L9a25 (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a25 at Knotilus! L9a25 is $9^2_{8}$ in the Rolfsen table of links.

Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(2) t(1)^2-2 t(1)^2+2 t(2)^2 t(1)-5 t(2) t(1)+2 t(1)-2 t(2)^2+2 t(2)}{t(1) t(2)}$ (db) Jones polynomial $-\frac{5}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{6}{q^{5/2}}-q^{3/2}+\frac{5}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-z a^7+z^3 a^5+2 z^3 a^3+2 z a^3+a^3 z^{-1} +z^3 a-z a-a z^{-1} -z a^{-1}$ (db) Kauffman polynomial $-z^6 a^8+4 z^4 a^8-4 z^2 a^8-2 z^7 a^7+8 z^5 a^7-9 z^3 a^7+3 z a^7-z^8 a^6+z^6 a^6+4 z^4 a^6-3 z^2 a^6-4 z^7 a^5+11 z^5 a^5-7 z^3 a^5+2 z a^5-z^8 a^4-z^6 a^4+5 z^4 a^4-z^2 a^4-2 z^7 a^3+6 z^3 a^3-5 z a^3+a^3 z^{-1} -3 z^6 a^2+3 z^4 a^2-z^2 a^2-a^2-3 z^5 a+3 z^3 a-3 z a+a z^{-1} -2 z^4+z^2-z^3 a^{-1} +z a^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
4         11
2        1 -1
0       31 2
-2      32  -1
-4     32   1
-6    34    1
-8   22     0
-10  13      2
-12 12       -1
-14 1        1
-161         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

Computer Talk

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