# L9a48

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a48 at Knotilus! L9a48 is $9^3_{5}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^2 w^2-u v^2 w-u v w^2+2 u v w-u v-u w+2 u-2 v^2 w^2+v^2 w+v w^2-2 v w+v+w-1}{\sqrt{u} v w}$ (db) Jones polynomial $-q^6+3 q^5-4 q^4+6 q^3-6 q^2+6 q-4+4 q^{-1} - q^{-2} + q^{-3}$ (db) Signature 2 (db) HOMFLY-PT polynomial $-z^4 a^{-4} -2 z^2 a^{-4} +z^6 a^{-2} +4 z^4 a^{-2} +a^2 z^2+5 z^2 a^{-2} +a^2 z^{-2} + a^{-2} z^{-2} +3 a^2+3 a^{-2} -2 z^4-7 z^2-2 z^{-2} -6$ (db) Kauffman polynomial $z^8 a^{-2} +z^8+a z^7+4 z^7 a^{-1} +3 z^7 a^{-3} +a^2 z^6+2 z^6 a^{-2} +4 z^6 a^{-4} -z^6-2 a z^5-10 z^5 a^{-1} -4 z^5 a^{-3} +4 z^5 a^{-5} -5 a^2 z^4-8 z^4 a^{-2} -4 z^4 a^{-4} +3 z^4 a^{-6} -6 z^4-3 a z^3+2 z^3 a^{-1} +z^3 a^{-3} -3 z^3 a^{-5} +z^3 a^{-7} +8 a^2 z^2+5 z^2 a^{-2} -2 z^2 a^{-6} +11 z^2+6 a z+6 z a^{-1} -5 a^2-3 a^{-2} + a^{-4} -8-2 a z^{-1} -2 a^{-1} z^{-1} +a^2 z^{-2} + a^{-2} z^{-2} +2 z^{-2}$ (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       21 -1
7      42  2
5     44   0
3    22    0
1   35     2
-1  11      0
-3  3       3
-511        0
-71         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=1$ $i=3$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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