# L11n162

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n162 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{t(1) t(2)^4+t(2)^4+t(1)^2 t(2)^3-2 t(1) t(2)^3-t(1)^2 t(2)^2+3 t(1) t(2)^2-t(2)^2-2 t(1) t(2)+t(2)+t(1)^2+t(1)}{t(1) t(2)^2}$ (db) Jones polynomial $q^{9/2}-2 q^{7/2}+3 q^{5/2}-\frac{2}{q^{5/2}}-4 q^{3/2}+\frac{3}{q^{3/2}}-\frac{1}{q^{11/2}}+4 \sqrt{q}-\frac{4}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $z a^5+a^5 z^{-1} -z^5 a-4 z^3 a-5 z a-2 a z^{-1} -z^5 a^{-1} -3 z^3 a^{-1} -z a^{-1} + a^{-1} z^{-1} +z^3 a^{-3} +2 z a^{-3}$ (db) Kauffman polynomial $a^5 z^7-7 a^5 z^5+14 a^5 z^3-9 a^5 z+a^5 z^{-1} +z^6 a^{-4} -2 a^4 z^4-4 z^4 a^{-4} +4 a^4 z^2+3 z^2 a^{-4} -a^4+2 z^7 a^{-3} -2 a^3 z^5-8 z^5 a^{-3} +6 a^3 z^3+7 z^3 a^{-3} -3 a^3 z-2 z a^{-3} +a^2 z^8+2 z^8 a^{-2} -7 a^2 z^6-8 z^6 a^{-2} +15 a^2 z^4+8 z^4 a^{-2} -10 a^2 z^2-5 z^2 a^{-2} +3 a^2+2 a^{-2} +a z^9+z^9 a^{-1} -6 a z^7-3 z^7 a^{-1} +13 a z^5-13 a z^3+2 z^3 a^{-1} +8 a z-2 a z^{-1} - a^{-1} z^{-1} +3 z^8-16 z^6+29 z^4-22 z^2+5$ (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 \
-6-5-4-3-2-1012345χ
10           1-1
8          1 1
6         21 -1
4        21  1
2       22   0
0     132    0
-2     23     1
-4   122      -1
-6    2       2
-8  11        0
-101           1
-121           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=-6$ ${\mathbb Z}$ ${\mathbb Z}$ $r=-5$ $r=-4$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ $r=-1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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