# L9a15

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a15 at Knotilus! L9a15 is $9^2_{15}$ 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 $q^{17/2}-2 q^{15/2}+4 q^{13/2}-6 q^{11/2}+7 q^{9/2}-7 q^{7/2}+5 q^{5/2}-5 q^{3/2}+2 \sqrt{q}-\frac{1}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -z^5 a^{-5} -3 z^3 a^{-5} -4 z a^{-5} -2 a^{-5} z^{-1} -z^5 a^{-3} -2 z^3 a^{-3} +z^3 a^{-1} +2 z a^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $z^4 a^{-10} -2 z^2 a^{-10} +2 z^5 a^{-9} -3 z^3 a^{-9} +3 z^6 a^{-8} -6 z^4 a^{-8} +6 z^2 a^{-8} -2 a^{-8} +3 z^7 a^{-7} -7 z^5 a^{-7} +10 z^3 a^{-7} -4 z a^{-7} + a^{-7} z^{-1} +z^8 a^{-6} +3 z^6 a^{-6} -11 z^4 a^{-6} +15 z^2 a^{-6} -5 a^{-6} +5 z^7 a^{-5} -12 z^5 a^{-5} +13 z^3 a^{-5} -7 z a^{-5} +2 a^{-5} z^{-1} +z^8 a^{-4} +2 z^6 a^{-4} -8 z^4 a^{-4} +7 z^2 a^{-4} -3 a^{-4} +2 z^7 a^{-3} -2 z^5 a^{-3} -3 z^3 a^{-3} +2 z^6 a^{-2} -4 z^4 a^{-2} + a^{-2} +z^5 a^{-1} -3 z^3 a^{-1} +3 z a^{-1} - a^{-1} z^{-1}$ (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 \
-2-101234567χ
18         1-1
16        1 1
14       31 -2
12      31  2
10     43   -1
8    33    0
6   24     2
4  33      0
2 14       3
0 1        -1
-21         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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