# L11n186

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n186 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{\left(t(1) t(2)^2-t(1) t(2)+t(2)+t(1)-1\right) \left(t(1) t(2)^2-t(2)^2-t(1) t(2)+t(2)-1\right)}{t(1) t(2)^2}$ (db) Jones polynomial $-q^{7/2}+3 q^{5/2}-5 q^{3/2}+7 \sqrt{q}-\frac{9}{\sqrt{q}}+\frac{8}{q^{3/2}}-\frac{8}{q^{5/2}}+\frac{5}{q^{7/2}}-\frac{3}{q^{9/2}}+\frac{1}{q^{11/2}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $a z^7-a^3 z^5+5 a z^5-z^5 a^{-1} -3 a^3 z^3+8 a z^3-3 z^3 a^{-1} -2 a^3 z+3 a z-2 z a^{-1} +a^3 z^{-1} -a z^{-1}$ (db) Kauffman polynomial $a^6 z^4-2 a^6 z^2+3 a^5 z^5-5 a^5 z^3+a^5 z+a^4 z^8-2 a^4 z^6+5 a^4 z^4-4 a^4 z^2+a^3 z^9-2 a^3 z^7+4 a^3 z^5+z^5 a^{-3} -2 z^3 a^{-3} -a^3 z^{-1} +4 a^2 z^8-11 a^2 z^6+3 z^6 a^{-2} +14 a^2 z^4-7 z^4 a^{-2} -4 a^2 z^2+2 z^2 a^{-2} +a^2+a z^9+2 a z^7+4 z^7 a^{-1} -10 a z^5-10 z^5 a^{-1} +14 a z^3+7 z^3 a^{-1} -4 a z-3 z a^{-1} -a z^{-1} +3 z^8-6 z^6+3 z^4$ (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 \
-5-4-3-2-101234χ
8         11
6        2 -2
4       31 2
2      42  -2
0     53   2
-2    45    1
-4   44     0
-6  25      3
-8 13       -2
-10 2        2
-121         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $r=-5$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=-2$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{5}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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