# L9a47

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a47 at Knotilus! L9a47 is $9^3_{2}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(3)^2 t(2)^2+t(1) t(2)^2-t(1) t(3) t(2)^2+2 t(3) t(2)^2-t(2)^2-t(1) t(3)^2 t(2)+2 t(3)^2 t(2)-2 t(1) t(2)+4 t(1) t(3) t(2)-4 t(3) t(2)+t(2)+t(1) t(3)^2-t(3)^2+t(1)-2 t(1) t(3)+t(3)}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $- q^{-8} +3 q^{-7} -5 q^{-6} +8 q^{-5} -8 q^{-4} +10 q^{-3} -7 q^{-2} +q+6 q^{-1} -3$ (db) Signature -2 (db) HOMFLY-PT polynomial $-a^8+3 z^2 a^6+a^6 z^{-2} +3 a^6-2 z^4 a^4-4 z^2 a^4-2 a^4 z^{-2} -5 a^4-z^4 a^2+z^2 a^2+a^2 z^{-2} +3 a^2+z^2$ (db) Kauffman polynomial $z^5 a^9-2 z^3 a^9+z a^9+3 z^6 a^8-7 z^4 a^8+5 z^2 a^8-2 a^8+3 z^7 a^7-3 z^5 a^7-4 z^3 a^7+3 z a^7+z^8 a^6+8 z^6 a^6-24 z^4 a^6+21 z^2 a^6+a^6 z^{-2} -8 a^6+7 z^7 a^5-10 z^5 a^5+5 z a^5-2 a^5 z^{-1} +z^8 a^4+10 z^6 a^4-26 z^4 a^4+24 z^2 a^4+2 a^4 z^{-2} -9 a^4+4 z^7 a^3-3 z^5 a^3-z^3 a^3+3 z a^3-2 a^3 z^{-1} +5 z^6 a^2-8 z^4 a^2+7 z^2 a^2+a^2 z^{-2} -4 a^2+3 z^5 a-3 z^3 a+z^4-z^2$ (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
3         11
1        2 -2
-1       41 3
-3      43  -1
-5     63   3
-7    46    2
-9   44     0
-11  25      3
-13 13       -2
-15 2        2
-171         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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