# L11n224

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n224 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1)^3 t(2)^3+t(1)^2 t(2)^3-t(1) t(2)^3-t(1)^2 t(2)^2+2 t(1) t(2)^2+2 t(1)^2 t(2)-t(1) t(2)-t(1)^2+t(1)+1}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}-\frac{2}{q^{13/2}}+\frac{3}{q^{15/2}}-\frac{3}{q^{17/2}}+\frac{3}{q^{19/2}}-\frac{3}{q^{21/2}}+\frac{2}{q^{23/2}}-\frac{1}{q^{25/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $z^3 a^{11}+z a^{11}+2 z^3 a^9+4 z a^9+a^9 z^{-1} -z^7 a^7-7 z^5 a^7-14 z^3 a^7-9 z a^7-a^7 z^{-1}$ (db) Kauffman polynomial $a^{15} z^5-3 a^{15} z^3+2 a^{15} z+2 a^{14} z^6-6 a^{14} z^4+3 a^{14} z^2+a^{13} z^7-a^{13} z^5-3 a^{13} z^3+a^{13} z+2 a^{12} z^6-3 a^{12} z^4-a^{12} z^2+3 a^{11} z^5-6 a^{11} z^3+3 a^{11} z+a^{10} z^4+a^{10} z^2-2 a^9 z^5+8 a^9 z^3-5 a^9 z+a^9 z^{-1} -2 a^8 z^4+5 a^8 z^2-a^8+a^7 z^7-7 a^7 z^5+14 a^7 z^3-9 a^7 z+a^7 z^{-1}$ (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 \
-10-9-8-7-6-5-4-3-2-10χ
-6          11
-8          11
-10       11  0
-12      2    2
-14     221   -1
-16    22     0
-18   121     0
-20  22       0
-22 12        1
-24 1         -1
-261          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb Z}$ $r=-9$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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