# L10n55

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n55 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{u^3 (-v)+u^2 v^3-3 u^2 v^2+3 u^2 v-2 u^2-2 u v^3+3 u v^2-3 u v+u-v^2}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $\frac{7}{q^{9/2}}-\frac{6}{q^{7/2}}+\frac{4}{q^{5/2}}-\frac{2}{q^{3/2}}-\frac{1}{q^{19/2}}+\frac{2}{q^{17/2}}-\frac{5}{q^{15/2}}+\frac{6}{q^{13/2}}-\frac{7}{q^{11/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $a^9 z+a^9 z^{-1} -2 a^7 z^3-3 a^7 z-a^7 z^{-1} +a^5 z^5+2 a^5 z^3+a^5 z-2 a^3 z^3-3 a^3 z$ (db) Kauffman polynomial $a^{11} z^5-3 a^{11} z^3+2 a^{11} z+2 a^{10} z^6-4 a^{10} z^4+a^{10} z^2+3 a^9 z^7-8 a^9 z^5+9 a^9 z^3-7 a^9 z+a^9 z^{-1} +a^8 z^8+3 a^8 z^6-10 a^8 z^4+6 a^8 z^2-a^8+5 a^7 z^7-11 a^7 z^5+12 a^7 z^3-7 a^7 z+a^7 z^{-1} +a^6 z^8+2 a^6 z^6-4 a^6 z^4+3 a^6 z^2+2 a^5 z^7-2 a^5 z^5+3 a^5 z^3-a^5 z+a^4 z^6+2 a^4 z^4-2 a^4 z^2+3 a^3 z^3-3 a^3 z$ (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 \
-8-7-6-5-4-3-2-10χ
-2        22
-4       31-2
-6      31 2
-8     43  -1
-10    33   0
-12   34    1
-14  23     -1
-16  3      3
-1812       -1
-201        1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}$ $r=-6$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-2$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=0$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$

### Computer Talk

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