L7a1 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L7a1 at Knotilus! L7a1 is $7^2_6$ in the Rolfsen table of links.

Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(v^2-v+1\right)}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-4 q^{5/2}+\frac{1}{q^{5/2}}+4 q^{3/2}-\frac{3}{q^{3/2}}-5 \sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^3 a^{-3} -z a^{-3} +z^5 a^{-1} -a z^3+3 z^3 a^{-1} -a z+2 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-5} +3 z^4 a^{-4} -2 z^2 a^{-4} +4 z^5 a^{-3} -5 z^3 a^{-3} +2 z a^{-3} +2 z^6 a^{-2} +a^2 z^4+z^4 a^{-2} -a^2 z^2-3 z^2 a^{-2} +3 a z^5+7 z^5 a^{-1} -6 a z^3-12 z^3 a^{-1} +2 a z+4 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +2 z^6-z^4-2 z^2-1$ (db)

Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-3-2-101234χ
10       11
8      2 -2
6     21 1
4    22  0
2   32   1
0  24    2
-2 11     0
-4 2      2
-61       -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.