# L9n7

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9n7 at Knotilus! L9n7 is $9^2_{48}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 u v^3-2 u v^2+u v+v^2-2 v+2}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{5/2}}+\frac{1}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{3}{q^{11/2}}-\frac{4}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{2}{q^{19/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +z^3 a^9+3 z a^9+2 a^9 z^{-1} -z^5 a^7-3 z^3 a^7-z a^7-z^5 a^5-4 z^3 a^5-4 z a^5-a^5 z^{-1}$ (db) Kauffman polynomial $3 a^{12} z^2-2 a^{12}+a^{11} z^5+a^{11} z^3-2 a^{11} z+a^{11} z^{-1} +2 a^{10} z^6-6 a^{10} z^4+11 a^{10} z^2-5 a^{10}+a^9 z^7-2 a^9 z^5+5 a^9 z^3-6 a^9 z+2 a^9 z^{-1} +3 a^8 z^6-8 a^8 z^4+7 a^8 z^2-3 a^8+a^7 z^7-2 a^7 z^5+a^6 z^6-2 a^6 z^4-a^6 z^2+a^6+a^5 z^5-4 a^5 z^3+4 a^5 z-a^5 z^{-1}$ (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 \
-7-6-5-4-3-2-10χ
-4       11
-6      110
-8     2  2
-10    11  0
-12   32   1
-14  12    1
-16 22     0
-18 1      1
-202       -2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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