# L11n223

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n223 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (v-1) \left(u^2 v+2 u v^2+2 u+v\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-2 \sqrt{q}+\frac{3}{\sqrt{q}}-\frac{6}{q^{3/2}}+\frac{7}{q^{5/2}}-\frac{8}{q^{7/2}}+\frac{8}{q^{9/2}}-\frac{7}{q^{11/2}}+\frac{4}{q^{13/2}}-\frac{2}{q^{15/2}}+\frac{1}{q^{17/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $-z^3 a^7-2 z a^7+z^5 a^5+2 z^3 a^5-a^5 z^{-1} +2 z^5 a^3+7 z^3 a^3+7 z a^3+3 a^3 z^{-1} -2 z^3 a-5 z a-2 a z^{-1}$ (db) Kauffman polynomial $a^{10} z^4-2 a^{10} z^2+2 a^9 z^5-3 a^9 z^3+3 a^8 z^6-5 a^8 z^4+3 a^8 z^2+4 a^7 z^7-11 a^7 z^5+15 a^7 z^3-4 a^7 z+3 a^6 z^8-8 a^6 z^6+12 a^6 z^4-5 a^6 z^2+a^6+a^5 z^9+a^5 z^7-6 a^5 z^5+7 a^5 z^3-a^5 z^{-1} +4 a^4 z^8-13 a^4 z^6+20 a^4 z^4-16 a^4 z^2+3 a^4+a^3 z^9-3 a^3 z^7+10 a^3 z^5-21 a^3 z^3+13 a^3 z-3 a^3 z^{-1} +a^2 z^8-2 a^2 z^6+2 a^2 z^4-6 a^2 z^2+3 a^2+3 a z^5-10 a z^3+9 a z-2 a z^{-1}$ (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         22
0        1 -1
-2       52 3
-4      43  -1
-6     43   1
-8    44    0
-10   34     -1
-12  14      3
-14 13       -2
-16 1        1
-181         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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