# L10a40

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a40 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 t(2)^3+4 t(1) t(2)^2-7 t(2)^2-7 t(1) t(2)+4 t(2)+2 t(1)}{\sqrt{t(1)} t(2)^{3/2}}$ (db) Jones polynomial $-\frac{5}{q^{9/2}}-q^{7/2}+\frac{6}{q^{7/2}}+2 q^{5/2}-\frac{8}{q^{5/2}}-4 q^{3/2}+\frac{9}{q^{3/2}}-\frac{1}{q^{13/2}}+\frac{2}{q^{11/2}}+6 \sqrt{q}-\frac{8}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a^7 z^{-1} -3 a^5 z-a^5 z^{-1} +2 a^3 z^3-a^3 z^{-1} -z a^{-3} +3 a z^3+z^3 a^{-1} +3 a z+2 a z^{-1} -z a^{-1} - a^{-1} z^{-1}$ (db) Kauffman polynomial $a^7 z^5-3 a^7 z^3+3 a^7 z-a^7 z^{-1} +2 a^6 z^6-4 a^6 z^4+a^6+2 a^5 z^7-a^5 z^5-6 a^5 z^3+3 a^5 z-a^5 z^{-1} +2 a^4 z^8-3 a^4 z^6+3 a^4 z^4-5 a^4 z^2+2 a^4+a^3 z^9+a^3 z^7-7 a^3 z^5+z^5 a^{-3} +13 a^3 z^3-3 z^3 a^{-3} -8 a^3 z+z a^{-3} +a^3 z^{-1} +5 a^2 z^8-17 a^2 z^6+2 z^6 a^{-2} +26 a^2 z^4-5 z^4 a^{-2} -13 a^2 z^2+z^2 a^{-2} +3 a^2+a z^9+2 a z^7+3 z^7 a^{-1} -15 a z^5-9 z^5 a^{-1} +28 a z^3+9 z^3 a^{-1} -14 a z-5 z a^{-1} +2 a z^{-1} + a^{-1} z^{-1} +3 z^8-10 z^6+14 z^4-7 z^2+1$ (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 \
-6-5-4-3-2-101234χ
8          11
6         1 -1
4        31 2
2       31  -2
0      53   2
-2     54    -1
-4    34     -1
-6   35      2
-8  23       -1
-10 14        3
-12 1         -1
-141          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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