# L11n219

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n219 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) \left(t(1)^2+t(1)+1\right) (t(2)-1)}{t(1)^{3/2} \sqrt{t(2)}}$ (db) Jones polynomial $q^{13/2}-q^{11/2}+q^{9/2}-2 q^{7/2}+q^{5/2}-q^{3/2}-\frac{1}{\sqrt{q}}$ (db) Signature -1 (db) HOMFLY-PT polynomial $z^3 a^{-5} +3 z a^{-5} + a^{-5} z^{-1} -z^5 a^{-3} -5 z^3 a^{-3} -7 z a^{-3} -3 a^{-3} z^{-1} +z^3 a^{-1} +4 z a^{-1} +2 a^{-1} z^{-1}$ (db) Kauffman polynomial $z^8 a^{-6} -7 z^6 a^{-6} +15 z^4 a^{-6} -10 z^2 a^{-6} + a^{-6} +z^9 a^{-5} -7 z^7 a^{-5} +16 z^5 a^{-5} -15 z^3 a^{-5} +6 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -13 z^6 a^{-4} +25 z^4 a^{-4} -16 z^2 a^{-4} +3 a^{-4} +z^9 a^{-3} -7 z^7 a^{-3} +17 z^5 a^{-3} -21 z^3 a^{-3} +13 z a^{-3} -3 a^{-3} z^{-1} +z^8 a^{-2} -6 z^6 a^{-2} +10 z^4 a^{-2} -7 z^2 a^{-2} +3 a^{-2} +z^5 a^{-1} -6 z^3 a^{-1} +7 z a^{-1} -2 a^{-1} z^{-1} -z^2$ (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 \
01234567χ
14       1-1
12        0
10     11 0
8    1   1
6    1   1
4  11    0
21       1
021      1
-21       1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-2$ $i=0$ $i=2$ $r=0$ ${\mathbb Z}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=3$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=6$ ${\mathbb Z}$ $r=7$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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