# L9a28

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a28 at Knotilus! L9a28 is $9^2_{20}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^2 t(2)^4-2 t(1)^2 t(2)^3+2 t(1) t(2)^3+2 t(1)^2 t(2)^2-3 t(1) t(2)^2+2 t(2)^2+2 t(1) t(2)-2 t(2)+1}{t(1) t(2)^2}$ (db) Jones polynomial $\frac{5}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{5}{q^{5/2}}+q^{3/2}-\frac{5}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{4}{q^{11/2}}-2 \sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^5 z^5+4 a^5 z^3+5 a^5 z+2 a^5 z^{-1} -a^3 z^7-6 a^3 z^5-13 a^3 z^3-12 a^3 z-3 a^3 z^{-1} +a z^5+4 a z^3+4 a z+a z^{-1}$ (db) Kauffman polynomial $a^9 z^3-a^9 z+2 a^8 z^4-a^8 z^2+3 a^7 z^5-3 a^7 z^3+2 a^7 z+3 a^6 z^6-3 a^6 z^4+a^6 z^2+3 a^5 z^7-7 a^5 z^5+9 a^5 z^3-7 a^5 z+2 a^5 z^{-1} +a^4 z^8+2 a^4 z^6-11 a^4 z^4+10 a^4 z^2-3 a^4+5 a^3 z^7-18 a^3 z^5+22 a^3 z^3-14 a^3 z+3 a^3 z^{-1} +a^2 z^8-10 a^2 z^4+12 a^2 z^2-3 a^2+2 a z^7-8 a z^5+9 a z^3-4 a z+a z^{-1} +z^6-4 z^4+4 z^2-1$ (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 \
-6-5-4-3-2-10123χ
4         1-1
2        1 1
0       21 -1
-2      31  2
-4     33   0
-6    32    1
-8   23     1
-10  23      -1
-12  2       2
-1412        -1
-161         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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