# L10a81

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a81 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{2 u^2 v^3-5 u^2 v^2+2 u^2 v+u v^4-6 u v^3+9 u v^2-6 u v+u+2 v^3-5 v^2+2 v}{u v^2}$ (db) Jones polynomial $-\sqrt{q}+\frac{3}{\sqrt{q}}-\frac{7}{q^{3/2}}+\frac{10}{q^{5/2}}-\frac{13}{q^{7/2}}+\frac{14}{q^{9/2}}-\frac{13}{q^{11/2}}+\frac{10}{q^{13/2}}-\frac{7}{q^{15/2}}+\frac{3}{q^{17/2}}-\frac{1}{q^{19/2}}$ (db) Signature -3 (db) HOMFLY-PT polynomial $z a^9+a^9 z^{-1} -3 z^3 a^7-5 z a^7-2 a^7 z^{-1} +2 z^5 a^5+5 z^3 a^5+5 z a^5+2 a^5 z^{-1} +z^5 a^3-3 z a^3-a^3 z^{-1} -z^3 a-z a$ (db) Kauffman polynomial $a^{11} z^5-2 a^{11} z^3+a^{11} z+3 a^{10} z^6-5 a^{10} z^4+a^{10} z^2+5 a^9 z^7-9 a^9 z^5+6 a^9 z^3-5 a^9 z+a^9 z^{-1} +5 a^8 z^8-8 a^8 z^6+5 a^8 z^4-2 a^8 z^2+2 a^7 z^9+6 a^7 z^7-22 a^7 z^5+26 a^7 z^3-12 a^7 z+2 a^7 z^{-1} +10 a^6 z^8-22 a^6 z^6+20 a^6 z^4-6 a^6 z^2+a^6+2 a^5 z^9+6 a^5 z^7-22 a^5 z^5+26 a^5 z^3-12 a^5 z+2 a^5 z^{-1} +5 a^4 z^8-8 a^4 z^6+5 a^4 z^4-2 a^4 z^2+5 a^3 z^7-9 a^3 z^5+6 a^3 z^3-5 a^3 z+a^3 z^{-1} +3 a^2 z^6-5 a^2 z^4+a^2 z^2+a z^5-2 a z^3+a z$ (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 \
-8-7-6-5-4-3-2-1012χ
2          11
0         2 -2
-2        51 4
-4       63  -3
-6      74   3
-8     76    -1
-10    67     -1
-12   47      3
-14  36       -3
-16 15        4
-18 2         -2
-201          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb Z}$ $r=-7$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-6$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=-5$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=-4$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=-3$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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