11a
346
(K11a
346
)
1
Arc Sequences
9 6 1 10 8 3 11 2 4 5 7
Solving Sequence
2,6
3
7,9
1 4 8 5 11 10
c
2
c
6
c
1
c
3
c
8
c
5
c
11
c
10
c
4
, c
7
, c
9
Representation Ideals
I =
5
\
i=1
I
u
i
I
u
1
= hu 1, b 1, 2a 1i
I
u
2
= h4a
2
+ 4a 1, b + 1, u + 1i
I
u
3
= h9a
4
+ 18a
3
+ 18a
2
+ 12a + 4, 9a
3
+ 9a
2
+ 6a + 2u + 2, 9a
3
12a
2
+ 2b 12a 6i
I
u
4
= hb
30
3b
29
+ ··· + 8b + 13, 7.08789 × 10
22
u 3.40058 × 10
23
b
29
+ ··· + 2.41364 × 10
25
b 1.16881 × 10
25
,
4.17087 × 10
23
b
29
+ 1.42208 × 10
24
b
28
+ ··· + 7.08789 × 10
22
a 1.48240 × 10
25
i
I
u
5
= hu
23
+ 2u
22
+ ··· + 71u + 8,
7.95000 × 10
18
u
22
1.97992 × 10
20
u
21
+ ··· + 1.71401 × 10
22
b + 1.09059 × 10
22
,
2.75649 × 10
21
u
22
+ 5.13543 × 10
21
u
21
+ ··· + 3.42803 × 10
22
a + 2.42288 × 10
23
i
There are 5 irreducible components with 60 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, b 1, 2a 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
7
=
1
0
a
9
=
1
2
1
a
1
=
3
2
1
a
4
=
1
4
1
2
a
8
=
1
2
1
a
5
=
1
4
1
2
a
11
=
1
2
1
a
10
=
1
2
1
a
10
=
1
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 = 0.500000
b = 1.00000
0 4.50000
3
II. I
u
2
= h4a
2
+ 4a 1, b + 1, u + 1i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
1
a
3
=
1
1
a
7
=
1
0
a
9
=
a
1
a
1
=
a + 1
1
a
4
=
2a +
3
4
a + 1
a
8
=
a + 1
1
a
5
=
a
5
4
a
a
11
=
a
1
a
10
=
1
2
a +
7
4
a +
1
2
a
10
=
1
2
a +
7
4
a +
1
2
(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 = 1.00000
a = 1.20711
b = 1.00000
4.93480 8.00000
u = 1.00000
a = 0.207107
b = 1.00000
4.93480 8.00000
5
III. I
u
3
=
h9a
4
+18a
3
+18a
2
+12a+4, 9a
3
+9a
2
+6a+2u+2, 9a
3
12a
2
+2b12a6i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
9
2
a
3
9
2
a
2
3a 1
a
3
=
1
9
2
a
3
9
2
a
2
3a 2
a
7
=
9
2
a
3
+
9
2
a
2
+ 3a + 1
9
2
a
3
9
2
a
2
3a
a
9
=
a
9
2
a
3
+ 6a
2
+ 6a + 3
a
1
=
3a
3
3a
2
3a 1
1
a
4
=
3
2
a
3
+
5
2
a
2
+ 2a + 2
15
2
a
3
9a
2
6a 3
a
8
=
9
2
a
3
6a
2
5a 3
9
2
a
3
+ 6a
2
+ 6a + 3
a
5
=
9
2
a
3
+
13
2
a
2
+ 5a + 3
21
2
a
3
12a
2
9a 4
a
11
=
3
2
a
3
+
9
2
a
2
+ 3a + 2
9a
3
27
2
a
2
12a 7
a
10
=
3
2
a
2
+ a +
1
3
3
2
a
2
a 1
a
10
=
3
2
a
2
+ a +
1
3
3
2
a
2
a 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 = 0.500000 + 0.866025I
a = 0.788675 0.211325I
b = 1.00000I
3.28987 2.02988I 10.00000 + 3.46410I
u = 0.500000 0.866025I
a = 0.788675 + 0.211325I
b = 1.00000I
3.28987 + 2.02988I 10.00000 3.46410I
u = 0.500000 0.866025I
a = 0.211325 0.788675I
b = 1.00000I
3.28987 + 2.02988I 10.00000 3.46410I
u = 0.500000 + 0.866025I
a = 0.211325 + 0.788675I
b = 1.00000I
3.28987 2.02988I 10.00000 + 3.46410I
7
IV.
I
u
4
= hb
30
3b
29
+· · ·+8b+13, 7.09×10
22
u3.40×10
23
b
29
+· · ·+2.41×10
25
b
1.17 × 10
25
, 4.17 × 10
23
b
29
+ 1.42 × 10
24
b
28
+ · · · + 7.09 × 10
22
a 1.48 × 10
25
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
4.79774b
29
16.1279b
28
+ ··· 340.531b + 164.902
a
3
=
1
0.732079b
29
+ 2.42387b
28
+ ··· + 52.3394b 24.4729
a
7
=
4.79774b
29
+ 16.1279b
28
+ ··· + 340.531b 164.902
7.04949b
29
23.8397b
28
+ ··· 505.629b + 244.604
a
9
=
5.88450b
29
20.0636b
28
+ ··· 439.100b + 209.145
b
a
1
=
2.41007b
29
+ 8.30531b
28
+ ··· + 162.069b 75.4985
b
2
a
4
=
1.29396b
29
+ 4.23482b
28
+ ··· + 128.276b 67.1735
0.929569b
29
+ 3.24141b
28
+ ··· + 57.2893b 24.9558
a
8
=
5.88450b
29
20.0636b
28
+ ··· 440.100b + 209.145
b
a
5
=
1.31203b
29
+ 4.54538b
28
+ ··· + 119.147b 60.7693
3.63651b
29
12.1141b
28
+ ··· 250.537b + 123.837
a
11
=
2.72764b
29
+ 9.34791b
28
+ ··· + 186.359b 88.3501
1
a
10
=
0.476015b
29
1.67468b
28
+ ··· 5.11740b 4.47311
1.04611b
29
3.34677b
28
+ ··· 73.7020b + 39.2535
a
10
=
0.476015b
29
1.67468b
28
+ ··· 5.11740b 4.47311
1.04611b
29
3.34677b
28
+ ··· 73.7020b + 39.2535
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
8
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.314203 0.295245I
a = 0.136655 0.231531I
b = 0.991485 0.204855I
5.47316 + 5.45324I 7.99532 6.35130I
u = 1.314203 + 0.295245I
a = 0.136655 + 0.231531I
b = 0.991485 + 0.204855I
5.47316 5.45324I 7.99532 + 6.35130I
u = 1.378138 + 0.316043I
a = 0.41954 1.81333I
b = 0.55486 1.65691I
12.8088 + 8.0168I 11.04132 4.89679I
u = 1.378138 0.316043I
a = 0.41954 + 1.81333I
b = 0.55486 + 1.65691I
12.8088 8.0168I 11.04132 + 4.89679I
u = 1.30332
a = 0.44659 1.82285I
b = 0.455272 1.273506I
8.81535 13.0394
u = 1.30332
a = 0.44659 + 1.82285I
b = 0.455272 + 1.273506I
8.81535 13.0394
u = 0.074720 0.708028I
a = 0.412232 0.904577I
b = 0.373462 0.000206I
1.11561 1.81248I 2.14381 + 4.33913I
u = 0.074720 + 0.708028I
a = 0.412232 + 0.904577I
b = 0.373462 + 0.000206I
1.11561 + 1.81248I 2.14381 4.33913I
u = 0.200931 0.760138I
a = 0.0025046 + 0.0745229I
b = 0.298420 1.343442I
7.81260 + 4.11725I 6.59688 3.71929I
u = 0.200931 + 0.760138I
a = 0.0025046 0.0745229I
b = 0.298420 + 1.343442I
7.81260 4.11725I 6.59688 + 3.71929I
9
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 0.339181
a = 3.50114 + 1.16755I
b = 0.150080 1.033234I
3.88780 12.6282
u = 0.339181
a = 3.50114 1.16755I
b = 0.150080 + 1.033234I
3.88780 12.6282
u = 1.224706 + 0.250895I
a = 0.81617 2.04457I
b = 0.116215 1.234426I
4.58881 + 1.64925I 5.60633 0.16522I
u = 1.224706 0.250895I
a = 0.81617 + 2.04457I
b = 0.116215 + 1.234426I
4.58881 1.64925I 5.60633 + 0.16522I
u = 0.897290 + 0.288232I
a = 3.17433 0.41297I
b = 0.085684 1.202646I
10.05371 + 0.15908I 9.79403 + 0.85194I
u = 0.897290 0.288232I
a = 3.17433 + 0.41297I
b = 0.085684 + 1.202646I
10.05371 0.15908I 9.79403 0.85194I
u = 0.074720 0.708028I
a = 0.009915 0.231974I
b = 0.186786 1.050899I
1.11561 1.81248I 2.14381 + 4.33913I
u = 0.074720 + 0.708028I
a = 0.009915 + 0.231974I
b = 0.186786 + 1.050899I
1.11561 + 1.81248I 2.14381 4.33913I
u = 0.897290 0.288232I
a = 0.32525 + 2.38302I
b = 0.312806 0.818632I
10.05371 0.15908I 9.79403 0.85194I
u = 0.897290 + 0.288232I
a = 0.32525 2.38302I
b = 0.312806 + 0.818632I
10.05371 + 0.15908I 9.79403 + 0.85194I
10
Solution to I
u
4
1(vol +
1CS) Cusp shape
u = 1.314203 + 0.295245I
a = 0.51084 1.84381I
b = 0.37957 1.41365I
5.47316 5.45324I 7.99532 + 6.35130I
u = 1.314203 0.295245I
a = 0.51084 + 1.84381I
b = 0.37957 + 1.41365I
5.47316 + 5.45324I 7.99532 6.35130I
u = 1.224706 0.250895I
a = 0.448867 + 0.146986I
b = 0.497171 0.372159I
4.58881 1.64925I 5.60633 + 0.16522I
u = 1.224706 + 0.250895I
a = 0.448867 0.146986I
b = 0.497171 + 0.372159I
4.58881 + 1.64925I 5.60633 0.16522I
u = 1.43385
a = 0.05645 1.56167I
b = 0.75703 1.64902I
17.0919 13.9771
u = 1.43385
a = 0.05645 + 1.56167I
b = 0.75703 + 1.64902I
17.0919 13.9771
u = 0.200931 + 0.760138I
a = 0.910945 + 0.772405I
b = 0.839538 0.236157I
7.81260 4.11725I 6.59688 + 3.71929I
u = 0.200931 0.760138I
a = 0.910945 0.772405I
b = 0.839538 + 0.236157I
7.81260 + 4.11725I 6.59688 3.71929I
u = 1.378138 0.316043I
a = 0.367182 0.457274I
b = 1.381213 0.203773I
12.8088 8.0168I 11.04132 + 4.89679I
u = 1.378138 + 0.316043I
a = 0.367182 + 0.457274I
b = 1.381213 + 0.203773I
12.8088 + 8.0168I 11.04132 4.89679I
11
V. I
u
5
=
hu
23
+2u
22
+· · ·+71u+8, 7.95×10
18
u
22
1.98×10
20
u
21
+· · ·+1.71×10
22
b+
1.09 × 10
22
, 2.76 × 10
21
u
22
+ 5.14 × 10
21
u
21
+ · · · + 3.43 × 10
22
a + 2.42 × 10
23
i
(i) Arc colorings
a
2
=
1
0
a
6
=
0
u
a
3
=
1
u
2
a
7
=
u
u
3
+ u
a
9
=
0.0804102u
22
0.149807u
21
+ ··· + 1.47745u 7.06786
0.000463824u
22
+ 0.0115513u
21
+ ··· + 2.33256u 0.636277
a
1
=
0.0791449u
22
+ 0.154837u
21
+ ··· 0.606289u + 7.33252
0.00348213u
22
+ 0.00104742u
21
+ ··· 1.07750u + 0.674619
a
4
=
0.0705113u
22
+ 0.140559u
21
+ ··· 0.178315u + 6.08243
0.000356169u
22
+ 0.00964749u
21
+ ··· + 0.118336u + 0.579177
a
8
=
0.0808741u
22
0.161358u
21
+ ··· 0.855105u 6.43158
0.000463824u
22
+ 0.0115513u
21
+ ··· + 2.33256u 0.636277
a
5
=
0.0719338u
22
0.138716u
21
+ ··· 1.07706u 4.45777
0.00246360u
22
0.0160908u
21
+ ··· + 1.28584u 0.605302
a
11
=
0.0685212u
22
+ 0.154326u
21
+ ··· + 0.0629189u + 7.33623
0.0194331u
22
0.0508356u
21
+ ··· 3.13401u + 0.505016
a
10
=
0.00394024u
22
+ 0.0385531u
21
+ ··· + 0.566623u + 1.38271
0.00190919u
22
0.00541351u
21
+ ··· 1.10632u 0.276903
a
10
=
0.00394024u
22
+ 0.0385531u
21
+ ··· + 0.566623u + 1.38271
0.00190919u
22
0.00541351u
21
+ ··· 1.10632u 0.276903
(ii) Obstruction class = 1
(iii) Cusp Shapes =unknown
12
(iv) Complex Volumes and Cusp Shapes
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.60687 0.59883I
a = 0.51743 1.35178I
b = 0.19561 1.40284I
10.15960 4.19504I 11.29932 + 2.35235I
u = 1.60687 + 0.59883I
a = 0.51743 + 1.35178I
b = 0.19561 + 1.40284I
10.15960 + 4.19504I 11.29932 2.35235I
u = 1.47860 0.53478I
a = 0.72895 1.60096I
b = 0.54587 1.52621I
18.3248 14.6898I 11.82186 + 6.64707I
u = 1.47860 + 0.53478I
a = 0.72895 + 1.60096I
b = 0.54587 + 1.52621I
18.3248 + 14.6898I 11.82186 6.64707I
u = 1.27993
a = 0.837329
b = 1.69273
6.97375 13.5441
u = 0.832315
a = 0.323128
b = 0.310175
1.09471 11.6781
u = 0.754919 0.676112I
a = 0.316248 0.543061I
b = 0.420512 0.719649I
2.03566 0.84759I 5.54570 1.19524I
u = 0.754919 + 0.676112I
a = 0.316248 + 0.543061I
b = 0.420512 + 0.719649I
2.03566 + 0.84759I 5.54570 + 1.19524I
u = 0.702926 0.643969I
a = 0.414128 + 0.935437I
b = 0.386436 + 0.666877I
2.01223 3.75776I 5.33334 + 8.56871I
u = 0.702926 + 0.643969I
a = 0.414128 0.935437I
b = 0.386436 0.666877I
2.01223 + 3.75776I 5.33334 8.56871I
13
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 0.12365 1.47479I
a = 0.102937 + 0.561813I
b = 0.109506 + 1.309224I
5.14680 3.47388I 11.56230 + 3.97702I
u = 0.12365 + 1.47479I
a = 0.102937 0.561813I
b = 0.109506 1.309224I
5.14680 + 3.47388I 11.56230 3.97702I
u = 0.113114
a = 6.92756
b = 0.880255
3.32968 1.46423
u = 0.146740 1.277214I
a = 0.284450 + 0.432546I
b = 0.33582 + 1.42592I
13.1520 + 8.3676I 10.67653 4.81997I
u = 0.146740 + 1.277214I
a = 0.284450 0.432546I
b = 0.33582 1.42592I
13.1520 8.3676I 10.67653 + 4.81997I
u = 0.474948 0.285362I
a = 0.003598 + 1.074414I
b = 0.565086 + 0.372468I
0.87976 + 1.13203I 3.15244 4.47998I
u = 0.474948 + 0.285362I
a = 0.003598 1.074414I
b = 0.565086 0.372468I
0.87976 1.13203I 3.15244 + 4.47998I
u = 1.04465
a = 1.26078
b = 0.0971819
6.54656 14.0365
u = 1.11050
a = 0.633591
b = 1.23682
0.342193 19.7628
u = 1.54005 0.52936I
a = 0.59381 1.52498I
b = 0.39680 1.43332I
10.7273 + 10.3813I 10.18639 6.73732I
14
Solution to I
u
5
1(vol +
1CS) Cusp shape
u = 1.54005 + 0.52936I
a = 0.59381 + 1.52498I
b = 0.39680 + 1.43332I
10.7273 10.3813I 10.18639 + 6.73732I
u = 1.54033 0.69917I
a = 0.586771 1.151979I
b = 0.04263 1.48432I
17.3595 1.0921I 13.73412 + 0.45411I
u = 1.54033 + 0.69917I
a = 0.586771 + 1.151979I
b = 0.04263 + 1.48432I
17.3595 + 1.0921I 13.73412 0.45411I
15
VI. u-Polynomials
Crossings u-Polynomials at each crossings
c
1
, c
11
(u 1)(u + 1)
2
(u
2
+ 1)
2
(u
23
u
22
+ ··· 14u + 1)
(u
30
+ 3u
29
+ ··· 8u + 13)
c
2
(u 1)(u + 1)
2
(u
2
u + 1)
2
(1 2u + 2u
2
2u
3
+ 4u
4
+ 6u
5
8u
6
+ 2u
7
u
8
14u
9
+ 8u
10
+ 14u
11
5u
12
6u
13
+ u
14
+ u
15
)
2
(u
23
+ 2u
22
+ ··· + 71u + 8)
c
3
(2u + 1)(4u
2
+ 4u 1)(9u
4
+ 18u
3
+ 9u
2
+ 1)(8u
23
20u
22
+ ··· u 1)
(u
30
+ 3u
29
+ ··· 67460u + 14279)
c
4
, c
9
, c
10
u(u
2
2)(u
4
u
2
+ 1)
(1 2u + 2u
2
+ 4u
3
14u
5
+ 2u
6
+ 26u
7
+ 11u
8
34u
9
16u
10
+ 24u
11
+ 7u
12
8u
13
u
14
+ u
15
)
2
(u
23
+ 3u
22
+ ··· + 32u 8)
c
5
(2u 1)(4u
2
4u 1)(9u
4
18u
3
+ 9u
2
+ 1)(8u
23
20u
22
+ ··· u 1)
(u
30
+ 3u
29
+ ··· 67460u + 14279)
c
6
(u 1)
2
(u + 1)(u
2
+ u + 1)
2
(1 2u + 2u
2
2u
3
+ 4u
4
+ 6u
5
8u
6
+ 2u
7
u
8
14u
9
+ 8u
10
+ 14u
11
5u
12
6u
13
+ u
14
+ u
15
)
2
(u
23
+ 2u
22
+ ··· + 71u + 8)
c
7
, c
8
(u 1)
2
(u + 1)(u
2
+ 1)
2
(u
23
u
22
+ ··· 14u + 1)
(u
30
+ 3u
29
+ ··· 8u + 13)
16
VII. Riley Polynomials
Crossings Riley Polynomials at each crossings
c
1
, c
7
, c
8
c
11
(y 1)
3
(y + 1)
4
(y
23
+ 19y
22
+ ··· + 112y 1)
(y
30
+ 23y
29
+ ··· 1572y + 169)
c
2
, c
6
(y 1)
3
(y
2
+ y + 1)
2
(1 + 8y + 12y
2
52y
3
18y
4
+ 168y
5
74y
6
254y
7
+ 295y
8
+ 104y
9
462y
10
+ 450y
11
237y
12
+ 74y
13
13y
14
+ y
15
)
2
(y
23
14y
22
+ ··· + 5473y 64)
c
3
, c
5
(4y 1)(16y
2
24y + 1)(81y
4
162y
3
+ 99y
2
+ 18y + 1)
(64y
23
1392y
22
+ ··· 37y 1)
(y
30
21y
29
+ ··· 2895344340y + 203889841)
c
4
, c
9
, c
10
y(y 2)
2
(y
2
y + 1)
2
(1 + 8y 20y
2
+ 76y
3
202y
4
+ 464y
5
1022y
6
+ 1778y
7
2561y
8
+ 2960y
9
2482y
10
+ 1418y
11
533y
12
+ 126y
13
17y
14
+ y
15
)
2
(y
23
23y
22
+ ··· + 672y 64)
17