# L10a99

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a99 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(2)^2 t(1)^3-2 t(2) t(1)^3+t(1)^3+t(2)^3 t(1)^2-3 t(2)^2 t(1)^2+3 t(2) t(1)^2-2 t(1)^2-2 t(2)^3 t(1)+3 t(2)^2 t(1)-3 t(2) t(1)+t(1)+t(2)^3-2 t(2)^2+t(2)}{t(1)^{3/2} t(2)^{3/2}}$ (db) Jones polynomial $-q^{9/2}+3 q^{7/2}-5 q^{5/2}+7 q^{3/2}-8 \sqrt{q}+\frac{8}{\sqrt{q}}-\frac{8}{q^{3/2}}+\frac{5}{q^{5/2}}-\frac{4}{q^{7/2}}+\frac{2}{q^{9/2}}-\frac{1}{q^{11/2}}$ (db) Signature 1 (db) HOMFLY-PT polynomial $a^5 z+a^5 z^{-1} -2 a^3 z^3-z^3 a^{-3} -4 a^3 z-a^3 z^{-1} -z a^{-3} +a z^5+z^5 a^{-1} +2 a z^3+2 z^3 a^{-1} +a z+z a^{-1}$ (db) Kauffman polynomial $a^5 z^7-5 a^5 z^5+8 a^5 z^3+z^3 a^{-5} -5 a^5 z+a^5 z^{-1} +2 a^4 z^8-9 a^4 z^6+11 a^4 z^4+3 z^4 a^{-4} -2 a^4 z^2-z^2 a^{-4} -a^4+a^3 z^9-13 a^3 z^5+5 z^5 a^{-3} +21 a^3 z^3-4 z^3 a^{-3} -9 a^3 z+z a^{-3} +a^3 z^{-1} +5 a^2 z^8-17 a^2 z^6+6 z^6 a^{-2} +14 a^2 z^4-8 z^4 a^{-2} -2 a^2 z^2+3 z^2 a^{-2} +a z^9+4 a z^7+5 z^7 a^{-1} -20 a z^5-7 z^5 a^{-1} +19 a z^3+z^3 a^{-1} -6 a z-z a^{-1} +3 z^8-2 z^6-8 z^4+4 z^2$ (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 \
-6-5-4-3-2-101234χ
10          11
8         2 -2
6        31 2
4       42  -2
2      43   1
0     55    0
-2    33     0
-4   25      3
-6  23       -1
-8 13        2
-10 1         -1
-121          1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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