# L10n27

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n27 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u v^3-4 u v^2+3 u v-u-v^3+3 v^2-4 v+1}{\sqrt{u} v^{3/2}}$ (db) Jones polynomial $\frac{6}{q^{9/2}}-\frac{7}{q^{7/2}}+\frac{5}{q^{5/2}}-\frac{5}{q^{3/2}}-\frac{1}{q^{15/2}}+\frac{3}{q^{13/2}}-\frac{5}{q^{11/2}}-\sqrt{q}+\frac{3}{\sqrt{q}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^7-2 z^3 a^5-2 z a^5+a^5 z^{-1} +z^5 a^3+2 z^3 a^3-a^3 z^{-1} -z^3 a-z a$ (db) Kauffman polynomial $a^9 z^3-a^9 z+3 a^8 z^4-2 a^8 z^2+a^7 z^7-a^7 z^5+4 a^7 z^3-a^7 z+a^6 z^8-a^6 z^6+3 a^6 z^4-a^6 z^2+4 a^5 z^7-8 a^5 z^5+6 a^5 z^3-2 a^5 z+a^5 z^{-1} +a^4 z^8+2 a^4 z^6-7 a^4 z^4+3 a^4 z^2-a^4+3 a^3 z^7-6 a^3 z^5+a^3 z^3-a^3 z+a^3 z^{-1} +3 a^2 z^6-7 a^2 z^4+2 a^2 z^2+a z^5-2 a z^3+a z$ (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
2        11
0       2 -2
-2      31 2
-4     33  0
-6    42   2
-8   23    1
-10  34     -1
-12 13      2
-14 2       -2
-161        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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