# L11a370

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a370 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(2)^2 t(1)^4-t(2) t(1)^4+2 t(2)^3 t(1)^3-5 t(2)^2 t(1)^3+5 t(2) t(1)^3-t(1)^3+t(2)^4 t(1)^2-5 t(2)^3 t(1)^2+9 t(2)^2 t(1)^2-5 t(2) t(1)^2+t(1)^2-t(2)^4 t(1)+5 t(2)^3 t(1)-5 t(2)^2 t(1)+2 t(2) t(1)-t(2)^3+t(2)^2}{t(1)^2 t(2)^2}$ (db) Jones polynomial $q^{5/2}-4 q^{3/2}+8 \sqrt{q}-\frac{12}{\sqrt{q}}+\frac{15}{q^{3/2}}-\frac{17}{q^{5/2}}+\frac{15}{q^{7/2}}-\frac{13}{q^{9/2}}+\frac{9}{q^{11/2}}-\frac{5}{q^{13/2}}+\frac{2}{q^{15/2}}-\frac{1}{q^{17/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^3+2 a^7 z+a^7 z^{-1} -a^5 z^5-2 a^5 z^3-3 a^5 z-a^5 z^{-1} -2 a^3 z^5-4 a^3 z^3-3 a^3 z-a z^5+z^3 a^{-1} +a z$ (db) Kauffman polynomial $-a^6 z^{10}-a^4 z^{10}-2 a^7 z^9-6 a^5 z^9-4 a^3 z^9-2 a^8 z^8-3 a^6 z^8-9 a^4 z^8-8 a^2 z^8-a^9 z^7+4 a^7 z^7+11 a^5 z^7-4 a^3 z^7-10 a z^7+8 a^8 z^6+15 a^6 z^6+22 a^4 z^6+7 a^2 z^6-8 z^6+5 a^9 z^5+3 a^7 z^5+a^5 z^5+20 a^3 z^5+13 a z^5-4 z^5 a^{-1} -10 a^8 z^4-14 a^6 z^4-9 a^4 z^4+4 a^2 z^4-z^4 a^{-2} +8 z^4-8 a^9 z^3-7 a^7 z^3-4 a^5 z^3-13 a^3 z^3-6 a z^3+2 z^3 a^{-1} +4 a^8 z^2+5 a^6 z^2-3 a^2 z^2-2 z^2+4 a^9 z-2 a^5 z+3 a^3 z+a z-a^6+a^7 z^{-1} +a^5 z^{-1}$ (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 \
-8-7-6-5-4-3-2-10123χ
6           1-1
4          3 3
2         51 -4
0        73  4
-2       96   -3
-4      86    2
-6     79     2
-8    68      -2
-10   37       4
-12  26        -4
-14 14         3
-16 1          -1
-181           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{9}$ ${\mathbb Z}^{9}$ $r=0$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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