# L9a9

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a9 at Knotilus! L9a9 is $9^2_{37}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^2+1\right) \left(t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-\frac{3}{q^{9/2}}-q^{7/2}+\frac{5}{q^{7/2}}+3 q^{5/2}-\frac{7}{q^{5/2}}-5 q^{3/2}+\frac{8}{q^{3/2}}+\frac{1}{q^{11/2}}+6 \sqrt{q}-\frac{9}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-3 a^3 z^3-2 a^3 z+a z^7+5 a z^5-z^5 a^{-1} +8 a z^3-3 z^3 a^{-1} +4 a z-2 z a^{-1} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-2 a^2 z^8-2 z^8-4 a^3 z^7-8 a z^7-4 z^7 a^{-1} -4 a^4 z^6-3 z^6 a^{-2} +z^6-3 a^5 z^5+7 a^3 z^5+22 a z^5+11 z^5 a^{-1} -z^5 a^{-3} -a^6 z^4+5 a^4 z^4+3 a^2 z^4+7 z^4 a^{-2} +4 z^4+4 a^5 z^3-8 a^3 z^3-24 a z^3-10 z^3 a^{-1} +2 z^3 a^{-3} +a^6 z^2-a^4 z^2-3 a^2 z^2-2 z^2 a^{-2} -3 z^2+4 a^3 z+8 a z+4 z a^{-1} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-5-4-3-2-101234χ
8         11
6        2 -2
4       31 2
2      32  -1
0     63   3
-2    45    1
-4   34     -1
-6  24      2
-8 13       -2
-10 2        2
-121         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{6}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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

### Modifying This Page

 Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.