# L9a34

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a34 at Knotilus! L9a34 is $9^2_{6}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)+t(2)-1) (t(1) t(2)+1) (t(2) t(1)-t(1)-t(2))}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $\frac{6}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{3/2}}+\frac{1}{q^{17/2}}-\frac{2}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{6}{q^{11/2}}-\sqrt{q}+\frac{2}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^7-2 z a^7+z^5 a^5+3 z^3 a^5+3 z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-z a^3-a^3 z^{-1} -z^3 a-2 z a$ (db) Kauffman polynomial $-z^4 a^{10}+2 z^2 a^{10}-2 z^5 a^9+4 z^3 a^9-2 z a^9-2 z^6 a^8+2 z^4 a^8-2 z^7 a^7+3 z^5 a^7-3 z^3 a^7-z a^7-z^8 a^6-z^2 a^6-4 z^7 a^5+10 z^5 a^5-12 z^3 a^5+6 z a^5-a^5 z^{-1} -z^8 a^4+2 z^4 a^4-z^2 a^4+a^4-2 z^7 a^3+4 z^5 a^3-2 z^3 a^3+3 z a^3-a^3 z^{-1} -2 z^6 a^2+5 z^4 a^2-2 z^2 a^2-z^5 a+3 z^3 a-2 z a$ (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
2         11
0        1 -1
-2       31 2
-4      32  -1
-6     32   1
-8    33    0
-10   33     0
-12  14      3
-14 12       -1
-16 1        1
-181         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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