# L11n226

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n226 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (u+1)^2 (v-1)}{u^{3/2} \sqrt{v}}$ (db) Jones polynomial $-\frac{1}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{1}{q^{13/2}}-\sqrt{q}$ (db) Signature -2 (db) HOMFLY-PT polynomial $-z a^7-z a^5-a^5 z^{-1} +z^5 a^3+5 z^3 a^3+6 z a^3+3 a^3 z^{-1} -z^3 a-4 z a-2 a z^{-1}$ (db) Kauffman polynomial $a^8 z^6-5 a^8 z^4+5 a^8 z^2+a^7 z^7-6 a^7 z^5+9 a^7 z^3-2 a^7 z-a^6 z^2+a^6-a^5 z^3+2 a^5 z-a^5 z^{-1} -a^4 z^6+6 a^4 z^4-10 a^4 z^2+3 a^4-a^3 z^7+7 a^3 z^5-15 a^3 z^3+11 a^3 z-3 a^3 z^{-1} +a^2 z^4-4 a^2 z^2+3 a^2+a z^5-5 a z^3+7 a z-2 a z^{-1}$ (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 \
-7-6-5-4-3-2-1012χ
2         11
0       1  1
-2      131 1
-4     111  1
-6    121   0
-8   121    0
-10   11     0
-12 111      1
-14          0
-161         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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