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Alexander polynomials of the prime knots with at most ten crossings
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Polynomials
$n$
$k$ 
$\Delta_{n_k}(t)$
0
1:
1
comment: $0_1$, the unknot; genus $0$; the first nontrivial knots with $\Delta=1$ are $11n_{34}$, the Conway knot, and $11n_{42}$, the Kinoshita–Terasaka knot
equals: One
3
1:
t^2 - t + 1
comment: $3_1$, the trefoil; the torus knot $T(2,3)$; determinant $3$, genus $1$, alternating
equals: $\Phi_{6}$
4
1:
t^2 - 3*t + 1
comment: $4_1$, the figure-eight knot; determinant $5$, genus $1$, alternating
5
1:
t^4 - t^3 + t^2 - t + 1
comment: $5_1$, the cinquefoil; the torus knot $T(2,5)$; determinant $5$, genus $2$, alternating; the same polynomial as $10_{132}$
equals: $\Phi_{10}$
5
2:
2*t^2 - 3*t + 2
comment: $5_2$, the three-twist knot; determinant $7$, genus $1$, alternating
6
1:
2*t^2 - 5*t + 2
comment: $6_1$, the stevedore knot; determinant $9$, genus $1$, alternating; the same polynomial as $9_{46}$
6
2:
t^4 - 3*t^3 + 3*t^2 - 3*t + 1
comment: $6_2$, the Miller Institute knot; determinant $11$, genus $2$, alternating
6
3:
t^4 - 3*t^3 + 5*t^2 - 3*t + 1
comment: $6_3$; determinant $13$, genus $2$, alternating
7
1:
t^6 - t^5 + t^4 - t^3 + t^2 - t + 1
comment: $7_1$; the torus knot $T(2,7)$; determinant $7$, genus $3$, alternating
equals: $\Phi_{14}$
7
2:
3*t^2 - 5*t + 3
comment: $7_2$; determinant $11$, genus $1$, alternating
7
3:
2*t^4 - 3*t^3 + 3*t^2 - 3*t + 2
comment: $7_3$; determinant $13$, genus $2$, alternating
7
4:
4*t^2 - 7*t + 4
comment: $7_4$, the endless knot; determinant $15$, genus $1$, alternating; the same polynomial as $9_2$
7
5:
2*t^4 - 4*t^3 + 5*t^2 - 4*t + 2
comment: $7_5$; determinant $17$, genus $2$, alternating; the same polynomial as $10_{130}$
7
6:
t^4 - 5*t^3 + 7*t^2 - 5*t + 1
comment: $7_6$; determinant $19$, genus $2$, alternating; the same polynomial as $10_{133}$
7
7:
t^4 - 5*t^3 + 9*t^2 - 5*t + 1
comment: $7_7$; determinant $21$, genus $2$, alternating
8
1:
3*t^2 - 7*t + 3
comment: $8_1$; determinant $13$, genus $1$, alternating
8
2:
t^6 - 3*t^5 + 3*t^4 - 3*t^3 + 3*t^2 - 3*t + 1
comment: $8_2$; determinant $17$, genus $3$, alternating
8
3:
4*t^2 - 9*t + 4
comment: $8_3$; determinant $17$, genus $1$, alternating; the same polynomial as $10_1$
8
4:
2*t^4 - 5*t^3 + 5*t^2 - 5*t + 2
comment: $8_4$; determinant $19$, genus $2$, alternating
8
5:
t^6 - 3*t^5 + 4*t^4 - 5*t^3 + 4*t^2 - 3*t + 1
comment: $8_5$; determinant $21$, genus $3$, alternating; the same polynomial as $10_{141}$
8
6:
2*t^4 - 6*t^3 + 7*t^2 - 6*t + 2
comment: $8_6$; determinant $23$, genus $2$, alternating
8
7:
t^6 - 3*t^5 + 5*t^4 - 5*t^3 + 5*t^2 - 3*t + 1
comment: $8_7$; determinant $23$, genus $3$, alternating
8
8:
2*t^4 - 6*t^3 + 9*t^2 - 6*t + 2
comment: $8_8$; determinant $25$, genus $2$, alternating; the same polynomial as $10_{129}$
8
9:
t^6 - 3*t^5 + 5*t^4 - 7*t^3 + 5*t^2 - 3*t + 1
comment: $8_9$; determinant $25$, genus $3$, alternating; the same polynomial as $10_{155}$
8
10:
t^6 - 3*t^5 + 6*t^4 - 7*t^3 + 6*t^2 - 3*t + 1
comment: $8_{10}$; determinant $27$, genus $3$, alternating; the same polynomial as $10_{143}$
8
11:
2*t^4 - 7*t^3 + 9*t^2 - 7*t + 2
comment: $8_{11}$; determinant $27$, genus $2$, alternating; the same polynomial as $10_{147}$
8
12:
t^4 - 7*t^3 + 13*t^2 - 7*t + 1
comment: $8_{12}$; determinant $29$, genus $2$, alternating
8
13:
2*t^4 - 7*t^3 + 11*t^2 - 7*t + 2
comment: $8_{13}$; determinant $29$, genus $2$, alternating
8
14:
2*t^4 - 8*t^3 + 11*t^2 - 8*t + 2
comment: $8_{14}$; determinant $31$, genus $2$, alternating; the same polynomial as $9_8$, $10_{131}$
8
15:
3*t^4 - 8*t^3 + 11*t^2 - 8*t + 3
comment: $8_{15}$; determinant $33$, genus $2$, alternating
8
16:
t^6 - 4*t^5 + 8*t^4 - 9*t^3 + 8*t^2 - 4*t + 1
comment: $8_{16}$; determinant $35$, genus $3$, alternating; the same polynomial as $10_{156}$
8
17:
t^6 - 4*t^5 + 8*t^4 - 11*t^3 + 8*t^2 - 4*t + 1
comment: $8_{17}$; determinant $37$, genus $3$, alternating
8
18:
t^6 - 5*t^5 + 10*t^4 - 13*t^3 + 10*t^2 - 5*t + 1
comment: $8_{18}$, the Carrick mat; determinant $45$, genus $3$, alternating; the same polynomial as $9_{24}$
8
19:
t^6 - t^5 + t^3 - t + 1
comment: $8_{19}$; the torus knot $T(3,4)$; determinant $3$, genus $3$, non-alternating; $\Delta=\Phi_6\Phi_{12}$
8
20:
t^4 - 2*t^3 + 3*t^2 - 2*t + 1
comment: $8_{20}$; determinant $9$, genus $2$, non-alternating; $\Delta=\Phi_6^2$, also the Alexander polynomial of the granny and square knots $3_1\#3_1$ and $3_1\#\bar{3}_1$; the same polynomial as $10_{140}$
8
21:
t^4 - 4*t^3 + 5*t^2 - 4*t + 1
comment: $8_{21}$; determinant $15$, genus $2$, non-alternating; the same polynomial as $10_{136}$
9
1:
t^8 - t^7 + t^6 - t^5 + t^4 - t^3 + t^2 - t + 1
comment: $9_1$; the torus knot $T(2,9)$; determinant $9$, genus $4$, alternating; $\Delta=\Phi_6\Phi_{18}$
9
2:
4*t^2 - 7*t + 4
comment: $9_2$; determinant $15$, genus $1$, alternating; the same polynomial as $7_4$
9
3:
2*t^6 - 3*t^5 + 3*t^4 - 3*t^3 + 3*t^2 - 3*t + 2
comment: $9_3$; determinant $19$, genus $3$, alternating
9
4:
3*t^4 - 5*t^3 + 5*t^2 - 5*t + 3
comment: $9_4$; determinant $21$, genus $2$, alternating
9
5:
6*t^2 - 11*t + 6
comment: $9_5$; determinant $23$, genus $1$, alternating
9
6:
2*t^6 - 4*t^5 + 5*t^4 - 5*t^3 + 5*t^2 - 4*t + 2
comment: $9_6$; determinant $27$, genus $3$, alternating
9
7:
3*t^4 - 7*t^3 + 9*t^2 - 7*t + 3
comment: $9_7$; determinant $29$, genus $2$, alternating
9
8:
2*t^4 - 8*t^3 + 11*t^2 - 8*t + 2
comment: $9_8$; determinant $31$, genus $2$, alternating; the same polynomial as $8_{14}$, $10_{131}$
9
9:
2*t^6 - 4*t^5 + 6*t^4 - 7*t^3 + 6*t^2 - 4*t + 2
comment: $9_9$; determinant $31$, genus $3$, alternating
9
10:
4*t^4 - 8*t^3 + 9*t^2 - 8*t + 4
comment: $9_{10}$; determinant $33$, genus $2$, alternating
9
11:
t^6 - 5*t^5 + 7*t^4 - 7*t^3 + 7*t^2 - 5*t + 1
comment: $9_{11}$; determinant $33$, genus $3$, alternating
9
12:
2*t^4 - 9*t^3 + 13*t^2 - 9*t + 2
comment: $9_{12}$; determinant $35$, genus $2$, alternating
9
13:
4*t^4 - 9*t^3 + 11*t^2 - 9*t + 4
comment: $9_{13}$; determinant $37$, genus $2$, alternating
9
14:
2*t^4 - 9*t^3 + 15*t^2 - 9*t + 2
comment: $9_{14}$; determinant $37$, genus $2$, alternating
9
15:
2*t^4 - 10*t^3 + 15*t^2 - 10*t + 2
comment: $9_{15}$; determinant $39$, genus $2$, alternating; the same polynomial as $10_{165}$
9
16:
2*t^6 - 5*t^5 + 8*t^4 - 9*t^3 + 8*t^2 - 5*t + 2
comment: $9_{16}$; determinant $39$, genus $3$, alternating
9
17:
t^6 - 5*t^5 + 9*t^4 - 9*t^3 + 9*t^2 - 5*t + 1
comment: $9_{17}$; determinant $39$, genus $3$, alternating
9
18:
4*t^4 - 10*t^3 + 13*t^2 - 10*t + 4
comment: $9_{18}$; determinant $41$, genus $2$, alternating
9
19:
2*t^4 - 10*t^3 + 17*t^2 - 10*t + 2
comment: $9_{19}$; determinant $41$, genus $2$, alternating
9
20:
t^6 - 5*t^5 + 9*t^4 - 11*t^3 + 9*t^2 - 5*t + 1
comment: $9_{20}$; determinant $41$, genus $3$, alternating; the same polynomial as $10_{149}$
9
21:
2*t^4 - 11*t^3 + 17*t^2 - 11*t + 2
comment: $9_{21}$; determinant $43$, genus $2$, alternating
9
22:
t^6 - 5*t^5 + 10*t^4 - 11*t^3 + 10*t^2 - 5*t + 1
comment: $9_{22}$; determinant $43$, genus $3$, alternating
9
23:
4*t^4 - 11*t^3 + 15*t^2 - 11*t + 4
comment: $9_{23}$; determinant $45$, genus $2$, alternating
9
24:
t^6 - 5*t^5 + 10*t^4 - 13*t^3 + 10*t^2 - 5*t + 1
comment: $9_{24}$; determinant $45$, genus $3$, alternating; the same polynomial as $8_{18}$
9
25:
3*t^4 - 12*t^3 + 17*t^2 - 12*t + 3
comment: $9_{25}$; determinant $47$, genus $2$, alternating
9
26:
t^6 - 5*t^5 + 11*t^4 - 13*t^3 + 11*t^2 - 5*t + 1
comment: $9_{26}$; determinant $47$, genus $3$, alternating
9
27:
t^6 - 5*t^5 + 11*t^4 - 15*t^3 + 11*t^2 - 5*t + 1
comment: $9_{27}$; determinant $49$, genus $3$, alternating
9
28:
t^6 - 5*t^5 + 12*t^4 - 15*t^3 + 12*t^2 - 5*t + 1
comment: $9_{28}$; determinant $51$, genus $3$, alternating; the same polynomial as $9_{29}$, $10_{163}$
9
29:
t^6 - 5*t^5 + 12*t^4 - 15*t^3 + 12*t^2 - 5*t + 1
comment: $9_{29}$; determinant $51$, genus $3$, alternating; the same polynomial as $9_{28}$, $10_{163}$
9
30:
t^6 - 5*t^5 + 12*t^4 - 17*t^3 + 12*t^2 - 5*t + 1
comment: $9_{30}$; determinant $53$, genus $3$, alternating
9
31:
t^6 - 5*t^5 + 13*t^4 - 17*t^3 + 13*t^2 - 5*t + 1
comment: $9_{31}$; determinant $55$, genus $3$, alternating
9
32:
t^6 - 6*t^5 + 14*t^4 - 17*t^3 + 14*t^2 - 6*t + 1
comment: $9_{32}$; determinant $59$, genus $3$, alternating
9
33:
t^6 - 6*t^5 + 14*t^4 - 19*t^3 + 14*t^2 - 6*t + 1
comment: $9_{33}$; determinant $61$, genus $3$, alternating
9
34:
t^6 - 6*t^5 + 16*t^4 - 23*t^3 + 16*t^2 - 6*t + 1
comment: $9_{34}$; determinant $69$, genus $3$, alternating
9
35:
7*t^2 - 13*t + 7
comment: $9_{35}$; determinant $27$, genus $1$, alternating
9
36:
t^6 - 5*t^5 + 8*t^4 - 9*t^3 + 8*t^2 - 5*t + 1
comment: $9_{36}$; determinant $37$, genus $3$, alternating
9
37:
2*t^4 - 11*t^3 + 19*t^2 - 11*t + 2
comment: $9_{37}$; determinant $45$, genus $2$, alternating
9
38:
5*t^4 - 14*t^3 + 19*t^2 - 14*t + 5
comment: $9_{38}$; determinant $57$, genus $2$, alternating; the same polynomial as $10_{63}$
9
39:
3*t^4 - 14*t^3 + 21*t^2 - 14*t + 3
comment: $9_{39}$; determinant $55$, genus $2$, alternating
9
40:
t^6 - 7*t^5 + 18*t^4 - 23*t^3 + 18*t^2 - 7*t + 1
comment: $9_{40}$; determinant $75$, genus $3$, alternating; the same polynomial as $10_{59}$
9
41:
3*t^4 - 12*t^3 + 19*t^2 - 12*t + 3
comment: $9_{41}$; determinant $49$, genus $2$, alternating
9
42:
t^4 - 2*t^3 + t^2 - 2*t + 1
comment: $9_{42}$; determinant $7$, genus $2$, non-alternating
9
43:
t^6 - 3*t^5 + 2*t^4 - t^3 + 2*t^2 - 3*t + 1
comment: $9_{43}$; determinant $13$, genus $3$, non-alternating
9
44:
t^4 - 4*t^3 + 7*t^2 - 4*t + 1
comment: $9_{44}$; determinant $17$, genus $2$, non-alternating
9
45:
t^4 - 6*t^3 + 9*t^2 - 6*t + 1
comment: $9_{45}$; determinant $23$, genus $2$, non-alternating
9
46:
2*t^2 - 5*t + 2
comment: $9_{46}$; determinant $9$, genus $1$, non-alternating; the same polynomial as $6_1$
9
47:
t^6 - 4*t^5 + 6*t^4 - 5*t^3 + 6*t^2 - 4*t + 1
comment: $9_{47}$; determinant $27$, genus $3$, non-alternating
9
48:
t^4 - 7*t^3 + 11*t^2 - 7*t + 1
comment: $9_{48}$; determinant $27$, genus $2$, non-alternating
9
49:
3*t^4 - 6*t^3 + 7*t^2 - 6*t + 3
comment: $9_{49}$; determinant $25$, genus $2$, non-alternating
10
1:
4*t^2 - 9*t + 4
comment: $10_1$; determinant $17$, genus $1$, alternating; the same polynomial as $8_3$
10
2:
t^8 - 3*t^7 + 3*t^6 - 3*t^5 + 3*t^4 - 3*t^3 + 3*t^2 - 3*t + 1
comment: $10_2$; determinant $23$, genus $4$, alternating
10
3:
6*t^2 - 13*t + 6
comment: $10_3$; determinant $25$, genus $1$, alternating
10
4:
3*t^4 - 7*t^3 + 7*t^2 - 7*t + 3
comment: $10_4$; determinant $27$, genus $2$, alternating
10
5:
t^8 - 3*t^7 + 5*t^6 - 5*t^5 + 5*t^4 - 5*t^3 + 5*t^2 - 3*t + 1
comment: $10_5$; determinant $33$, genus $4$, alternating
10
6:
2*t^6 - 6*t^5 + 7*t^4 - 7*t^3 + 7*t^2 - 6*t + 2
comment: $10_6$; determinant $37$, genus $3$, alternating
10
7:
3*t^4 - 11*t^3 + 15*t^2 - 11*t + 3
comment: $10_7$; determinant $43$, genus $2$, alternating
10
8:
2*t^6 - 5*t^5 + 5*t^4 - 5*t^3 + 5*t^2 - 5*t + 2
comment: $10_8$; determinant $29$, genus $3$, alternating
10
9:
t^8 - 3*t^7 + 5*t^6 - 7*t^5 + 7*t^4 - 7*t^3 + 5*t^2 - 3*t + 1
comment: $10_9$; determinant $39$, genus $4$, alternating
10
10:
3*t^4 - 11*t^3 + 17*t^2 - 11*t + 3
comment: $10_{10}$; determinant $45$, genus $2$, alternating; the same polynomial as $10_{164}$
10
11:
4*t^4 - 11*t^3 + 13*t^2 - 11*t + 4
comment: $10_{11}$; determinant $43$, genus $2$, alternating
10
12:
2*t^6 - 6*t^5 + 10*t^4 - 11*t^3 + 10*t^2 - 6*t + 2
comment: $10_{12}$; determinant $47$, genus $3$, alternating; the same polynomial as $10_{54}$
10
13:
2*t^4 - 13*t^3 + 23*t^2 - 13*t + 2
comment: $10_{13}$; determinant $53$, genus $2$, alternating
10
14:
2*t^6 - 8*t^5 + 12*t^4 - 13*t^3 + 12*t^2 - 8*t + 2
comment: $10_{14}$; determinant $57$, genus $3$, alternating
10
15:
2*t^6 - 6*t^5 + 9*t^4 - 9*t^3 + 9*t^2 - 6*t + 2
comment: $10_{15}$; determinant $43$, genus $3$, alternating
10
16:
4*t^4 - 12*t^3 + 15*t^2 - 12*t + 4
comment: $10_{16}$; determinant $47$, genus $2$, alternating
10
17:
t^8 - 3*t^7 + 5*t^6 - 7*t^5 + 9*t^4 - 7*t^3 + 5*t^2 - 3*t + 1
comment: $10_{17}$; determinant $41$, genus $4$, alternating
10
18:
4*t^4 - 14*t^3 + 19*t^2 - 14*t + 4
comment: $10_{18}$; determinant $55$, genus $2$, alternating; the same polynomial as $10_{24}$
10
19:
2*t^6 - 7*t^5 + 11*t^4 - 11*t^3 + 11*t^2 - 7*t + 2
comment: $10_{19}$; determinant $51$, genus $3$, alternating
10
20:
3*t^4 - 9*t^3 + 11*t^2 - 9*t + 3
comment: $10_{20}$; determinant $35$, genus $2$, alternating; the same polynomial as $10_{162}$
10
21:
2*t^6 - 7*t^5 + 9*t^4 - 9*t^3 + 9*t^2 - 7*t + 2
comment: $10_{21}$; determinant $45$, genus $3$, alternating
10
22:
2*t^6 - 6*t^5 + 10*t^4 - 13*t^3 + 10*t^2 - 6*t + 2
comment: $10_{22}$; determinant $49$, genus $3$, alternating
10
23:
2*t^6 - 7*t^5 + 13*t^4 - 15*t^3 + 13*t^2 - 7*t + 2
comment: $10_{23}$; determinant $59$, genus $3$, alternating; the same polynomial as $10_{52}$
10
24:
4*t^4 - 14*t^3 + 19*t^2 - 14*t + 4
comment: $10_{24}$; determinant $55$, genus $2$, alternating; the same polynomial as $10_{18}$
10
25:
2*t^6 - 8*t^5 + 14*t^4 - 17*t^3 + 14*t^2 - 8*t + 2
comment: $10_{25}$; determinant $65$, genus $3$, alternating; the same polynomial as $10_{56}$
10
26:
2*t^6 - 7*t^5 + 13*t^4 - 17*t^3 + 13*t^2 - 7*t + 2
comment: $10_{26}$; determinant $61$, genus $3$, alternating
10
27:
2*t^6 - 8*t^5 + 16*t^4 - 19*t^3 + 16*t^2 - 8*t + 2
comment: $10_{27}$; determinant $71$, genus $3$, alternating
10
28:
4*t^4 - 13*t^3 + 19*t^2 - 13*t + 4
comment: $10_{28}$; determinant $53$, genus $2$, alternating; the same polynomial as $10_{37}$
10
29:
t^6 - 7*t^5 + 15*t^4 - 17*t^3 + 15*t^2 - 7*t + 1
comment: $10_{29}$; determinant $63$, genus $3$, alternating
10
30:
4*t^4 - 17*t^3 + 25*t^2 - 17*t + 4
comment: $10_{30}$; determinant $67$, genus $2$, alternating
10
31:
4*t^4 - 14*t^3 + 21*t^2 - 14*t + 4
comment: $10_{31}$; determinant $57$, genus $2$, alternating; the same polynomial as $10_{68}$
10
32:
2*t^6 - 8*t^5 + 15*t^4 - 19*t^3 + 15*t^2 - 8*t + 2
comment: $10_{32}$; determinant $69$, genus $3$, alternating
10
33:
4*t^4 - 16*t^3 + 25*t^2 - 16*t + 4
comment: $10_{33}$; determinant $65$, genus $2$, alternating
10
34:
3*t^4 - 9*t^3 + 13*t^2 - 9*t + 3
comment: $10_{34}$; determinant $37$, genus $2$, alternating; the same polynomial as $10_{135}$
10
35:
2*t^4 - 12*t^3 + 21*t^2 - 12*t + 2
comment: $10_{35}$; determinant $49$, genus $2$, alternating
10
36:
3*t^4 - 13*t^3 + 19*t^2 - 13*t + 3
comment: $10_{36}$; determinant $51$, genus $2$, alternating
10
37:
4*t^4 - 13*t^3 + 19*t^2 - 13*t + 4
comment: $10_{37}$; determinant $53$, genus $2$, alternating; the same polynomial as $10_{28}$
10
38:
4*t^4 - 15*t^3 + 21*t^2 - 15*t + 4
comment: $10_{38}$; determinant $59$, genus $2$, alternating
10
39:
2*t^6 - 8*t^5 + 13*t^4 - 15*t^3 + 13*t^2 - 8*t + 2
comment: $10_{39}$; determinant $61$, genus $3$, alternating
10
40:
2*t^6 - 8*t^5 + 17*t^4 - 21*t^3 + 17*t^2 - 8*t + 2
comment: $10_{40}$; determinant $75$, genus $3$, alternating; the same polynomial as $10_{103}$
10
41:
t^6 - 7*t^5 + 17*t^4 - 21*t^3 + 17*t^2 - 7*t + 1
comment: $10_{41}$; determinant $71$, genus $3$, alternating
10
42:
t^6 - 7*t^5 + 19*t^4 - 27*t^3 + 19*t^2 - 7*t + 1
comment: $10_{42}$; determinant $81$, genus $3$, alternating; the same polynomial as $10_{75}$
10
43:
t^6 - 7*t^5 + 17*t^4 - 23*t^3 + 17*t^2 - 7*t + 1
comment: $10_{43}$; determinant $73$, genus $3$, alternating
10
44:
t^6 - 7*t^5 + 19*t^4 - 25*t^3 + 19*t^2 - 7*t + 1
comment: $10_{44}$; determinant $79$, genus $3$, alternating
10
45:
t^6 - 7*t^5 + 21*t^4 - 31*t^3 + 21*t^2 - 7*t + 1
comment: $10_{45}$; determinant $89$, genus $3$, alternating
10
46:
t^8 - 3*t^7 + 4*t^6 - 5*t^5 + 5*t^4 - 5*t^3 + 4*t^2 - 3*t + 1
comment: $10_{46}$; determinant $31$, genus $4$, alternating
10
47:
t^8 - 3*t^7 + 6*t^6 - 7*t^5 + 7*t^4 - 7*t^3 + 6*t^2 - 3*t + 1
comment: $10_{47}$; determinant $41$, genus $4$, alternating
10
48:
t^8 - 3*t^7 + 6*t^6 - 9*t^5 + 11*t^4 - 9*t^3 + 6*t^2 - 3*t + 1
comment: $10_{48}$; determinant $49$, genus $4$, alternating
10
49:
3*t^6 - 8*t^5 + 12*t^4 - 13*t^3 + 12*t^2 - 8*t + 3
comment: $10_{49}$; determinant $59$, genus $3$, alternating
10
50:
2*t^6 - 7*t^5 + 11*t^4 - 13*t^3 + 11*t^2 - 7*t + 2
comment: $10_{50}$; determinant $53$, genus $3$, alternating
10
51:
2*t^6 - 7*t^5 + 15*t^4 - 19*t^3 + 15*t^2 - 7*t + 2
comment: $10_{51}$; determinant $67$, genus $3$, alternating
10
52:
2*t^6 - 7*t^5 + 13*t^4 - 15*t^3 + 13*t^2 - 7*t + 2
comment: $10_{52}$; determinant $59$, genus $3$, alternating; the same polynomial as $10_{23}$
10
53:
6*t^4 - 18*t^3 + 25*t^2 - 18*t + 6
comment: $10_{53}$; determinant $73$, genus $2$, alternating
10
54:
2*t^6 - 6*t^5 + 10*t^4 - 11*t^3 + 10*t^2 - 6*t + 2
comment: $10_{54}$; determinant $47$, genus $3$, alternating; the same polynomial as $10_{12}$
10
55:
5*t^4 - 15*t^3 + 21*t^2 - 15*t + 5
comment: $10_{55}$; determinant $61$, genus $2$, alternating
10
56:
2*t^6 - 8*t^5 + 14*t^4 - 17*t^3 + 14*t^2 - 8*t + 2
comment: $10_{56}$; determinant $65$, genus $3$, alternating; the same polynomial as $10_{25}$
10
57:
2*t^6 - 8*t^5 + 18*t^4 - 23*t^3 + 18*t^2 - 8*t + 2
comment: $10_{57}$; determinant $79$, genus $3$, alternating
10
58:
3*t^4 - 16*t^3 + 27*t^2 - 16*t + 3
comment: $10_{58}$; determinant $65$, genus $2$, alternating
10
59:
t^6 - 7*t^5 + 18*t^4 - 23*t^3 + 18*t^2 - 7*t + 1
comment: $10_{59}$; determinant $75$, genus $3$, alternating; the same polynomial as $9_{40}$
10
60:
t^6 - 7*t^5 + 20*t^4 - 29*t^3 + 20*t^2 - 7*t + 1
comment: $10_{60}$; determinant $85$, genus $3$, alternating
10
61:
2*t^6 - 5*t^5 + 6*t^4 - 7*t^3 + 6*t^2 - 5*t + 2
comment: $10_{61}$; determinant $33$, genus $3$, alternating
10
62:
t^8 - 3*t^7 + 6*t^6 - 8*t^5 + 9*t^4 - 8*t^3 + 6*t^2 - 3*t + 1
comment: $10_{62}$; determinant $45$, genus $4$, alternating
10
63:
5*t^4 - 14*t^3 + 19*t^2 - 14*t + 5
comment: $10_{63}$; determinant $57$, genus $2$, alternating; the same polynomial as $9_{38}$
10
64:
t^8 - 3*t^7 + 6*t^6 - 10*t^5 + 11*t^4 - 10*t^3 + 6*t^2 - 3*t + 1
comment: $10_{64}$; determinant $51$, genus $4$, alternating
10
65:
2*t^6 - 7*t^5 + 14*t^4 - 17*t^3 + 14*t^2 - 7*t + 2
comment: $10_{65}$; determinant $63$, genus $3$, alternating; the same polynomial as $10_{77}$
10
66:
3*t^6 - 9*t^5 + 16*t^4 - 19*t^3 + 16*t^2 - 9*t + 3
comment: $10_{66}$; determinant $75$, genus $3$, alternating
10
67:
4*t^4 - 16*t^3 + 23*t^2 - 16*t + 4
comment: $10_{67}$; determinant $63$, genus $2$, alternating; the same polynomial as $10_{74}$
10
68:
4*t^4 - 14*t^3 + 21*t^2 - 14*t + 4
comment: $10_{68}$; determinant $57$, genus $2$, alternating; the same polynomial as $10_{31}$
10
69:
t^6 - 7*t^5 + 21*t^4 - 29*t^3 + 21*t^2 - 7*t + 1
comment: $10_{69}$; determinant $87$, genus $3$, alternating
10
70:
t^6 - 7*t^5 + 16*t^4 - 19*t^3 + 16*t^2 - 7*t + 1
comment: $10_{70}$; determinant $67$, genus $3$, alternating
10
71:
t^6 - 7*t^5 + 18*t^4 - 25*t^3 + 18*t^2 - 7*t + 1
comment: $10_{71}$; determinant $77$, genus $3$, alternating
10
72:
2*t^6 - 9*t^5 + 16*t^4 - 19*t^3 + 16*t^2 - 9*t + 2
comment: $10_{72}$; determinant $73$, genus $3$, alternating
10
73:
t^6 - 7*t^5 + 20*t^4 - 27*t^3 + 20*t^2 - 7*t + 1
comment: $10_{73}$; determinant $83$, genus $3$, alternating
10
74:
4*t^4 - 16*t^3 + 23*t^2 - 16*t + 4
comment: $10_{74}$; determinant $63$, genus $2$, alternating; the same polynomial as $10_{67}$
10
75:
t^6 - 7*t^5 + 19*t^4 - 27*t^3 + 19*t^2 - 7*t + 1
comment: $10_{75}$; determinant $81$, genus $3$, alternating; the same polynomial as $10_{42}$
10
76:
2*t^6 - 7*t^5 + 12*t^4 - 15*t^3 + 12*t^2 - 7*t + 2
comment: $10_{76}$; determinant $57$, genus $3$, alternating
10
77:
2*t^6 - 7*t^5 + 14*t^4 - 17*t^3 + 14*t^2 - 7*t + 2
comment: $10_{77}$; determinant $63$, genus $3$, alternating; the same polynomial as $10_{65}$
10
78:
t^6 - 7*t^5 + 16*t^4 - 21*t^3 + 16*t^2 - 7*t + 1
comment: $10_{78}$; determinant $69$, genus $3$, alternating
10
79:
t^8 - 3*t^7 + 7*t^6 - 12*t^5 + 15*t^4 - 12*t^3 + 7*t^2 - 3*t + 1
comment: $10_{79}$; determinant $61$, genus $4$, alternating
10
80:
3*t^6 - 9*t^5 + 15*t^4 - 17*t^3 + 15*t^2 - 9*t + 3
comment: $10_{80}$; determinant $71$, genus $3$, alternating
10
81:
t^6 - 8*t^5 + 20*t^4 - 27*t^3 + 20*t^2 - 8*t + 1
comment: $10_{81}$; determinant $85$, genus $3$, alternating
10
82:
t^8 - 4*t^7 + 8*t^6 - 12*t^5 + 13*t^4 - 12*t^3 + 8*t^2 - 4*t + 1
comment: $10_{82}$; determinant $63$, genus $4$, alternating
10
83:
2*t^6 - 9*t^5 + 19*t^4 - 23*t^3 + 19*t^2 - 9*t + 2
comment: $10_{83}$; determinant $83$, genus $3$, alternating
10
84:
2*t^6 - 9*t^5 + 20*t^4 - 25*t^3 + 20*t^2 - 9*t + 2
comment: $10_{84}$; determinant $87$, genus $3$, alternating
10
85:
t^8 - 4*t^7 + 8*t^6 - 10*t^5 + 11*t^4 - 10*t^3 + 8*t^2 - 4*t + 1
comment: $10_{85}$; determinant $57$, genus $4$, alternating
10
86:
2*t^6 - 9*t^5 + 19*t^4 - 25*t^3 + 19*t^2 - 9*t + 2
comment: $10_{86}$; determinant $85$, genus $3$, alternating
10
87:
2*t^6 - 9*t^5 + 18*t^4 - 23*t^3 + 18*t^2 - 9*t + 2
comment: $10_{87}$; determinant $81$, genus $3$, alternating; the same polynomial as $10_{98}$
10
88:
t^6 - 8*t^5 + 24*t^4 - 35*t^3 + 24*t^2 - 8*t + 1
comment: $10_{88}$; determinant $101$, genus $3$, alternating
10
89:
t^6 - 8*t^5 + 24*t^4 - 33*t^3 + 24*t^2 - 8*t + 1
comment: $10_{89}$; determinant $99$, genus $3$, alternating
10
90:
2*t^6 - 8*t^5 + 17*t^4 - 23*t^3 + 17*t^2 - 8*t + 2
comment: $10_{90}$; determinant $77$, genus $3$, alternating
10
91:
t^8 - 4*t^7 + 9*t^6 - 14*t^5 + 17*t^4 - 14*t^3 + 9*t^2 - 4*t + 1
comment: $10_{91}$; determinant $73$, genus $4$, alternating
10
92:
2*t^6 - 10*t^5 + 20*t^4 - 25*t^3 + 20*t^2 - 10*t + 2
comment: $10_{92}$; determinant $89$, genus $3$, alternating
10
93:
2*t^6 - 8*t^5 + 15*t^4 - 17*t^3 + 15*t^2 - 8*t + 2
comment: $10_{93}$; determinant $67$, genus $3$, alternating
10
94:
t^8 - 4*t^7 + 9*t^6 - 14*t^5 + 15*t^4 - 14*t^3 + 9*t^2 - 4*t + 1
comment: $10_{94}$; determinant $71$, genus $4$, alternating
10
95:
2*t^6 - 9*t^5 + 21*t^4 - 27*t^3 + 21*t^2 - 9*t + 2
comment: $10_{95}$; determinant $91$, genus $3$, alternating
10
96:
t^6 - 7*t^5 + 22*t^4 - 33*t^3 + 22*t^2 - 7*t + 1
comment: $10_{96}$; determinant $93$, genus $3$, alternating
10
97:
5*t^4 - 22*t^3 + 33*t^2 - 22*t + 5
comment: $10_{97}$; determinant $87$, genus $2$, alternating
10
98:
2*t^6 - 9*t^5 + 18*t^4 - 23*t^3 + 18*t^2 - 9*t + 2
comment: $10_{98}$; determinant $81$, genus $3$, alternating; the same polynomial as $10_{87}$
10
99:
t^8 - 4*t^7 + 10*t^6 - 16*t^5 + 19*t^4 - 16*t^3 + 10*t^2 - 4*t + 1
comment: $10_{99}$; determinant $81$, genus $4$, alternating
10
100:
t^8 - 4*t^7 + 9*t^6 - 12*t^5 + 13*t^4 - 12*t^3 + 9*t^2 - 4*t + 1
comment: $10_{100}$; determinant $65$, genus $4$, alternating
10
101:
7*t^4 - 21*t^3 + 29*t^2 - 21*t + 7
comment: $10_{101}$; determinant $85$, genus $2$, alternating
10
102:
2*t^6 - 8*t^5 + 16*t^4 - 21*t^3 + 16*t^2 - 8*t + 2
comment: $10_{102}$; determinant $73$, genus $3$, alternating
10
103:
2*t^6 - 8*t^5 + 17*t^4 - 21*t^3 + 17*t^2 - 8*t + 2
comment: $10_{103}$; determinant $75$, genus $3$, alternating; the same polynomial as $10_{40}$
10
104:
t^8 - 4*t^7 + 9*t^6 - 15*t^5 + 19*t^4 - 15*t^3 + 9*t^2 - 4*t + 1
comment: $10_{104}$; determinant $77$, genus $4$, alternating
10
105:
t^6 - 8*t^5 + 22*t^4 - 29*t^3 + 22*t^2 - 8*t + 1
comment: $10_{105}$; determinant $91$, genus $3$, alternating
10
106:
t^8 - 4*t^7 + 9*t^6 - 15*t^5 + 17*t^4 - 15*t^3 + 9*t^2 - 4*t + 1
comment: $10_{106}$; determinant $75$, genus $4$, alternating
10
107:
t^6 - 8*t^5 + 22*t^4 - 31*t^3 + 22*t^2 - 8*t + 1
comment: $10_{107}$; determinant $93$, genus $3$, alternating
10
108:
2*t^6 - 8*t^5 + 14*t^4 - 15*t^3 + 14*t^2 - 8*t + 2
comment: $10_{108}$; determinant $63$, genus $3$, alternating
10
109:
t^8 - 4*t^7 + 10*t^6 - 17*t^5 + 21*t^4 - 17*t^3 + 10*t^2 - 4*t + 1
comment: $10_{109}$; determinant $85$, genus $4$, alternating
10
110:
t^6 - 8*t^5 + 20*t^4 - 25*t^3 + 20*t^2 - 8*t + 1
comment: $10_{110}$; determinant $83$, genus $3$, alternating
10
111:
2*t^6 - 9*t^5 + 17*t^4 - 21*t^3 + 17*t^2 - 9*t + 2
comment: $10_{111}$; determinant $77$, genus $3$, alternating
10
112:
t^8 - 5*t^7 + 11*t^6 - 17*t^5 + 19*t^4 - 17*t^3 + 11*t^2 - 5*t + 1
comment: $10_{112}$; determinant $87$, genus $4$, alternating
10
113:
2*t^6 - 11*t^5 + 26*t^4 - 33*t^3 + 26*t^2 - 11*t + 2
comment: $10_{113}$; determinant $111$, genus $3$, alternating
10
114:
2*t^6 - 10*t^5 + 21*t^4 - 27*t^3 + 21*t^2 - 10*t + 2
comment: $10_{114}$; determinant $93$, genus $3$, alternating
10
115:
t^6 - 9*t^5 + 26*t^4 - 37*t^3 + 26*t^2 - 9*t + 1
comment: $10_{115}$; determinant $109$, genus $3$, alternating
10
116:
t^8 - 5*t^7 + 12*t^6 - 19*t^5 + 21*t^4 - 19*t^3 + 12*t^2 - 5*t + 1
comment: $10_{116}$; determinant $95$, genus $4$, alternating
10
117:
2*t^6 - 10*t^5 + 24*t^4 - 31*t^3 + 24*t^2 - 10*t + 2
comment: $10_{117}$; determinant $103$, genus $3$, alternating
10
118:
t^8 - 5*t^7 + 12*t^6 - 19*t^5 + 23*t^4 - 19*t^3 + 12*t^2 - 5*t + 1
comment: $10_{118}$; determinant $97$, genus $4$, alternating
10
119:
2*t^6 - 10*t^5 + 23*t^4 - 31*t^3 + 23*t^2 - 10*t + 2
comment: $10_{119}$; determinant $101$, genus $3$, alternating
10
120:
8*t^4 - 26*t^3 + 37*t^2 - 26*t + 8
comment: $10_{120}$; determinant $105$, genus $2$, alternating
10
121:
2*t^6 - 11*t^5 + 27*t^4 - 35*t^3 + 27*t^2 - 11*t + 2
comment: $10_{121}$; determinant $115$, genus $3$, alternating
10
122:
2*t^6 - 11*t^5 + 24*t^4 - 31*t^3 + 24*t^2 - 11*t + 2
comment: $10_{122}$; determinant $105$, genus $3$, alternating
10
123:
t^8 - 6*t^7 + 15*t^6 - 24*t^5 + 29*t^4 - 24*t^3 + 15*t^2 - 6*t + 1
comment: $10_{123}$; determinant $121$, genus $4$, alternating
10
124:
t^8 - t^7 + t^5 - t^4 + t^3 - t + 1
comment: $10_{124}$; the torus knot $T(3,5)$; determinant $1$, genus $4$, non-alternating
equals: $\Phi_{15}$
10
125:
t^6 - 2*t^5 + 2*t^4 - t^3 + 2*t^2 - 2*t + 1
comment: $10_{125}$; determinant $11$, genus $3$, non-alternating
10
126:
t^6 - 2*t^5 + 4*t^4 - 5*t^3 + 4*t^2 - 2*t + 1
comment: $10_{126}$; determinant $19$, genus $3$, non-alternating
10
127:
t^6 - 4*t^5 + 6*t^4 - 7*t^3 + 6*t^2 - 4*t + 1
comment: $10_{127}$; determinant $29$, genus $3$, non-alternating; the same polynomial as $10_{150}$
10
128:
2*t^6 - 3*t^5 + t^4 + t^3 + t^2 - 3*t + 2
comment: $10_{128}$; determinant $11$, genus $3$, non-alternating
10
129:
2*t^4 - 6*t^3 + 9*t^2 - 6*t + 2
comment: $10_{129}$; determinant $25$, genus $2$, non-alternating; the same polynomial as $8_8$
10
130:
2*t^4 - 4*t^3 + 5*t^2 - 4*t + 2
comment: $10_{130}$; determinant $17$, genus $2$, non-alternating; the same polynomial as $7_5$
10
131:
2*t^4 - 8*t^3 + 11*t^2 - 8*t + 2
comment: $10_{131}$; determinant $31$, genus $2$, non-alternating; the same polynomial as $8_{14}$, $9_8$
10
132:
t^4 - t^3 + t^2 - t + 1
comment: $10_{132}$; determinant $5$, genus $2$, non-alternating; the same polynomial as $5_1$
10
133:
t^4 - 5*t^3 + 7*t^2 - 5*t + 1
comment: $10_{133}$; determinant $19$, genus $2$, non-alternating; the same polynomial as $7_6$
10
134:
2*t^6 - 4*t^5 + 4*t^4 - 3*t^3 + 4*t^2 - 4*t + 2
comment: $10_{134}$; determinant $23$, genus $3$, non-alternating
10
135:
3*t^4 - 9*t^3 + 13*t^2 - 9*t + 3
comment: $10_{135}$; determinant $37$, genus $2$, non-alternating; the same polynomial as $10_{34}$
10
136:
t^4 - 4*t^3 + 5*t^2 - 4*t + 1
comment: $10_{136}$; determinant $15$, genus $2$, non-alternating; the same polynomial as $8_{21}$
10
137:
t^4 - 6*t^3 + 11*t^2 - 6*t + 1
comment: $10_{137}$; determinant $25$, genus $2$, non-alternating
10
138:
t^6 - 5*t^5 + 8*t^4 - 7*t^3 + 8*t^2 - 5*t + 1
comment: $10_{138}$; determinant $35$, genus $3$, non-alternating
10
139:
t^8 - t^7 + 2*t^5 - 3*t^4 + 2*t^3 - t + 1
comment: $10_{139}$; determinant $3$, genus $4$, non-alternating
10
140:
t^4 - 2*t^3 + 3*t^2 - 2*t + 1
comment: $10_{140}$; determinant $9$, genus $2$, non-alternating; $\Delta=\Phi_6^2$, also the Alexander polynomial of the granny and square knots $3_1\#3_1$ and $3_1\#\bar{3}_1$; the same polynomial as $8_{20}$
10
141:
t^6 - 3*t^5 + 4*t^4 - 5*t^3 + 4*t^2 - 3*t + 1
comment: $10_{141}$; determinant $21$, genus $3$, non-alternating; the same polynomial as $8_5$
10
142:
2*t^6 - 3*t^5 + 2*t^4 - t^3 + 2*t^2 - 3*t + 2
comment: $10_{142}$; determinant $15$, genus $3$, non-alternating
10
143:
t^6 - 3*t^5 + 6*t^4 - 7*t^3 + 6*t^2 - 3*t + 1
comment: $10_{143}$; determinant $27$, genus $3$, non-alternating; the same polynomial as $8_{10}$
10
144:
3*t^4 - 10*t^3 + 13*t^2 - 10*t + 3
comment: $10_{144}$; determinant $39$, genus $2$, non-alternating
10
145:
t^4 + t^3 - 3*t^2 + t + 1
comment: $10_{145}$; determinant $3$, genus $2$, non-alternating
10
146:
2*t^4 - 8*t^3 + 13*t^2 - 8*t + 2
comment: $10_{146}$; determinant $33$, genus $2$, non-alternating
10
147:
2*t^4 - 7*t^3 + 9*t^2 - 7*t + 2
comment: $10_{147}$; determinant $27$, genus $2$, non-alternating; the same polynomial as $8_{11}$
10
148:
t^6 - 3*t^5 + 7*t^4 - 9*t^3 + 7*t^2 - 3*t + 1
comment: $10_{148}$; determinant $31$, genus $3$, non-alternating
10
149:
t^6 - 5*t^5 + 9*t^4 - 11*t^3 + 9*t^2 - 5*t + 1
comment: $10_{149}$; determinant $41$, genus $3$, non-alternating; the same polynomial as $9_{20}$
10
150:
t^6 - 4*t^5 + 6*t^4 - 7*t^3 + 6*t^2 - 4*t + 1
comment: $10_{150}$; determinant $29$, genus $3$, non-alternating; the same polynomial as $10_{127}$
10
151:
t^6 - 4*t^5 + 10*t^4 - 13*t^3 + 10*t^2 - 4*t + 1
comment: $10_{151}$; determinant $43$, genus $3$, non-alternating
10
152:
t^8 - t^7 - t^6 + 4*t^5 - 5*t^4 + 4*t^3 - t^2 - t + 1
comment: $10_{152}$; determinant $11$, genus $4$, non-alternating
10
153:
t^6 - t^5 - t^4 + 3*t^3 - t^2 - t + 1
comment: $10_{153}$; determinant $1$, genus $3$, non-alternating
10
154:
t^6 - 4*t^4 + 7*t^3 - 4*t^2 + 1
comment: $10_{154}$; determinant $13$, genus $3$, non-alternating
10
155:
t^6 - 3*t^5 + 5*t^4 - 7*t^3 + 5*t^2 - 3*t + 1
comment: $10_{155}$; determinant $25$, genus $3$, non-alternating; the same polynomial as $8_9$
10
156:
t^6 - 4*t^5 + 8*t^4 - 9*t^3 + 8*t^2 - 4*t + 1
comment: $10_{156}$; determinant $35$, genus $3$, non-alternating; the same polynomial as $8_{16}$
10
157:
t^6 - 6*t^5 + 11*t^4 - 13*t^3 + 11*t^2 - 6*t + 1
comment: $10_{157}$; determinant $49$, genus $3$, non-alternating
10
158:
t^6 - 4*t^5 + 10*t^4 - 15*t^3 + 10*t^2 - 4*t + 1
comment: $10_{158}$; determinant $45$, genus $3$, non-alternating
10
159:
t^6 - 4*t^5 + 9*t^4 - 11*t^3 + 9*t^2 - 4*t + 1
comment: $10_{159}$; determinant $39$, genus $3$, non-alternating
10
160:
t^6 - 4*t^5 + 4*t^4 - 3*t^3 + 4*t^2 - 4*t + 1
comment: $10_{160}$; determinant $21$, genus $3$, non-alternating
10
161:
t^6 - 2*t^4 + 3*t^3 - 2*t^2 + 1
comment: $10_{161}$, the Perko pair, listed twice by Rolfsen as $10_{161}$ and $10_{162}$; determinant $5$, genus $3$, non-alternating
10
162:
3*t^4 - 9*t^3 + 11*t^2 - 9*t + 3
comment: $10_{162}$; determinant $35$, genus $2$, non-alternating; the same polynomial as $10_{20}$
10
163:
t^6 - 5*t^5 + 12*t^4 - 15*t^3 + 12*t^2 - 5*t + 1
comment: $10_{163}$; determinant $51$, genus $3$, non-alternating; the same polynomial as $9_{28}$, $9_{29}$
10
164:
3*t^4 - 11*t^3 + 17*t^2 - 11*t + 3
comment: $10_{164}$; determinant $45$, genus $2$, non-alternating; the same polynomial as $10_{10}$
10
165:
2*t^4 - 10*t^3 + 15*t^2 - 10*t + 2
comment: $10_{165}$; determinant $39$, genus $2$, non-alternating; the same polynomial as $9_{15}$
Definition
The Alexander polynomial $\Delta_K(t)\in\mathbb{Z}[t]$ of a knot $K$ [3], defined up to a factor $\pm t^m$ and normalised to a positive constant term, for the unknot $0_1$ and for every prime knot $n_k$ with at most ten crossings, named as in the Rolfsen table [1] with Perko's correction [2].
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)
$\Delta_K(t)\doteq\det(V-tV^{\mathsf T})$ for any Seifert matrix $V$ of $K$, where $\doteq$ is equality up to a factor $\pm t^m$ [3]. The polynomial listed is the representative with nonzero positive constant term.
(2)
$\Delta(t)=t^{2g}\Delta(1/t)$ with $2g=\deg\Delta$, $\Delta(1)=\pm1$, and $|\Delta(-1)|=\det K$, the determinant of the knot, the order of the first homology of its double branched cover.
(3)
$\Delta(s^2)=\epsilon\,s^{2g}\,\nabla(s-s^{-1})$, where $\nabla(z)\in\mathbb{Z}[z]$ is the Conway polynomial, $\nabla(0_1)=1$ and $\nabla(L_+)-\nabla(L_-)=z\,\nabla(L_0)$, and $\epsilon=\Delta(1)$ is the sign of the leading coefficient of $\nabla$ [3].
(4)
For the closure of a braid $\beta$ on $r$ strands, $\Delta(t)\doteq\det\bigl(I-\psi(\beta)\bigr)\dfrac{1-t}{1-t^r}$ with $\psi$ the reduced Burau representation of the braid group $B_r$ [5].
(5)
$\Delta_{T(p,q)}(t)=\dfrac{(t^{pq}-1)(t-1)}{(t^p-1)(t^q-1)} =\prod_{d\mid pq,\ d\nmid p,\ d\nmid q}\Phi_d(t)$ for the torus knot $T(p,q)$, $\gcd(p,q)=1$ [4], where $\Phi_d$ is the cyclotomic polynomial. 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 $10_{124}=T(3,5)$, with $\Delta=\Phi_6$, $\Phi_{10}$, $\Phi_{14}$, $\Phi_6\Phi_{18}$, $\Phi_6\Phi_{12}$ and $\Phi_{15}$.
(6)
$\Delta_{K_1\#K_2}=\Delta_{K_1}\Delta_{K_2}$ for the connected sum, and $\Delta_{\bar K}=\Delta_K$ for the mirror image; so the granny knot $3_1\#3_1$ and the square knot $3_1\#\bar 3_1$ both have $\Delta=\Phi_6^2$, the value of $8_{20}$ and $10_{140}$.
Comments
(7)
The Alexander polynomial is determined only up to a unit $\pm t^m$ of $\mathbb{Z}[t,t^{-1}]$. Rolfsen's table [1], the Knot Atlas [8], KnotInfo [9] and MathWorld [7] write it as a polynomial with nonzero positive constant term, and so does this table; by the symmetry $\Delta(t)=t^{2g}\Delta(1/t)$ the leading coefficient is then positive too. Sage's alexander_polynomial() [10] returns instead the Conway-normalised Laurent polynomial $\epsilon\,t^{-g}\Delta(t)$ with $\epsilon=\Delta(1)=\pm1$, which is symmetric under $t\mapsto 1/t$ and equal to $1$ at $t=1$: for $4_1$ it gives $-t^{-1}+3-t$ where this table has $t^2-3t+1$. Of the 249 prime knots here, 110 have $\epsilon=-1$.
(8)
The numbering is Rolfsen's after Perko's correction [2]: the knots Rolfsen listed as $10_{161}$ and $10_{162}$ are one knot, so there are $165$ prime knots with ten crossings rather than $166$ [11], and Rolfsen's $10_{163}$ to $10_{166}$ are $10_{162}$ to $10_{165}$ here [6]. Sage's knot table, the Knot Atlas and KnotInfo all use this numbering. Within each crossing number the alternating knots come first: the non-alternating ones are $8_{19}$ to $8_{21}$, $9_{42}$ to $9_{49}$ and $10_{124}$ to $10_{165}$, and every prime knot with fewer than eight crossings is alternating.
(9)
This is the one-variable Alexander polynomial of a knot, not the multivariable Alexander polynomial of a link. It does not depend on the orientation of the knot or on the choice between a knot and its mirror image, so it is well defined for the name $n_k$, which names a knot only up to mirror image; invariants that do see the mirror image, such as the signature and the Jones polynomial, are not listed here.
(10)
The degree of $\Delta$ is at most twice the genus $g$, with equality for alternating knots and for fibred knots [3], and $|\Delta(-1)|$ is the determinant of the knot. For every knot here the degree is exactly $2g$, and $\Delta$ is monic exactly when the knot is fibred. Each entry's comment gives the knot's common name where it has one, its determinant and genus, whether it is alternating, the torus knot it is if it is one, and every other knot in the table with the same polynomial.
(11)
The Alexander polynomial does not determine the knot: 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 $\Delta(5_1)=\Delta(10_{132})=\Phi_{10}$, $\Delta(6_1)=\Delta(9_{46})$, $\Delta(8_{20})=\Delta(10_{140})=\Phi_6^2$ and the triples $8_{14},9_8,10_{131}$ and $9_{28},9_{29},10_{163}$, where $\Phi_d$ is the cyclotomic polynomial. No knot here has $\Delta=1$ apart from the unknot; the first that do, the Conway knot $11n_{34}$ and the Kinoshita–Terasaka knot $11n_{42}$, have eleven crossings.
(12)
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 [9], which the two parameters here cannot express; KnotInfo carries the Alexander polynomials of all prime knots to thirteen crossings. The Jones and HOMFLY-PT polynomials of the same knots are not listed here.
Programs
(P1)
Sage
K = Knots().from_table(10, 132)
A = K.alexander_polynomial()        # t^-2 - t^-1 + 1 - t + t^2, Sage's Conway-normalised form
R.<t> = ZZ[]
D = R({e - A.valuation(): c for e, c in A.dict().items()})
D if D[0] > 0 else -D               # t^4 - t^3 + t^2 - t + 1, the form listed here
References
[1]
D. Rolfsen, Knots and Links, Publish or Perish, 1976, Appendix C: Table of knots and links.
[2]
K. A. Perko, On the classification of knots, Proc. Amer. Math. Soc. 45 (1974), 262–266.
Links
Similar tables
Cyclotomic polynomials —   the Alexander polynomial of a torus knot is a product of cyclotomic polynomials; $\Delta(3_1)$, $\Delta(5_1)$, $\Delta(7_1)$ and $\Delta(10_{124})$ are $\Phi_6$, $\Phi_{10}$, $\Phi_{14}$ and $\Phi_{15}$
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 alexander_polynomial() of Knots().from_table(n, k), the determinant $\det(V-tV^{\mathsf T})$ of a Seifert matrix of the braid closure, normalised.

more

Each was required to agree with $\det(I-\psi(\beta))(1-t)/(1-t^r)$ computed from the reduced Burau matrix of the same braid word on $r$ strands, with the matrix arithmetic and the determinant written out rather than taken from a library, and to satisfy $\Delta(1)=\pm1$, $\Delta(t)=t^{2g}\Delta(1/t)$ and $|\Delta(-1)|=\det K$. All 250 values agree with the alexander_polynomial column of KnotInfo (package database_knotinfo 2026.9.1, computed from KnotInfo's own diagrams); the six torus knots agree with their closed form and with the stored cyclotomic polynomials; every value agrees with Sage's conway_polynomial() through the Conway substitution; and the genus, alternating, fibred and torus facts in the comments agree with KnotInfo's columns.



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