713.276u 77.9898
6.92466u
36
23.4087u
35
+ ··· + 123.791u + 8.93263
a
3
=
1.92220u
36
+ 8.93616u
35
+ ··· + 58.9709u + 11.1452
10.5786u
36
+ 41.4198u
35
+ ··· 23.2581u + 5.52763
a
3
=
1.92220u
36
+ 8.93616u
35
+ ··· + 58.9709u + 11.1452
10.5786u
36
+ 41.4198u
35
+ ··· 23.2581u + 5.52763
(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 = 1.22045 0.81648I
a = 0.180721 + 0.726359I
b = 1.026124 + 0.886327I
0.93685 + 5.16140I 1.08220 5.99788I
u = 1.22045 + 0.81648I
a = 0.180721 0.726359I
b = 1.026124 0.886327I
0.93685 5.16140I 1.08220 + 5.99788I
u = 0.903161 0.961897I
a = 0.154482 1.188853I
b = 1.070019 0.832774I
0.20513 + 8.27502I 0.00412 7.41109I
u = 0.903161 + 0.961897I
a = 0.154482 + 1.188853I
b = 1.070019 + 0.832774I
0.20513 8.27502I 0.00412 + 7.41109I
u = 0.873231 0.478522I
a = 0.068350 0.944053I
b = 0.64511 1.54806I
6.92057 + 5.26122I 1.18539 5.90480I
u = 0.873231 + 0.478522I
a = 0.068350 + 0.944053I
b = 0.64511 + 1.54806I
6.92057 5.26122I 1.18539 + 5.90480I
u = 0.735831 0.950419I
a = 0.320806 + 1.281340I
b = 0.599605 + 0.263744I
1.39846 + 3.05009I 3.20830 2.79125I
u = 0.735831 + 0.950419I
a = 0.320806 1.281340I
b = 0.599605 0.263744I
1.39846 3.05009I 3.20830 + 2.79125I
u = 0.673472
a = 1.81257
b = 0.658844
2.54226 10.2456
u = 0.566787 0.031085I
a = 0.647528 + 0.421182I
b = 1.223990 + 0.280649I
2.40459 + 0.03773I 9.51019 + 3.78756I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.566787 + 0.031085I
a = 0.647528 0.421182I
b = 1.223990 0.280649I
2.40459 0.03773I 9.51019 3.78756I
u = 0.521259 0.344561I
a = 1.41314 1.35260I
b = 0.649301 0.015184I
2.53052 + 0.01603I 8.01901 + 0.30181I
u = 0.521259 + 0.344561I
a = 1.41314 + 1.35260I
b = 0.649301 + 0.015184I
2.53052 0.01603I 8.01901 0.30181I
u = 0.416603 0.420677I
a = 3.06528 0.37088I
b = 0.850079 + 0.415638I
9.44422 + 4.54412I 2.91144 6.06217I
u = 0.416603 + 0.420677I
a = 3.06528 + 0.37088I
b = 0.850079 0.415638I
9.44422 4.54412I 2.91144 + 6.06217I
u = 0.309800 0.111195I
a = 1.16330 + 2.20199I
b = 1.23107 + 0.72507I
0.70258 + 2.95006I 3.83378 3.59981I
u = 0.309800 + 0.111195I
a = 1.16330 2.20199I
b = 1.23107 0.72507I
0.70258 2.95006I 3.83378 + 3.59981I
u = 0.303136 0.820611I
a = 0.445886 + 0.283042I
b = 0.820454 0.302238I
0.44492 2.47536I 1.10953 + 5.85952I
u = 0.303136 + 0.820611I
a = 0.445886 0.283042I
b = 0.820454 + 0.302238I
0.44492 + 2.47536I 1.10953 5.85952I
u = 0.288731 1.234059I
a = 1.232391 + 0.101013I
b = 0.833991 0.353566I
9.78588 0.61235I 3.21523 0.70893I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.288731 + 1.234059I
a = 1.232391 0.101013I
b = 0.833991 + 0.353566I
9.78588 + 0.61235I 3.21523 + 0.70893I
u = 0.544271 0.980201I
a = 0.068431 + 0.971541I
b = 1.41428 + 1.03844I
12.51051 5.12964I 3.53026 + 4.63968I
u = 0.544271 + 0.980201I
a = 0.068431 0.971541I
b = 1.41428 1.03844I
12.51051 + 5.12964I 3.53026 4.63968I
u = 0.570654 0.353458I
a = 0.430660 0.883447I
b = 0.518070 1.102536I
1.17632 1.49794I 4.03696 + 6.33192I
u = 0.570654 + 0.353458I
a = 0.430660 + 0.883447I
b = 0.518070 + 1.102536I
1.17632 + 1.49794I 4.03696 6.33192I
u = 0.592564 0.572125I
a = 0.452319 0.687913I
b = 0.601373 0.797904I
0.33153 1.56965I 1.84729 + 2.87829I
u = 0.592564 + 0.572125I
a = 0.452319 + 0.687913I
b = 0.601373 + 0.797904I
0.33153 + 1.56965I 1.84729 2.87829I
u = 0.636852 0.765522I
a = 0.537623 1.221741I
b = 0.756589 0.697163I
0.52746 2.09006I 0.36189 + 3.60216I
u = 0.636852 + 0.765522I
a = 0.537623 + 1.221741I
b = 0.756589 + 0.697163I
0.52746 + 2.09006I 0.36189 3.60216I
u = 1.19358 1.18434I
a = 0.017218 0.946765I
b = 1.24432 0.89792I
9.0036 13.3479I 0.79476 + 6.55746I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 1.19358 + 1.18434I
a = 0.017218 + 0.946765I
b = 1.24432 + 0.89792I
9.0036 + 13.3479I 0.79476 6.55746I
u = 1.28095 0.96201I
a = 0.065594 + 0.777594I
b = 0.746755 + 0.717340I
3.60755 4.86664I 5.83345 + 5.56346I
u = 1.28095 + 0.96201I
a = 0.065594 0.777594I
b = 0.746755 0.717340I
3.60755 + 4.86664I 5.83345 5.56346I
u = 1.37112 1.76401I
a = 0.193445 + 0.015149I
b = 0.698049 0.369328I
9.27666 + 3.59094I 4.07946 9.46699I
u = 1.37112 + 1.76401I
a = 0.193445 0.015149I
b = 0.698049 + 0.369328I
9.27666 3.59094I 4.07946 + 9.46699I
u = 1.78574 0.46711I
a = 0.577972 + 0.024754I
b = 0.709146 + 0.450731I
8.92085 1.02999I 3.37249 2.08147I
u = 1.78574 + 0.46711I
a = 0.577972 0.024754I
b = 0.709146 0.450731I
8.92085 + 1.02999I 3.37249 + 2.08147I
8
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
11
+ u
10
+ ··· + 3u + 1)(u
37
+ 24u
35
+ ··· 4u + 1)
c
2
(u
11
+ 11u
10
+ ··· 3u 1)(u
37
+ 48u
36
+ ··· + 8u 1)
c
3
(u
11
5u
9
2u
8
+ 11u
7
+ 9u
6
12u
5
15u
4
+ 5u
3
+ 11u
2
+ u 3)
(u
37
+ u
36
+ ··· + 6u 1)
c
4
(u
11
2u
9
3u
8
+ 7u
7
+ 6u
6
3u
5
12u
4
+ 3u
3
+ 6u
2
u 1)
(u
37
+ 3u
36
+ ··· + 12u 1)
c
5
(u
11
3u
9
+ 3u
8
u
7
5u
6
+ 12u
5
15u
4
+ 13u
3
8u
2
+ 3u 1)
(u
37
+ u
36
+ ··· + 1746u 367)
c
6
(u
11
u
10
+ ··· + 3u 1)(u
37
+ 24u
35
+ ··· 4u + 1)
c
7
(u
11
+ u
10
+ u
9
3u
8
u
7
3u
6
+ 2u
5
5u
4
2u
2
+ u 1)
(u
37
+ 4u
36
+ ··· 8u + 1)
c
8
(u
11
u
10
+ 2u
9
+ 3u
8
4u
7
+ 4u
6
+ 5u
5
8u
4
+ u
3
+ 5u
2
4u 3)
(u
37
+ 10u
35
+ ··· 75u + 23)
c
9
(u
11
+ 7u
10
+ ··· + 8u + 1)(u
37
+ 6u
36
+ ··· 6703u + 583)
c
10
(u
11
5u
9
+ 2u
8
+ 11u
7
9u
6
12u
5
+ 15u
4
+ 5u
3
11u
2
+ u + 3)
(u
37
+ u
36
+ ··· + 6u 1)
c
11
(u
11
+ u
10
u
9
+ 7u
7
+ 6u
6
u
5
u
4
+ 5u
3
+ 2u
2
u 1)
(u
37
+ 2u
36
+ ··· 8u 11)
9
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
6
(y
11
+ 11y
10
+ ··· 3y 1)(y
37
+ 48y
36
+ ··· + 8y 1)
c
2
(y
11
17y
10
+ ··· + y 1)(y
37
112y
36
+ ··· 100y 1)
c
3
, c
10
(y
11
10y
10
+ ··· + 67y 9)(y
37
17y
36
+ ··· + 22y 1)
c
4
(y
11
4y
10
+ ··· + 13y 1)(y
37
3y
36
+ ··· + 24y 1)
c
5
(y
11
6y
10
+ ··· 7y 1)(y
37
53y
36
+ ··· + 2094316y 134689)
c
7
(y
11
+ y
10
+ ··· 3y 1)(y
37
+ 2y
36
+ ··· 56y
2
1)
c
8
(y
11
+ 3y
10
+ ··· + 46y 9)(y
37
+ 20y
36
+ ··· 44975y 529)
c
9
(y
11
7y
10
+ ··· + 2y 1)(y
37
66y
36
+ ··· + 4068905y 339889)
c
11
(y
11
3y
10
+ ··· + 5y 1)(y
37
10y
36
+ ··· + 2308y 121)
10