# L10n61

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10n61 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{-2 u^3 v^2+2 u^3 v-u^3+2 u^2 v^2-2 u^2 v-2 u v^2+2 u v-v^3+2 v^2-2 v}{u^{3/2} v^{3/2}}$ (db) Jones polynomial $-\frac{6}{q^{9/2}}+\frac{5}{q^{7/2}}-\frac{7}{q^{5/2}}+\frac{5}{q^{3/2}}+\frac{1}{q^{15/2}}-\frac{2}{q^{13/2}}+\frac{4}{q^{11/2}}+2 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-z a^7-2 a^7 z^{-1} +3 z^3 a^5+9 z a^5+7 a^5 z^{-1} -2 z^5 a^3-9 z^3 a^3-14 z a^3-7 a^3 z^{-1} +2 z^3 a+4 z a+2 a z^{-1}$ (db) Kauffman polynomial $a^8 z^6-4 a^8 z^4+5 a^8 z^2-2 a^8+2 a^7 z^7-7 a^7 z^5+7 a^7 z^3-4 a^7 z+2 a^7 z^{-1} +a^6 z^8+2 a^6 z^6-17 a^6 z^4+20 a^6 z^2-8 a^6+6 a^5 z^7-20 a^5 z^5+22 a^5 z^3-17 a^5 z+7 a^5 z^{-1} +a^4 z^8+5 a^4 z^6-24 a^4 z^4+29 a^4 z^2-13 a^4+4 a^3 z^7-12 a^3 z^5+18 a^3 z^3-16 a^3 z+7 a^3 z^{-1} +4 a^2 z^6-11 a^2 z^4+17 a^2 z^2-8 a^2+a z^5+3 a z^3-3 a z+2 a z^{-1} +3 z^2-2$ (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 \
-7-6-5-4-3-2-101χ
2        2-2
0       2 2
-2      43 -1
-4     31  2
-6    24   2
-8   43    1
-10  13     2
-12 13      -2
-14 1       1
-161        -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-7$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=1$ ${\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$

### Computer Talk

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