# L9a54

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a54 at Knotilus! L9a54 is $9^3_{9}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) (t(3)-1) \left(t(3)^2-t(3)+1\right)}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $-q^6+3 q^5-6 q^4+8 q^3+ q^{-3} -7 q^2-2 q^{-2} +9 q+5 q^{-1} -6$ (db) Signature 2 (db) HOMFLY-PT polynomial $z^6 a^{-2} +4 z^4 a^{-2} -z^4 a^{-4} -2 z^4+a^2 z^2+7 z^2 a^{-2} -2 z^2 a^{-4} -6 z^2+2 a^2+6 a^{-2} -2 a^{-4} -6+a^2 z^{-2} + a^{-2} z^{-2} -2 z^{-2}$ (db) Kauffman polynomial $z^8 a^{-2} +z^8+2 a z^7+7 z^7 a^{-1} +5 z^7 a^{-3} +a^2 z^6+11 z^6 a^{-2} +8 z^6 a^{-4} +4 z^6-6 a z^5-16 z^5 a^{-1} -4 z^5 a^{-3} +6 z^5 a^{-5} -4 a^2 z^4-35 z^4 a^{-2} -14 z^4 a^{-4} +3 z^4 a^{-6} -22 z^4+4 a z^3+3 z^3 a^{-1} -7 z^3 a^{-3} -5 z^3 a^{-5} +z^3 a^{-7} +6 a^2 z^2+30 z^2 a^{-2} +10 z^2 a^{-4} +26 z^2+2 a z+6 z a^{-1} +6 z a^{-3} +2 z a^{-5} -4 a^2-12 a^{-2} -4 a^{-4} -11-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-4-3-2-1012345χ
13         1-1
11        2 2
9       41 -3
7      42  2
5     34   1
3    64    2
1   47     3
-1  12      -1
-3 14       3
-5 1        -1
-71         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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