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Conway polynomials of the prime knots with at most ten crossings
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Polynomials
$n$
$k$ 
$\nabla_{n_k}(z)$
0
1:
1
comment: $0_1$, the unknot; $\nabla=1$; the first nontrivial knots with $\nabla=1$ are $11n_{34}$, the Conway knot, and $11n_{42}$, the Kinoshita–Terasaka knot
equals: One
3
1:
z^2 + 1
comment: $3_1$, the trefoil; the torus knot $T(2,3)$; $\nabla=F_{3}=\Phi_{4}$; $a_2=1$, Arf invariant $1$, determinant $3$
equals: $F_{3}$
4
1:
-z^2 + 1
comment: $4_1$, the figure-eight knot; $a_2=-1$, Arf invariant $1$, determinant $5$
5
1:
z^4 + 3*z^2 + 1
comment: $5_1$, the cinquefoil; the torus knot $T(2,5)$; $\nabla=F_{5}$; $a_2=3$, Arf invariant $1$, determinant $5$; the same polynomial as $10_{132}$
equals: $F_{5}$
5
2:
2*z^2 + 1
comment: $5_2$, the three-twist knot; $a_2=2$, Arf invariant $0$, determinant $7$
6
1:
-2*z^2 + 1
comment: $6_1$, the stevedore knot; $a_2=-2$, Arf invariant $0$, determinant $9$; the same polynomial as $9_{46}$
6
2:
-z^4 - z^2 + 1
comment: $6_2$, the Miller Institute knot; $a_2=-1$, Arf invariant $1$, determinant $11$
6
3:
z^4 + z^2 + 1
comment: $6_3$; $a_2=1$, Arf invariant $1$, determinant $13$
7
1:
z^6 + 5*z^4 + 6*z^2 + 1
comment: $7_1$; the torus knot $T(2,7)$; $\nabla=F_{7}$; $a_2=6$, Arf invariant $0$, determinant $7$
equals: $F_{7}$
7
2:
3*z^2 + 1
comment: $7_2$; $a_2=3$, Arf invariant $1$, determinant $11$
7
3:
2*z^4 + 5*z^2 + 1
comment: $7_3$; $a_2=5$, Arf invariant $1$, determinant $13$
7
4:
4*z^2 + 1
comment: $7_4$, the endless knot; $a_2=4$, Arf invariant $0$, determinant $15$; the same polynomial as $9_2$
7
5:
2*z^4 + 4*z^2 + 1
comment: $7_5$; $a_2=4$, Arf invariant $0$, determinant $17$; the same polynomial as $10_{130}$
7
6:
-z^4 + z^2 + 1
comment: $7_6$; $a_2=1$, Arf invariant $1$, determinant $19$; the same polynomial as $10_{133}$
7
7:
z^4 - z^2 + 1
comment: $7_7$; $\nabla=\Phi_{12}$; $a_2=-1$, Arf invariant $1$, determinant $21$
equals: $\Phi_{12}$
8
1:
-3*z^2 + 1
comment: $8_1$; $a_2=-3$, Arf invariant $1$, determinant $13$
8
2:
-z^6 - 3*z^4 + 1
comment: $8_2$; $a_2=0$, Arf invariant $0$, determinant $17$
8
3:
-4*z^2 + 1
comment: $8_3$; $a_2=-4$, Arf invariant $0$, determinant $17$; the same polynomial as $10_1$
8
4:
-2*z^4 - 3*z^2 + 1
comment: $8_4$; $a_2=-3$, Arf invariant $1$, determinant $19$
8
5:
-z^6 - 3*z^4 - z^2 + 1
comment: $8_5$; $a_2=-1$, Arf invariant $1$, determinant $21$; the same polynomial as $10_{141}$
8
6:
-2*z^4 - 2*z^2 + 1
comment: $8_6$; $a_2=-2$, Arf invariant $0$, determinant $23$
8
7:
z^6 + 3*z^4 + 2*z^2 + 1
comment: $8_7$; $a_2=2$, Arf invariant $0$, determinant $23$
8
8:
2*z^4 + 2*z^2 + 1
comment: $8_8$; $a_2=2$, Arf invariant $0$, determinant $25$; the same polynomial as $10_{129}$
8
9:
-z^6 - 3*z^4 - 2*z^2 + 1
comment: $8_9$; $a_2=-2$, Arf invariant $0$, determinant $25$; the same polynomial as $10_{155}$
8
10:
z^6 + 3*z^4 + 3*z^2 + 1
comment: $8_{10}$; $a_2=3$, Arf invariant $1$, determinant $27$; the same polynomial as $10_{143}$
8
11:
-2*z^4 - z^2 + 1
comment: $8_{11}$; $a_2=-1$, Arf invariant $1$, determinant $27$; the same polynomial as $10_{147}$
8
12:
z^4 - 3*z^2 + 1
comment: $8_{12}$; $a_2=-3$, Arf invariant $1$, determinant $29$
8
13:
2*z^4 + z^2 + 1
comment: $8_{13}$; $a_2=1$, Arf invariant $1$, determinant $29$
8
14:
-2*z^4 + 1
comment: $8_{14}$; $a_2=0$, Arf invariant $0$, determinant $31$; the same polynomial as $9_8$, $10_{131}$
8
15:
3*z^4 + 4*z^2 + 1
comment: $8_{15}$; $a_2=4$, Arf invariant $0$, determinant $33$
8
16:
z^6 + 2*z^4 + z^2 + 1
comment: $8_{16}$; $a_2=1$, Arf invariant $1$, determinant $35$; the same polynomial as $10_{156}$
8
17:
-z^6 - 2*z^4 - z^2 + 1
comment: $8_{17}$; $a_2=-1$, Arf invariant $1$, determinant $37$
8
18:
-z^6 - z^4 + z^2 + 1
comment: $8_{18}$, the Carrick mat; $a_2=1$, Arf invariant $1$, determinant $45$; the same polynomial as $9_{24}$
8
19:
z^6 + 5*z^4 + 5*z^2 + 1
comment: $8_{19}$; the torus knot $T(3,4)$; $a_2=5$, Arf invariant $1$, determinant $3$
8
20:
z^4 + 2*z^2 + 1
comment: $8_{20}$; $a_2=2$, Arf invariant $0$, determinant $9$; the same polynomial as $10_{140}$
8
21:
-z^4 + 1
comment: $8_{21}$; $a_2=0$, Arf invariant $0$, determinant $15$; the same polynomial as $10_{136}$
9
1:
z^8 + 7*z^6 + 15*z^4 + 10*z^2 + 1
comment: $9_1$; the torus knot $T(2,9)$; $\nabla=F_{9}$; $a_2=10$, Arf invariant $0$, determinant $9$
equals: $F_{9}$
9
2:
4*z^2 + 1
comment: $9_2$; $a_2=4$, Arf invariant $0$, determinant $15$; the same polynomial as $7_4$
9
3:
2*z^6 + 9*z^4 + 9*z^2 + 1
comment: $9_3$; $a_2=9$, Arf invariant $1$, determinant $19$
9
4:
3*z^4 + 7*z^2 + 1
comment: $9_4$; $a_2=7$, Arf invariant $1$, determinant $21$
9
5:
6*z^2 + 1
comment: $9_5$; $a_2=6$, Arf invariant $0$, determinant $23$
9
6:
2*z^6 + 8*z^4 + 7*z^2 + 1
comment: $9_6$; $a_2=7$, Arf invariant $1$, determinant $27$
9
7:
3*z^4 + 5*z^2 + 1
comment: $9_7$; $a_2=5$, Arf invariant $1$, determinant $29$
9
8:
-2*z^4 + 1
comment: $9_8$; $a_2=0$, Arf invariant $0$, determinant $31$; the same polynomial as $8_{14}$, $10_{131}$
9
9:
2*z^6 + 8*z^4 + 8*z^2 + 1
comment: $9_9$; $a_2=8$, Arf invariant $0$, determinant $31$
9
10:
4*z^4 + 8*z^2 + 1
comment: $9_{10}$; $a_2=8$, Arf invariant $0$, determinant $33$
9
11:
-z^6 - z^4 + 4*z^2 + 1
comment: $9_{11}$; $a_2=4$, Arf invariant $0$, determinant $33$
9
12:
-2*z^4 + z^2 + 1
comment: $9_{12}$; $a_2=1$, Arf invariant $1$, determinant $35$
9
13:
4*z^4 + 7*z^2 + 1
comment: $9_{13}$; $a_2=7$, Arf invariant $1$, determinant $37$
9
14:
2*z^4 - z^2 + 1
comment: $9_{14}$; $a_2=-1$, Arf invariant $1$, determinant $37$
9
15:
-2*z^4 + 2*z^2 + 1
comment: $9_{15}$; $a_2=2$, Arf invariant $0$, determinant $39$; the same polynomial as $10_{165}$
9
16:
2*z^6 + 7*z^4 + 6*z^2 + 1
comment: $9_{16}$; $a_2=6$, Arf invariant $0$, determinant $39$
9
17:
z^6 + z^4 - 2*z^2 + 1
comment: $9_{17}$; $a_2=-2$, Arf invariant $0$, determinant $39$
9
18:
4*z^4 + 6*z^2 + 1
comment: $9_{18}$; $a_2=6$, Arf invariant $0$, determinant $41$
9
19:
2*z^4 - 2*z^2 + 1
comment: $9_{19}$; $a_2=-2$, Arf invariant $0$, determinant $41$
9
20:
-z^6 - z^4 + 2*z^2 + 1
comment: $9_{20}$; $a_2=2$, Arf invariant $0$, determinant $41$; the same polynomial as $10_{149}$
9
21:
-2*z^4 + 3*z^2 + 1
comment: $9_{21}$; $a_2=3$, Arf invariant $1$, determinant $43$
9
22:
z^6 + z^4 - z^2 + 1
comment: $9_{22}$; $a_2=-1$, Arf invariant $1$, determinant $43$
9
23:
4*z^4 + 5*z^2 + 1
comment: $9_{23}$; $a_2=5$, Arf invariant $1$, determinant $45$
9
24:
-z^6 - z^4 + z^2 + 1
comment: $9_{24}$; $a_2=1$, Arf invariant $1$, determinant $45$; the same polynomial as $8_{18}$
9
25:
-3*z^4 + 1
comment: $9_{25}$; $a_2=0$, Arf invariant $0$, determinant $47$
9
26:
z^6 + z^4 + 1
comment: $9_{26}$; $a_2=0$, Arf invariant $0$, determinant $47$
9
27:
-z^6 - z^4 + 1
comment: $9_{27}$; $a_2=0$, Arf invariant $0$, determinant $49$
9
28:
z^6 + z^4 + z^2 + 1
comment: $9_{28}$; $a_2=1$, Arf invariant $1$, determinant $51$; the same polynomial as $9_{29}$, $10_{163}$
9
29:
z^6 + z^4 + z^2 + 1
comment: $9_{29}$; $a_2=1$, Arf invariant $1$, determinant $51$; the same polynomial as $9_{28}$, $10_{163}$
9
30:
-z^6 - z^4 - z^2 + 1
comment: $9_{30}$; $a_2=-1$, Arf invariant $1$, determinant $53$
9
31:
z^6 + z^4 + 2*z^2 + 1
comment: $9_{31}$; $a_2=2$, Arf invariant $0$, determinant $55$
9
32:
z^6 - z^2 + 1
comment: $9_{32}$; $a_2=-1$, Arf invariant $1$, determinant $59$
9
33:
-z^6 + z^2 + 1
comment: $9_{33}$; $a_2=1$, Arf invariant $1$, determinant $61$
9
34:
-z^6 - z^2 + 1
comment: $9_{34}$; $a_2=-1$, Arf invariant $1$, determinant $69$
9
35:
7*z^2 + 1
comment: $9_{35}$; $a_2=7$, Arf invariant $1$, determinant $27$
9
36:
-z^6 - z^4 + 3*z^2 + 1
comment: $9_{36}$; $a_2=3$, Arf invariant $1$, determinant $37$
9
37:
2*z^4 - 3*z^2 + 1
comment: $9_{37}$; $a_2=-3$, Arf invariant $1$, determinant $45$
9
38:
5*z^4 + 6*z^2 + 1
comment: $9_{38}$; $a_2=6$, Arf invariant $0$, determinant $57$; the same polynomial as $10_{63}$
9
39:
-3*z^4 + 2*z^2 + 1
comment: $9_{39}$; $a_2=2$, Arf invariant $0$, determinant $55$
9
40:
z^6 - z^4 - z^2 + 1
comment: $9_{40}$; $a_2=-1$, Arf invariant $1$, determinant $75$; the same polynomial as $10_{59}$
9
41:
3*z^4 + 1
comment: $9_{41}$; $a_2=0$, Arf invariant $0$, determinant $49$
9
42:
-z^4 - 2*z^2 + 1
comment: $9_{42}$; $a_2=-2$, Arf invariant $0$, determinant $7$
9
43:
-z^6 - 3*z^4 + z^2 + 1
comment: $9_{43}$; $a_2=1$, Arf invariant $1$, determinant $13$
9
44:
z^4 + 1
comment: $9_{44}$; $\nabla=\Phi_{8}$; $a_2=0$, Arf invariant $0$, determinant $17$
equals: $\Phi_{8}$
9
45:
-z^4 + 2*z^2 + 1
comment: $9_{45}$; $a_2=2$, Arf invariant $0$, determinant $23$
9
46:
-2*z^2 + 1
comment: $9_{46}$; $a_2=-2$, Arf invariant $0$, determinant $9$; the same polynomial as $6_1$
9
47:
z^6 + 2*z^4 - z^2 + 1
comment: $9_{47}$; $a_2=-1$, Arf invariant $1$, determinant $27$
9
48:
-z^4 + 3*z^2 + 1
comment: $9_{48}$; $a_2=3$, Arf invariant $1$, determinant $27$
9
49:
3*z^4 + 6*z^2 + 1
comment: $9_{49}$; $a_2=6$, Arf invariant $0$, determinant $25$
10
1:
-4*z^2 + 1
comment: $10_1$; $a_2=-4$, Arf invariant $0$, determinant $17$; the same polynomial as $8_3$
10
2:
-z^8 - 5*z^6 - 5*z^4 + 2*z^2 + 1
comment: $10_2$; $a_2=2$, Arf invariant $0$, determinant $23$
10
3:
-6*z^2 + 1
comment: $10_3$; $a_2=-6$, Arf invariant $0$, determinant $25$
10
4:
-3*z^4 - 5*z^2 + 1
comment: $10_4$; $a_2=-5$, Arf invariant $1$, determinant $27$
10
5:
z^8 + 5*z^6 + 7*z^4 + 4*z^2 + 1
comment: $10_5$; $a_2=4$, Arf invariant $0$, determinant $33$
10
6:
-2*z^6 - 6*z^4 - z^2 + 1
comment: $10_6$; $a_2=-1$, Arf invariant $1$, determinant $37$
10
7:
-3*z^4 - z^2 + 1
comment: $10_7$; $a_2=-1$, Arf invariant $1$, determinant $43$
10
8:
-2*z^6 - 7*z^4 - 3*z^2 + 1
comment: $10_8$; $a_2=-3$, Arf invariant $1$, determinant $29$
10
9:
-z^8 - 5*z^6 - 7*z^4 - 2*z^2 + 1
comment: $10_9$; $a_2=-2$, Arf invariant $0$, determinant $39$
10
10:
3*z^4 + z^2 + 1
comment: $10_{10}$; $a_2=1$, Arf invariant $1$, determinant $45$; the same polynomial as $10_{164}$
10
11:
-4*z^4 - 5*z^2 + 1
comment: $10_{11}$; $a_2=-5$, Arf invariant $1$, determinant $43$
10
12:
2*z^6 + 6*z^4 + 4*z^2 + 1
comment: $10_{12}$; $a_2=4$, Arf invariant $0$, determinant $47$; the same polynomial as $10_{54}$
10
13:
2*z^4 - 5*z^2 + 1
comment: $10_{13}$; $a_2=-5$, Arf invariant $1$, determinant $53$
10
14:
-2*z^6 - 4*z^4 + 2*z^2 + 1
comment: $10_{14}$; $a_2=2$, Arf invariant $0$, determinant $57$
10
15:
2*z^6 + 6*z^4 + 3*z^2 + 1
comment: $10_{15}$; $a_2=3$, Arf invariant $1$, determinant $43$
10
16:
-4*z^4 - 4*z^2 + 1
comment: $10_{16}$; $a_2=-4$, Arf invariant $0$, determinant $47$
10
17:
z^8 + 5*z^6 + 7*z^4 + 2*z^2 + 1
comment: $10_{17}$; $a_2=2$, Arf invariant $0$, determinant $41$
10
18:
-4*z^4 - 2*z^2 + 1
comment: $10_{18}$; $a_2=-2$, Arf invariant $0$, determinant $55$; the same polynomial as $10_{24}$
10
19:
2*z^6 + 5*z^4 + z^2 + 1
comment: $10_{19}$; $a_2=1$, Arf invariant $1$, determinant $51$
10
20:
-3*z^4 - 3*z^2 + 1
comment: $10_{20}$; $a_2=-3$, Arf invariant $1$, determinant $35$; the same polynomial as $10_{162}$
10
21:
-2*z^6 - 5*z^4 + z^2 + 1
comment: $10_{21}$; $a_2=1$, Arf invariant $1$, determinant $45$
10
22:
-2*z^6 - 6*z^4 - 4*z^2 + 1
comment: $10_{22}$; $a_2=-4$, Arf invariant $0$, determinant $49$
10
23:
2*z^6 + 5*z^4 + 3*z^2 + 1
comment: $10_{23}$; $a_2=3$, Arf invariant $1$, determinant $59$; the same polynomial as $10_{52}$
10
24:
-4*z^4 - 2*z^2 + 1
comment: $10_{24}$; $a_2=-2$, Arf invariant $0$, determinant $55$; the same polynomial as $10_{18}$
10
25:
-2*z^6 - 4*z^4 + 1
comment: $10_{25}$; $a_2=0$, Arf invariant $0$, determinant $65$; the same polynomial as $10_{56}$
10
26:
-2*z^6 - 5*z^4 - 3*z^2 + 1
comment: $10_{26}$; $a_2=-3$, Arf invariant $1$, determinant $61$
10
27:
2*z^6 + 4*z^4 + 2*z^2 + 1
comment: $10_{27}$; $a_2=2$, Arf invariant $0$, determinant $71$
10
28:
4*z^4 + 3*z^2 + 1
comment: $10_{28}$; $a_2=3$, Arf invariant $1$, determinant $53$; the same polynomial as $10_{37}$
10
29:
z^6 - z^4 - 4*z^2 + 1
comment: $10_{29}$; $a_2=-4$, Arf invariant $0$, determinant $63$
10
30:
-4*z^4 + z^2 + 1
comment: $10_{30}$; $a_2=1$, Arf invariant $1$, determinant $67$
10
31:
4*z^4 + 2*z^2 + 1
comment: $10_{31}$; $a_2=2$, Arf invariant $0$, determinant $57$; the same polynomial as $10_{68}$
10
32:
-2*z^6 - 4*z^4 - z^2 + 1
comment: $10_{32}$; $a_2=-1$, Arf invariant $1$, determinant $69$
10
33:
4*z^4 + 1
comment: $10_{33}$; $a_2=0$, Arf invariant $0$, determinant $65$
10
34:
3*z^4 + 3*z^2 + 1
comment: $10_{34}$; $a_2=3$, Arf invariant $1$, determinant $37$; the same polynomial as $10_{135}$
10
35:
2*z^4 - 4*z^2 + 1
comment: $10_{35}$; $a_2=-4$, Arf invariant $0$, determinant $49$
10
36:
-3*z^4 + z^2 + 1
comment: $10_{36}$; $a_2=1$, Arf invariant $1$, determinant $51$
10
37:
4*z^4 + 3*z^2 + 1
comment: $10_{37}$; $a_2=3$, Arf invariant $1$, determinant $53$; the same polynomial as $10_{28}$
10
38:
-4*z^4 - z^2 + 1
comment: $10_{38}$; $a_2=-1$, Arf invariant $1$, determinant $59$
10
39:
-2*z^6 - 4*z^4 + z^2 + 1
comment: $10_{39}$; $a_2=1$, Arf invariant $1$, determinant $61$
10
40:
2*z^6 + 4*z^4 + 3*z^2 + 1
comment: $10_{40}$; $a_2=3$, Arf invariant $1$, determinant $75$; the same polynomial as $10_{103}$
10
41:
z^6 - z^4 - 2*z^2 + 1
comment: $10_{41}$; $a_2=-2$, Arf invariant $0$, determinant $71$
10
42:
-z^6 + z^4 + 1
comment: $10_{42}$; $a_2=0$, Arf invariant $0$, determinant $81$; the same polynomial as $10_{75}$
10
43:
-z^6 + z^4 + 2*z^2 + 1
comment: $10_{43}$; $a_2=2$, Arf invariant $0$, determinant $73$
10
44:
z^6 - z^4 + 1
comment: $10_{44}$; $a_2=0$, Arf invariant $0$, determinant $79$
10
45:
-z^6 + z^4 - 2*z^2 + 1
comment: $10_{45}$; $a_2=-2$, Arf invariant $0$, determinant $89$
10
46:
-z^8 - 5*z^6 - 6*z^4 + 1
comment: $10_{46}$; $a_2=0$, Arf invariant $0$, determinant $31$
10
47:
z^8 + 5*z^6 + 8*z^4 + 6*z^2 + 1
comment: $10_{47}$; $a_2=6$, Arf invariant $0$, determinant $41$
10
48:
z^8 + 5*z^6 + 8*z^4 + 4*z^2 + 1
comment: $10_{48}$; $a_2=4$, Arf invariant $0$, determinant $49$
10
49:
3*z^6 + 10*z^4 + 7*z^2 + 1
comment: $10_{49}$; $a_2=7$, Arf invariant $1$, determinant $59$
10
50:
-2*z^6 - 5*z^4 - z^2 + 1
comment: $10_{50}$; $a_2=-1$, Arf invariant $1$, determinant $53$
10
51:
2*z^6 + 5*z^4 + 5*z^2 + 1
comment: $10_{51}$; $a_2=5$, Arf invariant $1$, determinant $67$
10
52:
2*z^6 + 5*z^4 + 3*z^2 + 1
comment: $10_{52}$; $a_2=3$, Arf invariant $1$, determinant $59$; the same polynomial as $10_{23}$
10
53:
6*z^4 + 6*z^2 + 1
comment: $10_{53}$; $a_2=6$, Arf invariant $0$, determinant $73$
10
54:
2*z^6 + 6*z^4 + 4*z^2 + 1
comment: $10_{54}$; $a_2=4$, Arf invariant $0$, determinant $47$; the same polynomial as $10_{12}$
10
55:
5*z^4 + 5*z^2 + 1
comment: $10_{55}$; $a_2=5$, Arf invariant $1$, determinant $61$
10
56:
-2*z^6 - 4*z^4 + 1
comment: $10_{56}$; $a_2=0$, Arf invariant $0$, determinant $65$; the same polynomial as $10_{25}$
10
57:
2*z^6 + 4*z^4 + 4*z^2 + 1
comment: $10_{57}$; $a_2=4$, Arf invariant $0$, determinant $79$
10
58:
3*z^4 - 4*z^2 + 1
comment: $10_{58}$; $a_2=-4$, Arf invariant $0$, determinant $65$
10
59:
z^6 - z^4 - z^2 + 1
comment: $10_{59}$; $a_2=-1$, Arf invariant $1$, determinant $75$; the same polynomial as $9_{40}$
10
60:
-z^6 + z^4 - z^2 + 1
comment: $10_{60}$; $a_2=-1$, Arf invariant $1$, determinant $85$
10
61:
-2*z^6 - 7*z^4 - 4*z^2 + 1
comment: $10_{61}$; $a_2=-4$, Arf invariant $0$, determinant $33$
10
62:
z^8 + 5*z^6 + 8*z^4 + 5*z^2 + 1
comment: $10_{62}$; $a_2=5$, Arf invariant $1$, determinant $45$
10
63:
5*z^4 + 6*z^2 + 1
comment: $10_{63}$; $a_2=6$, Arf invariant $0$, determinant $57$; the same polynomial as $9_{38}$
10
64:
-z^8 - 5*z^6 - 8*z^4 - 3*z^2 + 1
comment: $10_{64}$; $a_2=-3$, Arf invariant $1$, determinant $51$
10
65:
2*z^6 + 5*z^4 + 4*z^2 + 1
comment: $10_{65}$; $a_2=4$, Arf invariant $0$, determinant $63$; the same polynomial as $10_{77}$
10
66:
3*z^6 + 9*z^4 + 7*z^2 + 1
comment: $10_{66}$; $a_2=7$, Arf invariant $1$, determinant $75$
10
67:
-4*z^4 + 1
comment: $10_{67}$; $a_2=0$, Arf invariant $0$, determinant $63$; the same polynomial as $10_{74}$
10
68:
4*z^4 + 2*z^2 + 1
comment: $10_{68}$; $a_2=2$, Arf invariant $0$, determinant $57$; the same polynomial as $10_{31}$
10
69:
z^6 - z^4 + 2*z^2 + 1
comment: $10_{69}$; $a_2=2$, Arf invariant $0$, determinant $87$
10
70:
z^6 - z^4 - 3*z^2 + 1
comment: $10_{70}$; $a_2=-3$, Arf invariant $1$, determinant $67$
10
71:
-z^6 + z^4 + z^2 + 1
comment: $10_{71}$; $a_2=1$, Arf invariant $1$, determinant $77$
10
72:
-2*z^6 - 3*z^4 + 2*z^2 + 1
comment: $10_{72}$; $a_2=2$, Arf invariant $0$, determinant $73$
10
73:
z^6 - z^4 + z^2 + 1
comment: $10_{73}$; $a_2=1$, Arf invariant $1$, determinant $83$
10
74:
-4*z^4 + 1
comment: $10_{74}$; $a_2=0$, Arf invariant $0$, determinant $63$; the same polynomial as $10_{67}$
10
75:
-z^6 + z^4 + 1
comment: $10_{75}$; $a_2=0$, Arf invariant $0$, determinant $81$; the same polynomial as $10_{42}$
10
76:
-2*z^6 - 5*z^4 - 2*z^2 + 1
comment: $10_{76}$; $a_2=-2$, Arf invariant $0$, determinant $57$
10
77:
2*z^6 + 5*z^4 + 4*z^2 + 1
comment: $10_{77}$; $a_2=4$, Arf invariant $0$, determinant $63$; the same polynomial as $10_{65}$
10
78:
-z^6 + z^4 + 3*z^2 + 1
comment: $10_{78}$; $a_2=3$, Arf invariant $1$, determinant $69$
10
79:
z^8 + 5*z^6 + 9*z^4 + 5*z^2 + 1
comment: $10_{79}$; $a_2=5$, Arf invariant $1$, determinant $61$
10
80:
3*z^6 + 9*z^4 + 6*z^2 + 1
comment: $10_{80}$; $a_2=6$, Arf invariant $0$, determinant $71$
10
81:
-z^6 + 2*z^4 + 3*z^2 + 1
comment: $10_{81}$; $a_2=3$, Arf invariant $1$, determinant $85$
10
82:
-z^8 - 4*z^6 - 4*z^4 + 1
comment: $10_{82}$; $a_2=0$, Arf invariant $0$, determinant $63$
10
83:
2*z^6 + 3*z^4 + z^2 + 1
comment: $10_{83}$; $a_2=1$, Arf invariant $1$, determinant $83$
10
84:
2*z^6 + 3*z^4 + 2*z^2 + 1
comment: $10_{84}$; $a_2=2$, Arf invariant $0$, determinant $87$
10
85:
z^8 + 4*z^6 + 4*z^4 + 2*z^2 + 1
comment: $10_{85}$; $a_2=2$, Arf invariant $0$, determinant $57$
10
86:
-2*z^6 - 3*z^4 - z^2 + 1
comment: $10_{86}$; $a_2=-1$, Arf invariant $1$, determinant $85$
10
87:
-2*z^6 - 3*z^4 + 1
comment: $10_{87}$; $a_2=0$, Arf invariant $0$, determinant $81$; the same polynomial as $10_{98}$
10
88:
-z^6 + 2*z^4 - z^2 + 1
comment: $10_{88}$; $a_2=-1$, Arf invariant $1$, determinant $101$
10
89:
z^6 - 2*z^4 + z^2 + 1
comment: $10_{89}$; $a_2=1$, Arf invariant $1$, determinant $99$
10
90:
-2*z^6 - 4*z^4 - 3*z^2 + 1
comment: $10_{90}$; $a_2=-3$, Arf invariant $1$, determinant $77$
10
91:
z^8 + 4*z^6 + 5*z^4 + 2*z^2 + 1
comment: $10_{91}$; $a_2=2$, Arf invariant $0$, determinant $73$
10
92:
-2*z^6 - 2*z^4 + 2*z^2 + 1
comment: $10_{92}$; $a_2=2$, Arf invariant $0$, determinant $89$
10
93:
2*z^6 + 4*z^4 + z^2 + 1
comment: $10_{93}$; $a_2=1$, Arf invariant $1$, determinant $67$
10
94:
-z^8 - 4*z^6 - 5*z^4 - 2*z^2 + 1
comment: $10_{94}$; $a_2=-2$, Arf invariant $0$, determinant $71$
10
95:
2*z^6 + 3*z^4 + 3*z^2 + 1
comment: $10_{95}$; $a_2=3$, Arf invariant $1$, determinant $91$
10
96:
-z^6 + z^4 - 3*z^2 + 1
comment: $10_{96}$; $a_2=-3$, Arf invariant $1$, determinant $93$
10
97:
-5*z^4 + 2*z^2 + 1
comment: $10_{97}$; $a_2=2$, Arf invariant $0$, determinant $87$
10
98:
-2*z^6 - 3*z^4 + 1
comment: $10_{98}$; $a_2=0$, Arf invariant $0$, determinant $81$; the same polynomial as $10_{87}$
10
99:
z^8 + 4*z^6 + 6*z^4 + 4*z^2 + 1
comment: $10_{99}$; $a_2=4$, Arf invariant $0$, determinant $81$
10
100:
z^8 + 4*z^6 + 5*z^4 + 4*z^2 + 1
comment: $10_{100}$; $a_2=4$, Arf invariant $0$, determinant $65$
10
101:
7*z^4 + 7*z^2 + 1
comment: $10_{101}$; $a_2=7$, Arf invariant $1$, determinant $85$
10
102:
-2*z^6 - 4*z^4 - 2*z^2 + 1
comment: $10_{102}$; $a_2=-2$, Arf invariant $0$, determinant $73$
10
103:
2*z^6 + 4*z^4 + 3*z^2 + 1
comment: $10_{103}$; $a_2=3$, Arf invariant $1$, determinant $75$; the same polynomial as $10_{40}$
10
104:
z^8 + 4*z^6 + 5*z^4 + z^2 + 1
comment: $10_{104}$; $a_2=1$, Arf invariant $1$, determinant $77$
10
105:
z^6 - 2*z^4 - z^2 + 1
comment: $10_{105}$; $a_2=-1$, Arf invariant $1$, determinant $91$
10
106:
-z^8 - 4*z^6 - 5*z^4 - z^2 + 1
comment: $10_{106}$; $a_2=-1$, Arf invariant $1$, determinant $75$
10
107:
-z^6 + 2*z^4 + z^2 + 1
comment: $10_{107}$; $a_2=1$, Arf invariant $1$, determinant $93$
10
108:
2*z^6 + 4*z^4 + 1
comment: $10_{108}$; $a_2=0$, Arf invariant $0$, determinant $63$
10
109:
z^8 + 4*z^6 + 6*z^4 + 3*z^2 + 1
comment: $10_{109}$; $a_2=3$, Arf invariant $1$, determinant $85$
10
110:
z^6 - 2*z^4 - 3*z^2 + 1
comment: $10_{110}$; $a_2=-3$, Arf invariant $1$, determinant $83$
10
111:
-2*z^6 - 3*z^4 + z^2 + 1
comment: $10_{111}$; $a_2=1$, Arf invariant $1$, determinant $77$
10
112:
-z^8 - 3*z^6 - z^4 + 2*z^2 + 1
comment: $10_{112}$; $a_2=2$, Arf invariant $0$, determinant $87$
10
113:
2*z^6 + z^4 + 1
comment: $10_{113}$; $a_2=0$, Arf invariant $0$, determinant $111$
10
114:
-2*z^6 - 2*z^4 + z^2 + 1
comment: $10_{114}$; $a_2=1$, Arf invariant $1$, determinant $93$
10
115:
-z^6 + 3*z^4 + z^2 + 1
comment: $10_{115}$; $a_2=1$, Arf invariant $1$, determinant $109$
10
116:
-z^8 - 3*z^6 - 2*z^4 + 1
comment: $10_{116}$; $a_2=0$, Arf invariant $0$, determinant $95$
10
117:
2*z^6 + 2*z^4 + 2*z^2 + 1
comment: $10_{117}$; $a_2=2$, Arf invariant $0$, determinant $103$
10
118:
z^8 + 3*z^6 + 2*z^4 + 1
comment: $10_{118}$; $a_2=0$, Arf invariant $0$, determinant $97$
10
119:
-2*z^6 - 2*z^4 - z^2 + 1
comment: $10_{119}$; $a_2=-1$, Arf invariant $1$, determinant $101$
10
120:
8*z^4 + 6*z^2 + 1
comment: $10_{120}$; $a_2=6$, Arf invariant $0$, determinant $105$
10
121:
2*z^6 + z^4 + z^2 + 1
comment: $10_{121}$; $a_2=1$, Arf invariant $1$, determinant $115$
10
122:
-2*z^6 - z^4 + 2*z^2 + 1
comment: $10_{122}$; $a_2=2$, Arf invariant $0$, determinant $105$
10
123:
z^8 + 2*z^6 - z^4 - 2*z^2 + 1
comment: $10_{123}$; $a_2=-2$, Arf invariant $0$, determinant $121$
10
124:
z^8 + 7*z^6 + 14*z^4 + 8*z^2 + 1
comment: $10_{124}$; the torus knot $T(3,5)$; $a_2=8$, Arf invariant $0$, determinant $1$
10
125:
z^6 + 4*z^4 + 3*z^2 + 1
comment: $10_{125}$; $a_2=3$, Arf invariant $1$, determinant $11$
10
126:
z^6 + 4*z^4 + 5*z^2 + 1
comment: $10_{126}$; $a_2=5$, Arf invariant $1$, determinant $19$
10
127:
-z^6 - 2*z^4 + z^2 + 1
comment: $10_{127}$; $a_2=1$, Arf invariant $1$, determinant $29$; the same polynomial as $10_{150}$
10
128:
2*z^6 + 9*z^4 + 7*z^2 + 1
comment: $10_{128}$; $a_2=7$, Arf invariant $1$, determinant $11$
10
129:
2*z^4 + 2*z^2 + 1
comment: $10_{129}$; $a_2=2$, Arf invariant $0$, determinant $25$; the same polynomial as $8_8$
10
130:
2*z^4 + 4*z^2 + 1
comment: $10_{130}$; $a_2=4$, Arf invariant $0$, determinant $17$; the same polynomial as $7_5$
10
131:
-2*z^4 + 1
comment: $10_{131}$; $a_2=0$, Arf invariant $0$, determinant $31$; the same polynomial as $8_{14}$, $9_8$
10
132:
z^4 + 3*z^2 + 1
comment: $10_{132}$; $\nabla=F_{5}$; $a_2=3$, Arf invariant $1$, determinant $5$; the same polynomial as $5_1$
equals: $F_{5}$
10
133:
-z^4 + z^2 + 1
comment: $10_{133}$; $a_2=1$, Arf invariant $1$, determinant $19$; the same polynomial as $7_6$
10
134:
2*z^6 + 8*z^4 + 6*z^2 + 1
comment: $10_{134}$; $a_2=6$, Arf invariant $0$, determinant $23$
10
135:
3*z^4 + 3*z^2 + 1
comment: $10_{135}$; $a_2=3$, Arf invariant $1$, determinant $37$; the same polynomial as $10_{34}$
10
136:
-z^4 + 1
comment: $10_{136}$; $a_2=0$, Arf invariant $0$, determinant $15$; the same polynomial as $8_{21}$
10
137:
z^4 - 2*z^2 + 1
comment: $10_{137}$; $a_2=-2$, Arf invariant $0$, determinant $25$
10
138:
z^6 + z^4 - 3*z^2 + 1
comment: $10_{138}$; $a_2=-3$, Arf invariant $1$, determinant $35$
10
139:
z^8 + 7*z^6 + 14*z^4 + 9*z^2 + 1
comment: $10_{139}$; $a_2=9$, Arf invariant $1$, determinant $3$
10
140:
z^4 + 2*z^2 + 1
comment: $10_{140}$; $a_2=2$, Arf invariant $0$, determinant $9$; the same polynomial as $8_{20}$
10
141:
-z^6 - 3*z^4 - z^2 + 1
comment: $10_{141}$; $a_2=-1$, Arf invariant $1$, determinant $21$; the same polynomial as $8_5$
10
142:
2*z^6 + 9*z^4 + 8*z^2 + 1
comment: $10_{142}$; $a_2=8$, Arf invariant $0$, determinant $15$
10
143:
z^6 + 3*z^4 + 3*z^2 + 1
comment: $10_{143}$; $a_2=3$, Arf invariant $1$, determinant $27$; the same polynomial as $8_{10}$
10
144:
-3*z^4 - 2*z^2 + 1
comment: $10_{144}$; $a_2=-2$, Arf invariant $0$, determinant $39$
10
145:
z^4 + 5*z^2 + 1
comment: $10_{145}$; $a_2=5$, Arf invariant $1$, determinant $3$
10
146:
2*z^4 + 1
comment: $10_{146}$; $a_2=0$, Arf invariant $0$, determinant $33$
10
147:
-2*z^4 - z^2 + 1
comment: $10_{147}$; $a_2=-1$, Arf invariant $1$, determinant $27$; the same polynomial as $8_{11}$
10
148:
z^6 + 3*z^4 + 4*z^2 + 1
comment: $10_{148}$; $a_2=4$, Arf invariant $0$, determinant $31$
10
149:
-z^6 - z^4 + 2*z^2 + 1
comment: $10_{149}$; $a_2=2$, Arf invariant $0$, determinant $41$; the same polynomial as $9_{20}$
10
150:
-z^6 - 2*z^4 + z^2 + 1
comment: $10_{150}$; $a_2=1$, Arf invariant $1$, determinant $29$; the same polynomial as $10_{127}$
10
151:
z^6 + 2*z^4 + 3*z^2 + 1
comment: $10_{151}$; $a_2=3$, Arf invariant $1$, determinant $43$
10
152:
z^8 + 7*z^6 + 13*z^4 + 7*z^2 + 1
comment: $10_{152}$; $a_2=7$, Arf invariant $1$, determinant $11$
10
153:
z^6 + 5*z^4 + 4*z^2 + 1
comment: $10_{153}$; $a_2=4$, Arf invariant $0$, determinant $1$
10
154:
z^6 + 6*z^4 + 5*z^2 + 1
comment: $10_{154}$; $a_2=5$, Arf invariant $1$, determinant $13$
10
155:
-z^6 - 3*z^4 - 2*z^2 + 1
comment: $10_{155}$; $a_2=-2$, Arf invariant $0$, determinant $25$; the same polynomial as $8_9$
10
156:
z^6 + 2*z^4 + z^2 + 1
comment: $10_{156}$; $a_2=1$, Arf invariant $1$, determinant $35$; the same polynomial as $8_{16}$
10
157:
-z^6 + 4*z^2 + 1
comment: $10_{157}$; $a_2=4$, Arf invariant $0$, determinant $49$
10
158:
-z^6 - 2*z^4 - 3*z^2 + 1
comment: $10_{158}$; $a_2=-3$, Arf invariant $1$, determinant $45$
10
159:
z^6 + 2*z^4 + 2*z^2 + 1
comment: $10_{159}$; $a_2=2$, Arf invariant $0$, determinant $39$
10
160:
-z^6 - 2*z^4 + 3*z^2 + 1
comment: $10_{160}$; $a_2=3$, Arf invariant $1$, determinant $21$
10
161:
z^6 + 6*z^4 + 7*z^2 + 1
comment: $10_{161}$, the Perko pair, listed twice by Rolfsen as $10_{161}$ and $10_{162}$; $a_2=7$, Arf invariant $1$, determinant $5$
10
162:
-3*z^4 - 3*z^2 + 1
comment: $10_{162}$; $a_2=-3$, Arf invariant $1$, determinant $35$; the same polynomial as $10_{20}$
10
163:
z^6 + z^4 + z^2 + 1
comment: $10_{163}$; $a_2=1$, Arf invariant $1$, determinant $51$; the same polynomial as $9_{28}$, $9_{29}$
10
164:
3*z^4 + z^2 + 1
comment: $10_{164}$; $a_2=1$, Arf invariant $1$, determinant $45$; the same polynomial as $10_{10}$
10
165:
-2*z^4 + 2*z^2 + 1
comment: $10_{165}$; $a_2=2$, Arf invariant $0$, determinant $39$; the same polynomial as $9_{15}$
Definition
The Conway polynomial $\nabla_K(z)$ of a knot $K$ [8], defined by $\nabla(0_1)=1$ and the skein relation $\nabla(L_+)-\nabla(L_-)=z\,\nabla(L_0)$, for the unknot $0_1$ and every prime knot $n_k$ with at most ten crossings, named as in the Rolfsen table [2] with Perko's correction [3].
Parameters
$n$
—   crossing number ($n=0$ or $3\leq n\leq 10$)
$k$
—   index in the Rolfsen table ($1\leq k\leq N(n)$, where $N(n)$ is the number of knots listed with $n$ crossings, $1,1,1,2,3,7,21,49,165$ for $n=0,3,4,\ldots,10$)
Formulas
(1)
$\nabla(0_1)=1$ and $\nabla(L_+)-\nabla(L_-)=z\,\nabla(L_0)$ for three oriented link diagrams $L_+$, $L_-$ and $L_0$ that agree except at one crossing, which is positive in $L_+$, negative in $L_-$ and smoothed in $L_0$ [1] [8]. For a knot, $\nabla$ has only even powers of $z$ and constant term $1$; for a split link, $\nabla=0$.
(2)
$\Delta_K(t^2)=\epsilon\,t^{2d}\,\nabla_K(t-t^{-1})$, where $\Delta_K(t)\in\mathbb{Z}[t]$ is the Alexander polynomial with positive constant term, $2d=\deg\Delta_K=\deg\nabla_K$, and $\epsilon=\Delta_K(1)=\pm1$ is the sign of the leading coefficient of $\nabla_K$ [8]. Equivalently, $\nabla_K(t^{1/2}-t^{-1/2})$ is the Conway-normalised Alexander polynomial $\epsilon\,t^{-d}\Delta_K(t)$, which is symmetric under $t\mapsto t^{-1}$ and equal to $1$ at $t=1$.
(3)
$|\nabla_K(2i)|=\det K=|\Delta_K(-1)|$, the determinant of the knot, from (2) at $t=i$; since $\nabla_K$ is even, $\nabla_K(2i)=\sum_j a_{2j}(-4)^j$, where $a_{2j}$ is the coefficient of $z^{2j}$, and it is an odd integer.
(4)
$\operatorname{Arf}(K)\equiv a_2(K)\pmod 2$ for the coefficient $a_2$ of $z^2$ [4] [9], and $\operatorname{Arf}(K)=0$ exactly when $\det K\equiv\pm1\pmod 8$ [5] [10].
(5)
$V_K''(1)=-6\,a_2(K)$ [7] and $V_K(i)=(-1)^{\operatorname{Arf}(K)}$ [6] for the Jones polynomial $V_K(t)$, and $|V_K(-1)|=\det K=|\nabla_K(2i)|$ [14].
(6)
$\nabla_{T(2,q)}(z)=F_q(z)$ for the torus knot $T(2,q)$, $q$ odd, where $F_q$ is the Fibonacci polynomial, $F_0=0$, $F_1=1$, $F_q=zF_{q-1}+F_{q-2}$: the Alexander polynomial is $\Delta_{T(2,q)}(t)=(t^q+1)/(t+1)$ [11], and $F_q(z)=(\alpha^q-\beta^q)/(\alpha-\beta)$ for the roots $\alpha=t^{1/2}$, $\beta=-t^{-1/2}$ of $y^2-zy-1$ at $z=t^{1/2}-t^{-1/2}$ gives $F_q(t^{1/2}-t^{-1/2})=t^{-(q-1)/2}(t^q+1)/(t+1)$, the Conway-normalised form of (2). The torus knots here are $3_1=T(2,3)$, $5_1=T(2,5)$, $7_1=T(2,7)$, $9_1=T(2,9)$, $8_{19}=T(3,4)$ and $10_{124}=T(3,5)$, the last two with $\nabla=z^6+5z^4+5z^2+1$ and $z^8+7z^6+14z^4+8z^2+1$.
(7)
$\nabla_{K_1\#K_2}=\nabla_{K_1}\nabla_{K_2}$ for the connected sum, and $\nabla_{\bar K}(z)=\nabla_K(-z)$ for the mirror image $\bar K$, which for a knot is $\nabla_K(z)$ again [8]; so the granny knot $3_1\#3_1$ and the square knot $3_1\#\bar 3_1$ both have $\nabla=(z^2+1)^2$, the value of $8_{20}$ and $10_{140}$.
Comments
(8)
The Conway polynomial is unique: where the Alexander polynomial $\Delta_K$ is defined only up to a unit, $\nabla_K$ is fixed by the skein relation (1) with Conway's sign [1], the one used by Wikipedia [8], MathWorld [13], KnotInfo [16] and Sage [17], under which the positive Hopf link has $\nabla=z$. For a knot, $\nabla$ has only even powers of $z$ and constant term $1$. It does not depend on the orientation of the knot, and since the mirror image $\bar K$ has $\nabla_{\bar K}(z)=\nabla_K(-z)$ it does not depend on the choice between a knot and its mirror image either, so it is well defined for the name $n_k$, which names a knot only up to mirror image. The variable is Conway's $z$; Sage's conway_polynomial() prints the same coefficients in $t$.
(9)
$\nabla_K$ is a reparametrisation of the Alexander polynomial, $\Delta_K(t^2)=\epsilon\,t^{2d}\,\nabla_K(t-t^{-1})$ (2), so the two determine each other and the two tables hold the same information in different forms: $\nabla(5_1)=z^4+3z^2+1$ where $\Delta(5_1)=t^4-t^3+t^2-t+1$. $\nabla$ is the form the skein relation computes directly, and its coefficients are invariants in their own right: the coefficient $a_2$ of $z^2$ is the Casson invariant of the knot, the Vassiliev invariant of order two [9], and $a_2$ modulo $2$ is the Arf invariant [4] [10]. For every knot here the degree of $\nabla$ is twice the genus, and $a_2$ takes every integer value from $-6$ ($10_3$) to $10$ ($9_1$); 127 of the 249 prime knots have Arf invariant $1$. Each entry's comment gives the knot's common name where it has one, the torus knot it is if it is one, $a_2$, the Arf invariant, the determinant $|\nabla(2i)|$, and every other knot in the table with the same polynomial.
(10)
The torus knot $T(2,q)$ has $\nabla=F_q(z)$, the Fibonacci polynomial (6): $3_1$, $5_1$, $7_1$ and $9_1$ are $F_3$, $F_5$, $F_7$ and $F_9$, and $10_{132}$ shares $F_5$ with $5_1$. Three entries are cyclotomic polynomials: $\nabla(3_1)=\Phi_4$, $\nabla(7_7)=\Phi_{12}$ and $\nabla(9_{44})=\Phi_8$.
(11)
The numbering is Rolfsen's after Perko's correction [3]: the knots Rolfsen listed as $10_{161}$ and $10_{162}$ are one knot, so there are $165$ prime knots with ten crossings rather than $166$ [18], and Rolfsen's $10_{163}$ to $10_{166}$ are $10_{162}$ to $10_{165}$ here [12]. Sage's knot table, the Knot Atlas [15] and KnotInfo all use this numbering.
(12)
Since $\nabla$ determines $\Delta$, the knots that share a Conway polynomial are the knots that share an Alexander polynomial: the 249 prime knots here take 211 distinct values, and 36 of those values are shared by two or three knots each, 74 knots in all, among them $\nabla(5_1)=\nabla(10_{132})=F_5$, $\nabla(6_1)=\nabla(9_{46})=1-2z^2$, $\nabla(8_{20})=\nabla(10_{140})=(z^2+1)^2$ and the triples $8_{14},9_8,10_{131}$ and $9_{28},9_{29},10_{163}$. No knot here has $\nabla=1$ apart from the unknot; the first that do, the Conway knot $11n_{34}$ and the Kinoshita–Terasaka knot $11n_{42}$, have eleven crossings [8].
(13)
The Rolfsen table ends at ten crossings, and so do the names $n_k$. The 552 prime knots with eleven crossings are named $11a_1$ to $11a_{367}$ and $11n_1$ to $11n_{185}$ after Hoste and Thistlethwaite, alternating and non-alternating separately [16], which the two parameters here cannot express; KnotInfo carries the Conway polynomials of all prime knots to thirteen crossings. The Alexander polynomials of the knots listed here are in their own table, and so are the Jones polynomials.
Programs
(P1)
Sage
K = Knots().from_table(10, 132)
K.conway_polynomial()               # t^4 + 3*t^2 + 1, the same coefficients printed in t
R.<z> = ZZ[]
R(K.conway_polynomial().list())     # z^4 + 3*z^2 + 1, the entry listed here
References
[1]
J. H. Conway, An enumeration of knots and links, and some of their algebraic properties, in: Computational Problems in Abstract Algebra, Pergamon, 1970, 329–358.
[2]
D. Rolfsen, Knots and Links, Publish or Perish, 1976, Appendix C: Table of knots and links.
[3]
K. A. Perko, On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262–266.
[4]
L. H. Kauffman, On Knots, Annals of Mathematics Studies 115, Princeton University Press, 1987.
[5]
K. Murasugi, The Arf invariant for knot types, Proc. Amer. Math. Soc. 21 (1969), 69–72.
[6]
V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bull. Amer. Math. Soc. 12 (1985), 103–111.
[7]
H. Murakami, On derivatives of the Jones polynomial, Kobe J. Math. 3 (1986), 61–64.
Links
Similar tables
Alexander polynomials of the prime knots with at most ten crossings —   the same knots; $\Delta_K(t^2)=\epsilon\,t^{2d}\,\nabla_K(t-t^{-1})$, so each table determines the other
Jones polynomials of the prime knots with at most ten crossings —   the same knots; $V_K''(1)=-6a_2(K)$ and $V_K(i)=(-1)^{\operatorname{Arf}(K)}$
Fibonacci polynomials $F_n$ —   the Conway polynomial of the torus knot $T(2,q)$ is $F_q$; $3_1$, $5_1$, $7_1$, $9_1$ and $10_{132}$ are $F_3$, $F_5$, $F_7$, $F_9$ and $F_5$
Cyclotomic polynomials —   $\nabla(3_1)=\Phi_4$, $\nabla(7_7)=\Phi_{12}$ and $\nabla(9_{44})=\Phi_8$
Data properties
Entries are of type: integral polynomial
How well the digits are known: exact
Table is complete: yes
How they were obtained:

Each polynomial is Sage's conway_polynomial() of Knots().from_table(n, k), which substitutes into the determinant $\det(V-tV^{\mathsf T})$ of a Seifert matrix of the braid closure.

more

The generator requires it to agree with the polynomial obtained from $\det(I-\psi(\beta))(1-t)/(1-t^r)$, computed from the reduced Burau matrix $\psi(\beta)$ of the same braid word on $r$ strands, with the matrix arithmetic and the determinant written out rather than taken from a library, and converted to $\nabla$ by writing the symmetric Laurent polynomial $\epsilon t^{-d}\Delta(t)$ as a polynomial in $t+t^{-1}=z^2+2$; to have only even powers and constant term $1$; to satisfy $|\nabla(2i)|=\det K$; to have $a_2$ modulo $2$ equal to the Arf invariant both as Robertello's sum of Alexander coefficients and by Murasugi's criterion; and to equal $F_q$ for the torus knots $T(2,q)$ and $10_{132}$, and $\Phi_4$, $\Phi_{12}$, $\Phi_8$ for $3_1$, $7_7$, $9_{44}$. Outside the generator all 250 values were compared with the conway_polynomial column of KnotInfo (package database_knotinfo 2026.9.1, computed from KnotInfo's own diagrams) and the Arf invariants with its arf_invariant column; the stored Alexander polynomials satisfy $\Delta(s^2)=\epsilon s^{2d}\nabla(s-s^{-1})$ and the stored Jones polynomials $V''(1)=-6a_2$, $V(i)=(-1)^{\mathrm{Arf}}$ and $|V(-1)|=|\nabla(2i)|$ on every knot; the linked Fibonacci and cyclotomic entries hold the same polynomials, and no other entry of those two tables coincides with a value here. All agree.



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