# L9a13

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a13 at Knotilus! L9a13 is $9^2_{16}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{2 t(2)^3+4 t(1) t(2)^2-5 t(2)^2-5 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}+\frac{3}{q^{5/2}}-\frac{6}{q^{7/2}}+\frac{6}{q^{9/2}}-\frac{8}{q^{11/2}}+\frac{7}{q^{13/2}}-\frac{6}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{2}{q^{19/2}}+\frac{1}{q^{21/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^{11} z^{-1} +3 a^9 z+2 a^9 z^{-1} -2 a^7 z^3-a^7 z-3 a^5 z^3-4 a^5 z-a^5 z^{-1} -a^3 z^3$ (db) Kauffman polynomial $-z^6 a^{12}+4 z^4 a^{12}-5 z^2 a^{12}+2 a^{12}-2 z^7 a^{11}+7 z^5 a^{11}-7 z^3 a^{11}+2 z a^{11}-a^{11} z^{-1} -z^8 a^{10}-2 z^6 a^{10}+15 z^4 a^{10}-17 z^2 a^{10}+5 a^{10}-6 z^7 a^9+16 z^5 a^9-12 z^3 a^9+6 z a^9-2 a^9 z^{-1} -z^8 a^8-7 z^6 a^8+21 z^4 a^8-13 z^2 a^8+3 a^8-4 z^7 a^7+3 z^5 a^7+3 z^3 a^7-6 z^6 a^6+7 z^4 a^6-z^2 a^6-a^6-6 z^5 a^5+7 z^3 a^5-4 z a^5+a^5 z^{-1} -3 z^4 a^4-z^3 a^3$ (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 \
-9-8-7-6-5-4-3-2-10χ
-2         11
-4        31-2
-6       3  3
-8      33  0
-10     53   2
-12    34    1
-14   34     -1
-16  13      2
-18 13       -2
-20 1        1
-221         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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