# L11n443

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n443 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-u v w-u v x+u v-u w^2 x+u w^2+2 u w x-u w-v w x^2+2 v w x+v x^2-v x+w^2 x^2-w^2 x-w x^2}{\sqrt{u} \sqrt{v} w x}$ (db) Jones polynomial $q^{9/2}+\frac{1}{q^{9/2}}-2 q^{7/2}-\frac{2}{q^{7/2}}-q^{5/2}+\frac{2}{q^{5/2}}-q^{3/2}-\frac{3}{q^{3/2}}-q^{11/2}-3 \sqrt{q}+\frac{1}{\sqrt{q}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a z^5-z^5 a^{-1} -a^3 z^3+5 a z^3-7 z^3 a^{-1} +2 z^3 a^{-3} -2 a^3 z+9 a z-14 z a^{-1} +8 z a^{-3} -z a^{-5} -a^3 z^{-1} +6 a z^{-1} -11 a^{-1} z^{-1} +8 a^{-3} z^{-1} -2 a^{-5} z^{-1} +a z^{-3} -3 a^{-1} z^{-3} +3 a^{-3} z^{-3} - a^{-5} z^{-3}$ (db) Kauffman polynomial $z^7 a^{-5} -6 z^5 a^{-5} +11 z^3 a^{-5} - a^{-5} z^{-3} -10 z a^{-5} +5 a^{-5} z^{-1} +z^8 a^{-4} +a^4 z^6-5 z^6 a^{-4} -4 a^4 z^4+3 z^4 a^{-4} +3 a^4 z^2+7 z^2 a^{-4} +3 a^{-4} z^{-2} -a^4-9 a^{-4} +2 a^3 z^7+3 z^7 a^{-3} -9 a^3 z^5-21 z^5 a^{-3} +10 a^3 z^3+42 z^3 a^{-3} -3 a^{-3} z^{-3} -6 a^3 z-35 z a^{-3} +2 a^3 z^{-1} +14 a^{-3} z^{-1} +a^2 z^8+z^8 a^{-2} -2 a^2 z^6-5 z^6 a^{-2} -8 a^2 z^4-3 z^4 a^{-2} +14 a^2 z^2+24 z^2 a^{-2} +6 a^{-2} z^{-2} -6 a^2-21 a^{-2} +4 a z^7+4 z^7 a^{-1} -23 a z^5-29 z^5 a^{-1} +39 a z^3+60 z^3 a^{-1} -a z^{-3} -3 a^{-1} z^{-3} -30 a z-49 z a^{-1} +11 a z^{-1} +18 a^{-1} z^{-1} +z^8-3 z^6-10 z^4+28 z^2+3 z^{-2} -18$ (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 \
-5-4-3-2-10123456χ
12           11
10            0
8         21 1
6       311  3
4      141   2
2     422    4
0    251     2
-2   211      2
-4  131       1
-6 11         0
-8 1          1
-101           -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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