11a
12
(K11a
12
)
1
Arc Sequences
5 1 9 2 3 11 10 4 8 6 7
Solving Sequence
1,5
2 3
4,7
11 6 10 8 9
c
1
c
2
c
4
c
11
c
6
c
10
c
7
c
9
c
3
, c
5
, c
8
Representation Ideals
I =
2
\
i=1
I
u
i
I
u
1
= hu
2
+ u + 1, b + 1, a u 1i
I
u
2
= hu
53
+ 2u
52
+ ··· 3u 1, u
51
u
50
+ ··· + b + 1, u
52
+ 3u
51
+ ··· + a 1i
There are 2 irreducible components with 55 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
2
+ u + 1, b + 1, a u 1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u + 1
a
3
=
u
u + 1
a
4
=
u
u + 1
a
7
=
u + 1
1
a
11
=
u + 2
1
a
6
=
1
0
a
10
=
u + 1
1
a
8
=
u + 1
1
a
9
=
u + 1
1
a
9
=
u + 1
1
(ii) Obstruction class = 1
(iii) Cusp Shapes = 4u 1
2
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
1
1(vol +
1CS) Cusp shape
u = 0.500000 0.866025I
a = 0.500000 0.866025I
b = 1.00000
1.64493 + 2.02988I 3.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.500000 + 0.866025I
b = 1.00000
1.64493 2.02988I 3.00000 + 3.46410I
3
II.
I
u
2
= hu
53
+2u
52
+· · ·3u1, u
51
u
50
+· · ·+b+1, u
52
+3u
51
+· · ·+a1i
(i) Arc colorings
a
1
=
1
0
a
5
=
0
u
a
2
=
1
u
2
a
3
=
u
2
+ 1
u
2
a
4
=
u
u
3
+ u
a
7
=
u
52
3u
51
+ ··· + 3u + 1
u
51
+ u
50
+ ··· 2u 1
a
11
=
u
52
2u
51
+ ··· + 2u + 1
u
51
+ u
50
+ ··· 3u 1
a
6
=
u
5
2u
3
u
u
5
+ u
3
+ u
a
10
=
u
52
u
51
+ ··· 5u
3
+ 2u
2
u
51
+ u
50
+ ··· 3u 1
a
8
=
u
52
3u
51
+ ··· + 3u + 1
u
51
+ u
50
+ ··· 2u 1
a
9
=
u
52
2u
51
+ ··· + 4u
2
+ u
2u
51
+ u
50
+ ··· 3u 1
a
9
=
u
52
2u
51
+ ··· + 4u
2
+ u
2u
51
+ u
50
+ ··· 3u 1
(ii) Obstruction class = 1
(iii) Cusp Shapes = u
52
3u
51
+ ··· 7u 6
4
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.808210 0.423523I
a = 1.05829 1.42916I
b = 1.329343 + 0.384853I
1.26818 8.94141I 2.53984 + 5.45139I
u = 0.808210 + 0.423523I
a = 1.05829 + 1.42916I
b = 1.329343 0.384853I
1.26818 + 8.94141I 2.53984 5.45139I
u = 0.779449 0.450933I
a = 0.05385 + 1.77458I
b = 0.082170 0.858832I
5.69104 4.47611I 1.74448 + 3.16326I
u = 0.779449 + 0.450933I
a = 0.05385 1.77458I
b = 0.082170 + 0.858832I
5.69104 + 4.47611I 1.74448 3.16326I
u = 0.735188 0.481734I
a = 1.17639 1.50281I
b = 1.191388 + 0.413451I
2.28411 + 0.08613I 1.183312 0.169828I
u = 0.735188 + 0.481734I
a = 1.17639 + 1.50281I
b = 1.191388 0.413451I
2.28411 0.08613I 1.183312 + 0.169828I
u = 0.703476 0.173079I
a = 0.321386 0.743221I
b = 1.355442 + 0.136049I
5.74714 3.41063I 7.67550 + 3.71131I
u = 0.703476 + 0.173079I
a = 0.321386 + 0.743221I
b = 1.355442 0.136049I
5.74714 + 3.41063I 7.67550 3.71131I
u = 0.612898 1.110160I
a = 0.28728 + 2.57600I
b = 1.347191 0.390740I
0.7808 + 14.2618I 5.37238 9.48715I
u = 0.612898 + 1.110160I
a = 0.28728 2.57600I
b = 1.347191 + 0.390740I
0.7808 14.2618I 5.37238 + 9.48715I
5
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.610266 1.090290I
a = 0.98621 1.30128I
b = 0.108156 + 0.876063I
3.78899 + 9.71652I 1.28115 7.65601I
u = 0.610266 + 1.090290I
a = 0.98621 + 1.30128I
b = 0.108156 0.876063I
3.78899 9.71652I 1.28115 + 7.65601I
u = 0.598130 1.065379I
a = 0.900724 0.233711I
b = 1.163704 0.443994I
0.55249 + 5.00025I 4.21634 4.73509I
u = 0.598130 + 1.065379I
a = 0.900724 + 0.233711I
b = 1.163704 + 0.443994I
0.55249 5.00025I 4.21634 + 4.73509I
u = 0.491563 1.126401I
a = 1.13520 + 1.91058I
b = 1.397484 0.165043I
8.47430 + 7.85966I 10.86670 7.27765I
u = 0.491563 + 1.126401I
a = 1.13520 1.91058I
b = 1.397484 + 0.165043I
8.47430 7.85966I 10.86670 + 7.27765I
u = 0.483844 0.213394I
a = 0.526258 + 1.051301I
b = 0.340173 0.453801I
0.42373 1.39478I 2.79014 + 5.25225I
u = 0.483844 + 0.213394I
a = 0.526258 1.051301I
b = 0.340173 + 0.453801I
0.42373 + 1.39478I 2.79014 5.25225I
u = 0.472572 1.061221I
a = 0.667446 1.039929I
b = 0.430093 + 0.598904I
2.63045 + 5.30697I 7.17193 8.38740I
u = 0.472572 + 1.061221I
a = 0.667446 + 1.039929I
b = 0.430093 0.598904I
2.63045 5.30697I 7.17193 + 8.38740I
6
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.388754 1.027129I
a = 0.248402 + 0.014464I
b = 0.616308 0.457732I
3.24252 + 1.36437I 10.10455 0.49514I
u = 0.388754 + 1.027129I
a = 0.248402 0.014464I
b = 0.616308 + 0.457732I
3.24252 1.36437I 10.10455 + 0.49514I
u = 0.327705 1.125732I
a = 1.47765 + 0.42384I
b = 1.398165 + 0.082873I
9.55416 0.10492I 13.08079 0.09673I
u = 0.327705 + 1.125732I
a = 1.47765 0.42384I
b = 1.398165 0.082873I
9.55416 + 0.10492I 13.08079 + 0.09673I
u = 0.242746 0.870296I
a = 0.185705 1.369147I
b = 0.968116 + 0.283551I
2.32930 + 1.08642I 8.77370 + 0.57725I
u = 0.242746 + 0.870296I
a = 0.185705 + 1.369147I
b = 0.968116 0.283551I
2.32930 1.08642I 8.77370 0.57725I
u = 0.107146 1.137427I
a = 0.853208 0.669683I
b = 1.333881 + 0.341289I
4.03348 6.68828I 8.87296 + 4.89618I
u = 0.107146 + 1.137427I
a = 0.853208 + 0.669683I
b = 1.333881 0.341289I
4.03348 + 6.68828I 8.87296 4.89618I
u = 0.073654 1.073023I
a = 0.047762 + 0.488109I
b = 0.108842 0.774169I
0.49494 2.64295I 4.27164 + 3.21466I
u = 0.073654 + 1.073023I
a = 0.047762 0.488109I
b = 0.108842 + 0.774169I
0.49494 + 2.64295I 4.27164 3.21466I
7
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.114288 0.998949I
a = 1.20024 1.12645I
b = 1.218454 + 0.268078I
2.73834 + 1.08871I 7.57153 0.45228I
u = 0.114288 + 0.998949I
a = 1.20024 + 1.12645I
b = 1.218454 0.268078I
2.73834 1.08871I 7.57153 + 0.45228I
u = 0.411073
a = 1.51545
b = 1.18911
2.34833 2.97220
u = 0.422351 0.696988I
a = 0.445354 + 0.699170I
b = 0.020226 0.357518I
0.51548 1.38171I 2.20295 + 4.47540I
u = 0.422351 + 0.696988I
a = 0.445354 0.699170I
b = 0.020226 + 0.357518I
0.51548 + 1.38171I 2.20295 4.47540I
u = 0.432705 1.054622I
a = 2.30953 + 1.64012I
b = 1.294167 0.057870I
4.79813 3.38896I 8.56565 + 4.20601I
u = 0.432705 + 1.054622I
a = 2.30953 1.64012I
b = 1.294167 + 0.057870I
4.79813 + 3.38896I 8.56565 4.20601I
u = 0.482076 0.940480I
a = 0.761131 0.492556I
b = 0.219966 + 0.200182I
0.24387 2.48522I 1.64376 + 3.61634I
u = 0.482076 + 0.940480I
a = 0.761131 + 0.492556I
b = 0.219966 0.200182I
0.24387 + 2.48522I 1.64376 3.61634I
u = 0.598927 1.081494I
a = 0.42512 + 2.79867I
b = 1.291970 0.356069I
0.26166 7.85803I 4.27524 + 5.69230I
8
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.598927 + 1.081494I
a = 0.42512 2.79867I
b = 1.291970 + 0.356069I
0.26166 + 7.85803I 4.27524 5.69230I
u = 0.617590 1.056217I
a = 1.07508 1.24210I
b = 0.031004 + 0.805221I
4.38710 3.67589I 0.16278 + 2.56525I
u = 0.617590 + 1.056217I
a = 1.07508 + 1.24210I
b = 0.031004 0.805221I
4.38710 + 3.67589I 0.16278 2.56525I
u = 0.640661 1.028061I
a = 0.988481 0.371567I
b = 1.244293 0.351593I
0.641453 + 0.489898I 3.92823 1.19375I
u = 0.640661 + 1.028061I
a = 0.988481 + 0.371567I
b = 1.244293 + 0.351593I
0.641453 0.489898I 3.92823 + 1.19375I
u = 0.644462 0.810856I
a = 0.942392 0.961581I
b = 1.226730 + 0.035206I
2.80413 2.50478I 9.09308 + 4.30288I
u = 0.644462 + 0.810856I
a = 0.942392 + 0.961581I
b = 1.226730 0.035206I
2.80413 + 2.50478I 9.09308 4.30288I
u = 0.750238 0.453200I
a = 0.96810 1.62658I
b = 1.262355 + 0.367119I
2.12308 + 2.73219I 1.27274 1.03337I
u = 0.750238 + 0.453200I
a = 0.96810 + 1.62658I
b = 1.262355 0.367119I
2.12308 2.73219I 1.27274 + 1.03337I
u = 0.758578 0.509405I
a = 0.09687 + 1.75830I
b = 0.007145 0.820800I
6.01459 1.53976I 2.52644 + 2.51375I
9
Solution to I
u
2
1(vol +
1CS) Cusp shape
u = 0.758578 + 0.509405I
a = 0.09687 1.75830I
b = 0.007145 + 0.820800I
6.01459 + 1.53976I 2.52644 2.51375I
u = 0.768188 0.560407I
a = 1.21074 1.40785I
b = 1.273870 + 0.366638I
2.03469 5.80979I 1.88693 + 5.87696I
u = 0.768188 + 0.560407I
a = 1.21074 + 1.40785I
b = 1.273870 0.366638I
2.03469 + 5.80979I 1.88693 5.87696I
10
III. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
(u
2
+ u + 1)(u
53
+ 2u
52
+ ··· 3u 1)
c
2
(u
2
+ u + 1)(u
53
+ 24u
52
+ ··· + u 1)
c
3
, c
8
u
2
(u
53
+ u
52
+ ··· + 12u + 4)
c
4
(u
2
u + 1)(u
53
+ 2u
52
+ ··· 3u 1)
c
5
(u
2
+ u + 1)(u
53
+ 2u
52
+ ··· + 5u + 1)
c
6
(u 1)
2
(u
53
+ 3u
52
+ ··· 6u
2
1)
c
7
, c
9
u
2
(u
53
+ 15u
52
+ ··· 120u 16)
c
10
, c
11
(u + 1)
2
(u
53
+ 3u
52
+ ··· 6u
2
1)
11
IV. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
4
(y
2
+ y + 1)(y
53
+ 24y
52
+ ··· + y 1)
c
2
(y
2
+ y + 1)(y
53
+ 12y
52
+ ··· + 25y 1)
c
3
, c
8
y
2
(y
53
+ 15y
52
+ ··· 120y 16)
c
5
(y
2
+ y + 1)(y
53
+ 50y
51
+ ··· + 49y 1)
c
6
, c
10
, c
11
(y 1)
2
(y
53
43y
52
+ ··· 12y 1)
c
7
, c
9
y
2
(y
53
+ 43y
52
+ ··· 1248y 256)
12