# L11n66

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n66 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{u v^5-5 u v^4+8 u v^3-4 u v^2+u v+v^4-4 v^3+8 v^2-5 v+1}{\sqrt{u} v^{5/2}}$ (db) Jones polynomial $\frac{12}{q^{9/2}}-\frac{13}{q^{7/2}}+\frac{12}{q^{5/2}}+q^{3/2}-\frac{11}{q^{3/2}}-\frac{2}{q^{15/2}}+\frac{5}{q^{13/2}}-\frac{9}{q^{11/2}}-4 \sqrt{q}+\frac{7}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z^{-1} -a^7 z^3-4 a^7 z-3 a^7 z^{-1} +2 a^5 z^5+7 a^5 z^3+8 a^5 z+4 a^5 z^{-1} -a^3 z^7-4 a^3 z^5-6 a^3 z^3-6 a^3 z-2 a^3 z^{-1} +a z^5+2 a z^3$ (db) Kauffman polynomial $3 a^9 z^3-4 a^9 z+a^9 z^{-1} +a^8 z^6+4 a^8 z^4-4 a^8 z^2+a^8+5 a^7 z^7-13 a^7 z^5+25 a^7 z^3-16 a^7 z+3 a^7 z^{-1} +6 a^6 z^8-16 a^6 z^6+23 a^6 z^4-12 a^6 z^2+3 a^6+2 a^5 z^9+8 a^5 z^7-37 a^5 z^5+49 a^5 z^3-25 a^5 z+4 a^5 z^{-1} +11 a^4 z^8-29 a^4 z^6+23 a^4 z^4-9 a^4 z^2+2 a^4+2 a^3 z^9+7 a^3 z^7-35 a^3 z^5+35 a^3 z^3-15 a^3 z+2 a^3 z^{-1} +5 a^2 z^8-11 a^2 z^6+2 a^2 z^4+a^2+4 a z^7-11 a z^5+8 a z^3-2 a z+z^6-2 z^4+z^2$ (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 \
-6-5-4-3-2-10123χ
4         1-1
2        3 3
0       41 -3
-2      73  4
-4     65   -1
-6    76    1
-8   56     1
-10  47      -3
-12 26       4
-14 3        -3
-162         2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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