# L11n367

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n367 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(u-1) (w-1) \left(v^2 w-1\right)}{\sqrt{u} v w}$ (db) Jones polynomial $-q^4+2 q^3+2 q^{-3} -2 q^2- q^{-2} +3 q+3 q^{-1} -2$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^4 z^{-2} +a^4-a^2 z^4-z^4 a^{-2} -4 a^2 z^2-3 z^2 a^{-2} -2 a^2 z^{-2} -4 a^2- a^{-2} +z^6+5 z^4+7 z^2+ z^{-2} +4$ (db) Kauffman polynomial $-a^4 z^2-a^4 z^{-2} +3 a^4+z^7 a^{-3} -5 z^5 a^{-3} +6 z^3 a^{-3} -2 a^3 z-z a^{-3} +2 a^3 z^{-1} +a^2 z^8+2 z^8 a^{-2} -6 a^2 z^6-11 z^6 a^{-2} +12 a^2 z^4+17 z^4 a^{-2} -13 a^2 z^2-9 z^2 a^{-2} -2 a^2 z^{-2} +7 a^2+2 a^{-2} +a z^9+z^9 a^{-1} -5 a z^7-4 z^7 a^{-1} +6 a z^5+z^5 a^{-1} -a z^3+5 z^3 a^{-1} -4 a z-3 z a^{-1} +2 a z^{-1} +3 z^8-17 z^6+29 z^4-21 z^2- z^{-2} +7$ (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 \
-2-1012345χ
9       1-1
7      1 1
5     11 0
3    21  1
1  111   1
-1  32    1
-3112     2
-521      1
-72       2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $i=1$ $r=-2$ ${\mathbb Z}^{2}$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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