# L10n105

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n105 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1) t(4) t(3)^2-t(4)^2 t(3)-t(1) t(2) t(3)-t(1) t(4) t(3)+t(1) t(2) t(4) t(3)-t(2) t(4) t(3)+t(4) t(3)+t(2) t(4)}{\sqrt{t(1)} \sqrt{t(2)} t(3) t(4)}$ (db) Jones polynomial $-\frac{2}{q^{9/2}}+\frac{1}{q^{7/2}}-\frac{4}{q^{5/2}}-2 q^{3/2}+\frac{2}{q^{3/2}}+\sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $2 a^5 z^{-1} +a^5 z^{-3} -4 z a^3-7 a^3 z^{-1} -3 a^3 z^{-3} +2 z^3 a+6 z a+8 a z^{-1} +3 a z^{-3} -2 z a^{-1} -3 a^{-1} z^{-1} - a^{-1} z^{-3}$ (db) Kauffman polynomial $3 a^5 z^3-a^5 z^{-3} -8 a^5 z+5 a^5 z^{-1} +a^4 z^6-4 a^4 z^4+10 a^4 z^2+3 a^4 z^{-2} -10 a^4+a^3 z^7-5 a^3 z^5+15 a^3 z^3-3 a^3 z^{-3} -19 a^3 z+12 a^3 z^{-1} +2 a^2 z^6-8 a^2 z^4+20 a^2 z^2+6 a^2 z^{-2} -19 a^2+a z^7-5 a z^5+15 a z^3+3 z^3 a^{-1} -3 a z^{-3} - a^{-1} z^{-3} -19 a z-8 z a^{-1} +12 a z^{-1} +5 a^{-1} z^{-1} +z^6-4 z^4+10 z^2+3 z^{-2} -10$ (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 \
-4-3-2-1012χ
4      22
2     121
0    3  3
-2   13  2
-4  31   2
-6  3    3
-821     1
-102      2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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