# L9a53

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a53 at Knotilus! L9a53 is $9^3_{12}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) (t(3)-1)^3}{\sqrt{t(1)} \sqrt{t(2)} t(3)^{3/2}}$ (db) Jones polynomial $q^4-4 q^3+8 q^2-9 q+12-10 q^{-1} +10 q^{-2} -6 q^{-3} +3 q^{-4} - q^{-5}$ (db) Signature 0 (db) HOMFLY-PT polynomial $-a^4 z^2-a^4+2 a^2 z^4+z^4 a^{-2} +4 a^2 z^2+z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+ a^{-2} -z^6-3 z^4-4 z^2-2 z^{-2} -3$ (db) Kauffman polynomial $2 a^2 z^8+2 z^8+4 a^3 z^7+11 a z^7+7 z^7 a^{-1} +3 a^4 z^6+7 a^2 z^6+8 z^6 a^{-2} +12 z^6+a^5 z^5-5 a^3 z^5-17 a z^5-7 z^5 a^{-1} +4 z^5 a^{-3} -6 a^4 z^4-22 a^2 z^4-11 z^4 a^{-2} +z^4 a^{-4} -28 z^4-2 a^5 z^3-a^3 z^3+3 a z^3-2 z^3 a^{-3} +5 a^4 z^2+17 a^2 z^2+5 z^2 a^{-2} +17 z^2+a^5 z+3 a^3 z+3 a z+z a^{-1} -2 a^4-6 a^2-2 a^{-2} -5-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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 \
-5-4-3-2-101234χ
9         11
7        3 -3
5       51 4
3      43  -1
1     85   3
-1    68    2
-3   44     0
-5  26      4
-7 14       -3
-9 2        2
-111         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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