# L11n300

## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11n300 at Knotilus!

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $-\frac{(t(1)-1) (t(2)+t(3)-1) (t(3) t(2)-t(2)-t(3))}{\sqrt{t(1)} t(2) t(3)}$ (db) Jones polynomial $q^3-3 q^2+4 q-5+7 q^{-1} -5 q^{-2} +6 q^{-3} -3 q^{-4} +2 q^{-5}$ (db) Signature -2 (db) HOMFLY-PT polynomial $a^6 z^{-2} +a^4 z^2-2 a^4 z^{-2} -2 a^4-a^2 z^4+a^2 z^{-2} +z^2 a^{-2} +2 a^2-z^4-z^2$ (db) Kauffman polynomial $a^3 z^9+a z^9+a^4 z^8+4 a^2 z^8+3 z^8-3 a^3 z^7+3 z^7 a^{-1} -4 a^4 z^6-16 a^2 z^6+z^6 a^{-2} -11 z^6+a^5 z^5+3 a^3 z^5-9 a z^5-11 z^5 a^{-1} +8 a^4 z^4+21 a^2 z^4-3 z^4 a^{-2} +10 z^4+8 a z^3+8 z^3 a^{-1} +2 a^6 z^2-2 a^4 z^2-9 a^2 z^2+z^2 a^{-2} -4 z^2+2 a^5 z+2 a^3 z-2 a^6-3 a^4-2 a^2-2 a^5 z^{-1} -2 a^3 z^{-1} +a^6 z^{-2} +2 a^4 z^{-2} +a^2 z^{-2}$ (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 \
-4-3-2-101234χ
7        11
5       2 -2
3      21 1
1     32  -1
-1    42   2
-3   24    2
-5  43     1
-7 14      3
-912       -1
-112        2
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=-3$ $i=-1$ $r=-4$ ${\mathbb Z}^{2}$ ${\mathbb Z}$ $r=-3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{4}$ $r=-1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{4}$ $r=1$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=3$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=4$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

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