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Conway polynomials of the prime knots
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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 prime knot $K$ [8], defined by $\nabla(0_1)=1$ and the skein relation $\nabla(L_+)-\nabla(L_-)=z\,\nabla(L_0)$. The unknot $0_1$ is included as a reference case.
Parameters
$n$
—   crossing number ($n=0$ or $n\geq3$)
$k$
—   knot index ($1\leq k\leq N(n)$ for $n\geq3$, where $N(n)$ counts prime knots with $n$ crossings up to mirror image; $(n,k)=(0,1)$ denotes the unknot. The current entries use Rolfsen's index after Perko's correction)
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 existing entries retain Rolfsen's knot indices with Perko's correction [2] [3]. This catalogue covers prime knots through ten crossings. Higher-crossing catalogues use names that distinguish alternating and non-alternating knots, such as $11a_j$ and $11n_j$. Before adding such knots, an unambiguous extension of the $(n,k)$ indexing must be documented without changing any existing entry identifier. The present numbering convention limits the current implementation, not the mathematical family.
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 —   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 —   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: no (it currently holds the unknot and all 249 prime knots with at most ten crossings, in Rolfsen numbering after Perko's correction, with mirror images identified; these give 250 entries. The crossing-number bound describes the current coverage, not the definition of the family)
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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