# L10n109

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n109 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-t(2) t(3)^2+t(2) t(4) t(3)^2-t(4) t(3)^2+t(1) t(4)^2 t(3)-t(1) t(2) t(4)^2 t(3)+t(2) t(3)-t(1) t(4) t(3)+2 t(1) t(2) t(4) t(3)-t(2) t(4) t(3)+2 t(4) t(3)-t(3)-t(1) t(4)^2+t(1) t(4)-t(1) t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $-2 q^{3/2}+2 \sqrt{q}-\frac{6}{\sqrt{q}}+\frac{4}{q^{3/2}}-\frac{7}{q^{5/2}}+\frac{4}{q^{7/2}}-\frac{4}{q^{9/2}}+\frac{2}{q^{11/2}}-\frac{1}{q^{13/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^5 z^3+a^5 z^{-3} +2 a^5 z+2 a^5 z^{-1} -a^3 z^5-4 a^3 z^3-3 a^3 z^{-3} -8 a^3 z-7 a^3 z^{-1} +3 a z^3+3 a z^{-3} - a^{-1} z^{-3} +8 a z+8 a z^{-1} -2 z a^{-1} -3 a^{-1} z^{-1}$ (db) Kauffman polynomial $a^7 z^5-3 a^7 z^3+a^7 z+2 a^6 z^6-5 a^6 z^4+a^6 z^2+3 a^5 z^7-11 a^5 z^5+15 a^5 z^3-a^5 z^{-3} -12 a^5 z+5 a^5 z^{-1} +a^4 z^8+a^4 z^6-11 a^4 z^4+17 a^4 z^2+3 a^4 z^{-2} -10 a^4+5 a^3 z^7-20 a^3 z^5+36 a^3 z^3-3 a^3 z^{-3} -29 a^3 z+12 a^3 z^{-1} +a^2 z^8-8 a^2 z^4+23 a^2 z^2+6 a^2 z^{-2} -19 a^2+2 a z^7-8 a z^5+21 a z^3+3 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -23 a z-7 z a^{-1} +12 a z^{-1} +5 a^{-1} z^{-1} +z^6-2 z^4+7 z^2+3 z^{-2} -10$ (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 \
-6-5-4-3-2-1012χ
4        22
2       110
0      51 4
-2     35  2
-4    41   3
-6   25    3
-8  22     0
-10  2      2
-1212       -1
-141        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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