# L11n214

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n214 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^3 t(1)^3-t(2)^2 t(1)^3-3 t(2)^3 t(1)^2+5 t(2)^2 t(1)^2-4 t(2) t(1)^2+t(1)^2+t(2)^3 t(1)-4 t(2)^2 t(1)+5 t(2) t(1)-3 t(1)-t(2)+1}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{7}{q^{9/2}}+\frac{3}{q^{7/2}}-\frac{2}{q^{5/2}}+\frac{1}{q^{23/2}}-\frac{3}{q^{21/2}}+\frac{6}{q^{19/2}}-\frac{9}{q^{17/2}}+\frac{10}{q^{15/2}}-\frac{10}{q^{13/2}}+\frac{9}{q^{11/2}}$ (db) Signature -5 (db) HOMFLY-PT polynomial $a^9 \left(-z^5\right)-3 a^9 z^3-3 a^9 z-a^9 z^{-1} +a^7 z^7+5 a^7 z^5+10 a^7 z^3+10 a^7 z+3 a^7 z^{-1} -2 a^5 z^5-8 a^5 z^3-9 a^5 z-2 a^5 z^{-1}$ (db) Kauffman polynomial $a^{14} z^4-a^{14} z^2+3 a^{13} z^5-3 a^{13} z^3+a^{13} z+5 a^{12} z^6-6 a^{12} z^4+3 a^{12} z^2+5 a^{11} z^7-5 a^{11} z^5+a^{11} z^3+a^{11} z+3 a^{10} z^8-3 a^{10} z^4+a^{10}+a^9 z^9+3 a^9 z^7-2 a^9 z^5-6 a^9 z^3+4 a^9 z-a^9 z^{-1} +4 a^8 z^8-7 a^8 z^6+8 a^8 z^4-11 a^8 z^2+3 a^8+a^7 z^9-2 a^7 z^7+9 a^7 z^5-20 a^7 z^3+14 a^7 z-3 a^7 z^{-1} +a^6 z^8-2 a^6 z^6+4 a^6 z^4-7 a^6 z^2+3 a^6+3 a^5 z^5-10 a^5 z^3+10 a^5 z-2 a^5 z^{-1}$ (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 \
-9-8-7-6-5-4-3-2-10χ
-4         22
-6        21-1
-8       51 4
-10      42  -2
-12     65   1
-14    55    0
-16   45     -1
-18  25      3
-20 14       -3
-22 2        2
-241         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-6$ $i=-4$ $r=-9$ ${\mathbb Z}$ $r=-8$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-5$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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