# L9a27

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a27 at Knotilus! L9a27 is $9^2_{12}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u v-u-2 v+1) (u v-2 u-v+1)}{u v}$ (db) Jones polynomial $q^{11/2}-3 q^{9/2}+5 q^{7/2}-8 q^{5/2}+8 q^{3/2}-9 \sqrt{q}+\frac{7}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{1}{q^{7/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $z a^{-5} -2 z^3 a^{-3} +a^3 z-2 z a^{-3} - a^{-3} z^{-1} +z^5 a^{-1} -2 a z^3+2 z^3 a^{-1} -2 a z+3 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^8 a^{-2} -z^8-3 a z^7-6 z^7 a^{-1} -3 z^7 a^{-3} -3 a^2 z^6-7 z^6 a^{-2} -4 z^6 a^{-4} -6 z^6-a^3 z^5+4 a z^5+7 z^5 a^{-1} -z^5 a^{-3} -3 z^5 a^{-5} +7 a^2 z^4+13 z^4 a^{-2} +4 z^4 a^{-4} -z^4 a^{-6} +15 z^4+2 a^3 z^3+a z^3+z^3 a^{-1} +6 z^3 a^{-3} +4 z^3 a^{-5} -4 a^2 z^2-5 z^2 a^{-2} -z^2 a^{-4} +z^2 a^{-6} -7 z^2-a^3 z-a z-2 z a^{-1} -4 z a^{-3} -2 z a^{-5} - a^{-2} + a^{-1} z^{-1} + a^{-3} z^{-1}$ (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 \
-4-3-2-1012345χ
12         1-1
10        2 2
8       31 -2
6      52  3
4     44   0
2    54    1
0   35     2
-2  24      -2
-4 13       2
-6 2        -2
-81         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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