# L11n228

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n228 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u^3 v^3-u^3 v^2+u^2 v^3-u^2 v^2+2 u^2 v-u^2-u v^3+2 u v^2-u v+u-v+1}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-\frac{3}{q^{9/2}}+q^{7/2}+\frac{3}{q^{7/2}}-q^{5/2}-\frac{3}{q^{5/2}}+q^{3/2}+\frac{2}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}-\sqrt{q}-\frac{2}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+a^5 z+2 a^3 z^3+4 a^3 z+2 a^3 z^{-1} -a z^7-7 a z^5+z^5 a^{-1} -15 a z^3+5 z^3 a^{-1} -13 a z+6 z a^{-1} -3 a z^{-1} + a^{-1} z^{-1}$ (db) Kauffman polynomial $-a z^9-z^9 a^{-1} -a^2 z^8-z^8 a^{-2} -2 z^8-a^5 z^7+8 a z^7+7 z^7 a^{-1} -2 a^6 z^6-2 a^4 z^6+8 a^2 z^6+7 z^6 a^{-2} +15 z^6-a^7 z^5+a^5 z^5-2 a^3 z^5-19 a z^5-15 z^5 a^{-1} +6 a^6 z^4+4 a^4 z^4-18 a^2 z^4-15 z^4 a^{-2} -31 z^4+3 a^7 z^3+3 a^5 z^3+6 a^3 z^3+20 a z^3+14 z^3 a^{-1} -3 a^6 z^2+12 a^2 z^2+11 z^2 a^{-2} +20 z^2-2 a^7 z-6 a^3 z-15 a z-7 z a^{-1} -3 a^2- a^{-2} -3+2 a^3 z^{-1} +3 a z^{-1} + a^{-1} z^{-1}$ (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 \
-6-5-4-3-2-1012345χ
8           1-1
6            0
4         11 0
2       11   0
0      2 1   3
-2     231    0
-4    21      1
-6   121      0
-8  22        0
-10 12         1
-12 1          -1
-141           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}_2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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