# L9a38

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a38 at Knotilus! L9a38 is $9^2_{5}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) (t(1) t(2)+1)^2}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-2 q^{9/2}+3 q^{7/2}-\frac{1}{q^{7/2}}-5 q^{5/2}+\frac{2}{q^{5/2}}+5 q^{3/2}-\frac{3}{q^{3/2}}+q^{11/2}-6 \sqrt{q}+\frac{4}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $-z^7 a^{-1} +a z^5-6 z^5 a^{-1} +z^5 a^{-3} +4 a z^3-12 z^3 a^{-1} +4 z^3 a^{-3} +4 a z-8 z a^{-1} +4 z a^{-3} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-6} -2 z^2 a^{-6} +2 z^5 a^{-5} -4 z^3 a^{-5} +z a^{-5} +2 z^6 a^{-4} -3 z^4 a^{-4} +z^2 a^{-4} +2 z^7 a^{-3} +a^3 z^5-5 z^5 a^{-3} -3 a^3 z^3+8 z^3 a^{-3} +a^3 z-4 z a^{-3} +z^8 a^{-2} +2 a^2 z^6-z^6 a^{-2} -6 a^2 z^4+3 a^2 z^2+3 z^2 a^{-2} +2 a z^7+4 z^7 a^{-1} -6 a z^5-14 z^5 a^{-1} +6 a z^3+21 z^3 a^{-1} -4 a z-10 z a^{-1} +a z^{-1} + a^{-1} z^{-1} +z^8-z^6-2 z^4+3 z^2-1$ (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 \
-4-3-2-1012345χ
12         1-1
10        1 1
8       21 -1
6      31  2
4     22   0
2    43    1
0   24     2
-2  12      -1
-4 12       1
-6 1        -1
-81         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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