# L11n350

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n350 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{-t(3) t(2)^3+t(2)^3-t(1) t(3)^2 t(2)^2+t(3)^2 t(2)^2+t(1) t(3) t(2)^2-t(3) t(2)^2-t(2)^2+t(1) t(3)^3 t(2)+t(1) t(3)^2 t(2)-t(3)^2 t(2)-t(1) t(3) t(2)+t(3) t(2)-t(1) t(3)^3+t(1) t(3)^2}{\sqrt{t(1)} t(2)^{3/2} t(3)^{3/2}}$ (db) Jones polynomial $-q^5+2 q^4-2 q^3+2 q^2-q+2+ q^{-1} - q^{-2} +2 q^{-3} - q^{-4} + q^{-5}$ (db) Signature 0 (db) HOMFLY-PT polynomial $z^2 a^4+a^4 z^{-2} +2 a^4-z^4 a^2-4 z^2 a^2-2 a^2 z^{-2} -5 a^2+ z^{-2} +2+z^4 a^{-2} +3 z^2 a^{-2} +2 a^{-2} -z^2 a^{-4} - a^{-4}$ (db) Kauffman polynomial $z^5 a^{-5} -3 z^3 a^{-5} +z a^{-5} +a^4 z^8-7 a^4 z^6+2 z^6 a^{-4} +16 a^4 z^4-7 z^4 a^{-4} -15 a^4 z^2+4 z^2 a^{-4} -a^4 z^{-2} +6 a^4- a^{-4} +a^3 z^9-6 a^3 z^7+z^7 a^{-3} +9 a^3 z^5-3 z^5 a^{-3} -a^3 z^3-5 a^3 z+z a^{-3} +2 a^3 z^{-1} +3 a^2 z^8-21 a^2 z^6+z^6 a^{-2} +44 a^2 z^4-4 z^4 a^{-2} -36 a^2 z^2+2 z^2 a^{-2} -2 a^2 z^{-2} +13 a^2+a z^9-6 a z^7+z^7 a^{-1} +6 a z^5-7 z^5 a^{-1} +6 a z^3+10 z^3 a^{-1} -8 a z-3 z a^{-1} +2 a z^{-1} +2 z^8-15 z^6+31 z^4-23 z^2- z^{-2} +9$ (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
11           1-1
9          1 1
7         11 0
5       121  0
3      111   1
1     142    1
-1    124     3
-3   11       0
-5  111       1
-7 12         1
-9            0
-111           1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-1$ $i=1$ $i=3$ $r=-6$ ${\mathbb Z}$ $r=-5$ ${\mathbb Z}_2$ ${\mathbb Z}$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}_2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=0$ ${\mathbb Z}^{4}$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=2$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=4$ ${\mathbb Z}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=5$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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