# L9a17

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a17 at Knotilus! L9a17 is $9^2_{27}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)-1) \left(2 t(2)^2-t(2)+2\right)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $7 q^{9/2}-7 q^{7/2}+5 q^{5/2}-4 q^{3/2}+q^{17/2}-3 q^{15/2}+5 q^{13/2}-6 q^{11/2}+\sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $-z^5 a^{-3} -z^5 a^{-5} +z^3 a^{-1} -3 z^3 a^{-3} -2 z^3 a^{-5} +z^3 a^{-7} +3 z a^{-1} -4 z a^{-3} +z a^{-7} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1}$ (db) Kauffman polynomial $-z^8 a^{-4} -z^8 a^{-6} -z^7 a^{-3} -4 z^7 a^{-5} -3 z^7 a^{-7} -z^6 a^{-2} +z^6 a^{-4} -2 z^6 a^{-6} -4 z^6 a^{-8} -z^5 a^{-1} -z^5 a^{-3} +7 z^5 a^{-5} +4 z^5 a^{-7} -3 z^5 a^{-9} +z^4 a^{-2} -4 z^4 a^{-4} +2 z^4 a^{-6} +6 z^4 a^{-8} -z^4 a^{-10} +4 z^3 a^{-1} +7 z^3 a^{-3} -5 z^3 a^{-5} -4 z^3 a^{-7} +4 z^3 a^{-9} +3 z^2 a^{-2} +7 z^2 a^{-4} +z^2 a^{-6} -2 z^2 a^{-8} +z^2 a^{-10} -5 z a^{-1} -7 z a^{-3} +2 z a^{-7} -3 a^{-2} -3 a^{-4} - a^{-6} +2 a^{-1} z^{-1} +3 a^{-3} z^{-1} + a^{-5} z^{-1}$ (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 \
-2-101234567χ
18         1-1
16        2 2
14       31 -2
12      32  1
10     43   -1
8    33    0
6   24     2
4  23      -1
2 14       3
0          0
-21         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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