# L11a219

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a219 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1)^2 t(2)^4-3 t(1) t(2)^4+2 t(2)^4-3 t(1)^2 t(2)^3+7 t(1) t(2)^3-3 t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-3 t(1)^2 t(2)+7 t(1) t(2)-3 t(2)+2 t(1)^2-3 t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $q^{17/2}-3 q^{15/2}+7 q^{13/2}-11 q^{11/2}+14 q^{9/2}-17 q^{7/2}+15 q^{5/2}-14 q^{3/2}+10 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{3}{q^{3/2}}-\frac{1}{q^{5/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^3 a^{-7} +2 z a^{-7} + a^{-7} z^{-1} -2 z^5 a^{-5} -6 z^3 a^{-5} -6 z a^{-5} -3 a^{-5} z^{-1} +z^7 a^{-3} +4 z^5 a^{-3} +7 z^3 a^{-3} +7 z a^{-3} +2 a^{-3} z^{-1} -2 z^5 a^{-1} +a z^3-6 z^3 a^{-1} +2 a z-4 z a^{-1}$ (db) Kauffman polynomial $-z^{10} a^{-2} -z^{10} a^{-4} -3 z^9 a^{-1} -7 z^9 a^{-3} -4 z^9 a^{-5} -7 z^8 a^{-2} -11 z^8 a^{-4} -7 z^8 a^{-6} -3 z^8-a z^7+7 z^7 a^{-1} +13 z^7 a^{-3} -3 z^7 a^{-5} -8 z^7 a^{-7} +32 z^6 a^{-2} +33 z^6 a^{-4} +7 z^6 a^{-6} -6 z^6 a^{-8} +12 z^6+4 a z^5+2 z^5 a^{-1} +5 z^5 a^{-3} +22 z^5 a^{-5} +12 z^5 a^{-7} -3 z^5 a^{-9} -34 z^4 a^{-2} -24 z^4 a^{-4} +3 z^4 a^{-6} +7 z^4 a^{-8} -z^4 a^{-10} -15 z^4-5 a z^3-9 z^3 a^{-1} -12 z^3 a^{-3} -21 z^3 a^{-5} -11 z^3 a^{-7} +2 z^3 a^{-9} +11 z^2 a^{-2} +4 z^2 a^{-4} -7 z^2 a^{-6} -5 z^2 a^{-8} +z^2 a^{-10} +6 z^2+2 a z+2 z a^{-1} +5 z a^{-3} +10 z a^{-5} +5 z a^{-7} +3 a^{-4} +3 a^{-6} + a^{-8} -2 a^{-3} z^{-1} -3 a^{-5} z^{-1} - a^{-7} z^{-1}$ (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 \
-4-3-2-101234567χ
18           1-1
16          2 2
14         51 -4
12        62  4
10       85   -3
8      96    3
6     79     2
4    78      -1
2   48       4
0  26        -4
-2 14         3
-4 2          -2
-61           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=1$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{9}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{9}$ $r=3$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{8}$ ${\mathbb Z}^{8}$ $r=4$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=6$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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