# L9a16

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a16 at Knotilus! L9a16 is $9^2_{28}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3-4 u v^2+4 u v-2 u-2 v^3+4 v^2-4 v+1}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{5}{q^{3/2}}+\frac{6}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{7}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-a^9 z^{-1} +3 z a^7+3 a^7 z^{-1} -3 z^3 a^5-5 z a^5-2 a^5 z^{-1} +z^5 a^3+2 z^3 a^3+z a^3-z^3 a-z a$ (db) Kauffman polynomial $a^{10} z^4-2 a^{10} z^2+a^{10}+2 a^9 z^5-3 a^9 z^3+2 a^9 z-a^9 z^{-1} +2 a^8 z^6+a^8 z^4-5 a^8 z^2+3 a^8+2 a^7 z^7+a^7 z^5-5 a^7 z^3+7 a^7 z-3 a^7 z^{-1} +a^6 z^8+3 a^6 z^6-4 a^6 z^4-a^6 z^2+3 a^6+5 a^5 z^7-7 a^5 z^5+4 a^5 z-2 a^5 z^{-1} +a^4 z^8+4 a^4 z^6-11 a^4 z^4+5 a^4 z^2+3 a^3 z^7-5 a^3 z^5+3 a^2 z^6-7 a^2 z^4+3 a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0        2 -2
-2       31 2
-4      43  -1
-6     42   2
-8    34    1
-10   44     0
-12  14      3
-14 13       -2
-16 1        1
-181         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### 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.