11n
95
(K11n
95
)
1
Arc Sequences
5 1 9 7 2 11 3 1 7 5 9
Solving Sequence
1,5
2 3
6,8
9 7 10 4 11
c
1
c
2
c
5
c
8
c
7
c
9
c
4
c
11
c
3
, c
6
, c
10
Representation Ideals
I =
3
\
i=1
I
u
i
I
u
1
= hb
10
3b
9
+ 2b
8
+ 8b
7
24b
6
+ 16b
5
+ 25b
4
71b
3
+ 78b
2
40b + 7,
808b
9
+ 2123u + ··· + 23977b 3622, 1829b
9
4767b
8
+ ··· + 2123a 32502i
I
u
2
= hu
6
+ 2u
5
3u
3
2u
2
+ 2u + 1, u
5
+ 2u
4
2u
2
+ b u + 1, 3u
5
5u
4
+ 2u
3
+ 8u
2
+ a + 2u 6i
I
u
3
= hu
12
5u
11
+ 10u
10
4u
9
22u
8
+ 51u
7
48u
6
+ 10u
5
+ 23u
4
21u
3
+ 3u
2
+ 5u 2,
u
10
4u
9
+ 5u
8
+ 4u
7
20u
6
+ 24u
5
8u
4
8u
3
+ 7u
2
+ b 1,
u
11
7u
10
+ 18u
9
12u
8
36u
7
+ 95u
6
84u
5
2u
4
+ 59u
3
31u
2
+ 2a 9u + 9i
There are 3 irreducible components with 28 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
= hb
10
3b
9
+ · · · 40b + 7, 808b
9
+ 2123u + · · · + 23977b
3622, 1829b
9
4767b
8
+ · · · + 2123a 32502i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
0.380593b
9
+ 0.764013b
8
+ ··· 11.2939b + 1.70608
a
2
=
1
0.499293b
9
1.02120b
8
+ ··· + 15.8059b 2.73952
a
3
=
0.499293b
9
+ 1.02120b
8
+ ··· 15.8059b + 3.73952
0.499293b
9
1.02120b
8
+ ··· + 15.8059b 2.73952
a
6
=
0.380593b
9
0.764013b
8
+ ··· + 11.2939b 1.70608
0.141780b
9
+ 0.473858b
8
+ ··· 8.97268b + 1.84550
a
8
=
0.861517b
9
+ 2.24541b
8
+ ··· 48.5087b + 15.3095
b
a
9
=
0.861517b
9
+ 2.24541b
8
+ ··· 47.5087b + 15.3095
b
a
7
=
0.0414508b
9
+ 0.0617051b
8
+ ··· 7.85728b + 5.32454
0.141780b
9
+ 0.473858b
8
+ ··· 8.97268b + 1.84550
a
10
=
0.339143b
9
0.825718b
8
+ ··· + 19.1512b 7.03062
0.0744230b
9
+ 0.221856b
8
+ ··· 6.50212b + 2.58879
a
4
=
0.390956b
9
1.45313b
8
+ ··· + 39.8069b 16.1931
0.615167b
9
1.36317b
8
+ ··· + 22.5417b 4.73104
a
11
=
0.339143b
9
+ 0.825718b
8
+ ··· 19.1512b + 7.03062
b
2
a
11
=
0.339143b
9
+ 0.825718b
8
+ ··· 19.1512b + 7.03062
b
2
(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 = 0.935538 + 0.903908I
a = 2.70055 0.28054I
b = 1.72566 0.41266I
4.37856 + 3.33174I 10.08126 2.36228I
u = 0.935538 0.903908I
a = 2.70055 + 0.28054I
b = 1.72566 + 0.41266I
4.37856 3.33174I 10.08126 + 2.36228I
u = 0.645200
a = 3.94511
b = 0.340045
7.46192 19.6088
u = 0.758138 + 0.584034I
a = 1.87197 0.03044I
b = 0.452332 1.123840I
4.75993 2.21397I 11.11432 + 4.22289I
u = 0.758138 0.584034I
a = 1.87197 + 0.03044I
b = 0.452332 + 1.123840I
4.75993 + 2.21397I 11.11432 4.22289I
u = 0.758138 0.584034I
a = 0.87798 + 2.02319I
b = 0.81806 1.53771I
4.75993 + 2.21397I 11.11432 4.22289I
u = 0.758138 + 0.584034I
a = 0.87798 2.02319I
b = 0.81806 + 1.53771I
4.75993 2.21397I 11.11432 + 4.22289I
u = 0.935538 0.903908I
a = 1.88766 0.16400I
b = 0.868620 0.215856I
4.37856 3.33174I 10.08126 + 2.36228I
u = 0.935538 + 0.903908I
a = 1.88766 + 0.16400I
b = 0.868620 + 0.215856I
4.37856 + 3.33174I 10.08126 2.36228I
u = 0.645200
a = 4.07100
b = 1.83325
7.46192 19.6088
3
II. I
u
2
= hu
6
+ 2u
5
3u
3
2u
2
+ 2u + 1, u
5
+ 2u
4
2u
2
+ b u +
1, 3u
5
5u
4
+ 2u
3
+ 8u
2
+ a + 2u 6i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
8
=
3u
5
+ 5u
4
2u
3
8u
2
2u + 6
u
5
2u
4
+ 2u
2
+ u 1
a
9
=
2u
5
+ 3u
4
2u
3
6u
2
u + 5
u
5
2u
4
+ 2u
2
+ u 1
a
7
=
2u
5
+ 3u
4
u
3
5u
2
2u + 4
u
5
2u
4
u
3
+ 2u
2
+ 2u 1
a
10
=
2u
5
3u
4
+ u
3
+ 5u
2
+ 2u 4
u
5
+ u
4
u
3
3u
2
u + 2
a
4
=
3u
5
5u
4
+ 2u
3
+ 8u
2
+ 3u 7
u
5
+ 2u
4
2u
2
u + 2
a
11
=
2u
5
3u
4
+ u
3
+ 5u
2
+ 2u 4
u
2
u + 1
a
11
=
2u
5
3u
4
+ u
3
+ 5u
2
+ 2u 4
u
2
u + 1
(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 = 1.52247
a = 0.116304
b = 0.452275
11.4632 16.4313
u = 0.934823 0.946305I
a = 2.43499 0.27700I
b = 1.42809 0.28813I
5.64849 3.45368I 2.17386 + 2.96497I
u = 0.934823 + 0.946305I
a = 2.43499 + 0.27700I
b = 1.42809 + 0.28813I
5.64849 + 3.45368I 2.17386 2.96497I
u = 0.423796
a = 5.68327
b = 1.11543
6.80200 4.39680
u = 0.907957 0.227043I
a = 0.348496 0.361180I
b = 0.355765 + 0.898533I
1.45069 + 2.49752I 13.4121 4.8455I
u = 0.907957 + 0.227043I
a = 0.348496 + 0.361180I
b = 0.355765 0.898533I
1.45069 2.49752I 13.4121 + 4.8455I
5
III.
I
u
3
= hu
12
5u
11
+· · · +5u2, u
10
4u
9
+· · · +b1, u
11
7u
10
+· · · +2a+9i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
6
=
u
u
3
+ u
a
8
=
1
2
u
11
+
7
2
u
10
+ ··· +
9
2
u
9
2
u
10
+ 4u
9
5u
8
4u
7
+ 20u
6
24u
5
+ 8u
4
+ 8u
3
7u
2
+ 1
a
9
=
1
2
u
11
+
5
2
u
10
+ ··· +
9
2
u
7
2
u
10
+ 4u
9
5u
8
4u
7
+ 20u
6
24u
5
+ 8u
4
+ 8u
3
7u
2
+ 1
a
7
=
3
2
u
11
+
13
2
u
10
+ ··· +
11
2
u
7
2
u
11
4u
10
+ 6u
9
+ u
8
18u
7
+ 30u
6
22u
5
+ 2u
4
+ 9u
3
6u
2
+ 1
a
10
=
3
2
u
11
+
13
2
u
10
+ ··· +
11
2
u
7
2
u
11
5u
10
+ ··· 5u + 3
a
4
=
1
2
u
11
+
3
2
u
10
+ ··· +
1
2
u +
1
2
u
10
3u
9
+ 2u
8
+ 6u
7
14u
6
+ 11u
5
+ 2u
4
8u
3
+ 4u
2
+ 2u 1
a
11
=
3
2
u
11
13
2
u
10
+ ···
11
2
u +
7
2
u
11
+ 4u
10
+ ··· + 3u 1
a
11
=
3
2
u
11
13
2
u
10
+ ···
11
2
u +
7
2
u
11
+ 4u
10
+ ··· + 3u 1
(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 = 1.68330
a = 0.603416
b = 0.774996
10.8637 2.38840
u = 0.585422 0.102144I
a = 0.251345 + 0.127948I
b = 0.388763 + 1.056566I
0.76434 2.25567I 2.25761 + 1.65555I
u = 0.585422 + 0.102144I
a = 0.251345 0.127948I
b = 0.388763 1.056566I
0.76434 + 2.25567I 2.25761 1.65555I
u = 0.531194
a = 1.14054
b = 0.227398
0.766539 13.0250
u = 0.634055 0.761588I
a = 2.40907 + 0.56101I
b = 1.061816 0.403224I
2.28232 + 3.09531I 6.37045 4.07458I
u = 0.634055 + 0.761588I
a = 2.40907 0.56101I
b = 1.061816 + 0.403224I
2.28232 3.09531I 6.37045 + 4.07458I
u = 0.706953 1.119617I
a = 2.03790 + 1.24403I
b = 1.33004 0.60517I
1.55271 4.05634I 6.99284 + 2.54487I
u = 0.706953 + 1.119617I
a = 2.03790 1.24403I
b = 1.33004 + 0.60517I
1.55271 + 4.05634I 6.99284 2.54487I
u = 1.13630 0.87513I
a = 2.57645 0.49107I
b = 1.35172 + 1.03292I
2.90723 + 11.19714I 8.74463 6.08532I
u = 1.13630 + 0.87513I
a = 2.57645 + 0.49107I
b = 1.35172 1.03292I
2.90723 11.19714I 8.74463 + 6.08532I
7
Solution to I
u
3
1(vol +
1CS) Cusp shape
u = 1.184168 0.621257I
a = 1.075907 0.634900I
b = 0.732377 0.158790I
0.53240 + 2.26677I 5.92780 2.45213I
u = 1.184168 + 0.621257I
a = 1.075907 + 0.634900I
b = 0.732377 + 0.158790I
0.53240 2.26677I 5.92780 + 2.45213I
8
IV. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
5
u
4
+ u
2
+ u 1)
2
(u
6
+ 2u
5
3u
3
2u
2
+ 2u + 1)
(u
12
+ 5u
11
+ ··· 5u 2)
c
2
(u
5
+ u
4
+ 4u
3
+ 3u
2
+ 3u + 1)
2
(u
6
+ 4u
5
+ ··· + 8u + 1)(u
12
+ 5u
11
+ ··· + 37u + 4)
c
3
(u
6
+ u
5
2u
4
3u
3
5u
2
4u 1)
(u
10
u
9
4u
8
+ 4u
7
2u
6
2u
5
+ 29u
4
7u
3
48u
2
12u 1)
(u
12
+ u
11
+ ··· + 2u 1)
c
4
, c
10
(u
6
u
5
2u
4
+ 3u
3
5u
2
+ 4u 1)
(u
10
u
9
4u
8
+ 4u
7
2u
6
2u
5
+ 29u
4
7u
3
48u
2
12u 1)
(u
12
+ u
11
+ ··· + 2u 1)
c
5
(u
5
u
4
+ u
2
+ u 1)
2
(u
6
2u
5
+ 3u
3
2u
2
2u + 1)
(u
12
+ 5u
11
+ ··· 5u 2)
c
6
(u 1)
10
(u
6
+ u
5
+ ··· 2u + 1)(u
12
+ 12u
11
+ ··· + 240u + 32)
c
7
(u
6
4u
3
5u
2
2u 1)
(u
10
+ u
9
+ 2u
8
+ 6u
7
14u
6
+ 28u
5
59u
4
+ 53u
3
82u
2
+ 34u 13)
(u
12
+ 5u
10
+ ··· + 4u + 1)
c
8
(u
6
+ 2u
5
2u
3
2u
2
u + 1)
(u
10
+ 3u
9
+ 2u
8
8u
7
24u
6
16u
5
+ 25u
4
+ 71u
3
+ 78u
2
+ 40u + 7)
(u
12
+ 2u
11
+ ··· + 3u + 1)
c
9
(u
5
3u
4
+ 5u
2
u 3)
2
(u
6
+ 4u
5
+ 5u
4
+ 2u
3
u
2
u + 1)
(u
12
+ 7u
11
+ ··· + 15u 4)
c
11
(u
6
2u
5
+ 2u
3
2u
2
+ u + 1)
(u
10
+ 3u
9
+ 2u
8
8u
7
24u
6
16u
5
+ 25u
4
+ 71u
3
+ 78u
2
+ 40u + 7)
(u
12
+ 2u
11
+ ··· + 3u + 1)
9
V. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
5
(y
5
y
4
+ 4y
3
3y
2
+ 3y 1)
2
(y
6
4y
5
+ ··· 8y + 1)(y
12
5y
11
+ ··· 37y + 4)
c
2
(y
5
+ 7y
4
+ 16y
3
+ 13y
2
+ 3y 1)
2
(y
6
24y
4
+ ··· 32y + 1)(y
12
+ 7y
11
+ ··· 353y + 16)
c
3
, c
4
, c
10
(y
6
5y
5
+ ··· 6y + 1)(y
10
9y
9
+ ··· 48y + 1)
(y
12
19y
11
+ ··· + 2y + 1)
c
6
(y 1)
10
(y
6
5y
5
+ 2y
3
+ 4y
2
4y + 1)
(y
12
6y
11
+ ··· 7936y + 1024)
c
7
(y
6
10y
4
+ ··· + 6y + 1)(y
10
+ 3y
9
+ ··· + 976y + 169)
(y
12
+ 10y
11
+ ··· 10y + 1)
c
8
, c
11
(y
6
4y
5
+ ··· 5y + 1)(y
10
5y
9
+ ··· 508y + 49)
(y
12
6y
11
+ ··· 25y + 1)
c
9
(y
5
9y
4
+ 28y
3
43y
2
+ 31y 9)
2
(y
6
6y
5
+ ··· 3y + 1)(y
12
3y
11
+ ··· 689y + 16)
10