# L9a39

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a39 at Knotilus! L9a39 is $9^2_{2}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^3 t(1)^3-t(2)^2 t(1)^3-t(2)^3 t(1)^2+3 t(2)^2 t(1)^2-t(2) t(1)^2-t(2)^2 t(1)+3 t(2) t(1)-t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $\frac{4}{q^{9/2}}-\frac{5}{q^{7/2}}+\frac{4}{q^{5/2}}+q^{3/2}-\frac{4}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{2}{q^{13/2}}-\frac{3}{q^{11/2}}-2 \sqrt{q}+\frac{2}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^3 z^7+a^5 z^5-6 a^3 z^5+a z^5+4 a^5 z^3-12 a^3 z^3+4 a z^3+4 a^5 z-9 a^3 z+3 a z+a^5 z^{-1} -a^3 z^{-1}$ (db) Kauffman polynomial $-z^3 a^9+z a^9-2 z^4 a^8+2 z^2 a^8-2 z^5 a^7+z^3 a^7-2 z^6 a^6+2 z^4 a^6-z^2 a^6-2 z^7 a^5+5 z^5 a^5-7 z^3 a^5+5 z a^5-a^5 z^{-1} -z^8 a^4+z^6 a^4+3 z^4 a^4-4 z^2 a^4+a^4-4 z^7 a^3+16 z^5 a^3-20 z^3 a^3+10 z a^3-a^3 z^{-1} -z^8 a^2+2 z^6 a^2+3 z^4 a^2-4 z^2 a^2-2 z^7 a+9 z^5 a-11 z^3 a+4 z a-z^6+4 z^4-3 z^2$ (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 \
-6-5-4-3-2-10123χ
4         1-1
2        1 1
0       11 0
-2      31  2
-4     22   0
-6    32    1
-8   12     1
-10  23      -1
-12 12       1
-14 1        -1
-161         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.