# L11n229

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n229 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(2 u v-u-v+2) \left(u^2 v^2+1\right)}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-\frac{1}{q^{7/2}}+\frac{1}{q^{9/2}}-\frac{3}{q^{11/2}}+\frac{3}{q^{13/2}}-\frac{4}{q^{15/2}}+\frac{4}{q^{17/2}}-\frac{3}{q^{19/2}}+\frac{3}{q^{21/2}}-\frac{2}{q^{23/2}}$ (db) Signature -7 (db) HOMFLY-PT polynomial $-z a^{13}+z^5 a^{11}+5 z^3 a^{11}+5 z a^{11}-z^7 a^9-5 z^5 a^9-6 z^3 a^9-z a^9+a^9 z^{-1} -z^7 a^7-6 z^5 a^7-11 z^3 a^7-7 z a^7-a^7 z^{-1}$ (db) Kauffman polynomial $2 a^{15} z+a^{14} z^4+a^{13} z^7-3 a^{13} z^5+2 a^{13} z^3+a^{13} z+2 a^{12} z^8-10 a^{12} z^6+17 a^{12} z^4-11 a^{12} z^2+a^{11} z^9-4 a^{11} z^7+6 a^{11} z^5-9 a^{11} z^3+5 a^{11} z+3 a^{10} z^8-14 a^{10} z^6+18 a^{10} z^4-8 a^{10} z^2+a^9 z^9-4 a^9 z^7+3 a^9 z^5-a^9 z+a^9 z^{-1} +a^8 z^8-4 a^8 z^6+2 a^8 z^4+3 a^8 z^2-a^8+a^7 z^7-6 a^7 z^5+11 a^7 z^3-7 a^7 z+a^7 z^{-1}$ (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 \
-8-7-6-5-4-3-2-10χ
-6        11
-8       110
-10      2  2
-12     11  0
-14    32   1
-16   11    0
-18  23     -1
-20 11      0
-2212       -1
-242        2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-8$ $i=-6$ $r=-8$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}$

### Computer Talk

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