# L11n227

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n227 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v^2-u^2 v-u v^2-u v-u-v+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $5 q^{9/2}-6 q^{7/2}+6 q^{5/2}-\frac{1}{q^{5/2}}-7 q^{3/2}+\frac{2}{q^{3/2}}+q^{13/2}-3 q^{11/2}+5 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-3} -2 z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} +a z^3-8 z^3 a^{-1} +8 z^3 a^{-3} -4 z^3 a^{-5} +3 a z-7 z a^{-1} +6 z a^{-3} -3 z a^{-5} +z a^{-7} +a z^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $-z^9 a^{-1} -z^9 a^{-3} -5 z^8 a^{-2} -3 z^8 a^{-4} -2 z^8-a z^7-2 z^7 a^{-3} -3 z^7 a^{-5} +19 z^6 a^{-2} +9 z^6 a^{-4} -z^6 a^{-6} +9 z^6+5 a z^5+14 z^5 a^{-1} +17 z^5 a^{-3} +8 z^5 a^{-5} -17 z^4 a^{-2} -7 z^4 a^{-4} -z^4 a^{-6} -11 z^4-8 a z^3-22 z^3 a^{-1} -18 z^3 a^{-3} -7 z^3 a^{-5} -3 z^3 a^{-7} +3 z^2 a^{-2} +2 z^2 a^{-4} +z^2 a^{-6} -z^2 a^{-8} +3 z^2+5 a z+10 z a^{-1} +6 z a^{-3} +2 z a^{-5} +z a^{-7} +1-a z^{-1} - a^{-1} z^{-1}$ (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 \
-4-3-2-1012345χ
14         1-1
12        2 2
10       31 -2
8      32  1
6     33   0
4    43    1
2   35     2
0  12      -1
-2 13       2
-4 1        -1
-61         1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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