# L10a84

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a84 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{t(1)^2 t(2)^4-2 t(1) t(2)^4+t(2)^4-2 t(1)^2 t(2)^3+6 t(1) t(2)^3-3 t(2)^3+3 t(1)^2 t(2)^2-7 t(1) t(2)^2+3 t(2)^2-3 t(1)^2 t(2)+6 t(1) t(2)-2 t(2)+t(1)^2-2 t(1)+1}{t(1) t(2)^2}$ (db) Jones polynomial $q^{9/2}-\frac{3}{q^{9/2}}-4 q^{7/2}+\frac{7}{q^{7/2}}+7 q^{5/2}-\frac{11}{q^{5/2}}-11 q^{3/2}+\frac{13}{q^{3/2}}+\frac{1}{q^{11/2}}+13 \sqrt{q}-\frac{15}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $-a^3 z^5-3 a^3 z^3+z^3 a^{-3} -4 a^3 z+z a^{-3} -a^3 z^{-1} +a z^7+5 a z^5-2 z^5 a^{-1} +11 a z^3-6 z^3 a^{-1} +10 a z-6 z a^{-1} +3 a z^{-1} -2 a^{-1} z^{-1}$ (db) Kauffman polynomial $a^6 z^4-a^6 z^2+3 a^5 z^5-2 a^5 z^3+6 a^4 z^6+z^6 a^{-4} -7 a^4 z^4-2 z^4 a^{-4} +5 a^4 z^2+z^2 a^{-4} -a^4+8 a^3 z^7+4 z^7 a^{-3} -13 a^3 z^5-11 z^5 a^{-3} +13 a^3 z^3+8 z^3 a^{-3} -6 a^3 z-z a^{-3} +a^3 z^{-1} +6 a^2 z^8+5 z^8 a^{-2} -4 a^2 z^6-11 z^6 a^{-2} -7 a^2 z^4+3 z^4 a^{-2} +10 a^2 z^2+2 z^2 a^{-2} -3 a^2+2 a z^9+2 z^9 a^{-1} +11 a z^7+7 z^7 a^{-1} -35 a z^5-30 z^5 a^{-1} +33 a z^3+26 z^3 a^{-1} -15 a z-10 z a^{-1} +3 a z^{-1} +2 a^{-1} z^{-1} +11 z^8-22 z^6+6 z^4+5 z^2-3$ (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 \
-5-4-3-2-1012345χ
10          1-1
8         3 3
6        41 -3
4       73  4
2      75   -2
0     86    2
-2    68     2
-4   57      -2
-6  26       4
-8 15        -4
-10 2         2
-121          -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{5}$ ${\mathbb Z}^{5}$ $r=-1$ ${\mathbb Z}^{7}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{6}$ $r=0$ ${\mathbb Z}^{8}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{8}$ $r=1$ ${\mathbb Z}^{6}\oplus{\mathbb Z}_2^{7}$ ${\mathbb Z}^{7}$ $r=2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{6}$ ${\mathbb Z}^{7}$ $r=3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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